Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II: General Boundary Conditions on Riemannian Manifolds ... Equations and Their Applications, 98) 9783030886691, 9783030886707, 3030886697

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Progress in Nonlinear Differential Equations and Their Applications PNLDE Subseries in Control 98

Jérôme Le Rousseau Gilles Lebeau Luc Robbiano

Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II General Boundary Conditions on Riemannian Manifolds

Progress in Nonlinear Differential Equations and Their Applications PNLDE Subseries in Control Volume 98

Series Editor Jean-Michel Coron, Laboratory Jacques-Louis Lions, Pierre and Marie Curie University, Paris, France Editorial Board Members Viorel Barbu, Faculty of Mathematics, Alexandru Ioan Cuza University Ias¸i, Romania Piermarco Cannarsa, Department of Mathematics, University of Rome Tor Vergata, Rome, Italy Karl Kunisch, Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria Gilles Lebeau, Dieudonn´e Laboratory J.A., University of Nice Sophia Antipolis Nice, Paris, France Tatsien Li, School of Mathematical Sciences, Fudan University, Shanghai, China Shige Peng, Institute of Mathematics, Shandong University, Jinan, China Eduardo Sontag, Department of Electrical & Computer Engineering, Northeastern University, Boston, Massachusetts, USA Enrique Zuazua, Department of Mathematics, Autonomous University of Madrid, Madrid, Spain

More information about this series at https://link.springer.com/bookseries/15137

J´erˆome Le Rousseau • Gilles Lebeau • Luc Robbiano

Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II General Boundary Conditions on Riemannian Manifolds

J´erˆome Le Rousseau Laboratoire analyse, g´eom´etrie et applications Universit´e Sorbonne Paris-Nord, CNRS, Universit´e Paris 8 Villetaneuse, France

Gilles Lebeau Laboratoire Jean Dieudonn´e Universit´e de Nice Sophia-Antipolis Nice, France

Luc Robbiano Universit´e Paris-Saclay, UVSQ, CNRS Laboratoire de Math´ematiques de Versailles Versailles, France

ISSN 1421-1750 ISSN 2374-0280 (electronic) Progress in Nonlinear Differential Equations and Their Applications ISSN 2524-4639 ISSN 2524-4647 (electronic) PNLDE Subseries in Control ISBN 978-3-030-88669-1 ISBN 978-3-030-88670-7 (eBook) https://doi.org/10.1007/978-3-030-88670-7 Mathematics Subject Classification: 35Q93, 35B45, 35B60, 35L05, 35L20, 35K05, 35K10, 35K20, 35J15, 35J25, 35S15, 58J05, 58J32, 93B05, 93B07, 93D15 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkh¨auser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents Chapter 1. Introduction to Volume 2 1.1. Main Content 1.2. Outline 1.3. Acknowledgement

1 2 4 11

Part 1.

13

General Boundary Conditions

ˇ Chapter 2. Lopatinski˘ı–Sapiro Boundary Conditions 2.1. Introduction ˇ 2.2. Lopatinski˘ı–Sapiro Condition 2.3. First-Order Boundary Operator with Complex Coefficients 2.4. Notes

15 15 18 22 30

Chapter 3. Fredholm Properties of Second-Order Elliptic Operators 3.1. Setting and Main Result 3.2. Analysis in a Half-Space 3.3. Proof of the Fredholm Property ˇ 3.4. Necessity of the Lopatinski˘ı–Sapiro Conditions 3.5. Some Index Computations 3.6. Additional Regularity Results 3.7. Notes Appendix 3.A. Proof of Some Technical Results

33 34 36 43 48 51 64 68 68 68

Chapter 4. Selfadjoint Operators Under General Boundary Conditions 4.1. Introduction and Setting 4.2. Selfadjointness 4.3. Preliminary Result on Symmetry

87 87 88 90 V

VI

CONTENTS

4.4. Sufficient Conditions for Selfadjointness 4.5. A Green Formula 4.6. Spectral Properties ˇ 4.7. Lopatinski˘ı–Sapiro Elliptic Problem Part 2.

96 98 100 110

Carleman Estimates on Riemannian Manifolds

117

Chapter 5. Estimates on Riemannian Manifolds for Dirichlet Boundary Conditions 5.1. Setting 5.2. Estimates Away from the Boundary 5.3. Estimates at the Boundary 5.4. Global Estimations

119 119 121 123 126

Chapter 6. Pseudo-Differential Operators on a Half-Space 6.1. More on Tangential Symbols and Operators 6.2. Adapted Sobolev Norms and Continuity Results 6.3. Quadratic Forms in a Half-Space and G˚ arding Inequality 6.4. Estimates for First-Order Operators 6.5. Notes Appendix 6.A. Smooth Factorisation of Polynomials

129 130 133 136 143 147 148 148

Chapter 7. Sobolev Norms with a Large Parameter on a Manifold 7.1. Nonnegative Sobolev Orders on a Manifold 7.2. Manifold Without Boundary 7.3. Trace Norms 7.4. Carleman Weight Function and Sobolev-Norm Estimation 7.5. Notes

153 153 154 156 156 162

Chapter 8. Estimates for General Boundary Conditions 8.1. Introduction ˇ 8.2. Lopatinski˘ı–Sapiro Condition After Conjugation ˇ 8.3. Carleman Estimate Under the Lopatinski˘ı–Sapiro Condition 8.4. Estimates Without Any Prescribed Boundary Condition 8.5. Global Estimates 8.6. Notes Appendix 8.A. Some Technical Proofs

163 163 164 176 188 192 193 195 195

Part 3.

205

Applications

Chapter 9. Quantified Unique Continuation on a Riemannian Manifold 9.1. Unique Continuation Estimate Away from Boundaries 9.2. Unique Continuation Estimates Up to Boundaries

207 208 210

CONTENTS

9.3. Boundary Initiated Unique Continuation: Improved Estimates 9.4. Notes

VII

218 219

Chapter 10. Stabilization of Waves Under Neumann Boundary Damping 10.1. Setting 10.2. Preliminaries on the Boundary-Damped Wave Equation 10.3. Reduced Functional Space and Generator 10.4. Resolvent Estimate and Stabilization Result 10.5. Proof of the Resolvent Estimate 10.6. Notes Appendix 10.A. The Generator of the Boundary-Damped Wave Semigroup

221 221 222 229 231 232 235 236 236

Chapter 11. Stabilization of Waves Under General Boundary Damping 11.1. Setting 11.2. Strong Solutions and Energy 11.3. Reduced Functional Space and Generator 11.4. Resolvent Estimate and Stabilization Result 11.5. Proof of the Resolvent Estimate 11.6. Notes Appendix 11.A. The Generator of the Boundary-Damped Wave Semigroup

249 249 251 253 256 256 260 260 260

Chapter 12. Spectral Inequality for General Boundary Conditions and Application 12.1. Setting 12.2. Spectral Inequality 12.3. The Parabolic Semigroup 12.4. Null-Controllability of the Associated Parabolic Equation 12.5. Notes

271 271 273 275 278 280

Part 4.

Further Aspects of Carleman Estimates

281

Chapter 13. Carleman Estimates with Source Terms of Weaker Regularity 13.1. Setting and Main Result 13.2. Local Setting at the Boundary 13.3. A Microlocal Estimate 13.4. Patching Estimates Together 13.5. Shifted Estimate 13.6. Estimates Without Prescribed Boundary Conditions

283 284 288 290 301 305 305

VIII

CONTENTS

13.7. Global Estimates 13.8. Notes Appendix 13.A. Some Technical Proofs

324 325 326 326

Chapter 14. Optimal Estimates at the Boundary 14.1. Statement and Proof Scheme 14.2. Some Elements of H¨ormander Calculus 14.3. Proof of the First-Order Estimate 14.4. Proof of the Microlocal Estimate 14.5. Optimality Aspect 14.6. A Refined Estimate Without Any Prescribed Boundary Operator 14.7. Optimal Estimate with Source Terms of Weaker Regularity 14.8. Notes Appendix 14.A. H¨ormander Calculus Properties 14.B. Symbol Properties

358 362 368 368 368 371

Part 5.

385

Background Material: Geometry

333 334 337 344 348 353

Chapter 15.1. 15.2. 15.3. 15.4. 15.5. 15.6. 15.7.

15. Elements of Differential Geometry Manifolds Open Coverings and Partitions of Unity Tangent Space and Vector Fields Cotangent Vectors and Forms Submanifold Tensors and p-Forms Symplectic Structure of the Cotangent Bundle

387 388 390 396 402 405 406 411

Chapter 16.1. 16.2. 16.3.

16. Integration and Differential Operators on Manifolds Oriented Manifolds, Integration of d-Forms Densities on a Nonoriented Manifold Differential Operators on Manifolds

415 416 422 431

Chapter 17. Elements of Riemannian Geometry 17.1. Riemannian Structure on a Manifold 17.2. Gradient, Divergence, and Laplace–Beltrami Operators 17.3. Canonical Positive Density Function and Divergence Formula 17.4. Linear Connection and Covariant Derivatives 17.5. Geodesics and Geodesic Flows 17.6. Normal Geodesic Coordinates at the Boundary 17.7. Higher-Order Covariant Derivatives

437 438 442 445 447 451 455 460

CONTENTS

Chapter 18. Sobolev Spaces and Laplace Problems on a Riemannian Manifold 18.1. L2 and H 1 -Spaces 18.2. Sobolev Spaces 18.3. Transposition of the Laplace–Beltrami Operator and Action on H 1 Functions 18.4. The Laplace Problem on a Compact Manifold Without Boundary 18.5. Continuous Sobolev Scale and Traces 18.6. The Dirichlet-Laplace Problem 18.7. The Neumann-Laplace Problem 18.8. The Laplace Problem with Mixed Neumann-Dirichlet Boundary Conditions 18.9. Second-Order Elliptic Operators in the Euclidean Space Appendix 18.A. Traces Extension: Technical Aspects

IX

463 464 467 471 472 474 482 489 497 501 506 506

Bibliography

515

Index

537

Index of notation

543

CHAPTER 1

Introduction to Volume 2 Contents 1.1. Main Content 1.2. Outline 1.2.1. Part 1 1.2.2. Part 2 1.2.3. Part 3 1.2.4. Part 4 1.2.5. Part 5 1.3. Acknowledgement

2 4 5 5 7 8 10 11

In Volume 1, Carleman estimates for a second-order elliptic operators P were derived in a regular open set Ω of the Euclidean space, away from the boundary and at the boundary. For the latter case, derivation was made with Dirichlet boundary conditions.1 Such an estimation takes the form (1.0.1) τ 3/2 eτ ϕ uL2 (Ω) + τ 1/2 eτ ϕ ∇x uL2 (Ω) + τ 1/2 |eτ ϕ ∂ν u|∂Ω |L2 (∂Ω)   ≤ C eτ ϕ P uL2 (Ω) + τ 3/2 |eτ ϕ u|∂Ω |L2 (∂Ω) + τ 1/2 |eτ ϕ u|∂Ω |H 1 (∂Ω) , where u is smooth or H 2 supported in V a bounded neighborhood of a point of Ω. The function ϕ is smooth and is the so-called Carleman weight function. The parameter τ is positive and chosen such that τ ≥ τ∗ > 0, for τ∗ sufficiently large. The constants C > 0 and τ∗ do depend on the geometry, that is Ω and V , and the coefficients of P and the weight function. In the derivation of (1.0.1), the choice of the weight function is essential; two properties are important: (1) the sub-ellipticity of the conjugated operator Pϕ = eτ ϕ P e−τ ϕ and (2) the condition ∂ν ϕ|∂Ω < 0. Necessity and sufficiency of these conditions are discussed in Volume 1.

1Estimations involving both the Neumann and Dirichlet traces were also derived, but stronger estimates are obtained if one considers a particular boundary operator.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 1

1

2

1. INTRODUCTION TO VOLUME 2

Carleman estimates have been intensively used to obtain unique continuation properties. In Volume 1, we focused on this application, in particular on showing how Carleman estimates as in (1.0.1) are used to yield quantified versions of this property. Further applications were also given: stabilization of the damped wave equation and null-controllability of the heat equation. 1.1. Main Content In Volume 2, one of our goals is to extend Carleman estimates to more general boundary conditions, say Neumann, Robin, etc. A framework is ˇ given by the so-called Lopatinski˘ı–Sapiro condition. It encompasses all aforementioned conditions. ˇ The Lopatinski˘ı–Sapiro condition is of geometrical nature. It thus makes sense also to extend the analysis to Riemannian manifolds, for which the Laplace–Beltrami operator P0 = −Δg appears as a natural second-order elliptic operator. Since Carleman estimates are local in nature and can be patched together, their derivation will be carried out in local coordinates. For an operator like the Laplace–Beltrami operator, the change of coordiˇ nates is straightforward. Lopatinski˘ı–Sapiro boundary conditions are also appealing as they are key in the understanding of the resolvability of elliptic boundary problems. Basically, such problems are of Fredholm type if and ˇ only if boundary conditions fulfill the Lopatinski˘ı–Sapiro condition. In the case of Dirichlet boundary conditions, deriving a Carleman estimate on a Riemannian manifold is rather effortless if relying on the results of Chapter 3 in Volume 1. However, to address the derivation of such estiˇ mates with general Lopatinski˘ı–Sapiro conditions, techniques from Volume 1 do not work in general. If one carries out proofs as in Chapter 3 in Volume 1, some positivity arguments cannot be made, for instance, in the simple case of Neumann boundary conditions. Yet, an estimate can be proven if one relies on a microlocal partitioning of tangential phase space at the boundary. Each microlocal region then requires a different treatment. It is interesting to note that no boundary condition is needed in some microlocal regimes. In others, the boundary condition is crucial, and one sees that the Lopatinˇ ski˘ı–Sapiro condition is precisely the condition that allows one to reach an estimation. Let us introduce the following norm on ∂M, for s ∈ R+ : (1.1.1)

|u|τ,s  τ s |u|L2 (∂M) + |u|H s (∂M) .

For τ > 0 fixed, this gives a norm of H s (∂M) and one can obtain a dual norm |.|τ,s on H −s (∂M). For two functions u and f defined in M, and s ∈ R, one sets (1.1.2)

|f tr(u)|τ,1,s  |f u|∂M |τ,1+s + |f ∂ν u|∂M |τ,s .

1.1. MAIN CONTENT

3

Let V be a bounded open set of M. In some neighborhood of ∂M ∩ V , one can use (m , z) ∈ ∂M × [0, Z] for some Z > 0. The variable z gives a normal direction to ∂M. One can then introduce the following norms, for r ∈ N and s ∈ R+ : (1.1.3) Z 1/2 , uτ,0,s = ∫ |v(., z)|2τ,s dz 0

uτ,r,s 

 0≤j≤r

Dzj uτ,0,r+s−j .

If s = 0, one writes uτ,r in place of uτ,r,0 . For P with P0 = −Δg for principal part, B a boundary operator of order k, and ϕ a weight function, the local Carleman estimates we shall obtain take the form (1.1.4) τ −1/2 eτ ϕ uτ,2 + |eτ ϕ tr(u)|1,1/2  eτ ϕ P uL2 (M) + |eτ ϕ Bu|∂M |τ,3/2−k . In the case B is the Dirichlet boundary operator, such an estimate is stronger than (1.0.1), by several aspects. From Volume 1, we shall use the pseudo-differential techniques of Chapter 2, in particular positivity inequalities of G˚ arding type. Here, in the case of a half-space, we shall provide more material on pseudo-differential operators including such inequalities. We shall also often use the patching techniques for Carleman estimates presented in Sections 3.5 and 3.6 of Volume 1. On the application side, the subjects we cover are very similar to those treated in Volume 1: quantified unique continuation, logarithmic stabilization of the wave equation, and null-controllability of the heat equation. Yet, all these applications are treated by means of the new estimates derived in this volume. Quantified unique continuation is proven in the case of general ˇ Lopatinski˘ı–Sapiro conditions. Stabilization of the wave equation is treated in the case of a boundary damping, first, associated with a Neumann boundary condition and, second, for a general boundary operator that fulfills the ˇ Lopatinski˘ı–Sapiro condition and yields a selfadjoint Laplace–Beltrami operator. The case of inner damping as in Chapter 6 of Volume 1 is not treated here; yet it can be adapted from that chapter to the case of Riemannian ˇ manifolds and Lopatinski˘ı–Sapiro boundary conditions. Null-controllability ˇ of the heat equation is also proven in the case of a Lopatinski˘ı–Sapiro condition that yields a selfadjoint Laplace–Beltrami operator. In Volume 1, Chapter 4 focused on optimality aspects for Carleman estimates and one aspect was left untouched: the optimality of the norms associated with the boundary terms. Yet, the estimates obtained in Volume 1 and the estimate in (1.1.4) are not optimal with that respect. Here,

4

1. INTRODUCTION TO VOLUME 2

in Chap. 14, we present estimations with an improvement with respect to the boundary terms. They take the form (1.1.5) τ −1/2 eτ ϕ uτ,2 + τ −1/4 |eτ ϕ tr(u)|1,1/2  eτ ϕ P uL2 (M) + τ −1/4 |eτ ϕ Bu|∂M |τ,3/2−k . If compared to (1.1.4), there is an additional τ −1/4 in front of the boundary terms, in particular the term involving the boundary operator B on the right-hand side. The proof of such an estimate is quite delicate and requires the use of more advanced tools from pseudo-differential calculus. Here, we shall prove that this improvement is final as such estimates are optimal with respect to the boundary terms. 1.2. Outline Figure 1.1 describes the interrelation of the different parts and chapters in Volume 2. Part 1 Chapter 2

Part 2

Chapter 3

Chapter 8

Chapter 4

Part 4 Chapter 7

Chapter 13

Part 5 Chapter 15

Chapter 6

Chapter 16

Chapter 14 Chapter 17 Chapter 18

Chapter 5

Volume 1

Part 3

Chapter 10

Chapter 9

Chapter 12 Chapter 11

Figure 1.1. Interrelation of the parts and chapters in Volume 2

1.2. OUTLINE

5

1.2.1. Part 1. Chapters 2–4 form Part 1 that is devoted to the study of ˇ Lopatinski˘ı–Sapiro boundary conditions for the Laplace–Beltrami operator. They are presented in Chap. 2. In particular, a characterization is proved in the case of a first-order boundary operator, in all dimensions d ≥ 2. The case d = 2 appears quite different from the cases d = 3 and d ≥ 4. In Chap. 3, for P an elliptic operator with P0 = −Δg for principal part and B a boundary operator of order k, we consider the map L : u → (P u, Bu|∂M ) from H 2 (M) into L2 (M) ⊕ H 3/2−k (∂M). We prove that L is ˇ a Fredholm operator if and only if B fulfills the Lopatinski˘ı–Sapiro condition. ˇ We also show that the Lopatinski˘ı–Sapiro condition yields elliptic property for L, meaning that regularity of P u and Bu|∂M yields maximal regularity for u. We also perform some Fredholm index computations for L in the case of B is of order k ≤ 1. As mentioned above, the case d = 2 yields the most intricate case as topology plays an important rˆole. In Chap. 4, we characterize boundary operators B of order less than or equal to one for an elliptic operator P to be selfadjoint if P has P0 = −Δg for principal part. Useful tools such as a Green formula for functions with fairly low regularity are also provided. Spectral properties for P are given and the nonhomogeneous elliptic boundary value problem is described. 1.2.2. Part 2. Chapters 5–8 form Part 2 that is devoted to the derivation of Carleman estimates for the Laplace–Beltrami operator −Δg on a Riemannian manifold. In Chap. 5, we show how the results of Chapter 3 of Volume 1 yield estimates on a Riemannian manifold in the case of Dirichlet boundary conditions. Local and global estimations are given. Chapters 6 and 7 prepare for the derivation of Carleman estimates for general boundary conditions. The geometry considered in Chap. 6 is that of a half-space, that is, what one obtains when considering local coordinates near a boundary point. In this geometry, we introduce additional classes of pseudo-differential operators; they act like a differential operator in the normal variable and like a pseudo-differential operator in the tangential variables. Adapted Sobolev norms for the half-space are introduced. Quadratic forms built with these pseudo-differential operators are presented. A notion of principal symbol is given as well as a positivity G˚ arding-type estimation. The remaining of Chaps. 6 is devoted to the derivation of estimations for elliptic first-order pseudo-differential operators on a half-space. These estimations are obtained through Fourier-multiplier techniques. Their quality depends on a root localization in the complex plane. The case of sub-elliptic first-order operators, permitting a real root, is also given. In this latter case, the derivation is quite similar to the derivation of Carleman estimates at the boundary given in Chapter 3 of Volume 1.

6

1. INTRODUCTION TO VOLUME 2

In Chap. 7, Sobolev norms with a large parameter on manifolds are introduced. They are necessary for the formulation of Carleman estimates on a manifold, in particular in the case of fractional orders. This aspect was omitted in Volume 1 where Sobolev space on manifold (like a boundary) was rather treated intuitively (see Remark 3.21 in Volume 1). The norms introduced in Chap. 7 are based on the more systematic presentation of Sobolev spaces on Riemannian manifolds given in Chap. 18. They take the forms of the norms given in (1.1.1)–(1.1.3). ˇ Chapter 8 is the cornerstone of Volume 2. First, the Lopatinski˘ı–Sapiro τ ϕ −τ condition is adapted to the conjugated operators Bϕ = e Be ϕ and Pϕ = eτ ϕ P e−τ ϕ for which an estimation with the loss of a half-derivative is sought. The form of the condition after conjugation is very similar to ˇ that given in Chap. 2. If B fulfills the Lopatinski˘ı–Sapiro condition for P , we provide sufficient (and sometimes necessary) conditions on the Carleman ˇ condition for Pϕ . weight function ϕ for Bϕ to fulfill the Lopatinski˘ı–Sapiro ˇ Carleman estimates are derived under this Lopatinski˘ı–Sapiro condition and the now usual sub-ellipticity property. We prove local estimates using normal geodesic coordinates where  x = , τ ) ξ − principal symbol of P reads p (x, ξ, τ ) = ξ −γ (x, ξ (x , xd ). The ϕ ϕ 1 d d  γ2 (x, ξ  , τ ) . Depending on the sign of the imaginary parts of the roots γ1 and γ2 , different proof strategies are used. Tangential phase space is thus cut into various pieces. In particular, if γ1 = γ2 , then microlocally both roots are smooth and the principal part of Pϕ can be written as the product of two first-order operators. For one factor, the sub-elliptic estimation obtained in ˇ Chap. 6 is used. For the second factor, one exploits the Lopatinski˘ı–Sapiro condition. If roots cross, from choices made on the sign of ∂ν ϕ, this crossing happens in the lower complex half-plane. In such case, we rely on quadratic arding inequality forms in the half-space Rd+ and associated microlocal G˚ as introduced in Chap. 6 yielding an elliptic estimation. The microlocal estimates one obtains are patched together to produce a local estimate. These local estimates are in the form given by (1.1.4). In turn, a global estimate can be achieved by the patching of such local estimates. As in Chapter 3 of Volume 1, we also prove estimates without assuming any boundary condition. There, both the Dirichlet and Neumann traces were assumed known. In Chap. 8 in Volume 1, we consider known the Dirichlet trace and the trace Bu|∂M , for B = ∂ν + B  with B  some tangential differential operator of order one, under the assumption that B fulfills the ˇ Lopatinski˘ı–Sapiro condition.2 In Chapter 3, we obtained a local Carleman 2In that case, we do not ask the Lopatinski˘ı–Sapiro ˇ condition to hold for the conjugated

operators but the operators B and P .

1.2. OUTLINE

7

of the form (1.2.1) τ 1/2 eτ ϕ uτ,1  eτ ϕ P uL2 (M) + τ 1/2 |eτ ϕ ∂ν u|∂M |L2 (∂M) + τ 3/2 |eτ ϕ u|∂M |L2 (∂M) + τ 1/2 |eτ ϕ u|∂M |H 1 (∂M) , with the left-hand side formulated with the norm introduced in (1.1.3). Here, a similar estimation is obtained. Yet, through the microlocal arguments we use, we obtain (1.2.2) τ −1/2 eτ ϕ uτ,2  eτ ϕ P uL2 (M) +|eτ ϕ Bu|∂M |τ,1/2 +τ 3/2 |eτ ϕ u|∂M |L2 (∂M) , which in turn can be weakened to the form τ 1/2 eτ ϕ uτ,1  eτ ϕ P uL2 (M) + τ 1/2 |eτ ϕ Bu|∂M |L2 (∂M) + τ 3/2 |eτ ϕ u|∂M |L2 (∂M) . In the case B is the Neumann operator, note the improvement: on the right-hand side, τ 3/2 |eτ ϕ u|∂M |L2 (∂M) + τ 1/2 |eτ ϕ u|∂M |H 1 (∂M) is replaced by simply τ 3/2 |eτ ϕ u|∂M |L2 (∂M) . Such an improvement is useful in applications, for instance, in Chaps. 10 and 11 that deal with the stabilization of the wave equation through boundary damping. 1.2.3. Part 3. Chapters 9–12 form Part 3 that is devoted to applications of Carleman estimates. In Chap. 9, we address quantified unique continuation issues. As opposed to the counterpart chapter of Volume 1, Chapter 5, the analysis in Chap. 9 is performed on a Riemannian manifold. Away from boundaries, results from Volume 1 can be adapted. At boundaries, with the estimates of Chap. 8, we ˇ are able to derive unique continuation estimates under Lopatinski˘ı–Sapiro boundary conditions. As far as we know, the results of Chap. 9 at the boundaries are not available in the literature in such generality. In Chap. 10, we prove the logarithmic stabilization of the following boundary-damped wave equation ⎧ 2 ⎪ in (0, +∞) × M, ⎨ ∂ t y − Δg y = 0 (1.2.3) on (0, +∞) × ∂M, ∂ν y + α∂t y = 0 ⎪ ⎩ 0 1 y|t=0 = y , ∂t y|t=0 = y in M. As in Chapter 6 of Volume 1, stabilization is proven through a resolvent estimate for the generator of the associated semigroup that itself relies on the quantified unique continuation results of Chap. 9. Here, the generator is defined on H = H 1 (M) ⊕ L2 (M). Observe that constant functions are trivial solutions of (1.2.3). Associated with these constant functions is the one-dimensional kernel N of the generator. For the proof of the stabilization

8

1. INTRODUCTION TO VOLUME 2

result, one has to reduce the analysis to an invariant subspace of H that is in direct sum with N . In Chap. 11, the previous result is extended to the wave equation with more general boundary damping. The nature of the wave operator is also extended. One considers ⎧ 2 ⎪ in (0, +∞) × M, ⎨∂ t y + P y = 0 By + α∂t y = 0 on (0, +∞) × ∂M, ⎪ ⎩ 0 1 y|t=0 = y , ∂t y|t=0 = y in M, with P = −Δg + R1 where R1 is a first-order differential operator and ˇ B = ∂ν + B  both chosen such that B and P fulfill the Lopatinski˘ı–Sapiro condition and such that P is selfadjoint (additional assumptions are needed in the case d = 2). The strategy follows also that of Chapter 6 in Volume 1. Like in Chap. 10, the analysis is carried out in a reduced space excluding the kernel of the generator of the associated semigroup. Of course, the result of Chap. 10 is a particular case of Chap. 11. We however chose to consider separately the case of a Neumann boundary damping as it is treated in publications. As far as we know, the result of Chap. 11 is not available in the literature in such generality for a first-order boundary damping condition. Chapter 12 is the counterpart of Chapter 7 of Volume 1. We prove a spectral inequality for the operator P = −Δg + R1 where R1 is as above along with a boundary operator B = ∂ν + B  such that P is selfadjoint. It reads (1.2.4)

wL2 (M) ≤ KeK

√

μ

wL2 (ω) ,

w ∈ span{φj ; μj ≤ μ},

where (φj )j∈N is a Hilbert basis of L2 (M) formed by eigenfunctions of P , with 0 < μ0 ≤ μ1 ≤ · · · the associated real and positive eigenvalues. Like in Chapter 7 in Volume 1, we deduce from this spectral inequality a nullcontrollability result for the associated heat equation. The spectral inequality (1.2.4) with such a general boundary condition is an original result. 1.2.4. Part 4. Chapters 13 and 14 form Part 4 that is devoted to more advanced results on Carleman estimates. It is quite usual to face elliptic problems with a solution u that only lies in 1 H (M). Then, P0 u ∈ H −1 (M), with P0 = −Δg . Up to Chap. 13, Carleman estimates were devoted to the case P u ∈ L2 (M). If P u ∈ H −1 (M) on a ˇ manifold, along with boundary conditions that fulfill the Lopatinski˘ı–Sapiro condition, a Carleman estimate can also be derived. If B = B k−1 ∂ν u|∂M + B k u|∂M , where B k−1 and B k are smooth differential operators on ∂M of order k − 1 and k, respectively, it takes the form τ 1/2 eτ ϕ uτ,1 + τ |eτ ϕ u|∂M |τ,1/2  eτ ϕ F0 L2 (M) + τ eτ ϕ F L2 V (M)   + τ |eτ ϕ|∂M B k−1 (∂ν u + g(F, ν))|∂M + B k u|∂M |τ,1/2−k ,

1.2. OUTLINE

9

where P u = F0 + divg F . The norms .τ,n and |.|τ,/2 are as introduced in (1.1.1)–(1.1.3) (see Chap. 7 for details). Their version on a half-space is given in Sect. 6.2. The proof of this estimation has some similarities with those performed in Chap. principal symbol of Pϕ reading pϕ (x, ξ, τ ) =  8. With the     ξd − γ1 (x, ξ , τ ) ξd − γ2 (x, ξ , τ ) , strategies vary depending on the position of the two roots in the complex plane. Microlocal estimates are derived based in part on the first-order estimations obtained in Chap. 6. Here, as in (1.2.2), we also prove an estimate without assuming any boundary condition, that is, considering known the Dirichlet trace and a trace Bu|∂M , for B = ∂ν + B  with B  some tangential differential operator of order one. One assumes ˇ that this boundary operator B fulfills the Lopatinski˘ı–Sapiro condition.3 −1 We believe that Carleman estimates with source terms in H in Chap. 13 were only known in the literature for Dirichlet and Neumann boundary conditions. In Chap. 14, Carleman inequalities are improved with respect to the estimation of the volume norm. On the one hand, in Chap. 8, one finds an estimate of the form τ −1/2 eτ ϕ uτ,2 + |eτ ϕ|∂M tr(u)|τ,1,1/2  eτ ϕ P uL2 (M) + |eτ ϕ Bu|∂M |τ,3/2−k . On the other hand, in Chap. 14, one obtains (1.2.5) τ −1/2 eτ ϕ uτ,2 + τ −1/4 |eτ ϕ|∂M tr(u)|τ,1,1/2  eτ ϕ P uL2 (M) + τ −1/4 |eτ ϕ Bu|∂M |τ,3/2−k . ˇ Here also, the Lopatinski˘ı–Sapiro condition is assumed. The second estimate is better in the sense that the volume norm eτ ϕ uτ,2 on the left-hand side is estimated thanks to a boundary term with an additional factor τ −1/4 . The proof of this second estimate relies on an improved version of an inequality obtained for a first-order factor under the sub-ellipticity condition in Chap. 6. The improvement one obtains is precisely a factor τ −1/4 for the boundary term. The proof of this first-order operator estimate relies on the use of more advanced pseudo-differential calculus, namely the H¨ormander calculus, and a fine Fefferman–Phong inequality proved by J.-M. Bony. The application of that inequality generalizes the use of the G˚ arding inequality we have made elsewhere for the proof of this type of estimate. Here, we also prove that the τ −1/4 factor gain in (1.2.5) is optimal, thus completing the optimality results of Chapter 4 of Volume 1. Carleman estimates as in (1.2.5) were only known in the literature for Dirichlet boundary conditions. The proof of their optimality is moreover original.

3As for (1.2.2), we do not ask the Lopatinski˘ı–Sapiro ˇ condition to hold for the conjugated operators but the operators B and P .

10

1. INTRODUCTION TO VOLUME 2

In addition, in Chap. 14, Carleman estimates with source terms in H −1 as in Chap. 13 are also considered, and we also prove estimates without assuming any boundary condition. 1.2.5. Part 5. Chapters 15–18 form Part 5 that is devoted to background material on geometry and basic analysis on manifolds. Chapter 15 provides the necessary background on differential geometry: manifolds, submanifolds, partitions of unity, tangent and cotangent vectors, tensors and associated bundles. We also provide elements of symplectic geometry. In Chap. 16, we provide theories of integrations on manifolds: integration of d-forms on oriented manifolds, in particular with the Stokes formula, and integration of densities on non-orientable manifolds. We also present some aspects of Radon measures and distribution densities and of differential operators on manifolds. In Chap. 17, we provide basic aspect of Riemannian geometry, like gradient and divergence associated with the metric and thus the Laplace–Beltrami operator. In connection with Chap. 16, we present how the metric yields a natural density that is very useful to integrate functions. Geodesics can be defined locally as distance minimizing curves. Here, as is also done classically, first, we use the Levi-Civita connection to define geodesics and the geodesic flow, and, second, we prove the distance minimizing property. The Levi-Civita connection is also used to define intrinsic high-order derivatives, including the Hessian of a function and of a tensor field. Normal geodesic coordinates play an important role in proofs of Carleman estimates in Volumes 1 and 2. Yet, in Section 9.4 of Volume 1, they are only defined locally near a point of the boundary. In Chap. 17, we show how such coordinates can be introduced in a neighborhood of a bounded part of the boundary, which can be handy, for instance, for compact manifolds, as it then yields a global parametrization of neighborhood of the boundary. In Chap. 18, we review Sobolev spaces on Riemannian manifolds. Intrinsic derivatives based on the Levi-Civita connection provide intrinsic norms, without relying on partition of unity and local charts. Traces theorems are presented. We also consider the Laplace problem with, first, Dirichlet boundary conditions, second, Neumann boundary conditions, and third, mixed conditions, that is, Dirichlet boundary conditions on some connected components of the boundary and Neumann boundary conditions on the others. This third case is useful in the Fredholm index computations performed in Chap. 3. Lifting maps are studied in relation with these three boundary value problems as well as some spaces with traces properties. Finally, we show at the end of Chap. 18 how properties obtained for operators on a Riemannian manifold can be used for second-order elliptic operators in an open set in the Euclidean space.

1.3. ACKNOWLEDGEMENT

11

1.3. Acknowledgement In addition to their institutions, the authors wish to thank the Institut Henri-Poincar´e in Paris and the Laboratoire de Math´ematiques d’Orsay for their hospitality on numerous occasions during the preparation of this book. They wish to thank R´emi Buffe, Camille Laurent, Matthieu L´eautaud, Kim-Dang Phung, and Emmanuel Zongo for useful suggestions and discussions on some chapters.

Part 1

General Boundary Conditions

CHAPTER 2

ˇ Lopatinski˘ı–Sapiro Boundary Conditions Contents 2.1. Introduction 2.1.1. Observations Concerning the Need for Boundary Conditions 2.1.2. Outline ˇ 2.2. Lopatinski˘ı–Sapiro Condition 2.3. First-Order Boundary Operator with Complex Coefficients 2.4. Notes

15 16 18 18 22 30

2.1. Introduction In Volume 1, on some bounded smooth open set Ω, we considered a second-order elliptic operator P = P (x, D) with a principal part of the form   Di (pij (x)Dj ), with pij (x)ξi ξj ≥ C|ξ|2 , 1≤i,j≤d

C ∞ (Ω; R)

1≤i,j≤d

∈ are such that = pji , 1 ≤ i, j ≤ d. Along with such where an elliptic operator we considered Dirichlet boundary conditions. In the present chapter, and in this second volume for the most part, we provide a more comprehensive and geometrical point of view by working directly on a Riemannian manifold. The goal of the present short chapter is to introduce the broad class of boundary conditions known under the name Lopatinski˘ı– ˇ Sapiro condition. Here, (M, g) will thus denote a d-dimensional smooth compact Riemannian manifold. We refer to Chaps. 15–17 for the notions of differential and Riemannian geometry that are needed in what follows. In the case Ω pij

pij

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 2

15

ˇ 2. LOPATINSKI˘I–SAPIRO BOUNDARY CONDITIONS

16

is a smooth open set in Rd , setting M = Ω allows one to cover this case in the presentation below; see Sect. 18.9. The elliptic boundary value problem under consideration is thus the following (2.1.1)

P u = f ∈ L2 (M),

Bu|∂M = f∂ ,

where P = −Δg + R1 , with Δg the Laplace–Beltrami operator on M, with R1 a first-order differential operator on M with bounded coefficients, and with B a smooth differential boundary operator on ∂M. The source term f∂ lies in a prescribed space depending on the nature of the operator B. Remark 2.1. Note that it can be convenient to consider the operator B as a differential operator defined in a neighborhood of the boundary. Both points of view are equivalent as one can perform a smooth extension of the coefficients of B. Near a point m0 ∈ ∂M, if one uses normal geodesic coordinates (x , xd ) where ∂M is locally given by {xd = 0} (see Sect. 17.6) the Laplace–Beltrami operator takes the form (2.1.2)

−Δg = Dd2 + R(x, Dx ) + Q1 (x, D),

where R(x, Dx ) is a second-order differential operator and Q1 (x, D) is a first-order differential operator. Thus, formally one can use the equation P u = f to reduce the form of the operator B to the case where the order of Dd is at most one. We shall thus assume that B is an operator of order β ∈ N with yet differentiation transverse to ∂M of order less than or equal to one. Denoting ν the unique unitary outward pointing vector field along ∂M orthogonal to Tm ∂M, one then finds that B takes the form (2.1.3)

B = B k + B k−1 ∂ν ,

on every connected component N of ∂M, with B k and B k−1 smooth differential operators on N of order k and k − 1, respectively, with k ≤ β. Naturally, if B is of order zero on N one has B k−1 = 0. 2.1.1. Observations Concerning the Need for Boundary Conditions. On (0, +∞) consider the first-order operator L = Dt − λρ, with ρ ∈ C and λ > 0. The parameter λ is meant to become large. Let v ∈ S (Rd+ ). One wishes to investigate whether one can achieve the estimate (2.1.4)

vL2 (R+ )  LvL2 (R+ ) .

First, assume that Im ρ < 0. One computes (2.1.5)

Re(Lv, iv)L2 (R+ ) = Re(Dt v, iv)L2 (R+ ) − λ Im ρv2L2 (R+ ) =−

1 ∫ ∂t |v|2 dt + λ| Im ρ|v2L2 (R+ ) 2 R+

1 = |v(0)|2 + λ| Im ρ|v2L2 (R+ ) . 2

2.1. INTRODUCTION

17

With the Cauchy–Schwarz inequality and the Young inequality we obtain for any ε > 0 λv2L2 (R+ ) + |v(0)|2  ελv2L2 (R+ ) + (ελ)−1 Lv2L2 (R+ ) . With ε > 0 chosen sufficiently small one concludes that (2.1.6)

λvL2 (R+ ) + λ1/2 |v(0)|  LvL2 (R+ ) .

In this first case, not only do we obtain (2.1.4) but, better yet, we also estimate the trace of v at t = 0+ . Second, assume that Im ρ > 0. Consider χ ∈ Cc∞ (R) with χ(t) = 1 for |t| ≤ 1 and χ(t) = 0 for t ≥ 2. Set v(t) = χ(t)eiλρt and observe that Lv(t) = −iχ (t)eiλρt . On the one hand one has 2

v2L2 (R+ ) = ∫ χ2 (t)e−2λ Im ρt dt 0 1

≥ ∫ e−2λ Im ρt dt = 0

 1  1 − e−2λ Im ρ . 2λ Im ρ

On the other hand, one has 2 2 Lv2L2 (R+ ) = ∫ χ (t) e−2λ Im ρ t dt 0 2

 ∫ e−2λ Im ρ t dt = 1

 1  −2λ Im ρ e − e−4λ Im ρ . 2λ Im ρ

If λ → +∞, one thus finds vL2 (R+ )  (2λ Im ρ)−1/2 and LvL2 (R+ )  (2λ Im ρ)−1/2 e−λ Im ρ . This ruins any hope of having an estimate of the form of (2.1.4). Yet, one has (2.1.7)

λvL2 (R+ )  LvL2 (R+ ) + λ1/2 |v(0)|.

For a proof, compute Re(Lv, −iv)L2 (R+ ) and argue as above for (2.1.5) and (2.1.6). Consider now the Laplace–Beltrami operator −Δg in the normal geodesic coordinates provided above, that is, as in (2.1.2), −Δg = Dd2 + R(x, Dx ) + Q1 (x, D). To simplify, assume here that R(x, Dx ) is a constant coefficient operator. Then, up to a Fourier transformation in the x variables, one obtains for the principal part the operator ˜  ), Pˆ = D2 + R(ξ  ) = D2 + |ξ  |2 R(ξ d

d

˜  ) = R(ξ  /|ξ  |), which we write with R(ξ ˜  )1/2 . L± = Dd ∓ |ξ  |R(ξ Pˆ = L+ L− , With λ = |ξ  |, xd = t, and the two cases considered above, one finds that for the factor L− an estimation as in (2.1.6) can be obtained. For the factor

ˇ 2. LOPATINSKI˘I–SAPIRO BOUNDARY CONDITIONS

18

L+ one can only obtain an estimation as in (2.1.7). Combined together one obtains an estimation either of the form λ2 vL2 (R+ ) + λ1/2 |Dd v(0)|  Pˆ vL2 (R+ ) + λ3/2 |v(0)|, or the form λ2 vL2 (R+ ) + λ3/2 |v(0)|  Pˆ vL2 (R+ ) + λ1/2 |Dd v(0)|. This simple example shows that if an elliptic operator can be written as a product of several factors as above, factors of the form Dd −λρ with Im ρ < 0 yield an estimation without requiring any boundary term; factors of the form Dd − λρ with Im ρ > 0 yet require a boundary term. The number of factors of the second kind yields the number of required boundary conditions. In that framework, if given some boundary operators, the Lopatinski˘ı– ˇ Sapiro condition states their compatibility with the different factors Dd − λρ with Im ρ > 0. If one considers again the Laplace–Beltrami, one sees that only one boundary condition is needed, as is classically known. As we shall see beˇ low the Lopatinski˘ı–Sapiro condition gives a relation between the chosen boundary operator and the factor L+ given above. ˇ 2.1.2. Outline. In Sect. 2.2 we introduce the Lopatinski˘ı–Sapiro boundary condition. Then, in Sect. 2.3 we focus our interest to the case where B is a first-order operator with complex coefficients and provide a full description ˇ of such operators that fulfill the Lopatinski˘ı–Sapiro conditions. The actual analysis of the elliptic boundary value problem (2.1.1) is, however, postponed to the next chapter. ˇ 2.2. Lopatinski˘ı–Sapiro Condition ∗ (∂M) denotes the conormal space at For m ∈ ∂M, the vector space Nm m given by (see Sect. 15.5 for details) ∗ ∗ ∂M = {ω ∈ Tm M; ω(u) = ω, u = 0, for all u ∈ Tm ∂M}. Nm

The conormal bundle of ∂M is given by ∗ {m} × Nm ∂M. N ∗ ∂M = m∈∂M

Consider ν to be the unique outward pointing vector field on M along ∂M such that, for all m ∈ ∂M, gm (νm , νm ) = 1 and gm (νm , u) = 0 for all u ∈ Tm ∂M, that is, ν is unitary and orthogonal to Tm ∂M in the sense of g. We then set n = ν the associated outward pointing one-form on M along ∂M. We refer to Sect. 17.1.3 for the musical isomorphisms and notation. It is a nonvanishing section of N ∗ ∂M. In particular note that ∗ ∂M = Rn and thus T ∗ M = Rn ⊕ T ∗ ∂M (see Sect. 15.5). Moreover, Nm m m m m ∗ ∂M is orthogonal to T ∗ ∂M for the inner product (., .) for cotangent Nm g m vectors associated with the metric at m, see (17.1.4) in Sect. 17.1.

ˇ 2.2. LOPATINSKI˘I–SAPIRO CONDITION

19

We recall that p(m, ω) = |ω|2g = gm (ω  , ω  ) is the principal symbol of the (positive) Laplace–Beltrami operator P = −Δg (see Sect. 17.2 with the notation introduced in Sect. 17.1). In a local chart C = (O, κ), its representative reads  g C,ij (x)ξi ξj . pC (x, ξ) = 1≤i,j≤d

For

(m, ω  )

∈

T ∗ ∂M,

we set

pˇ(m, ω  , z) = p(m, ω  − znm ) = |ω  − znm |g , 2

that is a monic polynomial function of degree two in z by Proposition 16.13 and (16.3.1). As we have (nm , ω  )g = νm , ω   = 0 we find pˇ(m, ω  , z) = z 2 |nm |2g + |ω  |g = z 2 + |ω  |g = − (z)+ (z), 2

2

with + = (z − i|ω  |g ) and − = (z + i|ω  |g ). We denote by b(m, ω) the principal symbol of the boundary operator B defined on ∂M. On a connected component N of ∂M where B is of order k ≤ β, with B in the form given in (2.1.3), and bk (m, ω  ) and bk−1 (m, ω  ) the respective principal symbols of B k and B k−1 , we have b(m, ω) = bk (m, ω  ) + ibk−1 (m, ω  )ω, νm , ∗ (M) with ω = ω  + ω n  ∗ for ω ∈ Tm n m for some ω ∈ Tm (∂M) and ωn = ω, νm . For (m, ω  ) ∈ T ∗ ∂M, we set

ˇb(m, ω  , z) = b(m, ω  − znm ) = bk (m, ω  ) − ibk−1 (m, ω  )z, that is a polynomial function in z of order less than or equal to one. ˇ We may now state the Lopatinski˘ı–Sapiro condition first microlocally, second at one point, and third locally. ˇ Definition 2.2 (Lopatinski˘ı–Sapiro Condition for the Laplace–Beltrami  ∗ Operator). Let (m, ω ) ∈ T ∂M with ω  = 0. One says that the Lopatinski˘ı– ˇ Sapiro condition holds for (P, B) at (m, ω  ) if for any polynomial function f (z) with complex coefficients there exists c ∈ C and a polynomial function g(z) with complex coefficients such that, for all z ∈ C, (2.2.1)

f (z) = c ˇb(m, ω  , z) + g(z)+ (z).

ˇ One says that the Lopatinski˘ı–Sapiro condition holds for (P, B) at m ∈ ∂M   ∗ if it holds at (m, ω ) for all ω ∈ Tm ∂M with ω  = 0. If Γ ⊂ ∂M, one says ˇ that the Lopatinski˘ı–Sapiro condition holds for (P, B) on Γ if it holds at m for all m ∈ Γ.

20

ˇ 2. LOPATINSKI˘I–SAPIRO BOUNDARY CONDITIONS

The choice of the polynomial + (z) = z − i|ω  |g is in connection with the nonnegative sign of the imaginary part of its roots. This can be motivated by the observations made in Sect. 2.1.1. The formulation of the Lopatinˇ ski˘ı–Sapiro condition for general operators makes this more explicit (see Remark 2.7 below). With the Euclidean division of polynomials, we see that it suffices to consider the polynomial function f (z) to be of degree zero in (2.2.1). Obˇ serve also that the Lopatinski˘ı–Sapiro condition holds if and only if for any f (z) the complex number i|ω  |g is a root of the polynomial function f (z) − cˇb(m, ω  , z) for some c ∈ C. We thus have the following proposition. Proposition 2.3. Let (m, ω  ) ∈ T ∗ ∂M with ω  = 0. The Lopatinski˘ı– ˇ Sapiro condition holds for (P, B) at (m, ω  ) if and only if ˇb(m, ω  , i|ω  | ) = bk (m, ω  ) + bk−1 (m, ω  )|ω  | = 0. g g Remark 2.4. ˇ (1) With the equivalent formulation of the Lopatinski˘ı–Sapiro condition given by Proposition 2.3 one sees that if it holds at (m, ω  ) ∈ T ∗ ∂M with ω  = 0, then there exists a conic neighborhood of (m, ω  ) in T ∗ ∂M where it holds also. Similarly, if the Lopatinˇ ski˘ı–Sapiro condition holds at m ∈ ∂M (meaning it holds for all ∗ ∂M \ {0}), then there exists a neighborhood (m, ω  ) with ω  ∈ Tm of m ∈ ∂M where it holds also. ˇ (2) Observe that the Lopatinski˘ı–Sapiro condition is written here without any use of local coordinates. It is thus a geometrical condition. As the principal parts of the operators P and B are geometrical objects (see Sect. 16.3.1), if a particular local chart is chosen, then ˇ the Lopatinski˘ı–Sapiro condition for (P, B) can be equivalently expressed in this chart. We list examples of classical boundary conditions that fit the Lopatinˇ ski˘ı–Sapiro framework. Examples 2.5. (1) The Dirichlet boundary condition. In this case, B is a zero-order operator given by Bu = u, that is, a zero-order operator. Then, b(m, ω) = 1 and ˇb(m, ω  , z) = 1. By ˇ Proposition 2.3, we see that the Lopatinski˘ı–Sapiro condition holds  at any (m, ω ). If now B is of order 0, that is, b(m, ω) = b(m), if the Lopatinˇ ski˘ı–Sapiro condition holds at some (m, ω  ), then b(m) cannot vanˇ ish by Proposition 2.3. If the Lopatinski˘ı–Sapiro condition holds on some open set Γ of ∂M we thus see that up to dividing by a nonvanishing function we recover the Dirichlet boundary condition.

ˇ 2.2. LOPATINSKI˘I–SAPIRO CONDITION

21

(2) The Neumann and Robin boundary conditions. In the Neumann case, Bu = ∂ν u = ν(u). Then, b(m, ω) = iω, νm  and ˇb(m, ω  , z) = −iz. As ˇb(m, ω  , i|ω  |g ) = |ω  |g , we see ˇ that the Lopatinski˘ı–Sapiro condition holds at any (m, ω  ) if ω  = 0. The Robin boundary condition is a natural generalization with Bu(m) = (∂ν u)(m) + a(m)u(m). As the principal symbol of B is the same as that in the Neumann case, the conclusion follows. (3) An oblique boundary condition. This is a generalization of the Robin boundary condition. In this case Bu(m) = gm (∇g u, v) + a(m)u(m) = du(m)(v) + a(m)u(m) for v a real vector field on M along ∂M. This precisely covers the case of a first-order boundary operator with a purely imaginary principal symbol. We have b(m, ω) = iω, vm . Let us assume moreover that nm , vm  = 0 at any m ∈ ∂M. We have ˇb(m, ω  , z) = iω  , vm  − iznm , vm  and ˇb(m, ω  , i|ω  | ) = iω  , vm  + |ω  | nm , vm . g

g

|ω  |

  ˇ Since nm , vm  = 0 and g = 0, we see that b(m, ω , i|ω |g ) = ˇ 0. The Lopatinski˘ı–Sapiro condition thus holds at any (m, ω  ) if  ω = 0. However, observe that if nm , vm  = 0 for some m ∈ ∂M, that is, vm ∈ Tm ∂M, we then have ˇb(m, ω  , i|ω  |g ) = iω  , vm  that ∗ ∂M if d ≥ 3. The condition n , v  = 0 vanishes for some ω  ∈ Tm m m is thus necessary for the oblique boundary operator to fulfill the ˇ Lopatinski˘ı–Sapiro condition if d ≥ 3. (4) Ventcel boundary condition. This is an example of a second-order boundary condition. In this case Bu|∂M = ∂ν u|∂M − hΔg∂ u|∂M , where Δg∂ is the Laplace– Beltrami operator on ∂M associated with g∂ , the induced metric on ∂M, and h is some function defined on ∂M. The principal symbol of B is b(m, ω) = h(m)|ω  |2g∂ m and thus ˇb(m, ω  , z) = h(m)|ω  |2g∂ m . ˇ Consequently, by Proposition 2.3, the Lopatinski˘ı–Sapiro condition holds for (P, B) at m ∈ ∂M if and only if h(m) = 0. ˇ Example 2.6. A setting where the Lopatinski˘ı–Sapiro condition does not hold is given by the following classical example. On the unit disc D consider P = −Δ = −∂x2 − ∂y2 that reads P = −r−1 ∂r (r∂r ) − r−2 ∂θ2 in polar coordinates. In the latter coordinates, define the boundary operator ˇ Bu|∂D = ∂r u|∂D + i∂θ u|∂D . As one can readily check the Lopatinski˘ı–Sapiro condition holds nowhere on ∂D. In Chap. 3 we prove that the Lopatinski˘ı– ˇ Sapiro condition is equivalent to having the operator L : H 2 (D) → L2 (D) ⊕ 1/2 H (∂D) given by Lu = (P u, Bu|∂D ) of Fredholm type; see Theorem 3.1. Definition and basic properties of Fredholm operators are given in Section 11.5 of Volume 1. One can see that the present operator is not Fredholm ˇ in agreement with the Lopatinski˘ı–Sapiro condition not holding. In fact,

22

ˇ 2. LOPATINSKI˘I–SAPIRO BOUNDARY CONDITIONS

the operator reads P = −4∂∂ with ∂ = ∂z = (∂x − i∂y )/2 and ∂ = ∂z = (∂x + i∂y )/2. Moreover, we have 1 ∂ = eiθ (∂r + ir−1 ∂θ ). 2 Consequently any H 2 -holomorphic function f on a neighborhood of D, solution to ∂f = 0, is also solution to P f = 0 and Bf|∂D = 0. As the space of such functions is of infinite dimension, the operator L is not Fredholm. ˇ Remark 2.7 (Lopatinski˘ı–Sapiro Condition for General Elliptic Differential Operators). For a general elliptic differential operator Q of degree 2k on M, with principal symbol q(m, ω) one defines the following polynomial in z qˇ(m, ω  , z) = q(m, ω  − znm ), and one denotes its complex roots by γj (m, ω  ), 1 ≤ j ≤ 2k. One sets

(z − γj (m, ω  )). qˇ+ (m, ω  , z) = Im γj (m,ω  )≥0

Given boundary operators B1 , . . . , Bk in a neighborhood of ∂M, with principal symbols bj (m, ω), j = 1, . . . , k, one also sets ˇbj (m, ω  , z) = bj (m, ω  − znm ). ˇ Let (m, ω  ) ∈ T ∗ ∂M with ω  = 0. One says that the Lopatinski˘ı–Sapiro  condition holds for (Q, B1 , . . . , Bk ) at (m, ω ) if for any polynomial function f (z) with complex coefficients there exists c1 , . . . , ck ∈ C and a polynomial function g(z) with complex coefficients such that, for all z ∈ C,  (2.2.2) cj ˇbj (m, ω  , z) + g(z)ˇ q + (m, ω  , z). f (z) = 1≤j≤k

Definition 2.2 gives precisely this condition in the case of the Laplace– Beltrami operator. In Examples 2.5-(2) and 2.5-(3), the boundary operator is of order one and we note that the principal symbol b satisfies b(m, ω) ∈ cR for some fixed c ∈ C. We now consider the general case of boundary operator of order one. 2.3. First-Order Boundary Operator with Complex Coefficients The following proposition provides a classification of first-order differˇ ential boundary operators B that yield Lopatinski˘ı–Sapiro conditions along with the operator P , depending on the dimension of M. Proposition 2.8. Let the boundary operator B be of order β = 1 with nonvanishing principal symbol b(m, ω) = ω, tm  + iω, vm ,

2.3. FIRST-ORDER BOUNDARY OPERATOR WITH COMPLEX COEFFICIENTS 23 ν ν + where t, v are two real vector fields on M along ∂M. We write vm = vm m  ν  ν ν   vm , tm = tm νm + tm , with vm , tm ∈ R and vm , tm ∈ Tm ∂M. Depending on the dimension d ≥ 2 of M we have the following results.

ˇ Case d = 2. The Lopatinski˘ı–Sapiro condition holds at m if and only if (2.3.1)

 ν = i(tνm + ivm )X  , tm + ivm

for any X  ∈ Tm ∂M with |X  |g∂ = 1. ˇ Case d = 3. The Lopatinski˘ı–Sapiro condition holds at m if and only if (2.3.2)

ν  | or |vm |g∂ < |tνm |, |tm |g∂ < |vm

or (2.3.3)

   2  2 ν 2 gm (tm , vm )2 = |tm |g∂ − (vm ) |vm |g∂ − (tνm )2 .

ˇ Case d ≥ 4. The Lopatinski˘ı–Sapiro condition holds at m if and only if (2.3.4)

ν  | or |vm |g∂ < |tνm |, |tm |g∂ < |vm

or (2.3.5)

   2  2 ν 2 gm (tm , vm )2 > |tm |g∂ − (vm ) |vm |g∂ − (tνm )2 .

Observe that one only assumes that the operator B does not degenerate into a zero-order operator at some point m: one thus excludes that tm and vm vanish simultaneously. ∗ ∂M one has Proof. For ω  ∈ Tm

ˇb(m, ω  , z) = ω  , t  + iω  , v   − z(tν + iv ν ). m m m m ˇ condition does not hold By proposition 2.3, if ω  = 0, the Lopatinski˘ı–Sapiro    ˇ for (P, B) at (m, ω ) if and only if b(m, ω , i|ω |g∂ ) = 0. By homogeneity, note ∗ ∂M, that is, |ω  | that it suffices to consider ω  ∈ Sm g∂ = 1. In such case, we ˇ find that the Lopatinski˘ı–Sapiro condition does not hold at (m, ω  ) if and only if (2.3.6)

ν  and ω  , vm  = tνm . ω  , tm  = −vm

ν ω  Case 1: d = 2. As dim Tm ∂M = 1, from (2.3.6) we find tm = −vm  ν   ∗ and vm = tm ω . Here ω ∈ Sm ∂M. This precisely means that the Lopatinˇ ski˘ı–Sapiro holds at m if and only if one has (2.3.1).

24

ˇ 2. LOPATINSKI˘I–SAPIRO BOUNDARY CONDITIONS

ν | or |v  | ν Case 2: d = 3. If either |tm |g∂ < |vm m g∂ < |tm |, then (2.3.6) ∗ ∂M. Thus, in this case, the Lopatinski˘ ˇ cannot hold for any ω  ∈ Sm ı–Sapiro condition holds at m. ν | and |v  | ν Consider now the case |tm |g∂ ≥ |vm m g∂ ≥ |tm |. Each equation in (2.3.6) has nontrivial solutions. We seek conditions so that such solutions can coincide. ν = 0 and then there exists such a common First, if tm = 0, one has vm solution; moreover one has   2   2 ν 2 0 = gm (tm , vm )2 = |tm |g∂ − (vm ) |vm |g∂ − (tνm )2 .

ˇ Hence, in such case where the Lopatinski˘ı–Sapiro condition does not hold at m conditions (2.3.2) and (2.3.3) are not fulfilled. The same reasoning  = 0. applies to the case vm  = 0. Assume that there Second, we consider the case tm = 0 and vm  ∗ ∗ ∂M so that ˜  ∈ Sm exists ω ∈ Sm ∂M such that (2.3.6) holds. We pick ω  ∗ (∂M). Then, ω  , ω (ω  , ω ˜  ) forms an orthonormal basis of Tm ˜  forms an orthonormal basis of Tm (∂M).  /|v  | θ  as We set T  = tm /|tm |g∂ and V  = vm m g∂ and we set θT  and   V      ˜ . the angles between ω and T and V , respectively, in the frame ω , ω Then, θ = θV  − θT  is the angle between T  and V  . From (2.3.6) we have cos θT  = −

ν vm tνm  = and cos θ . V  | |tm |g∂ |vm g∂

This is illustrated in Fig. 2.1. We have  ν 2 1/2  ν 2 1/2   vm t and sin θV  = ± 1 − m 2 , sin θT  = ± 1 − 2  | |tm |g∂ |vm g∂ and thus (2.3.7) (g∂ )m (T  , V  ) = cos θ

 ν 2   ν 2 

ν tν 1/2 1/2 t vm vm m 1− m2 ± 1− , =−  2   | |tm |g∂ |vm |g∂ |tm |g∂ |vm g∂

that is,

 ν 2   ν 2 

1/2 1/2 t vm gm (tm , vm ) 1− m2 =± 1− . 2    | |tm |g∂ |vm |g∂ |tm |g∂ |vm g∂

This is precisely the negation of condition (2.3.3). We have thus found that ν | and |v  | ν  if |tm |g∂ ≥ |vm m g∂ ≥ |tm |, and if (2.3.3) holds (implying that tm = 0  ˇ and vm = 0), then the Lopatinski˘ı–Sapiro condition holds at m.  ν |, |v  | ν  Conversely, let us assume that |tm |g∂ ≥ |vm m g∂ ≥ |tm |, tm = 0, and  vm = 0 and moreover   2   2 ν 2 (2.3.8) ) |vm |g∂ − (tνm )2 gm (tm , vm )2 = |tm |g∂ − (vm

2.3. FIRST-ORDER BOUNDARY OPERATOR WITH COMPLEX COEFFICIENTS 25

ω ˜ V

T θT θ

−

ν vm

|tm |g∂M

θV

ω tνm

0

|vm |g∂M

Sm ∂M

Figure 2.1. Geometry associated with a solution of (2.3.6)  = 0 in the case tm = 0 and vm  /|v  | . We also pick holds. As above, we set T  = tm /|tm |g∂ and V  = vm m g∂    U ∈ Tm ∂M such that (T , U ) is an orthonormal basis of Tm ∂M. Then, we have V  = cos(θ)T  + sin(θ)U  for some θ ∈ [0, 2π). We also set tν vν a =  m and b =  m . |tm |g∂ |vm |g∂  2 Condition (2.3.8) reads cos(θ) + ab = (1 − a2 )(1 − b2 ) or equivalently

(2.3.9)

a2 + b2 + 2ab cos(θ) = sin2 (θ).

From (2.3.6) we seek ω  of the form ω  = −a(T  ) + y(U  ) for some y ∈ R such that a2 + y 2 = 1 and ω  , V   = −a cos(θ)  + y sin(θ) = b. If sin(θ) = 0. We then set y = sin(θ)−1 b + a cos(θ) . With (2.3.9) we find a2 + y 2 = 1. We thus found a solution to (2.3.6). If sin(θ) = 0. If θ = 0, then V  = T  and (2.3.9) yields a = −b. If θ = π, then V  = −T  and (2.3.9) yields a = b. In both cases, ω  = √ −a(T  ) + 1 − a2 (U  ) is a solution to (2.3.6) such that |ω  |g∂ = 1. ˇ In both cases, the Lopatinski˘ı–Sapiro condition does not hold. This concludes the case d = 3. ν | or Case 3: d ≥ 4. As for the previous case, if either |tm |g∂ < |vm  | ν  ∗ |vm g∂ < |tm |, then (2.3.6) cannot hold for any ω ∈ Sm ∂M. Thus, in this ˇ case, the Lopatinski˘ı–Sapiro condition holds at m.

26

ˇ 2. LOPATINSKI˘I–SAPIRO BOUNDARY CONDITIONS

ν | and |v  | ν Consider now the case |tm |g∂ ≥ |vm m g∂ ≥ |tm |.  ν First, if tm = 0, we have vm = 0 and then there exists ω  solution to both equations in (2.3.6); moreover, we have    2  2 ν 2 ) |vm |g∂ − (tνm )2 . 0 = gm (tm , vm )2 = |tm |g∂ − (vm

ˇ Hence in such case where the Lopatinski˘ı–Sapiro condition does not hold at m conditions (2.3.4)–(2.3.5) are not fulfilled. The same reasoning applies to  = 0. the case vm ν = 0. Then, as dim T ∂M ≥ 3, Second, we consider the case tνm = vm m  ∗    = 0. In such case there exists ω ∈ Sm ∂M such that ω , tm  = ω  , vm ˇ the Lopatinski˘ı–Sapiro condition does not hold. Observe that both conν = 0 (2.3.5) reads dition in (2.3.4) clearly do not hold. With tνm = vm 2 2   2   (g∂ )m (tm , vm ) > |tm |g∂ |vm |g∂ , contradicting the Cauchy–Schwarz inequalˇ ity. We have thus found that in this case where the Lopatinski˘ı–Sapiro condition does not hold at m conditions (2.3.4) and (2.3.5) are not fulfilled.  = 0 and (tν , v ν ) = (0, 0). Third, we consider the case tm = 0 and vm m m ∗ ∂M such that both equations in (2.3.6) Assume that there exists ω  ∈ Sm ∗ ∂M as the orthogonal projection of ω  onto hold. We then define α ∈ Tm 



span((tm ) , (vm ) ). We then have |α |g∂ ≤ 1 and ν  α , tm  = −vm and α , vm  = tνm . ν ) = (0, 0) we have α = 0. We set T  = t /|t | , V  = v  /|v  | , As (tνm , vm m m g∂ m m g∂  and A = α /|α |g∂ . We have vν tν and A , V   =  m  . A , T   = −  m |α |g∂ |tm |g∂ |α |g∂ |vm |g∂

The analysis that led to (2.3.7) applies here with ω  replaced by A yielding

ab 2 b2  a2  cos θ +  2 (2.3.10) 1−  2 , = 1−  2 |α |g∂ |α |g∂ |α |g∂ where cos θ = (g∂ )m (T  , V  ) and a=

ν vm  |tm |g∂

and b =

tνm .  | |vm g∂

With some simple computations we then find a2 + b2 + 2ab cos(θ) = |α |2g∂ sin2 (θ) ≤ sin2 (θ). Performing the above computations backward (formally with |α |g∂ now equal to one) we obtain (cos θ + ab)2 ≤ (1 − a2 )(1 − b2 ), that in turn reads

  2   2 ν 2 gm (tm , vm )2 ≤ |tm |g∂ − (vm ) |vm |g∂ − (tνm )2 .

2.3. FIRST-ORDER BOUNDARY OPERATOR WITH COMPLEX COEFFICIENTS 27

This is precisely the negation of condition (2.3.5). We have thus found that ν | and |v  | ν   ν ν if |tm |g∂ ≥ |vm m g∂ ≥ |tm |, if tm = 0, vm = 0 and (tm , vm ) = (0, 0), ˇ and if (2.3.5) holds, then the Lopatinski˘ı–Sapiro condition holds at m.  ν |, |v  | ν  Conversely, let us assume that |tm |g∂ ≥ |vm m g∂ ≥ |tm |, tm = 0,  ν ν vm = 0, (tm , vm ) = (0, 0), and moreover    2  2 ν 2 ) |vm |g∂ − (tνm )2 . (2.3.11) gm (tm , vm )2 ≤ |tm |g∂ − (vm  /|v  | . We also set As above, we set T  = tm /|tm |g∂ and V  = vm m g∂ ν vm tνm and b = .  | |tm |g∂ |vm g∂  2 Condition (2.3.11) reads cos(θ) + ab ≤ (1 − a2 )(1 − b2 ) or equivalently

a=

(2.3.12)

a2 + b2 + 2ab cos(θ) ≤ sin2 (θ).

The argument is different depending on the dimension of span(T  , V  ). Case dim span(T  , V  ) = 1. If V  = T  , then cos(θ) = 1 and thus (2.3.12) yields (a + b)2 ≤ 0, that is, a = −b. If V  = −T  , then cos(θ) = −1 and thus (2.3.12) yields (a − b)2 ≤ 0, that is a = b. In both cases, ˜  is a solution to (2.3.6) such that |ω  |g∂ = 1 if ω ˜  is chosen ω  = −a(T  ) + ω orthogonal to T  (and thus V  ) and |˜ ω  |2g∂ = 1 − a2 .   Case dim span(T , V ) = 2. We pick U  ∈ span(T  , V  ) such that (T  , U  ) is an orthonormal basis of span(T  , V  ). Then, we have V  = ˜ cos(θ)T  + sin(θ)U  for some θ ∈ [0, 2π). From (2.3.6) we seek ω  = α + ω  with α of the form α = −a(T  ) + y(U  ) , for some y ∈ R such that a2 + y 2 ≤ 1 and ω  , V   = −a cos(θ) + y sin(θ) = b, and ω ˜  orthogonal to span(T  , V  ) with |˜ ω  |2g∂ = 1 − a2 − y 2 . We then set   y = sin(θ)−1 b + a cos(θ) . With (2.3.12) we find a2 + y 2 ≤ 1. We thus found a solution to (2.3.6). ˇ In both cases, the Lopatinski˘ı–Sapiro condition does not hold. This concludes the case d = 4.  ˇ The Lopatinski˘ı–Sapiro condition can be very restrictive in the choices of the operator B. For instance, cases of purely tangential operators B are very limited if d ≥ 3 as presented in the following remark and proposition. Remark 2.9 (Cases of Boundary Operators with Purely Tangential Action). ν ) = (0, 0), then condition (2.3.1) always (1) In the case d = 2, if (tνm , vm   holds since tm +ivm does not vanish. A typical example is M given by the closed unit disc D, equipped with the flat metric, and with polar coordinates (r, θ) the operator B = v with v given by v = ∂θ ν = 0 and v  does not vanish. The proposition shows we see that vm m ˇ that in such a case the Lopatinski˘ı–Sapiro condition holds.

ˇ 2. LOPATINSKI˘I–SAPIRO BOUNDARY CONDITIONS

28

ν ) = (0, 0), then one can prove that (2) In the case d = 3, if (tνm , vm necessarily the connected component of m in ∂M is a torus. This result is in fact contained in Proposition 2.10 below. (3) In the case d ≥ 4, the form of conditions (2.3.4) and (2.3.5) shows that the occurrence of a boundary operators with purely tangenˇ tial action is prohibited under the Lopatinski˘ı–Sapiro condition. In     | fact, in such case we find that |g∂ (tm , vm )| > |tm |g∂ |vm g∂ in contradiction with the Cauchy–Schwarz inequality.

Proposition 2.10. Let M be of dimension d = 3. Let the boundary operator B be of order β = 1 with nonvanishing principal symbol b(m, ω) = ω, tm  + iω, vm , where t, v are two real vector fields on M along ∂M. We write vm = ν ν + v  , t = tν ν + t , with v ν , tν ∈ R and v  , t ∈ T ∂M. vm m m m m m m m m m m m Let N be a connected component of ∂M. Assume that the Lopatinski˘ı– ˇ Sapiro condition holds for (P, B) at every point of the closed surface1 N and moreover there exists m0 ∈ N such that (2.3.13) and (2.3.14)

ν  | and |vm | ≥ |tνm0 | |tm0 |g∂ ≥ |vm 0 0 g∂

   2  2 ν 2 ) |vm | − (tνm0 )2 . gm0 (tm0 , vm0 )2 < |tm0 |g − (vm 0 0 g ∂

∂

Then, N is a torus and conditions (2.3.13) and (2.3.14) hold at every point  )=T M of N . Moreover, t and v  form a frame for N , that is, span(tm , vm m for all m ∈ N . Example 2.11. An example of manifold that fits the framework of Proposition 2.10 is given by, for 0 < R < r0 , M = {(r, θ, z); (r − r0 )2 + z 2 ≤ R2 } ⊂ R3 , in cylindrical coordinates, equipped with the Euclidean metric inherited from R3 . Then ∂M is a torus, and for B = v + it with vm = ∂θ and ν = tν = 0 and dim span(t , v  ) = 2. tm = −z∂r + (r − r0 )∂z we see that vm m m m Proof of Proposition 2.10. We proceed by contradiction. Assume that m1 ∈ N is such that ν  | or |vm | < |tνm1 | |tm1 |g∂ < |vm 1 1 g∂

or

   2  2 ν 2 ν 2 ) | − (t ) |v . gm1 (tm1 , vm1 )2 ≥ |tm1 |g − (vm m m 1 1 g 1 ∂

∂

1By closed surface one means a compact manifold without boundary of dimension two.

2.3. FIRST-ORDER BOUNDARY OPERATOR WITH COMPLEX COEFFICIENTS 29

Let now γ : [0, 1] → N be a continuous path such that γ(0) = m0 and γ(1) = m1 . Set S = {s ∈ [0, 1]; (2.3.13)and(2.3.14) holds at γ(s ) for all s ∈ [0, s]}. Set s = sup S. We have s < 1 and, by a continuity argument, (2.3.13) holds at m = γ(s) and    2  2 ν 2 ) |vm |g∂ − (tνm )2 . gm (tm , vm )2 ≤ |tm |g∂ − (vm ˇ Since the Lopatinski˘ı–Sapiro condition holds everywhere on N , from Proposition 2.8 we deduce that    2  2 ν 2 ) |vm |g∂ − (tνm )2 . (2.3.15) gm (tm , vm )2 < |tm |g∂ − (vm Thus condition (2.3.14) holds for s ≤ s ≤ s + ε for some ε > 0. For any s < s ≤ s + ε since (2.3.13) and (2.3.14) does not hold this means that ν  | or |vm |g∂ < |tνm | either |tm |g∂ < |vm

for m = γ(s).

Thus, either there exists (sn )n with s < sn ≤ s + ε and sn → s such that ν | for m = γ(s ) or there exists (s ) with s < s ≤ s + ε |tmn |g∂ < |vm n n n n n n   | ν   and sn → s such that |vm  g∂ < |tm | for mn = γ(sn ). Since we have n n ν  | and |vm |g∂ ≥ |tνm |, |tm |g∂ ≥ |vm

this implies that ν  | or |vm |g∂ = |tνm |. either |tm |g∂ = |vm

Then the r.h.s. of (2.3.15) vanishes at m yielding a contradiction. Hence, conditions (2.3.13) and (2.3.14) hold at every point of N . By Lemma 2.12 given below, used in the case s = 0, one finds for every m∈N   )| < |tm |g∂ |vm | g∂ . |(g∂ )m (tm , vm

This strict estimate in the Cauchy–Schwarz inequality yields  ) = 2. dim span(tm , vm  With t and v  , the closed surface N exhibits nonvanishing real vector fields. By a classical result of differential topology the Euler characteristic of N is zero [310, Corollary 39.8]. By the classification of closed surfaces [245, page 30], this leaves two possibilities: N is either a torus or a Klein bottle.  ) = T M for all m ∈ N , then the vector fields t and v  As span(tm , vm m form a frame. In particular, this yields an orientation on the surface N ; see Sect. 16.1.1. This rules out the possibility of N being a Klein bottle as this latter surface is not orientable.  Lemma 2.12. Under the assumptions of Proposition 2.10, for s ∈ [0, 1] define the real vector fields t(s) and v(s) by (t(s) )m = stνm νm + tm ,

ν  (v(s) )m = svm νm + vm .

30

ˇ 2. LOPATINSKI˘I–SAPIRO BOUNDARY CONDITIONS

Then, for any m ∈ N and any s ∈ [0, 1]  2    2  2 ν 2 ) |vm |g∂ − (stνm )2 . gm (t(s) )m , (v(s) )m < |tm |g∂ − (svm (2.3.16) Proof. After simple computations, condition (2.3.14) reads ν 2  ν  ) |vm |g∂ + (tνm )2 |tm |g∂ + 2tνm vm (g∂ )m (tm , vm ) (vm 2

2

  2 |g∂ − (g∂ )m (tm , vm ) . < |tm |g∂ |vm 2

2

With the Cauchy–Schwarz inequality one can estimate the l.h.s. from below as follows ν 2  ν  ) |vm |g∂ + (tνm )2 |tm |g∂ + 2tνm vm (g∂ )m (tm , vm ) (vm 2

2

ν 2  ν  ≥ (vm ) |vm |g∂ + (tνm )2 |tm |g∂ − 2|tνm vm ||vm |g∂ |tm |g∂  2 ν  ≥ |tνm ||tm |g∂ − |vm ||vm | g∂ 2

2

≥ 0. Thus, for s ∈ [0, 1]  ν 2  2  2 ν  s2 (vm ) |vm |g∂ + (tνm )2 |tm |g∂ + 2tνm vm (g∂ )m (tm , vm )   2 |g∂ − (g∂ )m (tm , vm ) , < |tm |g∂ |vm 2

2



which is precisely (2.3.16). 2.4. Notes

ˇ The now called Lopatinski˘ı–Sapiro conditions (with various spellings and author orders) originate from the 1953 articles of Y. B. Lopatinski˘ı [238] ˇ and Z. Y. Sapiro [323]. The conditions where further studied in the works of S. Agmon, A. Douglis and L. Nirenberg [3, 4] and M. S. Agranoviˇc and A. S. Dynin [5]. In fact, elliptic problems on a bounded open set or a manifold are well-posed or at least of Fredholm type (finite dimension kernel and co-kernel) if proper boundary conditions are imposed. These conditions ˇ are precisely the Lopatinski˘ı–Sapiro boundary conditions. This is recalled in Chap. 3. Dirichlet, Neumann, Robin-type boundary conditions are all ˇ particular cases of Lopatinski˘ı–Sapiro conditions in the elliptic case. A good reference for the study of general elliptic operators of arbitrary order along ˇ with Lopatinski˘ı–Sapiro conditions is the work of L. H¨ ormander [175, Section 20.1]. ˇ Note that for nonelliptic problems, Lopatinski˘ı–Sapiro conditions do not provide all conditions under which a problem is well-posed. For example, the mixed boundary value problem for the wave equation with Cauchy data at t = 0 and a homogeneous Neumann boundary condition for (t, m) ∈ [0, +∞) × ∂M is well-posed. However, the Neumann boundary condition ˇ criterium in the ∂ν u|[0,+∞)×∂M = 0 fails to satisfy the Lopatinski˘ı–Sapiro 2 case of the wave operator ∂t − Δg .

2.4. NOTES

31

Here, we only treat the case of second-order elliptic operators as in the ˇ whole book. Lopatinski˘ı–Sapiro conditions are stated in Definition 2.2 in the form of an algebraic criterium and given in the simple form ˇb(m, ω  , i|ω  |g ) = 0 in Proposition 2.3. In Sect. 2.3 we fully describe first-order boundary ˇ operators that fulfill the Lopatinski˘ı–Sapiro conditions. For higher-order ˇ elliptic operators the Lopatinski˘ı–Sapiro conditions are stated in Remark 2.7. The conditions given therein can in fact be written as a rank condition for some matrix built from the principal symbols of the elliptic operator and the boundary operator.

CHAPTER 3

Fredholm Properties of Second-Order Elliptic Operators Contents 3.1. Setting and Main Result 3.2. Analysis in a Half-Space 3.2.1. Local Setting Near a Point of the Boundary 3.2.2. Action of a Parametrix on a Half-Space 3.2.3. The Calder´ on Projector 3.2.4. Pseudo-Differential Symbol Computations 3.2.5. Expressing All Traces from the Boundary Condition 3.3. Proof of the Fredholm Property 3.3.1. Charts Away from the Boundary 3.3.2. Charts at the Boundary 3.3.3. Conclusion of the Proof of the Fredholm Property ˇ 3.4. Necessity of the Lopatinski˘ı–Sapiro Conditions 3.5. Some Index Computations 3.5.1. Dimension d ≥ 4 3.5.2. Dimension d = 3 3.5.3. Dimension d = 2 3.6. Additional Regularity Results 3.7. Notes Appendix 3.A. Proof of Some Technical Results 3.A.1. An A Priori Estimate Consequence of the Fredholm Property 3.A.2. Fredholm Index Independent of the Regularity Level 3.A.3. Sobolev Regularity of the Parametrix Action 3.A.4. Properties of the Calder´ on Projector

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 3

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33

34

3. FREDHOLM PROPERTIES

3.A.5. 3.A.6. 3.A.7.

Pseudo-Differential Form of the Action on the Traces Recovery of the Traces up to Regularizing Operators Local Right and Left Inverses up to a Compact Operator

74 77 79

3.1. Setting and Main Result On a smooth compact Riemannian manifold (M, g) with boundary, we set P = −Δg + R1 , where Δg denotes the Laplace–Beltrami operator whose form is recalled in Sect. 17.2 and R1 is a first-order differential operator on M. We consider a differential boundary operator B of order β ∈ N defined on ∂M (or equivalently, in a neighborhood of ∂M, see Remark 2.1) with differentiation transverse to ∂M of order less than or equal to one. If ∂M is not connected, the order of the operator may vary from one connected component to the other. Recall that the number of connected components of ∂M is finite since M is compact. Consider N a connected component of ∂M where B is of order k; then on N the boundary operator takes the form B = B k + B k−1 ∂ν , where both B k and B k−1 are differential operators on N of order k and k − 1, respectively. If u ∈ H 2 (∂M), then u|N ∈ H 3/2 (N ) and ∂ν u|N ∈ H 1/2 (N ) by the trace formulae of Theorem 18.25. With the usual continuity result for differential operators recalled in Proposition 18.23, one then finds (3.1.1)

Bu|N = B k u|N + B k−1 ∂ν u|N ∈ H 3/2−k (N ).

∗ M, its principal symbol. We denote by b(m, ω), m ∈ ∂M and ω ∈ Tm ˇ We shall be interested in cases where the Lopatinski˘ı–Sapiro condition of Definition 2.2 holds on the whole ∂M. If B is of order zero on a connected component of ∂M, then the condition is of Dirichlet type by Example 2.5(1). The elliptic problem with this type of condition on the whole ∂M is treated in Sect. 18.6, partly with  the analysis based on that of Section 10.1 of Volume 1 in the case P = 1≤i,j≤d Di (pij (x)Dj ) in a regular open set of Rd . There, we invoke results proven by means of a variational formulation of the elliptic problem. The analysis proposed in the present chapter applies ˇ to this case and to all boundary conditions that fit the Lopatinski˘ı–Sapiro framework. However, we do not prove well-posedness here (as it may not hold) but rather consider Fredholm properties. On each connected component of ∂M, the differential boundary operator B is of order k ∈ {0, . . . , β}. For each k ∈ {0, . . . , β}, we denote by k∂M the union of all the connected components of ∂M where the order of B is exactly k. For r ∈ R, we set  (r) (3.1.2) H r−k (k∂M). HB (∂M) = 0≤k≤β

Observe that k∂M may possibly be empty for some k ∈ {0, . . . , β}.

3.1. SETTING AND MAIN RESULT

35

For m ∈ N, we study the Fredholm property of the operator (3.1.3)

(m+3/2)

L : H m+2 (M) → H m (M) ⊕ HB

(∂M)

u → (P u, Bu|0∂M , . . . , Bu|β∂M ), which is bounded by (3.1.1). Fredholm operators are recalled in Section 11.5 of Volume 1. Having the case of the Neumann boundary condition in mind, this is quite natural since in such case the map L is neither injective nor surjective, but rather Fredholm of index zero, and well-posedness only makes sense after applying quotients with respect to finite dimensional subspaces; see Sect. 18.7. If 1∂M = ∅, basic examples of boundary operators of order one for which ˇ the Lopatinski˘ı–Sapiro condition holds are given in Examples 2.5-(2) to 2.5(3) and the general case is analyzed in Sect. 2.3. If 2∂M = ∅, an example of a boundary operator of order two is given in Example 2.5-(4). The main result we prove in this chapter is the following. Theorem 3.1. The operator L is Fredholm if and only if (P, B) fulfills ˇ the Lopatinski˘ı–Sapiro condition on ∂M. To prove Theorem 3.1, we shall first establish the following result. By ˇ Theorem 11.7 of Volume 1, this implies that the Lopatinski˘ı–Sapiro condition is sufficient for the Fredholm property of L to hold. Proposition 3.2. Let m ∈ N. Assume that (P, B) fulfills the Lopatinˇ ski˘ı–Sapiro condition on ∂M. There exists a bounded linear operator (m+3/2)

M : H m (M) ⊕ HB

(∂M) → H m+2 (M)

such that M L = IdH m+2 (M) +K  and LM = IdH m (M)⊕H (m+3/2) (∂M) +K r , B

where both operators K  : H m+2 (M) → H m+3 (M) (m+3/2)

K r : H m (M) ⊕ HB

(m+5/2)

(∂M) → H m+1 (M) ⊕ HB

(∂M)

are bounded. By the Rellich–Kondrachov theorem (see Theorem 18.7), K  is compact (m+3/2) from H m+2 (M) into itself and K r is compact from H m (M)⊕HB (∂M) into itself. The proof of Proposition 3.2 is performed in Sect. 3.3 based on the analysis in a half-space of Sect. 3.2. The second part of Theorem 3.1, that ˇ is, the necessity of Lopatinski˘ı–Sapiro conditions, is proven in Sect. 3.4. Theorem 3.1 states in particular that L is not Fredholm if the Lopatinˇ ski˘ı–Sapiro condition does not hold. This is illustrated in Example 2.6 A corollary of Theorem 3.1 is the following one.

36

3. FREDHOLM PROPERTIES

ˇ Corollary 3.3. Assume that (P, B) fulfills the Lopatinski˘ı–Sapiro condition on ∂M, and let m ∈ N. Then, there exists C > 0 such that   uH m+2 (M) ≤ C P uH m (M) + |Bu|∂M |H (m+3/2) (∂M) + uL2 (M) , B

for all u ∈ H m+2 (M). A proof is given in Appendix 3.A.1. Note that this result gives a precise form for operator K1 and the space Z1 in the first inequality in Proposition 11.10 of Volume 1 if applied to the present case. The following proposition states that the Fredholm index of L is independent of the chosen value of m. This explains why we chose to write L in place of L(m) . Proposition 3.4. The values of nul L, def L, and ind(L) are independent of the chosen value m ∈ N. Note that in fact ker(L) is independent of m. We refer to Appendix 3.A.2 for a proof. In Sect. 3.5 in the case β ≤ 1, we compute the index of L in some cases: in dimension d ≥ 3 the index is zero, and in dimension d = 2 this is not always the case. 3.2. Analysis in a Half-Space 3.2.1. Local Setting Near a Point of the Boundary. At a point m0 ∈ ∂M, we consider a local chart C = (O, κ) characterized by normal geodesic coordinates; see Sect. 17.6. We recall that κ(O) is an open set of Rd+ and κ(∂M ∩ O) = {xd = 0} ∩ κ(O), with x = (x , xd ), x ∈ Rd−1 and xd ∈ R. The local representative of P takes the form P C (x, D) = Dd2 + R(x, D ) + r0 (x)Dd , where R(x, D ) ∈ DT2 is a tangential differential operator with xd as a parameter and r0 is a smooth function. Note that O meets only one component of ∂M. Thus, O ∩ ∂M ⊂ k∂M for some k ∈ {0, . . . , β}. In what follows, we drop the superscript C since there is no possible confusion. We thus write P (x, D) and B(x, D) in place of P C (x, D) and B C (x, D), respectively. If we denote by r(x, ξ  ) the principal symbol of R(x, D ), in the sense of standard operators (see Section 2.12 of Volume 1), then ξ  → r(x, ξ  ) is a positive definite quadratic form and, for some C > 0, r(x, ξ  ) ≥ C|ξ  |2 ,

x ∈ κ(O), ξ  ∈ Rd−1 .

3.2. ANALYSIS IN A HALF-SPACE

37

For convenience, the coefficients of the operator R(x, D ) and the function r0 (x) are extended to Rd , yet preserving the elliptic property1 of R(x, D ). We pick ρ(x, ξ  ) ∈ ST1 such that ρ(x, ξ  ) > 0 and ρ(x, ξ  ) = r(x, ξ  )1/2

for |ξ  | ≥ 1.

Remark 3.5. As a reminder, on the manifold level, for |ξ  | ≥ 1, we have ρ(x, ξ  ) = |ω  |g , if x = (x , 0) = κ(m) with m ∈ ∂M and if (ξ  , 0) is ∗ (∂M). This is useful below to write the the local representative of ω  ∈ Tm ˇ Lopatinski˘ı–Sapiro condition in the local coordinates. The principal symbol of P (x, D) is given by p(x, ξ) = ξd2 + r(x, ξ  ), and one has p(x, ξ)  |ξ|2 for x ∈ Rd , ξ ∈ Rd . For |ξ  | ≥ 1, we have    (3.2.1) p(x, ξ) = ξd − iρ(x, ξ  ) ξd + iρ(x, ξ  ) . As mentioned above, O ∩ ∂M ⊂ k∂M for some k ∈ {0, . . . , β}, meaning that the boundary operator B is exactly of order k in O ∩∂M. In the chosen local coordinates, we may write, in a neighborhood of (or at) the boundary, B(x, D) = B k (x, D ) − iB k−1 (x, D )Dd ,

(3.2.2)

with B k (x, D ) and B k−1 (x, D ) differential operators in the x variables of order k and k − 1, respectively, with principal symbols bk (x, ξ  ) ∈ STk and bk−1 (x, ξ  ) ∈ STk−1 . Note that if k = 0, we take b1 (x, ξ  ) = 1 and b−1 (x, ξ  ) = 0 as it may be obtained upon division by a suitable nonvanishing function ˇ since the Lopatinski˘ı–Sapiro condition is assumed to hold; see Example 2.5(1). ˇ According to Proposition 2.3 and Remark 3.5, the Lopatinski˘ı–Sapiro condition reads   k (3.2.3) b + bk−1 ρ |x =0+ (x , ξ  ) = 0, d

for all

(x , 0)

∈ κ(O ∩ ∂M), and all ξ  ∈ Rd−1 , with |ξ  | ≥ 1.

3.2.2. Action of a Parametrix on a Half-Space. Let u ∈ H m+2 (Rd+ ) and set f = P (x, D)u. Denoting by Y = Y (xd ) the Heaviside function in the xd variable, we set u = Y u and f = Y f ; we have u, f ∈ L2 (Rd ). We also set γ D (u) = u|xd =0 and γ N (u) = ∂ν u|xd =0 = −∂d u|xd =0 = −iDd u|xd =0 . Observing that Dd u = −iδ|xd =0 ⊗ γ D (u) + Y Dd u Dd2 u = −δx d =0 ⊗ γ D (u) + δxd =0 ⊗ γ N (u) + Y Dd2 u,

1Such an extension is carried out explicitly at the beginning of the proof of Theorem 2.28 in Section 2.A.6.1 of Volume 1.

38

3. FREDHOLM PROPERTIES

we obtain

  P (x, D)u = f − δx d =0 ⊗ γ D (u) + δxd =0 ⊗ γ N (u) − ir0 γ D (u) ,

in the sense of distributions in Rd . This leads us to define γ˜ N = γ N − ir0 γ D , which is bounded from H m+2 (Rd+ ) into H m+1/2 (Rd−1 ). We thus have P (x, D)u = f − δx d =0 ⊗ γ D (u) + δxd =0 ⊗ γ˜ N (u). We now use a parametrix of P (x, D) in Rd . By Proposition 2.33 of Volume 1 (adapted to standard operators), as p(x, ξ)  |ξ|2 for (x, ξ) ∈ Rd × Rd , there exists q(x, ξ) ∈ S −2 (Rd × Rd ), with p(x, ξ)−1 as principal symbol for |ξ| ≥ 1, such that Q = Op(q) ∈ Ψ−2 (Rd ) and QP = Id −R and P Q = Id −R ,

(3.2.4)

with R, R ∈ Ψ−∞ (Rd ) = ∩N ∈N Ψ−N (Rd ). This yields

 u = Qf − Q(δx d =0 ⊗ γ D (u)) + Q δxd =0 ⊗ γ˜ N (u)) + Ru.

(3.2.5)

We define Q : L2 (Rd+ ) → H 2 (Rd+ )   f → Qf |Rd .

(3.2.6)

+

and the restriction of a It is well defined and continuous since Qf ∈ 2 d d H -function in R to R+ yields (continuously) a function in H 2 (Rd+ ). Thus, H 2 (Rd )

Q(f )H 2 (Rd )  f L2 (Rd ) . +

+

For h ∈ S  (Rd−1 ) and j ∈ {−1, 0}, we also define   (j+1) (3.2.7) Q (j) (h) = Q(δxd =0 ⊗ h) |Rd ∈ D  (Rd+ ), +

where δxd =0 = δxd =0 and δxd =0 = δx d =0 . As Q maps S  (Rd ) into itself, the operators Q (j) are well defined. From (3.2.5), we have   (3.2.8) γ N (u)) + Ru |Rd . u = Q(f ) − Q (0) (γ D (u)) + Q (−1) (˜ (0)

(1)

+

The following lemma improves the bound of Q and provides bounds for the operators Q (j) , for j = −1, 0. Lemma 3.6. Let m ∈ N. (1) The map Q is bounded from H m (Rd+ ) into H m+2 (Rd+ ) and P (x, D)Q = Id +K, with K bounded from H m (Rd+ ) into H N (Rd+ ) for any N ∈ N.

3.2. ANALYSIS IN A HALF-SPACE

39

(2) If j ∈ {−1, 0}, the map Q (j) is bounded from H m+j−1/2 (Rd−1 ) into H m (Rd+ ) and P (x, D)Q (j) = K (j) , with K (j) bounded from H m+j−1/2 (Rd−1 ) into H N (Rd+ ) for any N ∈ N. We refer to Appendix 3.A.3 for a proof. Remark 3.7. Note that we shall first use that Q (j) is bounded from into H m+2 (Rd+ ). However, the finer result of Lemma 3.6 is of use in Sect. 3.6. H m+j+3/2 (Rd−1 )

3.2.3. The Calder´ on Projector. Let φ ∈ H m (Rd+ ) and h ∈ H m+j+3/2 (Rd−1 ) for j ∈ {−1, 0}. With the regularity obtained in Lemma 3.6, that is, Q(φ), Q (j) (h) ∈ H m+2 (Rd−1 ), the traces of Q(φ), Q (j) (h) can be considered at xd = 0+ . With the trace formula of Theorem 18.25, we may thus introduce the following bounded maps: (3.2.9) d−1 ) (R QD : H m (Rd+ ) → H m+3/2   D φ → γ Q(φ) ,

d−1 ) (R QN : H m (Rd+ ) → H m+1/2   N φ → γ˜ Q(φ) ,

and, for j ∈ {−1, 0}, (j)

(3.2.10)

d−1 QD : H m+j+3/2 (Rd−1 ) → H m+3/2  (j)(R  ) D h → γ Q (h) ,

and (j+1)

(3.2.11)

QN

: H m+j+3/2 (Rd−1 ) → H m+1/2 (Rd−1   ) N (j) h → γ˜ Q (h) . (j+1)

is to be justified in Sect. 3.2.4. The shift j → j + 1 in the definition of QN (k) k d−1 In fact, there one proves that QN ∈ Ψ (R ), k = 0, 1. We also have (j) QD ∈ Ψj (Rd−1 ), j = −1, 0. We also introduce the following operators: (0)

(3.2.12) CD,D = −QD , and the matrix operator (3.2.13)

(−1)

CD,N = QD

(1)

, CN,D = −QN ,

(0)

CN,N = QN ,

 D,D CD,N C , C= CN,D CN,N

which is bounded from H m+3/2 (Rd−1 ) ⊕ H m+1/2 (Rd−1 ) into itself. From (3.2.8), computing the Dirichlet and Neumann traces at xd = 0+ , we obtain the following identity:     γ D (u) QD (f ) (3.2.14) = Id −C + RC (u), QN (f ) γ˜ N (u)

40

3. FREDHOLM PROPERTIES

for u ∈ H m+2 (Rd+ ) and f = P (x, D)u, and where RC is bounded from  2 H m+2 (Rd+ ) into H N (Rd−1 ) , for any N ∈ N. The following properties hold. Proposition 3.8. Let m ∈ N. The matrix operator C2 − C maps  2 H m+3/2 (Rd−1 ) ⊕ H m+1/2 (Rd−1 ) into H N (Rd−1 ) , for any N ∈ N. The map  QD (φ) (3.2.15) φ → C QN (φ)  2 is bounded from H m (Rd+ ) into H N (Rd−1 ) , for any N ∈ N. Since C2 = C up to a regularizing operator, one says that C is a projector. It is often referred to as the Calder´ on projector associated with the elliptic operator P (x, D). Observe that the regularizing aspect of the map (3.2.15) can easily be perceived upon applying the operator C to identity (3.2.14) and using the projector property of C. We refer to Appendix 3.A.4 for a proof of Proposition 3.8. From Proposition 3.8, computing C2 − C, we find that the operators (3.2.16)

CD,N CN,D + CD,D (CD,D − Id) : H m+3/2 (Rd−1 ) → H N (Rd−1 )

and (3.2.17)

CD,D CD,N + CD,N (CN,N − Id) : H m+1/2 (Rd−1 ) → H N (Rd−1 )

are bounded for any N ∈ N. 3.2.4. Pseudo-Differential Symbol Computations. With the following lemma, we further detail the structures of the operators Q (j) , for j = −1, 0. Lemma 3.9. Let h ∈ S (Rd−1 ). For xd ≥ 0, we have   Q (−1) (h)(x) = Q(δxd =0 ⊗ h)(xd , x ) = Op q (−1) (xd ) h(x ), and

  Q (0) (h)(x) = Q(δx d =0 ⊗ h)(xd , x ) = Op q (0) (xd ) h(x ),

with q (−1) (xd ) = t(−1) (xd )e(xd ) and q (0) (xd ) = t(0) (xd )e(xd ) such that 

(1) e(xd )(x , ξ  ) = e−xd ρ(x,ξ ) . (2) t(−1) (xd ) and t(0) (xd ) are smooth with values in S −1 (Rd−1 × Rd−1 ) and S 0 (Rd−1 × Rd−1 ), respectively. (3) Their principal symbols are 1 (−1) t−1 (xd )(x , ξ  ) = ρ−1 (x, ξ  ), 2 respectively.

1 (0) t0 (xd )(x , ξ  ) = − , 2

3.2. ANALYSIS IN A HALF-SPACE

41

A proof of Lemma 3.9 is given in Appendix 3.A.5. Observe that e(xd ) ∈ S 0 (Rd−1 × Rd−1 ). However, differentiation with respect to xd affects the symbol order; e(xd )(x , ξ  ) is thus not a tangential symbol in the sense given in Section 2.12 of Volume 1. The same applies to q (−1) (xd ) and q (0) (xd ). There is however no obstruction  (0) to con(−1) sider the pseudo-differential operators Op q (xd ) and Op q (xd ) as in Lemma 3.9. Lemma 3.10. We have the following properties: (1) For xd ≥ 0, xd → e(xd ) is bounded with values in S 0 (Rd−1 × Rd−1 ) and continuous with values in S 1 (Rd−1 × Rd−1 ). (2) Moreover, there exists C0 > 0 such that the map xd → eC0 xd e(xd ) with values in S 0 (Rd−1 × Rd−1 ) is bounded. (3) For n ∈ N and xd ≥ 0, ∂dn e(xd ) = a(xd )e(xd ) and xd → a(xd ) is smooth with values in S n (Rd−1 × Rd−1 ) and xd → ∂dn e(xd ) is bounded with values in S n (Rd−1 × Rd−1 ). In Sect. 3.2.3, we saw that with the regularity obtained in Lemma 3.6, the traces of Q (−1) (h) and Q (0) (h) can be considered at xd = 0+ . With oscillatory integrals as defined in Section 2.4 of Volume 1 and with Lemma 3.9, the action of the operator Q (j) , j = −1, 0, reads   Q (j) (h)(x) = Op q (j) (xd ) h(x ) = (2π)1−d



∫∫

Rd−1 ×Rd−1





ei(x −y )·ξ q (j) (xd )(x , ξ  )h(y  )dy  dξ  .

With Remark 2.13 in Volume 1, oscillatory integral regularization allows one to perform limits or differentiation under the sum sign. With Lemma 3.10, we thus obtain (3.2.18)      (j)  (j) QD (h) = Op q (j) (xd ) h |x =0+ (x ) = Op qD h(x ), j = −1, 0, d

(j)

with qD = t(j) (0) = q (j) (0), with principal symbols given by 1 (−1)   −1 qD,−1 (0)(x , ξ  ) = ρ−1 (Rd−1 × Rd−1 ), + (x , ξ ) ∈ S 2 |xd =0 1 (0) qD,−0 (0)(x , ξ  ) = − ∈ S 0 (Rd−1 × Rd−1 ). 2 We also obtain  (k) QN (h) = γ˜ N Op q (k−1) (xd )h (3.2.19)     = −i Dd Op q (k−1) (xd ) h |x =0+ (x ) d     − ir0 Op q (k−1) (xd ) h |x =0+ (x ) d  (k)   = Op qN h(x ),

42

3. FREDHOLM PROPERTIES (k)

for k = 0, 1, where qN ∈ S k (Rd−1 × Rd−1 ) with principal symbols given by 1 (0) qN,0 (x , ξ  ) = ∈ S 0 (Rd−1 × Rd−1 ), 2 1 (1) qN,1 (x , ξ  ) = − ρ|xd =0+ (x , ξ  ) ∈ S 1 (Rd−1 × Rd−1 ). 2 With the above characterizations, we find that the Calder´on projector C is a matrix pseudo-differential operator. Its matrix symbol is given by   (0) (−1) −qD (0) qD (0) (x , ξ  ), (1) (0) −qN (0) qN (0) yielding the principal symbol 1 c(x , ξ  ) = 2



1 ρ|xd =0+

ρ−1 |xd =0+ 1

 (x , ξ  ),

and one can check that c(x , ξ  )2 = c(x , ξ  ), in agreement with Proposition 3.8. 3.2.5. Expressing All Traces from the Boundary Condition. In the considered chart C = (O, κ), the boundary operator B of order k and P ˇ fulfill the Lopatinski˘ı–Sapiro condition in κ(∂M∩O). We consider B k (x, D ) k−1  and B (x, D ) as in (3.2.2). We denote B k (x , D ) = B k (x , xd = 0, D ) k−1 (x , D ) = B k−1 (x , xd = 0, D ), with coefficients smoothly exand B tended in the whole Rd−1 , preserving the operator orders. Similarly, we set r0 (x ) = r0 (x , xd = 0). We then set  −CD,N Id −CD,D L∂ = (3.2.20) . B k (x , D ) + iB k−1 (x , D )r0 (x ) B k−1 (x , D ) We set H r,s (Rd−1 ) = H r (Rd−1 ) ⊕ H s (Rd−1 ). We see that L∂ is a bounded operator from H s,s−1 (Rd−1 ) into H s,s−k (Rd−1 ), for any s ∈ R. We choose χ∂ ∈ Cc∞ (Rd−1 ) such that supp(χ∂ ) ⊂ κ(∂M ∩ O). Lemma 3.11. There exist operators M∂ , K∂ , and K∂r M∂ : H s,s−k (Rd−1 ) → H s,s−1 (Rd−1 ), K∂ : H s,s−1 (Rd−1 ) → H s+1,s (Rd−1 ), K∂r : H s,s−k (Rd−1 ) → H s+1,s+1−k (Rd−1 ), which are bounded for any s ∈ R, and such that the following hold: (1) For U∂ ∈ H s,s−1 (Rd−1 ) and F∂ ∈ H s,s−k (Rd−1 ), we have M∂ L∂ U∂ = χ∂ U∂ + K∂ U∂ , and L∂ M∂ F∂ = χ∂ F∂ + K∂r F∂ . (2) The operators M∂ , K∂ , and K∂r are 2 × 2 matrix operators  11  •,11 M∂ M∂12 K∂ K∂•,12 • , K∂ = , • = , r, M∂ = M∂21 M∂22 K∂•,21 K∂•,22

3.3. PROOF OF THE FREDHOLM PROPERTY

43

where (a) M∂11 ∈ Ψ0 (Rd−1 ), M∂12 ∈ Ψ−k (Rd−1 ), M∂21 ∈ Ψ1 (Rd−1 ), and M∂22 ∈ Ψ1−k (Rd−1 ); (b) K∂,11 , K∂,22 ∈ Ψ−1 (Rd−1 ), K∂,12 ∈ Ψ−2 (Rd−1 ), and K∂,21 ∈ Ψ0 (Rd−1 ); (c) K∂r,11 , K∂r,22 ∈ Ψ−1 (Rd−1 ), K∂r,12 ∈ Ψ−k−1 (Rd−1 ), and K∂r,21 ∈ Ψk−1 (Rd−1 ). We refer to Appendix 3.A.6 for a proof of Lemma 3.11. For u ∈ H m+2 (Rd+ ), if we consider the first row in (3.2.14) and set g to be the first component of t (QD f, QN f ) + RC (u), we obtain F∂ = L∂ U∂ , where  D γ (u) U∂ = ∈ H m+3/2,m+1/2 (Rd−1 ) γ˜ N (u) and

 F∂ =

g B(x, D)u|xd =0+

∈ H m+3/2,m+3/2−k (Rd−1 ).

If we apply Lemma 3.11, we thus see that we can recover all the traces of ˇ u expressed in U∂ , in the region where χ∂ ≡ 1, from the Lopatinski˘ı–Sapiro boundary data and g up to a regularizing operator. Remark 3.12. In the case of Dirichlet boundary conditions, with B 1 = 1 and B 0 = 0, observe that the second row of the operator M∂ yields the principal part of the so-called Dirichlet-to-Neumann map, here in the case of a interior source term g. 3.3. Proof of the Fredholm Property Here, we consider the case k∂M = ∅ for all k ∈ {0, . . . , β}. Other cases for which k∂M = ∅ for k in a subset of {0, . . . , β} can be treated similarly and yield simpler matrix operators in the proof. Recall that the notation k ∂M, k = 0, . . . , β, is introduced in Sect. 3.1. For each m ∈ M \ ∂M, there exists a chart C m = (κm , Om ) such that m ∈ Om and Om ∩ ∂M = ∅. For each m ∈ ∂M, there exists a chart C m = (κm , Om ) such that m ∈ Om and Om meets only one of the connected components of ∂M, and κm yields normal geodesic coordinates as introduced in Sect. 17.6. Because of the compactness of M, there exists (mj )j∈J with J finite such that (1) M = ∪j∈J Omj . (2) The set J is partitioned into J = Jint ∪ J∂ . (3) One has j ∈ Jint if and only if Omj ∩ ∂M = ∅. (4) the set J∂ reads J∂ = 0J∂ ∪ · · · ∪ βJ∂ , and j ∈ kJ∂ if and only if Omj ∩ k∂M = ∅.

44

3. FREDHOLM PROPERTIES

To ease notation, we write C j = (κj , Oj ) in place of C mj = (κmj , Omj ). With Theorem 15.14, we introduce a C ∞ -partition of unity (ψ j )j∈J subordinated to the open covering (Oj )j∈J , that is, (1) For all j ∈ J, we have ψ j ∈ C ∞ (M), 0 ≤ ψ j ≤ 1, and supp(ψ j ) ⊂ Oj .  (2) We have j∈J ψ j = 1 on M. For each j ∈ J, we also consider ϕj ∈ C ∞ (M) such that supp(ϕj ) ⊂ Oj and ϕj ≡ 1 on a neighborhood of supp(ψ j ). In each chart C j , the local representatives of ψ j and ϕj are denoted by ψˇj and ϕˇj , respectively, that is,   ∗ ∗ ψˇj = (κj )−1 ψ j , ϕˇj = (κj )−1 ϕj . Because of the supports of ψ j and ϕj , we have supp(ψˇj ) ⊂ supp(ϕˇj ) ⊂ κj (Oj ) and ϕˇj ≡ 1 on a neighborhood of supp(ψˇj ). (m+3/2) We now construct the operator M : H m (M) ⊕ HB (∂M) → H m+2 (M) of Proposition 3.2, in the form  j j M= (3.3.1) ψ M , j∈J

Mj

with constructed in Sects. 3.3.1 and 3.3.2 below, first in the local chart C j using the analysis performed in a half-space in Sect. 3.2 and, second, lifted back to the manifold. (m+3/2) (∂M). For We consider f ∈ H m (M) and h = (0 h, . . . , β h) ∈ HB each j ∈ J, we let f j be the local representative of f in C j , that is,   ∗  f j = (κj )−1 f ∈ H m κj (Oj ) . (3.3.2) If moreover j ∈ J∂ and j ∈ kJ∂ , that is, Oj meets k∂M, we also let k hj be the local representative of k h in C j , that is,   ∗  k j (3.3.3) h = (κj )−1 k h ∈ H m+3/2−k κj (∂M ∩ Oj ) . 3.3.1. Charts Away from the Boundary. Here, j ∈ Jint . The chart C j does not meet the boundary. j If P C (x, D) is the representative of P in κj (Oj ), we extend its coefficients to the whole Rd to form a second-order elliptic operator. We denote by Q ∈ Ψ−2 (Rd ) a parametrix as given by Proposition 2.33 of Volume 1, that is, (3.3.4)

QP C (x, D) = Id +R, j

P C (x, D)Q = Id +R , j

with R, R ∈ Ψ−∞ (Rd ) = ∩N ∈N Ψ−N (Rd ). We define the operator M C as j

(3.3.5)

M C = ϕˇj Qϕˇj , j

which is bounded from H m (Rd ) into H m+2 (Rd ). We then define the 1 × (β + 2) matrix operator

    j M j = κj ∗ M C (κj )−1 ∗ 0 · · · 0 , (3.3.6)

3.3. PROOF OF THE FREDHOLM PROPERTY

45

(m+3/2)

which is bounded from H m (M) ⊕ HB (∂M) into H m+2 (M). With the map L defined in (3.1.3), we have the following lemma. Lemma 3.13. Let j ∈ Jint . We have ψ j LM j = ψ j Id +K r,j and ψ j M j L = ψ j + K ,j ,

(3.3.7)

(m+3/2)

(∂M) into H m+1 (M) ⊕ where K r,j is bounded from H m (M) ⊕ HB (m+5/2) (∂M) and K ,j is bounded from H m+2 (M) into H m+3 (M). HB Note that because of the support of ψ j , one has ψ j h = 0 implying that the first equality in (3.3.7) actually reads    j j f j f r,j f ψ LM =ψ +K . h 0 h Proof. We compute

 ∗ j j ψ j P M j = κj ψˇj P C (x, D) M C

0 ···

 ∗ (κj )−1 .

0

j Using that P C (x, D) is a local operator and the support properties of ψˇj and ϕˇj and (3.3.4), we find that j j j j ψˇj P C (x, D)M C = ψˇj P C (x, D)ϕˇj Qϕˇj = ψˇj P C (x, D)Qϕˇj . j

j

= ψˇj ϕˇj + ψˇj K1r,C ϕˇj = ψˇj + ψˇj K1r,C ϕˇj , j

where K1r,C is bounded from H m (Rd ) into H N (Rd ) for any N ∈ N. We then define the operator  ∗  ∗ j r,j = κj ψˇj K1r,C ϕˇj (κj )−1 K1,1 that is bounded from H m (M) into H m+1 (M). Because of the support of ϕj , we have   j f  = 0. BM  h ∂M Setting the (β + 2) × (β + 2) matrix operator ⎛ r,j K1,1 0 · · · ⎜ 0 0 ··· ⎜ K r,j = ⎜ . .. . . ⎝ .. . . 0 0 ···

⎞ 0 0⎟ ⎟ , .. ⎟ .⎠ 0

we obtain the first result. For the second result, we write   ∗ ∗ j ψ j M j L = ψ j κj ϕˇj Qϕˇj P C (x, D) (κj )−1  ∗  ∗ j = κj ψˇj Qϕˇj P C (x, D) (κj )−1 .

46

3. FREDHOLM PROPERTIES

Let ϕˆj ∈ Cc∞ (κ(Oj )) be such that ϕˆj ≡ 1 on a neighborhood of supp(ψˇj ) and ϕˇj ≡ 1 on a neighborhood of supp(ϕˆj ). We then have j j ψˇj Qϕˇj P C (x, D) = ψˇj Qϕˇj P C (x, D)ϕˆj + K1,j , j with K1,j = ψˇj Qϕˇj P C (x, D)(1 − ϕˆj ) ∈ Ψ−N (Rd ) for any N ∈ N, by symbol j calculus using the support properties of ψˇj and ϕˆj . Using that P C (x, D) is a local operator and (3.3.4), we then write j j ψˇj Qϕˇj P C (x, D)ϕˆj = ψˇj QP C (x, D)ϕˆj = ψˇj ϕˆj + ψˇj K2,j ϕˆj

= ψˇj + ψˇj K2,j ϕˆj , where K2,j ∈ Ψ−N (Rd ) for any N ∈ N. We then conclude as above.



3.3.2. Charts at the Boundary. Here, j ∈ kJ∂ for some k ∈ {0, . . . , β} that gives the order of the boundary operator equal to k. We choose χ∂ ∈ Cc∞ (Rd−1 ) as introduced above Lemma 3.11 such that χ∂ ≡ 1 on a neighborhood of supp(ϕˇj ). With the operator QD introduced in (3.2.9), we define the following 3×2 matrix operator: ⎛ ⎞ 1 0 Ma = ⎝QD 0⎠ , 0 1 which is bounded from H m (Rd+ ) ⊕ H m+3/2−k (Rd−1 ) into H m (Rd+ ) ⊕ H m+3/2 (Rd−1 ) ⊕ H m+3/2−k (Rd−1 ). With the 2 × 2 matrix operator M∂ given by Lemma 3.11, we define the following 3 × 3 matrix operator: Im(ξd )

•

−R0 ξ

0

iρ(x, ξ )

Γ0

R0 ξ

Re(ξd )

which is bounded from H m (Rd+ ) ⊕ H m+3/2 (Rd−1 ) ⊕ H m+3/2−k (Rd−1 ) into H m (Rd+ )⊕H m+3/2 (Rd−1 )⊕H m+1/2 (Rd−1 ). With the operator Q introduced in (3.2.6) and the operators Q (−1) , Q (0) introduced in (3.2.7), we define the 1 × 3 matrix operator   (3.3.8) Mc = Q −Q (0) Q (−1) that is bounded from H m (Rd+ ) ⊕ H m+3/2 (Rd−1 ) ⊕ H m+1/2 (Rd−1 ) into H m+2 j (Rd+ ). We then define the operator M C as (3.3.9)

M C = ϕˇj Mc Mb Ma ϕˇj , j

3.3. PROOF OF THE FREDHOLM PROPERTY

47

which is bounded from H m (Rd+ ) ⊕ H m+3/2−k (Rd−1 ) into H m+2 (Rd+ ). Set the 2 × (β + 2) matrices  1 0 ··· 0 0 0 ··· 0 k , W = 0 0 ··· 0 1 0 ··· 0 where the ‘1’ in the second row lies in the (k + 2)th column. We then set the 1 × (β + 2) matrix operator ∗  ∗ j (3.3.10) M j = κj M C (κj )−1 W k (m+3/2)

that is bounded from H m (M) ⊕ HB (∂M) into H m+2 (M). Recall that the value of k is given by the chart and corresponds to the order of the boundary operator in that chart. Observe that M j is well defined as an operator on functions defined in M because of the two occurrences of the j cutoff function ϕˇj in the definition of M C . With the map L defined in (3.1.3), we have the following lemma. Lemma 3.14. Let j ∈ kJ∂ . We have ψ j LM j = ψ j Id +K r,j and ψ j M j L = ψ j + K ,j , (m+3/2)

where K r,j is bounded from H m (M) ⊕ HB (∂M) into H m+1 (M) ⊕ (m+5/2) (∂M) and K ,j is bounded from H m+2 (M) into H m+3 (M). HB We refer to Appendix 3.A.7 for a proof of Lemma 3.14. 3.3.3. Conclusion of the Proof of the Fredholm Property. Here, we prove Proposition 3.2. With Lemmata 3.13 and 3.14, we write   j ψ LM j + [L, ψ j ]M j = Id +K r , LM = recalling that



j∈J

ψj

j∈J K r,j are

≡ 1, with K r =







r,j + [L, ψ j ]M j , where j∈J K (m+3/2) H m (M) ⊕ HB (∂M) into

bounded maps from the operators (m+5/2) (∂M). If j ∈ Jint , we have H m+1 (M) ⊕ HB  [P, ψ j ]v j . [L, ψ ]v = 0 If j ∈ kJ∂ , we have

[P, ψ j ]v . [B, ψ j ]v|∂M

 j

[L, ψ ]v =

As [P, ψ j ] is a differential operator of order 1 and as [B, ψ j ] is of order k − 1 if k ≥ 1 and vanishes if k = 0, we find that [L, ψ j ]M j also maps (m+3/2) (m+5/2) H m (M) ⊕ HB (∂M) into H m+1 (M) ⊕ HB (∂M). Similarly, with Lemmata 3.13 and 3.14, we write  j j ψ M L = Id +K  , ML = j∈J

48

3. FREDHOLM PROPERTIES

with K  =

 j∈J

K ,j that is bounded from H m+2 (M) into H m+3 (M). 

ˇ 3.4. Necessity of the Lopatinski˘ı–Sapiro Conditions Here, we prove that if the map L defined in (3.1.3) is Fredholm, then ˇ (P, B) fulfills the Lopatinski˘ı–Sapiro condition. We consider the bounded map LP : H m+2 (M) → H m (M) ⊕ H m+3/2 (∂M)

D

u → (P u, u|∂M ), ˇ that is, the case of Dirichlet boundary conditions. As the Lopatinski˘ı–Sapiro condition holds on ∂M for such boundary conditions, by Proposition 3.2, there exists2 a bounded linear operator DMP : H m (M) ⊕ H m+3/2 (∂M) → H m+2 (M) such that D

MP DLP = IdH m+2 (M) +DKP and

D

LP DMP = IdH m (M)⊕H m+3/2 (∂M) +DKPr ,

where both operators KP : H m+2 (M) → H m+3 (M),

D

KPr : H m (M) ⊕ H m+3/2 (∂M) → H m+1 (M) ⊕ H m+5/2 (∂M)

D

are bounded. Let k ∈ {0, . . . , β} and m0 ∈ k∂M, the connected component of ∂M where B is of order k. Let C = (O, κ) be a local chart such that m0 ∈ O. We also consider U ⊂ O an open set of M with m0 ∈ U , and ϕ, ϕ˜ ∈ C ∞ (M) with supp(ϕ) ˜ ⊂ O, ϕ˜ ≡ 1 on a neighborhood of ϕ, ϕ ≡ 1 on a neighborhood of U . We consider h ∈ H m+3/2 (∂M) supported in ∂M ∩ U , and we define  0 D D u = mP (h) = MP ∈ H m+2 (M), h and we also consider the operators   D r 0 D r,1 , kP = 1 0 KP 1

D r,2 kP

  = 0 1 DKPr

 0 , 1

which are bounded from H m+3/2 (∂M) into H m+1 (M) and from H m+3/2 (∂M) into H m+5/2 (∂M), respectively. We have P u = DkPr,1 (h). We set v = ϕu = ϕ DmP (h), and we compute ˜ 1 ϕ˜|∂M h, P v = [P, ϕ] DmP (h) + ϕDkPr,1 h = ϕK

2Note that this follows also from Theorem 18.40 and Theorem 11.15 of Volume 1 as P + Δg is a first-order differential operator.

ˇ 3.4. NECESSITY OF THE LOPATINSKI˘I–SAPIRO CONDITIONS

49

where K1 : H m+3/2 (∂M) → H m+1 (M) is bounded. We also have v|∂M = h + ϕ˜|∂M DkPr,2 ϕ˜|∂M h. To ease notation, we also denote by h and v their local representatives in the chart C. We then write v = mh with m = (κ−1 )∗ ϕ DmP ϕκ∗ : H m+3/2 (Rd−1 ) → H m+2 (Rd+ ) bounded. We set ˜ 1 ϕ˜|∂M κ∗ , K = (κ−1 )∗ ϕK

k = (κ−1 )∗ ϕ˜|∂M DkPr,2 ϕ˜|∂M κ∗ ,

and we have (3.4.1)

P (x, D)v = Kh, and v|xd =0+ = h + kh,

with K : H m+3/2 (Rd−1 ) → H m+1 (Rd+ ) and k : H m+3/2 (Rd−1 ) → H m+5/2 (Rd−1 ) bounded, and where P (x, D) denotes P C (x, D), the local representative of P in C. Note that K is compact from H m+3/2 (Rd−1 ) into H m (Rd+ ) and k is compact from H m+3/2 (Rd−1 ) into itself because of the cutoff functions in its definition and the Rellich–Kondrachov theorem (see Theorem 18.7). Similarly, we denote by B(x, D) the local representative of B in C, which takes the form B(x, D) = B k (x, D ) − iB k−1 (x, D )Dd , with B k (x, D ) and B k−1 (x, D ) as described in (3.2.2). From the analysis of Sects. 3.2.2 and 3.2.3, we find with identity (3.2.14)    γ D (v) (3.4.2) = Kh, Id −C γ˜ N (v) where

 QD ◦ K + RC ◦ m : H m+3/2 (Rd−1 ) → H m+5/2 (Rd−1 ) K= QN ⊕ H m+3/2 (Rd−1 )

is bounded. With K1 denoting the first component of the 2 × 1 matrix operator K, from (3.4.2), we have (Id −CD,D )γ D (v) − CD,N γ˜ N (v) = K1 h. We recall that the principal symbol of CD,N is ρ−1 /2, which is an elliptic |xd =0+ symbol of order −1. With a parametrix as given by Proposition 2.33 of Volume 1 adapted to standard operators, we find γ˜ N (v) = Gγ D (v) + K1 h, where G ∈ Ψ1 (Rd−1 ) with principal symbol equal to ρ|xd =0+ (x , ξ  ) and where K1 : H m+3/2 (Rd−1 ) → H m+3/2 (Rd−1 ) is bounded.

50

3. FREDHOLM PROPERTIES

We now compute B(x, D)v|xd =0+ = B k (x , 0, D )γ D (v) − iB k−1 (x , 0, D )Dd v|xd =0+   = B k (x , 0, D ) + iB k−1 (x , 0, D )r0 (x , 0) γ D (v) + B k−1 (x , 0, D )˜ γ N (v) = Eγ D (v) + B k−1 (x , 0, D )K1 h,   with E = B k (x , 0, D ) + iB k−1 (x , 0, D )r0 (x , 0) + B k−1 (x , 0, D )G ∈ Ψk (Rd−1 ) with symbol e = bk (x , 0, ξ  ) + bk−1 (x , 0, ξ  )ρ|xd =0+ (x , ξ  ). With (3.4.1), we write B(x, D)v|xd =0+ = Eh + K1 h, where K1 : H m+3/2 (Rd−1 ) → H m+5/2−k (Rd−1 ) is bounded. Because of the ˜ we have support of v, for ϕˇ = (κ−1 )∗ ϕ, B(x, D)v|xd =0+ = ϕB(x, ˇ D)v|xd =0+ = ϕEh ˇ + ϕK ˇ 1 h. Note that ϕK ˇ 1 is compact from H m+3/2 (Rd−1 ) into H m+3/2−k (Rd−1 ) by the Rellich–Kondrachov theorem. Since L = (P, B) is Fredholm, by Proposition 11.10 and Remark 11.11 of Volume 1, there exists K  : H m+2 (M) → H m+2 (M) compact such that wH m+2 (M)  P (x, D)wH m (M) + |B(x, D)w|∂M |H (m+3/2) (∂M) B

+ K wH m+2 (M) , 

for w ∈ H m+2 (M). Applied to w = κ∗ v, with the notation abuses mentioned above, this yields vH m+2 (Rd )  P (x, D)vH m (Rd ) + |B(x, D)v|xd =0+ |H m+3/2−k (Rd−1 ) +

+

 ∗

+ K κ vH m+2 (M) . As we have |h|H m+3/2 (Rd−1 )  vH m+2 (Rd ) + |kh|H m+3/2 (Rd−1 ) , +

this gives, with (3.4.1), ˇ |h|H m+3/2 (Rd−1 )  |ϕEh| H m+3/2−k (Rd−1 ) + KhH m (Rd ) + |kh|H m+3/2 (Rd−1 ) +

 ∗

+ K κ mhH m+2 (Rd ) + +

ϕK ˇ 1 hH m+3/2−k (Rd−1 ) .

Here, h is any smooth function supported in κ(∂M ∩ U ). Note that ϕK ˇ 1 is m+3/2 d−1 m+3/2−k d−1 (R ) into H (R ) by the Rellich– compact from H 0 0 Kondrachov theorem. As m ∈ U , for x = κ(m0 ), by Theorem 2.57 of Volume 1, this implies that e(x0 , ξ  )  |ξ  |k , for |ξ  | ≥ 1 as ϕˇ ≡ 1 in a neighˇ borhood of x0 . This precisely means that the Lopatinski˘ı–Sapiro condition  holds for (P, B) at m0 as recalled in (3.2.3).

3.5. SOME INDEX COMPUTATIONS

51

3.5. Some Index Computations Let P = −Δg + R1 and B be as in Sect. 3.1 and L be the map from (m+3/2) into H m (M) ⊕ HB (∂M) defined in (3.1.3), with m ∈ N. By Proposition 3.4, for index computation, it suffices to only consider the case m = 0. Since R1 is a first-order differential operator, it is compact from H 2 (M) into L2 (M); thus, by Theorem 11.15 of Volume 1 for the computation of the index, we simply take R1 = 0. In the present section, we restrict the analysis to the case β ≤ 1. Recalling the notation of Sect. 3.1, we denote by k∂M the union of the connected components of ∂M where B is of order k, k = 0, 1. For better reading, in this section, we shall write H m+2 (M)

H 3/2,1/2 (∂M) = H 3/2 (0∂M) ⊕ H 1/2 (1∂M) (3/2)

in place of HB (∂M), compare with (3.1.2). For the computation of the index, dimension is of importance in connection with the result of Proposition 2.8. 3.5.1. Dimension d ≥ 4. This is the simplest case to analyze. Theorem 3.15. Let d ≥ 4 and β ≤ 1. If (P, B) is such that the Lopatinˇ ski˘ı–Sapiro condition holds at ∂M, then ind(L) = 0. Case 1: Dirichlet-Like Boundary Condition. If 0∂M = ∅, as the ˇ Lopatinski˘ı–Sapiro condition holds there, then it is equivalent to consider Dirichlet boundary conditions upon division by a nonvanishing function on 0 ∂M; see Example 2.5-(1). Case 2: Neumann-Like Boundary Condition. If 1∂M = ∅, there, we write the principal symbol of B as (3.5.1)

b(m, ω) = ω, tm  + iω, vm ,

where t and v are two real vector fields on M along ∂M. We write vm = ν ν + v  , t = tν ν + t , with v ν , tν ∈ R and v  , t ∈ T ∂M. Recall vm m m m m m m m m m m m ν and tν cannot vanish simultaneously as pointed out in Remark 2.9that vm m (3). With tν + iv ν nowhere vanishing, upon division by this term, we may reduce the analysis to the case where tν = 1 and v ν = 0 everywhere. We  for principal symbol. By then have B = ∂ν + B  with B  with tm + ivm ˇ Proposition 2.8, the Lopatinski˘ı–Sapiro condition holding for (P, B) on 1∂M reads  2   2   2 |vm (3.5.2) |g∂ < 1 or g∂ (tm , vm ) > |tm |g∂ |vm | g∂ − 1 , for all m ∈ 1∂M. Proof of Theorem 3.15. On 0∂M, if it is nonempty, we have a Dirichlet-like case, and we define the operator B(s) to be constant equal to B(s) u|0∂M = u|0∂M .

52

3. FREDHOLM PROPERTIES

On 1∂M, if it is nonempty, we have a Neumann-like case, we have B = ∂ν + B  , and we define the operator B(s) to be B(s) = ∂ν + sB  ,

s ∈ [0, 1],

 )  for principal symbol, which has b(s) (m, ω) = νm , ω + ω, (t(s) )m + i(v(s) m     with t(s) = st and v(s) = sv . We define

L(s) : H 2 (M) → L2 (M) ⊕ H 3/2,1/2 (∂M) u → (P u, B(s) u|0∂M , B(s) u|1∂M ). ˇ We claim that the Lopatinski˘ı–Sapiro condition holds for (P, B(s) ) for all s ∈ [0, 1]. Thus, L(s) is Fredholm by Theorem 3.1. As s → L(s) is continu  ous on [0, 1] with values in B H 2 (M), L2 (M) ⊕ H 3/2,1/2 (∂M)  2 , then 2every L(s) lies in one connected component of the open set F B H (M), L (M)   ⊕H 3/2,1/2 (∂M) , the set of Fredholm operators in B H 2 (M), L2 (M) ⊕  H 3/2,1/2 (∂M) ; see Theorem 11.13 of Volume 1. By Theorem 11.14 (also in Volume 1), the index of L(s) is independent of s ∈ [0, 1]. If s = 0, L(s) coincides with (1) either a purely Dirichlet–Laplace boundary problem if ∂M = 0∂M; then, the index is zero by Theorem 18.40; (2) or a purely Neumann–Laplace boundary problem if ∂M = 1∂M; then, the index is zero by Theorem 18.56; (3) or a mixed Dirichlet–Neumann boundary problem otherwise; then, the index is zero by Theorem 18.63. We now prove the claim formulated above. For the Dirichlet-like case, it is obvious. For the Neumann-like case on 1∂M, we argue as follows.  | First, consider m ∈ 1∂M where |vm g∂ < 1. Then, for all s ∈ [0, 1],  ˇ we have |(v(s) )m |g∂ < 1. Thus, the Lopatinski˘ı–Sapiro condition holds for (P, B(s) ) at m for all s ∈ [0, 1] as (3.5.2) holds at such a point.    )2 > |t |2 |v  |2 − 1 . Then, Second, consider the case where g∂ (tm , vm m g∂ m g∂ for s ∈ (0, 1], one has  2  2  2   2  )m = s4 g∂ (tm , vm ) > |(t(s) )m | |(v(s) )m | − s2 , g∂ (t(s) )m , (v(s) g∂

g∂

implying   2 2  2   )m > |(t(s) )m | |(v(s) )m | − 1 . g∂ (t(s) )m , (v(s) g∂

g∂

ˇ Thus, the Lopatinski˘ı–Sapiro condition holds for (P, B(s) ) at m for all s ∈ (0, 1] as (3.5.2) holds at such a point. Observe now that for s0 > 0 close to  )m | < 1, meaning that we can conclude using the first 0, we have |(v(s 0) case for s ∈ [0, s0 ].

g∂



3.5. SOME INDEX COMPUTATIONS

53

3.5.2. Dimension d = 3. This case has some similarity with the case d ≥ 4. Its analysis is however more involved. Theorem 3.16. Let d = 3 and β ≤ 1. If (P, B) is such that the Lopatinˇ ski˘ı–Sapiro condition holds at ∂M, then ind(L) = 0. Below, we face three disjoint cases. Connected components of 0∂M are treated as in the case d ≥ 4. For the connected components of 1∂M, the analysis needs to be split into two subcases. Case 1: Dirichlet-Like Boundary Condition. If 0∂M = ∅, there the boundary operator B can be replaced by the Dirichlet boundary operator; see Sect. 3.5.1 in the case d ≥ 4. We now consider 1∂M if it is nonempty. Let N be one of the connected components of 1∂M. There, we write the principal symbol of B as in (3.5.1). ˇ As the Lopatinski˘ı–Sapiro condition holds for (P, B) on N , according to Propositions 2.8 and 2.10, we now face two disjoint cases. Case 2: Neumann-Like Boundary Condition. For all m ∈ N , one has either (3.5.3) or (3.5.4)

ν  | or |vm |g∂ < |tνm | |tm |g∂ < |vm

   2  2 ν 2 ) |vm |g∂ − (tνm )2 . gm (tm , vm )2 > |tm |g∂ − (vm

Observe that (3.5.3)–(3.5.4) coincide exactly with the characterization of ˇ the Lopatinski˘ı–Sapiro condition in the case d ≥ 4 in Proposition 2.8. Thus, the procedure of the proof of Theorem 3.15 applies. The principal symbol of B|N given in (3.5.1) reads   n ν  . b(m, ω) = tνm + ivm ωm + ω  , tm  + iω  , vm ν cannot vanish, and upon a division by this term, we take Then, tνm + ivm B|N of the form

(3.5.5)

 . B|N = ∂ν + B|N

If we set (3.5.6)

 , B(s) = ∂ν + sB|N

s ∈ [0, 1],

we reach the conclusion of the following lemma. ˇ Lemma 3.17. The Lopatinski˘ı–Sapiro condition holds for (P, B(s) ) on N and all s ∈ [0, 1]. Observe that the Neumann boundary conditions fall into this case and are given by B(0) . Hence, the name “Neumann-like” we use to describe this case.

54

3. FREDHOLM PROPERTIES

Case 3: Tangential-Like Boundary Condition on a Torus. For all m ∈ N , one has (3.5.7) and (3.5.8)

ν  | and |vm |g∂ ≥ |tνm | |tm |g∂ ≥ |vm

   2  2 ν 2 gm (tm , vm )2 < |tm |g∂ − (vm ) |vm |g∂ − (tνm )2 ,

and in this case N is a torus. On N , the operator B reads (3.5.9)

ν  )∂ν + B|N , B|N = (tνm + ivm

 meaning that B|N is a first-order differential operator on N with principal symbol  , b (m, ω  ) = ω  , tm  + iω  , vm

(m, ω  ) ∈ T ∗ N .

Consider now the boundary operator, for s ∈ [0, 1] (3.5.10)

ν  )∂ν + B|N . B(s) = s(tνm + ivm

ˇ Lemma 3.18. The Lopatinski˘ı–Sapiro condition holds for (P, B(s) ) on N for all s ∈ [0, 1]. ν ν + Proof. For s ∈ [0, 1], set (t(s) )m = stνm νm + tm and (v(s) )m = svm m  . One then finds |t | ν ν  ν ν vm m g∂ > |vm | ≥ |svm |, |vm |g∂ > |tm | ≥ |stm |, and  2    2  2 ν 2 ) |vm |g∂ − (stνm )2 , gm (t(s) )m , (t(s) )m < |tm |g∂ − (svm

by Lemma 2.12. Together, these three conditions yield the conclusion by Proposition 2.8.  The path s → B(s) initiated from B has the purely tangential operator B  for end point. Hence, the name ‘Tangential-like’ we use to describe this case.  ) = T N , meaning that the By Proposition 2.10, one has span(tm , vm m  first-order operator B is elliptic on N .

Lemma 3.19. Let N be a torus. Consider the first-order differential operator W = X + iY + V , where X and Y are smooth real vector fields and V ∈ C ∞ (N ; C). Assume that span(Xm , Ym ) = Tm N for every m ∈ N . Consider W as a bounded operator from H s+1 (N ) into H s (N ), s ∈ R. Then, W is Fredholm and ind(W ) = 0. Proof. As W is elliptic and N does not have a boundary, by using a parametrix in every chart of an atlas, as is done in Sect. 3.3.1, one can adapt the proof of the Fredholm property of Sect. 3.3.3 without having to deal with boundaries. The order of the operator plays essentially no role. A (smooth) torus is by definition diffeomorphic to S1 × S1 . We may thus use global coordinates and use N = (R/2πZ)2 . We denote the coordinates on N by (x, y) and W = X + iY = (a∂x + b∂y ) + i(c∂x + d∂y ), where

3.5. SOME INDEX COMPUTATIONS

55

a, b, c, and d are smooth periodic functions. As V : H s+1 (N ) → H s (N ) is a compact operator, by Theorem 11.15 of Volume 1, we may simply take V = 0 to compute the index of W . As dim span(Xm , Ym ) = 2, one has a+ic = 0 everywhere. Upon dividing by this nonvanishing function, we may reduce the computation of the index to that of W  = ∂x + (α + iβ)∂y , where α and β are smooth real valued periodic functions and β = 0 everywhere. As β = 0, we have β > 0 everywhere or β < 0 everywhere. If β < 0, the change of variables (x, y) → (x, −y) yields β > 0. We thus simply assume that β > 0. Let Wt = ∂x + (tα + i(tβ + (1 − t)))∂y for t ∈ [0, 1], then we have W1 = W  , W0 = ∂x + i∂y , and tβ + (1 − t) ≥ min(β, 1) > 0. Then, Wt : H s+1 (N ) → H s (N ) is an elliptic operator for every t ∈ [0, 1]. It is Fredholm and, by Theorem 11.14 in Volume 1, W  and ∂x + i∂y have the same index. To compute the index of ∂x + i∂y , it is more convenient to consider Wμ = ∂x + i∂y + μ with μ ∈ R \ Z. The two operators share the same index by Theorem 11.15 in Volume 1. Working with Fourier series, we have   (in − k + μ)un,k ei(nx+ky) , u= un,k ei(nx+ky) . Wμ u = (n,k)∈Z2

(n,k)∈Z2

As (in − k + μ) = 0 for all (n, k) ∈ Z2 , we deduce that Wμ is invertible from  H s+1 (N ) to H s (N ), which concludes the proof. An example of a geometry where the three cases, “Dirichlet-like,” “Neumann-like,” and “Tangential-like” can occur is given in Fig. 3.1.

Figure 3.1. Example of a manifold of dimension 3 with a boundary with several connected components. Here, two solid tori are removed from a ball. One component of ∂M is a sphere and two other components are tori Proof of Theorem 3.16. We write 1∂M = N∂M ∪ T∂M, with N∂M and T∂M as follows:

56

3. FREDHOLM PROPERTIES

(1) The manifold N∂M is the union of the connected components of ∂M characterized by “Neumann-like” boundary conditions, that is, where (3.5.3)–(3.5.4) hold. (2) The manifold T∂M is the union of the connected components of ∂M that are tori characterized by “tangential-like” boundary conditions, that is, where (3.5.7)–(3.5.8) hold. On 0∂M, if it is nonempty, we define the operator B(s) to be constant equal to B(s) u|0∂M = u|0∂M . On N∂M, if it is nonempty, on each of its connected components, we define B(s) , s ∈ [0, 1], by first reducing B|N∂M to the form given in (3.5.5) and then setting B(s) according to (3.5.6). On T∂M, if it is nonempty, on each of its connected components, we define B(s) , s ∈ [0, 1], according to (3.5.10). The boundary operator B(s) , defined in that manner on each connected component of ∂M, fulfills the following properties:  (1) The map s → B(s) is continuous from [0, 1] into B H 2 (M),  H 3/2,1/2 (∂M) . ˇ (2) The Lopatinski˘ı–Sapiro condition holds for (P, B(s) ) on ∂M for all s ∈ [0, 1]; (3) The end point s = 1 is such that B(1) = B. (4) The end point s = 0 is such that (a) On 0∂M, we have the Dirichlet boundary operator B(0) u = u|0∂M ; (b) On N∂M, we have the Neumann boundary operator B(0) u = ∂ν u|N∂M ; (c) On each connected component of T∂M, we have a first-order  tangential boundary operator B(0) u = B  u|T∂M with B as defined in (3.5.9). We set L(s) : H 2 (M) → L2 (M) ⊕ H 3/2,1/2 (∂M) u → (P u, B(s) u) and find that L(s) is Fredholm by Theorem 3.1. As s → L(s) is continuous   on [0, 1] with values in B H 2 (M), L2 (M) ⊕ H 3/2,1/2 (∂M) , then every L(s)  lies in one connected component of the open set F B H 2 (M), L2 (M) ⊕   H 3/2,1/2 (∂M) , the set of Fredholm operators in B H 2 (M), L2 (M) ⊕  H 3/2,1/2 (∂M) ; see Theorem 11.13 of Volume 1. By Theorem 11.14 also in Volume 1, the index of L(s) is independent of s ∈ [0, 1]. The conclusion thus follows from Lemma 3.20 below. 

3.5. SOME INDEX COMPUTATIONS

57

Lemma 3.20. The index of the limit operator L(0) : H 2 (M) → L2 (M) ⊕ H 3/2,1/2 (∂M) u → (P u, B(0) u) is zero. Proof. If 0∂M ∪ T∂M = ∅, then the limit operator L(0) corresponds to a pure Neumann problem. By Theorem 18.56, its index is zero. We now assume that 0∂M ∪ T∂M = ∅. We set D∂M = 0∂M ∪ T∂M. We consider an auxiliary boundary value elliptic problem where Neumann boundary conditions are imposed on N∂M, if nonempty, and Dirichlet boundary conditions on D∂M. We denote by M the mixed Dirichlet–Neumann lifting map of Definition 18.60 associated with this auxiliary problem. This map is well defined precisely because D∂M = ∅; see Sect. 18.8. It has the regularity given by Proposition 18.62. If T∂M is nonempty, we denote by T∂M1 , . . . , T∂Mn its connected components, which are all tori. On T∂Mj , denote Bj = B(0) |T∂Mj . This is a tangential boundary operator. By Lemma 3.19, the bounded operator Bj : H 3/2 (T∂Mj ) → H 1/2 (T∂Mj ) is Fredholm and ind(Bj ) = 0. We then set 3/2,1/2

K∂ = {(Dh, Nh) ∈ HDN

(∂M); Dh|0∂M = 0, h|T∂Mj ∈ ker(Bj ), j = 1, . . . , n, and Nh = 0}

D

3/2,1/2

(∂M) is defined in (18.8.2). We have and K =  M(K∂ ). The space HDN dim K = 1≤j≤n dim ker(Bj ). We claim that K = ker(L(0) ). Let h ∈ K∂ and u = M(h), then we find that L(0) u = 0; in fact, using that Bj is   a tangential differential operator, note that B(0) u |T∂Mj = Bj u|T∂Mj = 0. Thus, K ⊂ ker(L(0) ). Conversely, let u ∈ ker(L(0) ). Then, P u = 0,   u|0∂M = 0, and ∂ν u|N∂M = 0. We also have B(0) u |T∂Mj = Bj u|T∂Mj = 0, j = 1, . . . , n. Thus, u|T∂Mj ∈ ker(Bj ), meaning that u ∈ K. We now consider the range of L(0) . We define the following subspace of ⊕ H 3/2,1/2 (∂M):

L2 (M)

R∂ = {(f, h); f ∈ L2 (M), h = (0 h, 1 h) ∈ H 3/2,1/2 (∂M) with 0 h ∈ H 3/2 (0∂M), 1 h|N∂M ∈ H 1/2 (N∂M), and1 h|T∂Mj ∈ Ran(Bj ), j = 1, . . . , n}. We have Ran(L(0) ) ⊂ R∂ . Conversely, let (f, h) ∈ R∂ . We set Nh = 1 h|N∂M ∈ H 1/2 (N∂M) and Thj = 1 h|T∂Mj ∈ Ran(Bj ), that is, Thj = Bj (uj ) with 3/2,1/2

uj ∈ H 3/2 (T∂Mj ). We now define h = (Dh, Nh) ∈ HDN

(∂M) given by

58

3. FREDHOLM PROPERTIES

h|0∂M = 0 h ∈ H 3/2 (0∂M),

D

h|T∂M = uj ∈ H 3/2 (T∂Mj ),

D

h|N∂M = Nh ∈ H 1/2 (N∂M).

N

We denote by ML the map 3/2,1/2

L : H 2 (M) → L2 (M) ⊕ HDN (∂M)   u → P u, u|D∂M , ∂ν u|N∂M .

M

It is a homeomorphism by Theorem 18.63. Thus, there exists a unique u ∈ H 2 (M) such that MLu = h. We then find the L(0) u = (f, h). Hence, R∂ = Ran(L(0) ) and codim Ran(L(0) ) =

 1≤j≤n

codim Ran(Bj ) =

 1≤j≤n

dim ker(Bj )

= dim ker(L(0) ), since ind(Bj ) = 0 by Lemma 3.19. This concludes the proof.



3.5.3. Dimension d = 2. This is the most intricate case as topology plays an important role. We exhibit cases where the index is zero and cases where this index is nonzero. As above, we restrict our analysis to the case β ≤ 1. Recalling the notation of Sect. 3.1, we denote by 0∂M the union of the connected components of ∂M where the order of the boundary operator B is k = 0 and by 1∂M the union of the connected components of ∂M where k = 1. As d = 2, the manifold 1∂M is a finite union of one-dimensional manifolds without boundaries all diffeomorphic to S1 . Let N be one of these connected components, and let Y  ∈ C ∞ V (N ) be a unitary vector field (in the sense of the metric g∂ ). We write the principal symbol of B on N as (3.5.11)

b(m, ω) = ω, tm  + iω, vm ,

where t and v are two real vector fields on M along N . We write vm = ν ν + v , t ν  ν ν   vm m m = tm νm + tm , with vm , tm ∈ R and vm , tm ∈ Tm ∂M. By m ˇ Proposition 2.8, the Lopatinski˘ı–Sapiro condition holds on N if and only if  ν = ±i(tνm + ivm )Ym , tm + ivm

m ∈ N.

We shall present some geometrical reduction that may allow one to compute the index of L.

3.5. SOME INDEX COMPUTATIONS

59

3.5.3.1. Some Admissible Deformations. A function f ∈ C ∞ (N ; C) is called a Morse function if it has no degenerate critical point. Morse functions form a dense subset of C ∞ (N ) in the C k -topology [260, Corollary 6.8], ν in the k ∈ N. Thus, there exists a Morse function ρ close to m → tνm + ivm 0 C -topology such that    ν = ±i s(tνm + ivm ) + (1 − s)ρ(m) Ym , s ∈ [0, 1], m ∈ N . tm + ivm Thus, if we replace t and v by t + (Re ρ)ν and v  + (Im ρ)ν, the operator ˇ (P, B) is also Fredholm as the Lopatinski˘ı–Sapiro condition holds and its index is unchanged by Theorem 11.14 of Volume 1. Now, if m0 ∈ N is such that ρ(m0 ) = 0, then such a point is isolated3. Since ρ is complex valued, there exists ρ˜ ∈ C ∞ (N ; C) close to ρ in the C 0 -topology such that    = ±i sρ(m) + (1 − s)˜ ρ(m) Ym , s ∈ [0, 1], m ∈ N , tm + ivm and moreover ρ˜ does not vanish on N . Arguing as above, we may thus replace t and v by t + (Re ρ˜)ν and v  + (Im ρ˜)ν. Since ρ˜ = 0 upon dividing by ρ˜, we may reduce the analysis to only considering boundary operators that take the form B = ∂ν + B  ,

(3.5.12)

with B  a first-order tangential differential operator with principal symbol  . The Lopatinski˘ ˇ ı–Sapiro condition given by b (m, ω  ) = ω  , tm  + iω  , vm then reads  = ±iYm , tm + ivm 

Since Y does not vanish, we may write ˇ Lopatinski˘ı–Sapiro condition is then (3.5.13)

α(m) + iβ(m) = ±i

m ∈ N. t

= αY  and v  = βY  . The m ∈ N.

We define the closed curve γN : N → C given by γN (m) = α(m) + iβ(m). This reduction can be performed on all connected components of ∂M. On the one hand, the following lemma allows one to deform the boundary condition toward Neumann boundary conditions on N under favorable topological conditions. Lemma 3.21 (Deformation Toward Neumann Boundary Conditions). Assume that the close curve γN lies in C∗∗ = C \ {−i, i} and is homotopic to 0 in C∗∗ . Then, there exists a continuous family of first-order differential operators B(s) on N , s ∈ [0, 1] such that (1) B(1) = B and B(0) = ∂ν .

3In fact, one faces two cases. Either m0 is not a critical point; then since dim N =

1, the function ρ cannot vanish near m0 . Or m0 is a critical point; then, as ρ is a Morse function, it is nondegenerate; then, either Re ρ or Im ρ is nondegenerate, and since dim N = 1 again, this implies that the either Re ρ or Im ρ cannot vanish near m0 .

60

3. FREDHOLM PROPERTIES

Im z i

γN Re z

−i

Figure 3.2. Example of a curve γN that does not fit the assumption of Lemma 3.21. However, both the winding numbers of i and −i are zero ˇ (2) The Lopatinski˘ı–Sapiro condition holds on N for (P, B(s) ) for all s ∈ [0, 1]. Proof. There exists γ(s),N : [0, 1] × N → C∗∗ such that γ(1),N = γN and γ(0),N (m) = 0 for all m ∈ N . One sees that the boundary operator ˇ condition holds on B(s) = ∂ν + γ(s),N Y  is such that the Lopatinski˘ı–Sapiro N for (P, B(s) ) for all s ∈ [0, 1]. With the definition of B(s) for s ∈ [0, 1], all the listed properties are fulfilled.  Remark 3.22. Note that for the assumption of this lemma to hold, it is not sufficient that both the winding numbers of i and −i with respect to the closed curve γN be zero. Figure 3.2 provides an example. On the other hand, the following lemma allows one to deform the boundary condition toward the case of tangential boundary conditions. Lemma 3.23 (Deformation Toward Tangential Boundary Conditions). Assume that the close curve γN is homotopic to μ : S1  z → 2z k , for some k ∈ Z, then there exists a continuous family of first-order differential operators B(s) on N , s ∈ [0, 1] such that (1) B(1) = B and B(0) is a nonvanishing tangential operator on N . ˇ (2) The Lopatinski˘ı–Sapiro condition holds on N for (P, B(s) ) for all s ∈ [0, 1]. Proof. By assumption, there exists γ(s),N : [1/2, 1] × N → C \ {−i, i} such that γ(1),N = γN and γ(1/2),N = μ; one sees that the boundary operator ˇ condition holds on B(s) = ∂ν + γ(s),N Y  is such that the Lopatinski˘ı–Sapiro N for (P, B(s) ) for all s ∈ [1/2, 1]. Next, for s ∈ [0, 1/2], we set B(s) = 2s∂ν + μY  . One sees that the ˇ Lopatinski˘ı–Sapiro condition holds on N for (P, B(s) ) for all s ∈ [0, 1/2]. With the definition of B(s) for s ∈ [0, 1], all the listed properties are fulfilled. 

3.5. SOME INDEX COMPUTATIONS

61

Lemma 3.24. Consider the first-order differential operator W = X + iY + V , where X and Y are smooth real vector fields and V ∈ C ∞ (N ; C). Assume that X + iY is nonvanishing. Consider W as a bounded operator from H s+1 (N ) into H s (N ), s ∈ R. Then, W is Fredholm and ind(W ) = 0. Proof. As N is diffeomorphic to S1 parameterized by θ ∈ R/2πZ, upon division by a smooth nonvanishing function, we may consider W = ∂θ + q. Upon perturbation by a compact operator, by Theorem 11.5 of Volume 1, it is equivalent to prove that ∂θ + 1/2 is Fredholm with index equal to zero. By Fourier series, one finds that this operator is in fact invertible.  3.5.3.2. Computation of the Index in Some Topological Situations. Theorem 3.25. We use the notation of the previous section. We assume that for every connected component N of 1∂M, we either have (1) The close curve γN lies in C∗∗ = C \ {−i, i} and is homotopic to 0 in C∗∗ . (2) The close curve γN is homotopic to μ : S1  z → 2z k . Then, the index of L = (P, B) : H 2 (M) → L2 (M) ⊕ H 3/2,1/2 (∂M) is zero. Proof. On each connected component of 1∂M, we use the continuous deformation given by either Lemma 3.21 or Lemma 3.23. As the Lopatinˇ ski˘ı–Sapiro condition holds along that deformation, the continuously deformed operator L remains Fredholm and its index remains unchanged by Theorem 11.4 of Volume 1. We thus end up with either Neumann boundary conditions or tangential boundary conditions on the different connected components of 1∂M. On 0∂M, as in the case d = 3, we may simply replace the boundary operator by the Dirichlet operator. One then sees that the setting is exactly the same setting as we have obtained in the case d = 3. The proof of Lemma 3.20 can be adapted mutatis mutandis, for instance, by replacing the use of Lemma 3.19 by that of Lemma 3.24.  The topological assumption in the previous theorem is far from being trivial. The following result shows that in the case d = 2, any value in Z can be obtained as the index of the operator L for a properly chosen boundary operator B. This example shows also the importance of the winding number of +i and −i with respect to γN for the value of the index. Proposition 3.26. Let D be the unit disc in C, and let k ∈ Z. We denote by r the complex modulus and by θ the complex argument. Let Δ be the Laplace operator with the flat metric on D, that is, Δ = ∂x2 + ∂y2 = 4∂∂ and set Bu|∂D = (∂r u + α∂θ u)|r=1 with α(θ) = i + eikθ . Then, L = (P, B) is Fredholm and ind(L) = −k. In this example, the closed path γN for N = ∂D is given by γN (θ) = α(θ). The closed path thus goes k times around +i counter clockwise. The ˇ Lopatinski˘ı–Sapiro condition holds by (3.5.13), and thus L is Fredholm by Theorem 3.1.

62

3. FREDHOLM PROPERTIES

For the proof of Proposition 3.26, it is convenient to use the following lemma. ˜ : w → B D(w|∂D ) that is bounded Lemma 3.27. Define the operator B 3/2 1/2 ˜ is Fredholm. Moreover, from H (∂D) into H (∂D). The operator B ˜ ˜ nul L = nul B and def L = def B. Recall that D is the Dirichlet lifting map. On D, if h ∈ H 3/2 (∂D), then D(h) is the unique H 2 solution of Δu = 0,

u|∂D = h;

see Definition 18.37 and Proposition 18.39. Proof. Denote by R0 the resolvent map associated with the homogeneous Dirichlet problem, that is, if f ∈ L2 (D), then v = R0 f is the unique solution in H 2 (D) of Δv = f,

v|∂D = 0.

˜ We see that D(ker B) ˜ = ker L. First, we consider the kernels of L and B. ˜ = As D is injective and ker L is finite dimensional, we obtain dim ker B dim ker L. Second, let us exhibit the connection between the ranges of the two maps. Let (f, h) ∈ L2 (D) × H 1/2 (∂D) be in Ran(L). There exists u ∈ H 2 (D) such that Lu = (f, h). We have u = R0 f + Du|∂D , leading to h = Bu|∂D = ˜ |∂D ). We thus find h − BR0 f|∂D ∈ Ran(B). ˜ BR0 f|∂D + B(u Conversely, let (f, h) ∈ L2 (D) × H 1/2 (∂D) be such that h − BR0 f|∂D ∈ ˜ = h − BR0 f|∂D . If we set ˜ Let then w ∈ H 3/2 (∂D) be such that Bw Ran(B). u = R0 f + Dw, then P u = f and ˜ = h. Bu|∂D = BR0 f|∂D + BDw|∂D = BR0 f|∂D + Bw Thus, (f, h) ∈ Ran(L). We have thus found that ˜ (f, h) ∈ Ran(L) ⇔ h − BR0 f|∂D ∈ Ran(B), ˜ or equivalently k ∈ Ran(B) ⇔ (f, k + BR0 f|∂D ) ∈ Ran(L), for all 2 ˜ closed in H 1/2 (D). f ∈ L (D). In particular, this gives Ran(B) Assume that (g, ) ∈ L2 (D) ⊕ H −1/2 (∂D) is orthogonal to Ran(L) in the sense of duality. Then, f, gL2 (D),L2 (D) + k + BR0 f|∂D , H 1/2 (D),H −1/2 (D) = 0, ˜ This is equivalent to having for all f ∈ L2 (D) and all k ∈ Ran(B). ˜ g = −R∗0 B ∗  and  ⊥ Ran(B). ˜ This yields codim(Ran(L)) = codim(Ran(B)).



3.5. SOME INDEX COMPUTATIONS

63

Proof of Proposition 3.26. For the computation of nul L, we char˜ and acterize ker L. For the computation of def L, we compute in fact def B we use Lemma 3.27. Let u ∈ ker L, that is, u ∈ H 2 (D) with Δu = 0 and Bu|∂D = 0. As u ∈ H 2 (D), then u|∂D ∈ H 3/2 (∂D). We may thus write u|∂D in the form    u|∂D = u(n) einθ , with n3/2 u(n) ∈ 2 (Z). n∈Z

Then, u is of the form    u= u(n) z n + u(−n) z n = u(n) r|n| einθ , n≥0

n≥1

n∈Z

as this is solution of Δu = 0 with the Dirichlet trace given above and this solution is unique. To see this, use that Δ = 4∂∂, with ∂ = ∂z = (∂x −i∂y )/2 and ∂ = ∂z = (∂x + i∂y )/2. We now compute   nu(n) einθ (1 + iα) + nu(−n) e−inθ (1 − iα) Bu|∂D = n≥1

=i



n≥1

nu(n) e

i(n+k)θ

+2

n≥1

=i





nu(−n) e−inθ − i

n≥1

nu(n) e

i(n+k)θ

−2

n∈Z





nu(−n) ei(k−n)θ

n≥1

(n + k)u(n+k) ei(n+k)θ .

n≤−k−1

Having Bu|∂D = 0 reads nu(n) = 0 for n ≥ −k,

inu(n) − 2(n + k)u(n+k) = 0 for n ≤ −k − 1.

Consider the case k ≥ 0. With the first condition, we have u(n) = n ≥ −k and if n = 0. With the second equation, we conclude that u(n) for all n = 0. Hence, ker(L) = {u = u(0) } and nul L = 1. Consider now the case k ≤ −1. With the first condition, we have u(n) if n ≥ −k = |k|. With the second equation, for n = 0, we find that u(k) and then u(jk) = 0, j ∈ N, by induction. If |k| ≥ 2, we also find u(n+jk) =

0 if =0 =0 =0

 i j n u 2 n + 2k (n)

for 1 ≤ n ≤ |k|−1 and j ∈ N. An element of ker L is thus fully characterized by the values of u(0) , . . . , u(|k|−1) , and the resulting values of (u(n) ) yield a converging series and u ∈ H 2 (D). Hence, nul L = |k|. ˜ = k + 1 if k ≥ 0 and def(B) ˜ = 0 if Second, we prove that def(B) 2 ˜ k ≤ −1. We compute the adjoint ofB using L (∂D) as a pivot space. For  w = n∈Z w(n) einθ with n3/2 w(n) ∈ 2 (Z) using the computations made above, we have ˜ (n) (n) =  w(n) (−in(n+k) ) +  w(n) (−2n(n) ) ˜ ) =  (Bw) (Bw, n∈Z

n∈Z

n≤−1

64

3. FREDHOLM PROPERTIES

   for  = einθ ∈ H −1/2 (∂D), that is, n−1/2 (n) ∈ 2 (Z). This n∈Z (n) inθ with ˜ ∗ = v =  gives B n∈Z v(n) e v(n) = −in(n+k) if n ≥ 0, and v(n) = −in(n+k) − 2n(n) if n ≤ −1. ˜∗ Observe that these explicit formulae give the expected boundedness of B from H −1/2 (∂D) into H −3/2 (∂D). ˜ ∗  = 0, then  is fully determined by Consider the case k = 0. If B ˜ ∗  = 0, then  is fully determined by (0) . Consider the case k > 0. If B (0) , . . . , (k) . In fact, (n) = 0 for n ≥ k + 1 and (−kr+j) = (−i/2)r (j) , j = 0, . . . , k − 1, r ∈ N, leading to a convergent series. ˜ = (ker B ˜ ∗ )⊥ , ˜ ∗ ) = k + 1. As Ran(B) For k ≥ 0, we thus have dim ker(B ˜ = Ran(B) ˜ since B ˜ is Fredholm, we find that def(B) ˜ = and as here Ran(B) k + 1. ˜ ∗  = 0, then (n) = 0 for n ≥ −|k|+1. Consider now the case k ≤ −1. If B We also find that (3.5.14)

(−j|k|) = (2i)(j−1) (−|k|) ,

j ∈ N∗ ,

and if |k| ≥ 2, n−j|k| = 0 for 1 ≤ n ≤ |k| − 1 and j ∈ N. As we have n−1/2 (n) ∈ 2 (Z), condition (3.5.14) implies (−|k|) = 0 and thus  = 0. ˜ = 0.  ˜ ∗ ) = 0. Arguing as above, this gives def(B) We thus have dim ker(B 3.6. Additional Regularity Results Here, we consider a function u ∈ L2 (M) such that P u ∈ H m−1 (M) with m ≥ 1. Then, u ∈ WP (M), and from Lemma 18.32, we know that both the Dirichlet and Neumann traces can be defined and γ D (u) ∈ H −1/2 (∂M) and γ N (u) ∈ H −3/2 (∂M). Here, we wish to prove that additional regularity can be reached in the case of smoother boundary data by means of a boundary operator B that ˇ fulfills the Lopatinski˘ı–Sapiro conditions. Theorem 3.28. Let ,  ∈ N with  ≥ 1, and let u ∈ L2 (M) be such that P u ∈ H −1 (M). Let B be a boundary operator as in Sect. 3.1 such ˇ that Lopatinski˘ı–Sapiro condition holds for (P, B) on ∂M. If Bu|∂M ∈ ( +1/2)

HB



(∂M), then u ∈ H min(, )+1 (M).

This result is consistent with Corollary 18.41 obtained in the case P = −Δg along with Dirichlet boundary conditions. Proof. From standard elliptic theory, the function u is H +1 away from the boundary. We thus only consider the issue of its regularity in some neighborhood of ∂M.

3.6. ADDITIONAL REGULARITY RESULTS

65

First, we use normal geodesic coordinates as given by Theorem 17.22: there exist V an open set of M neighborhood of ∂M, z0 > 0, and a diffeomorphism Φ, such that Φ : ∂M × [0, z0 ) → V (m , z) → Φ(m , z). The pullback of the metric takes the form  Φ∗ g(m ,z) = gm  (z) ⊗ 1z + dz ⊗ dz ,

where g  (z) is a Riemannian metric on ∂M that smoothly depends on z. In these coordinates, by Lemma 18.68, the function u has the following regularity: (3.6.1)       u ∈ H 2 [0, z0 ); H −2 (∂M) ∩ H 1 [0, z0 ); H −1 (∂M) ∩ L2 (0, z0 ) × ∂M . Second, let m0 ∈ ∂M, and let C∂ = (O∂ , κ∂ ) be a local chart of ∂M such 0 C = (O, κ) with O = that  m ∈ O∂ . This allows us to  form a local chart  Φ O∂ × [0, z0 ) and κ(m) = (κ(m ), z) if m = Φ(m , z). With ψ ∈ C ∞ (M) with supp(ψ) ⊂ O, we set v = ψu. By (3.6.1), we have     (3.6.2) v ∈ H 2 [0, z0 ); H −2 (∂M) ∩ H 1 [0, z0 ); H −1 (∂M) and γ D (v) = ψ|z=0 γ D (u) ∈ H −1/2 (∂M) and γ N (v) ∈ H −3/2 (∂M) by the Leibniz rule given in Lemma 18.32. Since [P, ψ] is a first-order differential operator on M, with (3.6.1), we find   f = P v = ψP u + [P, Ψ]v ∈ L2 [0, z0 ); H −1 (∂M) . We still denote by v and f the representative of v and f in the chart C. As we did in Sect. 3.2, we write P (x, D) and B(x, D) in place of P C (x, D) and B C (x, D), respectively. We observe that the regularity properties of v given in (3.6.2) allow one to carry out the analysis of Sect. 3.2.2. We may thus write     v = Qf − Q δx d =0 ⊗ γ D (v) + Q δxd =0 ⊗ γ˜ N (v) + Rv.   Note that f ∈ L2 R; H −1 (Rd−1 ) ⊂ H −1 (Rd ). As Q ∈ Ψ−2 (Rd ), we have Qf ∈ H 1 (Rd ). Note also that   Q δx d =0 ⊗ γ D (v) |Rd = Q (0) (γ D (v)) and +   N γ N (v)), Q δxd =0 ⊗ γ˜ (v) |Rd = Q (−1) (˜ +

since

Q (0)

(3.6.3)

are both well defined on S  (Rd−1 ). We thus have    γ N (v)) + Rv |Rd . v = Qf |Rd − Q (0) (γ D (v)) + Q (−1) (˜

and

Q (−1) 

+

+

Q (0) (γ D (v))

Q (−1) (˜ γ N (v))

L2 (Rd+ )

and in and By Lemma 3.6, we have (0) D N d (−1) N N d (˜ γ (v)) ∈ H (R+ ) for any N ∈ N. P Q (γ (v)) ∈ H (R+ ) and P Q Observe that the proof of Lemma 18.32 in Sect. 18.A.1 can be adapted

66

3. FREDHOLM PROPERTIES

mutatis mutandis in the case ∂M is replaced4 by Rd−1 . The Dirichlet γ N (v)) are thus well defined, and they lie traces of Q (0) (γ D (v)) and Q (−1) (˜ in H −1/2 (Rd−1 ). From the regularity of Qf , we thus have     γ D (v) + γ D Q (0) (γ D (v)) − γ D Q (−1) (˜ γ N (v)) ∈ H 1/2 (Rd−1 ). As QD ∈ Ψ0 (Rd−1 ) and QD ∈ Ψ−1 (Rd−1 ) as shown in Lemma 3.9, their action on function in H −1/2 (Rd−1 ) and H −3/2 (Rd−1 ) makes sense, and we find  (0)  (−1) N γ (v)) ∈ H 1/2 (Rd−1 ), Id +QD (γ D (v)) − QD (˜ (0)

which reads (3.6.4)

(−1)



 Id −CD,D (γ D (v)) − CD,N (˜ γ N (v)) ∈ H 1/2 (Rd−1 ). ( +1/2)

(∂M), if m0 ∈ k∂M, that is, the operator As we have Bu|∂M ∈ HB B is of order k on the connected component of ∂M where m0 lies, we have  Bv|k∂M ∈ H  +1/2−k (k∂M). In local coordinates, this reads (3.6.5)  H  +1/2−k (Rd−1 )  B k (x , 0, D )v|xd =0+ − iB k−1 (x , 0, D )Dd v|xd =0+   = B k (x , 0, D ) + iB k−1 (x , 0, D )r0 (x , xd = 0) γ D (v) + B k−1 (x , 0, D )˜ γ N (v). Together, (3.6.4) and (3.6.5) read  D   γ (v) ∈ H 1/2 (Rd−1 ) ⊕ H  +1/2−k (Rd−1 ) = H 1/2, +1/2−k (Rd−1 ), L∂ γ˜ N (v) with the notation introduced in the beginning of Sect. 3.2.5. By Lemma 3.11, we then obtain that γ D (v) ∈ H 1/2 (Rd−1 ) and γ˜ N (v) ∈ H −1/2 (Rd−1 ). γ N (u)) in H 1 (Rd+ ). Thus, by Lemma 3.6, we have Q (0) (γ D (v)) and Q (−1) (˜ 1 d  Thus, by (3.6.3), we find that v ∈ H (R+ ). If  = 0, this result is precisely the result we sought. We may now assume  ≥ 1. Starting  the beginning, we now find  from that f = P v ∈ L2 (R). Then, Q(f ) = Qf |Rd ∈ H 2 (Rd+ ) by Lemma 3.6 + and   (3.6.6) γ N (v)) + Rv |Rd . v = Q(f ) − Q (0) (γ D (v)) + Q (−1) (˜ +

Arguing as above, we find that  D   γ (v) L∂ ∈ H 3/2 (Rd−1 ) ⊕ H  +1/2−k (Rd−1 ) = H 3/2, +1/2−k (Rd−1 ), γ˜ N (v)

4Observe that we did mention this replacement in one of the key results: Lemma 18.69.

3.6. ADDITIONAL REGULARITY RESULTS

67

implying by Lemma 3.11 that, as  + 1/2 ≥ 3/2, γ D (v) ∈ H 3/2 (Rd−1 ) and γ˜ N (v) ∈ H 1/2 (Rd−1 ). This yields v ∈ H 2 (Rd+ ) by (3.6.6). If min( , ) = 1, this result is precisely the result we sought. If now min( , ) ≥ 2, we iterate the argument we used above. This leads    to v ∈ H min( ,)+1 (Rd+ ), and thus we find u ∈ H min( ,)+1 (M). The result of Theorem 3.28 may very well fail to hold if the Lopatinski˘ı– ˇ Sapiro condition is not fulfilled. To see this, let us use Example 2.6 with M = D, the unit disc, P = −Δ = −∂x2 − ∂y2 and Bu|∂D = ∂r u|∂D + i∂θ u|∂D . There ˇ we saw that the Lopatinski˘ı–Sapiro condition does not hold for (P, B). Let us use polar coordinates on D and consider the function given by u(r, θ) = log(1 − reiθ ) = log(1 − z). This function is in L2 (D) (see below). The function u is also holomorphic in D. As a result, it satisfies P u = 0 in D  (D) since P = −4∂∂ with ∂ = ∂z = (∂x +i∂y )/2. Consequently, u ∈ WP (D) with this space defined in (18.6.4). By Lemma 18.32, the Dirichlet and Neumann traces of u are well defined in H −1/2 (∂D) and H −3/2 (∂D), respectively. Thus, Bu|∂D is well defined in H −3/2 (∂D). The following proposition shows that a regularity result as in Theorem 3.28 does not hold. Proposition 3.29. The function u satisfies P u = 0 and Bu|∂D = 0, meaning that Bu|∂D ∈ ∩s∈R H s (∂D). Yet, u ∈ L2 (D) \ H 1 (D). Proof. The L2 -norm is given by 2π 1

u2L2 (D) = ∫ ∫ r| log(1 − reiθ )|2 drdθ. 0 0

The only singularity in the integrand occurs at r = 1 and θ = 0. There, we have r| log(1 − reiθ )|2 ∼ log(1 − r)2 that is integrable. Thus, uL2 (D) < ∞. We have 2π 1   u2H 1 (D)  u2L2 (D) + ∫ ∫ |∂r u|2 + r−2 |∂θ u|2 rdrdθ. 0 0

This can be seen by observing that the Riemannian metric associated with Δ is g = dr2 +r2 dθ2 in the (r, θ) coordinates. Then, ∇g u = (∂r u)∂r +r−2 (∂θ u)∂θ by (17.2.2). We compute ∂r u(r, θ) = −eiθ /(1 − reiθ ). Observe then that 1 as(r, θ) → (1, 0), r|∂r u(r, θ)|2 ∼ (1 − r)2 implying that ∫02π ∫01 |∂r u|2 rdrdθ = +∞. We now prove that ∂r u|r=1 + i∂θ u|r=1 = 0. First, observe that these traces make sense by Theorem 18.32 as u ∈ WP (D). As we have ∂r + ir−1 ∂θ = 2e−iθ ∂ and as u is holomorphic in D, we find (∂r + ir−1 ∂θ )u = 0 for r < 1. From (18.A.3) and (18.A.4) in the proof of Lemma 18.32 given in Sect. 18.A.1, we obtain ∂r u|r=1 + i∂θ u|r=1 = 0 by letting r → 1− . 

68

3. FREDHOLM PROPERTIES

3.7. Notes ˇ The Lopatinski˘ı–Sapiro boundary conditions presented in Chap. 2 are here studied from the point of view of the associated boundary value problem ˇ as is done in the seminal works of Y. B. Lopatinski˘ı [238] and Z. Y. Sapiro [323]. Here, we only treat the case of second-order elliptic operators as in the whole book. A good reference for the study of general elliptic operators ˇ of arbitrary order along with Lopatinski˘ı–Sapiro conditions is the work of L. H¨ormander [175, Section 20.1]. Here, we have chosen to consider this analysis only for a second order which makes it simpler to expose. For further details, we refer the reader to Chapter 20 in [175] and to the bibliographical notes therein. ˇ Lopatinski˘ı–Sapiro conditions are necessary and sufficient for the boundary value problem to be Fredholm. Both the necessary and the sufficient parts of the equivalence are treated with care here. Moreover, as presented ˇ in Sect. 3.6, the Lopatinski˘ı–Sapiro conditions give the proper framework to study how, for the boundary value problem, the regularity of the source terms, in the interior or at the boundary, gives the regularity of the solution. Concerning the value of the index of a boundary value problem, we do not attempt to make any connection with the celebrated Atiyah–Singer theorem [43, 44]. For this connection, we again refer to Section 20.3 in [175]. However, in the case a boundary operator of order at most one, we show how a continuous deformation of the boundary operator can lead to a boundary problem with the same index and for which the index is easily computed, e.g., in the case of Dirichlet or Neumann boundary conditions. In dimension d ≥ 3, we then find a zero index. In dimension d = 2, examples of boundary value problems with a nonzero index are provided.

Appendix 3.A. Proof of Some Technical Results 3.A.1. An A Priori Estimate Consequence of the Fredholm Property. Here, we prove Corollary 3.3. By Theorem 3.1, if the Lopatinˇ ski˘ı–Sapiro condition holds, then the operator L given in (3.1.3) is Fredholm. First, ker(L) is finite dimensional, say dim ker(L) = nul L = J, and is given by ker(L) = span{uj ; 1 ≤ j ≤ J} for some uj ∈ H m+2 (M). Second, Ran(L) has a finite codimension, say codim Ran(L) = def L = N , and is given by (m+3/2) m (∂M). We then Ran(L) = ∩1≤n≤N φ⊥ n for some φn ∈ H (M) ⊕ HB define ˜ : H m+2 (M) ⊕ CN → H m (M) ⊕ H (m+3/2) (∂M) ⊕ CJ L B    (u, α) → Lu + αn φn , (u, u1 )L2 (M) , . . . , (u, uJ )L2 (M) . 1≤n≤N

3.A. PROOF OF SOME TECHNICAL RESULTS

69

  ˜ is bijective. In fact, if L(u, ˜ α) = 0, then u ∈ ker(L) ⊥ We claim that L  since (u, uj )L2 (M) = 0, j = 1, . . . , J. As Lu and 1≤n≤N αn φn lie in or(m+3/2)

(∂M), we thus have Lu = 0 and thogonal subspaces of H m (M) ⊕ HB  α φ = 0 implying that u = 0 and α1 = · · · = αN = 0. We thus 1≤n≤N n n ˜ is injective. see that L ˜ α) = (0, ej ) with e1 , . . . , eJ forming If u = uj and α = 0, then L(u, (m+3/2) (∂M), it takes the the canonical basis of CJ . If F ∈ H m (M) ⊕ HB form F = F1 + F2 with F1 ∈ Ran(L) and F2 ∈ span{φn ; 1 ≤ n ≤ N }. Thus, there exist u ∈ H m+2 (M) and α ∈ CN such that Lu = F1 and  J ˜ 1≤n≤N αn φn = F2 ; that is, L(u, α) = (F, z) with some z ∈ C . By ˜ is surjective. linearity, we see that L ˜ −1 is bounded yielding By the closed graph theorem, the inverse map L ˜ α) m uH 2+m (M) + |α|CN  L(u, (m+3/2) H (M)⊕HB (∂M)⊕CJ       Lu + αn φ n  1≤n≤N

H m (M)⊕H

(m+3/2)

B    + (u, u1 )L2 (M) , . . . , (u, uJ )L2 (M) 

(∂M)

CJ

.

If we now choose α = 0, the conclusion follows by the Cauchy–Schwarz inequality.  3.A.2. Fredholm Index Independent of the Regularity Level. Here, we prove Proposition 3.4. Here, we denote by L(m) the operator from H m+2 (M) into H m (M) ⊕ (m+3/2) HB (∂M) defined in (3.1.3). We also denote by M (m) , K (m), , K (m),r the operators given by Proposition 3.2. The operator K (m), is bounded from H m+2 (M) into H m+3 (M). Note also that since P and B are differential operators, we have (3.A.1)

L(m) |H m+3 (M) = L(m+1) .

Let m ∈ N. With (3.A.1), one has ker(L(m+1) ) ⊂ ker(L(m) ). Let now u ∈ ker(L(m) ), meaning u ∈ H m+2 (M), P u = 0, and Bu = 0. By Proposition 3.2, we have u = −K (m), u. With the above observation, we find that u ∈ H m+3 (M). This means that u ∈ ker(L(m+1) ). We have thus obtained that ker(L(m+1) ) = ker(L(m) ). Consequently, nul L(m+1) = nul L(m) . With (3.A.1), one has Ran(L(m+1) ) ⊂ Ran(L(m) ). As n = def L(m+1) < ∞, there exist φ1 , . . . , φn linearly independent continuous forms on H m+1 (m+5/2) (M) ⊕ HB (∂M) such that Ran(L(m+1) ) = ∩1≤j≤n ker(φj ). As H m+1 (m+5/2) (m+3/2) (M) ⊕ HB (∂M) is dense in H m (M) ⊕ HB (∂M), the maps φ1 , . . . , φn can be uniquely extended as continuous forms on this latter space; we denote by φ˜1 , . . . , φ˜n these extensions. Let now u ∈ H m+2 (M) and (un )n ⊂ H m+3 (M) be a sequence that converges to u in H m+2 (M). Then, for j = 1, . . . , n, we have 0 = φj (P un , Bun ) = φ˜j (P un , Bun ) → φ˜j (P u, Bu). We thus find that Ran(L(m) ) ⊂ ∩1≤j≤n ker(φ˜j ). Thus, def L(m) ≥ def L(m+1) .

70

3. FREDHOLM PROPERTIES

As n = def L(m) < ∞, there exist ψ1 , . . . , ψn linearly independent con(m+3/2) tinuous forms on H m (M) ⊕HB (∂M) such that Ran(L(m) ) = ∩1≤j≤n (m+5/2) ker(ψj ). Since H m+1 (M) ⊕ HB (∂M) is dense in (m+3/2) m ˆ (∂M), the restrictions ψ1 , . . . , ψˆn of these forms to H (M) ⊕ HB (m+5/2) (∂M) are also linearly independent. We thus find H m+1 (M) ⊕ HB (m+1) ) ⊂ ∩1≤j≤n ker(ψˆj ). Thus, def L(m+1) ≥ def L(m) . that Ran(L 3.A.3. Sobolev Regularity of the Parametrix Action. Here, we prove Lemma 3.6. First, we consider the operator Q. Let φ ∈ H m (Rd+ ) and set v = Qφ. Let ψ, ψ˜ ∈ C ∞ (Rd+ ) both vanishing for 0 ≤ xd ≤ 1 be such that ψ˜ ≡ 1 on a neighborhood of supp(ψ). We can consider ψv as a function on the whole Rd that vanishes on Rd− ; we have ˜ + ψQ(1 − ψ)φ. ˜ ψv = ψQφ = ψQ(ψφ) ˜ ∈ H m+2 (Rd ). As ˜ ∈ H m (Rd ) and Q ∈ Ψ−2 (Rd ), we have ψQ(ψφ) As ψφ −N d ˜ ∈ Ψ (R ) for any N ∈ N by symbol calculus, we find that ψQ(1 − ψ) ˜ ψQ(1 − ψ)φ ∈ H m+2 (Rd ). Moreover, the map φ → ψQφ is bounded from H m (Rd+ ) into H m+2 (Rd+ ). We thus see that the result holds away from {xd = 0}. In what follows, we show how this Sobolev regularity holds up to the boundary by means of an iterative procedure that relies on the structure of the operator P (x, D). As Q ∈ Ψ−2 (Rd ), by Proposition 2.52 of Volume 1 (adapted to standard operators), we have m Λm T QφH 2 (Rd )  ΛT φL2 (Rd )  φH m (Rd ) . +

2 d Consequently, we have Λm T v ∈ H (R+ ), that is,    v∈ (3.A.2) H k R+ ; H m+2−k (Rd−1 ) . 0≤k≤2



Moreover, we have 0≤k≤2 vH k (R+ ;H m+2−k (Rd−1 ))  φH m (Rd ) . + With (3.2.4), we have P (x, D)Qφ = φ + R φ,   where R ∈ Ψ−∞ (Rd × Rd ). For w ∈ S  (Rd ), we set R  w = R w |Rd . +

For any N ∈ N, we have R  φH N (Rd )  φL2 (Rd ) = φL2 (Rd ) . Since the +

+

operator P (x, D) is local, we obtain   P (x, D)v = P (x, D)Qφ |Rd = φ + R  φ. +

We set Kφ = R  φ.

3.A. PROOF OF SOME TECHNICAL RESULTS

71

We set w = (1 − ψ)v, that is, we localize v near the boundary. From (3.A.2), we have    w∈ (3.A.3) H k R+ ; H m+2−k (Rd−1 ) . 0≤k≤2



0≤k≤2 wH k (R+ ;H m+2−k (Rd−1 ))  φH m (Rd+ ) . We set xd  α(x) = ∫0 r0 (x , σ)dσ, and we have P (x, D) = Dd2 + r0 Dd + R(x, D ) = e−α Dd eα Dd ) + R(x, D ). We have

Moreover, we have

P (x, D)w = (1 − ψ)P (x, D)v − [P (x, D), ψ]v. Above we saw that v is H m+2 in the support of ψ, we thus find that P (x, D)w ∈ H m (Rd+ ) and P (x, D)wH m (Rd )  φH m (Rd ) . + + From (3.A.3), we obtain     Dd eα Dd w) ∈ H k R+ ; H m−k (Rd−1 ) , 0≤k≤2

as eα is bounded in the support of w. Moreover, we have   Dd eα Dd )H k (R+ ;H m−k (Rd−1 ))  φH m (Rd ) . +

0≤k≤2

With Lemma 18.69 applied twice, we conclude that    H k R+ ; H m+2−k (Rd−1 ) v∈ 0≤k≤4

and



0≤k≤4 vH k (R+ ;H m+2−k (Rd−1 ))  φH m (Rd+ ) . If we iterate this argument j times then we find    H k R+ ; H m+2−k (Rd−1 ) v∈ 0≤k≤2+2j

and



then v

0≤k≤2+2j vH k (R+ ;H m+2−k (Rd−1 ))  φH m (Rd+ ) . ∈ H m+2 (Rd+ ) and vH m+2 (Rd )  φH m (Rd ) . + +

If 2 + 2j ≥ m + 2,

Second, we consider the operator Q (j) for some j ∈ {−1, 0}. Let h ∈ H m+j−1/2 (Rd−1 ) and set v (j) = Q (j) h. With ψ, ψ˜ ∈ C ∞ (Rd+ ) chosen as at the beginning of the proof, we write   (j+1) ˜ (j+1) ⊗ h . ψv (j) = ψQ(δxd =0 ⊗ h) = ψQ (1 − ψ)δ xd =0 ˜ ∈ Ψ−N (Rd ) for any N ∈ N by symbol calculus, we find As ψQ(1 − ψ) that ψv (j) ∈ H m (Rd ), and moreover the map h → ψQ (j) h is bounded from H m+j−1/2 (Rd−1 ) into H m (Rd+ ).

72

3. FREDHOLM PROPERTIES

ˆ  ). For Set k (j) = δxd =0 ⊗ h. Its Fourier transform is given by (iξd )j+1 h(ξ r > j − 1/2 and s ∈ R, we compute (j+1)

2

(j) Λs+r H −r−2 (Rd ) = ∫ ξ  2(s+r) ξ−2r−4 ξd T k

2(j+1)

Rd

ˆ  )|2 dξ |h(ξ

ˆ  )|2 dξ  = |h|2 s+j−1/2 d−1 ,  ∫ ξ  2(s+j)−1 |h(ξ (R ) H Rd−1

using that, for a > 0, ∫ (a2 + ξd2 )−r−2 ξd

2(j+1)

R

dξd = a2(j−r)−1 ∫ (1 + η 2 )−r−2 η 2(j+1) ∂η  a2(j−r)−1 , R

if r > j − 1/2. In particular, with 0 < ε < 1/2, we have r = j − 1/2 + ε < 0, j−1/2+ε (j) and with s = 0, we find that ΛT k ∈ H −j−3/2−ε (Rd ) and (3.A.4)

j−1/2+ε (j)

k (j) H −2 (Rd ) ≤ ΛT

k

H −j−3/2−ε (Rd )  |h|H j−1/2 (Rd−1 ) .

As Q ∈ Ψ−2 (Rd ), by Proposition 2.52 of Volume 1 (adapted to standard operators), we obtain Qk (j) H −r (Rd )  Λm+r k (j) H −r−2 (Rd )  |h|H m+j−1/2 (Rd−1 ) . Λm+r T T Since r > j − 1/2 with j = −1, 0, we may pick r = 0, and we obtain v (j) L2 (R+ ;H m (Rd−1 ))  |h|H m+j−1/2 (Rd−1 ) . Arguing as above, we have

  P (x, D)v (j) = P (x, D)Qk (j) |Rd = R  (k (j) ), +  (j)  using that k |Rd = 0. With (3.A.4), for any N ∈ N, we obtain +

R  k (j) H N (Rd )  |h|H m+j−1/2 (Rd−1 ) . +

K (j) h

R  k (j) .

= We set Choosing N ≥ m, we obtain P (x, D)v (j) ∈ H m (Rd ). We set w(j) = (1 − ψ)v (j) . Arguing as we did for w = (1 − ψ)v, and using the structure of P (x, D), we obtain    Dd eα Dd w(j) ) ∈ L2 R+ ; H m−2 (Rd−1 ) ,  with, moreover, Dd eα Dd w(j) )L2 (R+ ;H m−2 (Rd−1 ))  |h|H m+j−1/2 (Rd−1 ) . With   Lemma 18.69 applied twice, we conclude that w(j) ∈ H 2 R+ ; H m−2 (Rd−1 ) and w(j) H 2 (R+ ;H m−2 (Rd−1 ))  |h|H m+j+3/2 (Rd−1 ) . With an interpolation argument [74, 236], we obtain that    H k R+ ; H m−k (Rd−1 ) w(j) ∈ 

0≤k≤2

and 0≤k≤2 w(j) H k (R+ ;H m−k (Rd−1 ))  |h|H m+j−1/2 (Rd−1 ) . From that point, we proceed as we did for v from (3.A.2). 

3.A. PROOF OF SOME TECHNICAL RESULTS

73

3.A.4. Properties of the Calder´ on Projector. Here, we prove Proposition 3.8. Let h(0) ∈ H m+3/2 (Rd−1 ) and h(−1) ∈ H m+1/2 (Rd−1 ). We set v (j) = (j) Q h(j) for j = −1, 0. We have v (j) ∈ H m+2 (Rd+ ) by Lemma 3.6 and P (x, D)v (j) = K (j) h(j) , with K (j) bounded from H m+3/2+j (Rd−1 ) into H N (Rd+ ), for any N ∈ N. By (3.2.8), we then have       v (j) = −Q (0) γ D (v (j) ) + Q (−1) γ˜ N (v (j) ) + Q(K (j) h(j) ) + Rv (j) |Rd , +  (j)  (j) (j) (j) m+3/2+j where h → Q(K h ) + Rv is bounded from H (Rd−1 ) |Rd +

into H N (Rd+ ), for any N ∈ N. Computing both the Dirichlet γ D and the modified Neumann traces γ˜ N at xd = 0+ , we obtain  (0)  (−1) N (j) (j) Id +QD (γ D (v (j) )) − QD (˜ γ (v )) = YD (h(j) ),   (1)  D (j) (0)  N (j) (j) γ (v )) = YN (h(j) ), QN (γ (v )) + Id −QN (˜ (j)

(j)

with YD and YN both bounded from H m+3/2+j (Rd−1 ) into H N (Rd−1 ), for (j) (j+1) any N ∈ N. As γ D (v (j) ) = QD (h(j) ) and γ˜ N (v (j) ) = QN (h(j) ) and using the definition of the entries of the matrix operator C, we have, in the case j = −1,   (−1) Id −CD,D CD,N − CD,N CN,N = YD ,   (−1) − CN,D CD,N + Id −CN,N CN,N = YN , and, in the case j = 0,   (0) Id −CD,D CD,D − CD,N CN,D = −YD ,   (0) − CN,D CD,D + Id −CN,N CN,D = −YN . We then observe that this reads    D,D  (0) (−1) −YD CD,N YD C Id −CD,D −CD,N = (Id −C)C = (0) (−1) . −CN,D Id −CN,N CN,D CN,N −YN YN As the operator on the r.h.s. maps H m+3/2 (Rd−1 ) ⊕ H m+1/2 (Rd−1 ) into  N d−1 2 H (R ) , we have reached the first result. We now consider φ ∈ H m (Rd+ ). Set v = Q(φ). We have v ∈ H m+2 (Rd+ ) by Lemma 3.6 and P (x, D)v = φ+Kφ with φ → Kφ bounded from H m (Rd+ ) into H N (Rd+ ), for any N ∈ N. Applying identity (3.2.14), we obtain      γ D (v) QD (Kφ) QD (φ) + + RC (v). = Id −C QN (φ) QN (Kφ) γ N (v) Observe that

 QD (Kφ) φ → + RC (v) QN (Kφ)

74

3. FREDHOLM PROPERTIES

 2 maps H m (Rd+ ) into H N (Rd−1 ) for any N ∈ N. Since γ D (v) = QD (φ) and γ˜ N (v) = QN (φ), the result follows.  3.A.5. Pseudo-Differential Form of the Action on the Traces. Here, we prove Lemma 3.9. We set w = δxd =0 ⊗ h and wε (x) = ψε (xd )h(x ) ∈ S (Rd ), with ψε (s) = ε−1 ψ(ε−1 s) with ε > 0 and ψ ∈ Cc∞ (R) with supp(ψ) ⊂ R− and ∫ ψ = 1. As wε converges to w in S  (Rd ) and as Q maps S  (Rd ) into itself continuously by Proposition 2.19 of Volume 1 (see Remark 2.55 also in Volume 1), we have in S  (Rd ).

Q(δxd =0 ⊗ h) = lim Q(wε ) ε→0

As Q maps S (Rd ) into itself, we have Qwε ∈ S (Rd ), and, in the sense of oscillatory integrals (see Theorem 2.11 of Volume 1), we have (3.A.5) Q(wε )(x) = (2π)−d ∫∫

Rd ×Rd

= (2π)

−(d−1)

ei(x−y)·ξ q(x, ξ)wε (y) dydξ ∫∫



Rd−1 ×Rd−1 (−1)

where qε





ei(x −y )·ξ qε(−1) (xd )(x , ξ  )h(y  ) dy  dξ  ,

(xd )(x , ξ  ) is a function given by

qε(−1) (xd )(x , ξ  ) = (2π)−1 ∫∫ ei(xd −yd )ξd q(x, ξ)ψε (yd ) dyd dξd , R×R

using the Fubini theorem that applies to oscillatory integrals. Observe that the integrand has a compact support in yd and behaves like ξd −2 as |ξd | → ∞ and thus this formula is well defined in the sense of the Lebesgue integral. In particular, we have continuity with respect to (xd , x , ξ  ) by the Lebesgue dominated-convergence theorem. Moreover, differentiation with respect to (x , ξ  ) under the integral sign can be performed yielding a smooth function with respect to these tangential variables. We can regularize the oscillatory (−1) integral defining qε (xd )(x , ξ  ). Using that i(1 + ∂yd )ei(xd −yd )ξd = (ξd + i)ei(xd −yd )ξd , we obtain, with integration by parts, qε(−1) (xd )(x , ξ  )

  + ∂yd ) ψε (yd ) dyd dξd = i(2π) ∫∫ e q(x, ξ) (ξd + i) R×R   t (1 + ∂ )k ψ (y ) yd ε d k −1 i(xd −yd )ξd = i (2π) ∫∫ e q(x, ξ) dyd dξd , (ξd + i)k R×R −1

i(xd −yd )ξd

t (1

for k ∈ N. With this last form, we see that the integrand now behaves like ξd −2−k as |ξd | → ∞. This allows for differentiations with respect to xd . (−1) We conclude that qε (xd )(x , ξ  ) is smooth with respect to (xd , x , ξ  ).

3.A. PROOF OF SOME TECHNICAL RESULTS

75

Knowing now that qε (xd )(x , ξ  ) is smooth with respect to all its variables, we now study its symbol properties. For that, it suffices to consider |ξ  | ≥ 1. In such case, by (3.2.1), we have (−1)

q(x, ξ) = p(x, ξ)−1 q˜(x, ξ) = 

(3.A.6)

q˜(x, ξ)  , ξd − iρ(x, ξ  ) ξd + iρ(x, ξ  )

where q˜(x, ξ) has 1 for principal symbol. The iterative construction of q˜ in the proof of Proposition 2.34 in Appendix 2.A.7 of Volume 1 shows that here q˜(x, ξ) − 1 is a rational function in the variable ξ. We then observe that the integrand is meromorphic in the upper complex half-plane with a pole at ξd = iρ(x, ξ  ). We consider xd ≥ 0. Because of the support of ψ, we have xd − yd > 0 and thus Re(i(xd − yd )ξd ) ≤ 0 for Im ξd ≥ 0. A classical change of contour of integration then gives qε(−1) (xd )(x , ξ  ) = (2π)−1 ∫ ∫ ei(xd −yd )ξd q(x, ξ)ψε (yd ) dξd dyd , R Γ0

where Γ0 is the closed path in C, oriented counter-clockwise, composed with (1) the real interval [−R0 ξ  , R0 ξ  ]; (2) the upper half-circle centered at 0 of radius R0 ξ  , with R0 ≥ 1 + |ρ(x, ξ  )|/ξ  , for x ∈ Rd+ and ξ  ∈ Rd−1 . Im(ξd )

•

−R0 ξ

Γ0

iρ(x, ξ )

Re(ξd )

R0 ξ

0

With (3.A.6), the residue theorem yields qε(−1) (xd )(x , ξ  ) = t(−1) (xd )(x , ξ  )eε (xd )(x , ξ  ), where t(−1) (xd ) is smooth with values in S −1 (Rd−1 ×Rd−1 ), with 12 ρ−1 (x , xd , ξ  ) as principal symbol, and 



eε (xd )(x , ξ  ) = ∫ e−(xd −yd )ρ(x,ξ ) ψε (yd )dyd = ∫ e−(xd −εs)ρ(x,ξ ) ψ(s)ds. R

R

εsρ(x, ξ  )

≤ 0 since ρ(x, ξ  ) ≥ 0, yielding In supp(ψ), we have s < 0 and thus ) εsρ(x,ξ | ≤ 1. The Lebesgue dominated-convergence theorem then gives |e 

eε (xd )(x , ξ  ) → e(xd )(x , ξ  ) = e−xd ρ(x,ξ ) ,

as ε → 0,

and we obtain 

q (−1) (xd )(x , ξ  ) = lim qε(−1) (xd )(x , ξ  ) = t(−1) (xd )(x , ξ  )e−xd ρ(x,ξ ) . ε→0

We see that

q (−1)

is a smooth function.

76

3. FREDHOLM PROPERTIES

   Observe that ∂xα ∂ξβ e−(xd −εs)ρ(x,ξ ) is a linear combination of terms of the form 

(xd − εs)k e−(xd −εs)ρ(x,ξ ) ∂xα1 ∂ξβ1 ρ(x, ξ  ) · · · ∂xαk ∂ξβk ρ(x, ξ  ), with α1 + · · · + αk = α and β1 + · · · + βk = β. Let xd ≥ 0. As s < 0 and ρ(x, ξ  )  ξ  , we find that  α β  −(x −εs)ρ(x,ξ )    (xd − εs)k ξ  k−|β| e−(xd −εs) ξ  ξ  −|β| , ∂  ∂  e d x ξ where the estimation is independent of ε > 0. We thus find that xd → eε (xd ) is a bounded map with values in S 0 (Rd−1 × Rd−1 ). Similarly, we have |∂xα ∂ξβ e(xd )(x , ξ  )|  ξ  −|β| . We compute (e − eε )(xd )(x , ξ  ) = e(xd )(x , ξ  )σε (xd )(x , ξ  ) with 

σε (xd )(x , ξ  ) = ∫ (1 − eεsρ(x,ξ ) )ψ(s) ds. R

We have |1 − e thus have

εsρ(x,ξ  )

|  ερ(x, ξ  )  εξ  , since |1 − eθ | ≤ |θ| if θ ≤ 0. We |σε (xd )(x , ξ  )|  εξ  .

With a computation similar to that performed above, |∂xα ∂ξβ σε (xd )(x , ξ  )|  εξ  1−|β| .   For xd ≥ 0, we thus find that the sequence σε (xd ) ε with values in S 0 (Rd−1 × Rd−1 ) converges to 1 for the topology of the symbol class S 1 . This implies  (−1)  that the sequence qε (xd ) ε with values in S −1 (Rd−1 × Rd−1 ) converges to q (−1) (xd ), itself in the same symbol space for the topology of the symbol class S 0 . (−1) Since the oscillatory integral (3.A.5) is continuous with respect to qε (xd ) lying in S 0 (Rd−1 × Rd−1 ) by Theorem 2.11 of Volume 1, we find that we may let ε → 0 in (3.A.5), and we obtain (3.A.7) Q(w)(x) = (2π)−(d−1) 

= Op q



∫∫

Rd−1 ×Rd−1

(−1)





ei(x −y )·ξ q (−1) (xd )(x , ξ  )h(y  ) dy  dξ 

 (xd ) h(x ).

Similarly, we set w = δx d =0 ⊗ h and wε (x) = ψε (xd )h(x ) ∈ S (Rd ), with ψε (s) = ε−2 ψ  (ε−1 s) with ε > 0. As wε converges to w in S  (Rd ), we have Q(δx d =0 ⊗ h) = lim Q(wε ) ε→0

in S  (Rd ),

3.A. PROOF OF SOME TECHNICAL RESULTS

and we obtain (3.A.8) Q(wε )(x) = (2π)−d ∫∫

Rd ×Rd

77

ei(x−y)·ξ q(x, ξ)wε (y) dydξ

= (2π)−(d−1)



∫∫

Rd−1 ×Rd−1





ei(x −y )·ξ qε(0) (xd )(x , ξ  )h(y  ) dy  dξ  ,

where qε (x , xd , ξ  ) is a function given by (0)

qε(0) (xd )(x , ξ  ) = (2π)−1 ∫∫ ei(xd −yd )ξd q(x, ξ)ψε (yd ) dyd dξd . R×R

Regularizing the oscillatory integral as is done above for qε (xd )(x , ξ  ) in (0) the analysis of the funtion Q(wε ) defined in (3.A.5), we obtain qε (xd )(x , ξ  ) is smooth with respect to all variables. We then consider |ξ  | ≥ 1, allowing one to use (3.A.6) and to perform the same change of contour of integration as above and obtain (−1)

qε(0) (xd )(x , ξ  ) = (2π)−1 ∫ ∫ ei(xd −yd )ξd q(x, ξ)ψε (yd ) dξd dyd . R Γ0

An integration by parts gives qε(0) (xd )(x , ξ  ) = i(2π)−1 ∫ ∫ ei(xd −yd )ξd q(x, ξ)ξd ψε (yd ) dξd dyd . R Γ0

Proceeding as above, first with the residue theorem, and second letting ε go to zero, we obtain that   Q(δx d =0 ⊗ h) = Op q (0) (xd ) h(x ), with 

q (0) (xd )(x , ξ  ) = lim qε(0) (x , xd , ξ  ) = t(0) (xd )(x , ξ  )e−xd ρ(x,ξ ) , ε→0

t(0) (x

where symbol.

d)

is smooth with values in S 0 (Rd−1 ×Rd−1 ), with − 12 as principal 

3.A.6. Recovery of the Traces up to Regularizing Operators. Here, we prove Lemma 3.11. With ΛsT = OpT (ξ  s ), s ∈ R, we define   1 0 1 0 ˜= L∂ , L 0 Λ1T 0 Λ−k T which is a 2×2 matrix operator with each entry in Ψ0 (Rd−1 ). The associated principal symbol is   −1  , ξ  )ξ  /2 1/2 −ρ (x + |xd =0 ˜ , ξ) = , (x (x , ξ  ) ξ  −k bk|xd =0+ (x , ξ  ) ξ  1−k bk−1 |x =0+ d

with each entry made of a symbol in × Observe that  k      1  1−k −1   k−1 ˜ , ξ ) = ξ  ρ ρ |x =0+ (x , ξ  ) = 0, det (x + (x , ξ ) b + b |x =0 d d 2 S 0 (Rd−1

Rd−1 ).

78

3. FREDHOLM PROPERTIES

ˇ for x ∈ κ(∂M ∩ O) and |ξ  | ≥ 1, by the Lopatinski˘ı–Sapiro property for ˜ (respec(P, B) written in (3.2.3) in the local coordinates. As a Result, L   ˜ tively, (x , ξ )) is elliptic in κ(∂M ∩ O).    −1 ˜ ,ξ ) ˜ = Op(m) and M ˜ that is a 2 × 2 We set m(x ˜  , ξ  ) = χ∂ (x ) (x 0 d−1 matrix operator with entries in Ψ (R ). By symbol calculus, we have  r ˜L ˜ = χ∂ + R−1 ˜M ˜ = χ∂ + R−1 M and L ,

(3.A.9)

 and Rr are two 2×2 matrix operators with entries in Ψ−1 (Rd−1 ). where R−1 −1 We set   1 0 1 0 ˜ . M M∂ = 0 Λ1T 0 Λ−k T

As we have

  0 1 0 ˜ 1 , L∂ = L 0 ΛkT 0 Λ−1 T

we obtain

     1 0 1 0 1 0  1 0  ˜ ˜ = χ∂ + R−1 . M∂ L ∂ U ∂ = ML 0 Λ1T 0 Λ1T 0 Λ−1 0 Λ−1 T T

By symbol calculus, we have    1 0 χ∂ 1 0 χ∂ −1 = 1 0 ΛT 0 ΛT 0 with  K1,∂

0 1 ΛT χ∂ Λ−1 T

 = χ∂ + K1,∂ ,

 0 0 = ,22 , 0 K1,∂

,22 ∈ Ψ−1 (Rd−1 ). We also have with K1,∂



   ,11 ,12 K K 1 0 1 0   2,∂ 2,∂ = R−1 = K2,∂ ,21 ,22 , 0 Λ1T 0 Λ−1 K2,∂ K2,∂ T

,11 ,22 ,12 ,21 where K2,∂ , K2,∂ ∈ Ψ−1 (Rd−1 ), K2,∂ ∈ Ψ−2 (Rd−1 ), and K2,∂ ∈ Ψ0 (Rd−1 ).  + K  , we see that K  has the sought operator matrix Setting K∂ = K1,∂ 2,∂ ∂ structure, and

M∂ L∂ U∂ = χ∂ U∂ + K∂ U∂ , for U∂ ∈ H s,s−1 (Rd−1 ). We note that the operator K∂ maps H s,s−1 (Rd−1 ) in H s+1,s (Rd−1 ). We also have      1 0 0 1 0  1 0 ˜ ˜ 1 r L M = . + R χ L ∂ M∂ = ∂ −1 0 ΛkT 0 ΛkT 0 Λ−k 0 Λ−k T T

3.A. PROOF OF SOME TECHNICAL RESULTS

79

Proceeding as above, we find L∂ M∂ = χ∂ + K∂r , with K∂r given by    0 0 1 0 1 0 r + R−1 . 0 ΛkT 0 ΛkT χ∂ Λ−k 0 Λ−k T − χ∂ T We thus find that K∂r has the matrix structure described in the lemma and  maps H s,s−k (Rd−1 ) in H s+1,s+1−k (Rd−1 ). 3.A.7. Local Right and Left Inverses up to a Compact Operator. Here, we prove Lemma 3.14. We compute ∗  ∗ j j ψ j P M j = κj ψˇj P C (x, D)M C (κj )−1 W k . j Using that P C (x, D) is a local operator and the support properties of ψˇj and ϕˇj , we find that j j j j ψˇj P C (x, D)M C = ψˇj P C (x, D)ϕˇj Mc Mb Ma ϕˇj = ψˇj P C (x, D)Mc Mb Ma ϕˇj .

By Lemma 3.6, we have

  j P C (x, D)Mc = Id +K −K (0) K (−1) ,

where the operators K : H m (Rd+ ) → H m+1 (Rd+ ), K (j) : H m+j+3/2 (Rd−1 ) → H m+1 (Rd+ ), j = −1, 0, are all bounded. As Mb Ma ϕˇj is a bounded map from H m (Rd+ ) ⊕ H m+3/2−k (Rd−1 ) into H m (Rd+ ) ⊕ H m+3/2 (Rd−1 ) ⊕ H m+1/2 (Rd−1 ), we conclude that, if (f, h) ∈ H m (Rd+ ) ⊕ H m+3/2−k (Rd−1 ), we have   j Cj Cj f j j r,C j j f ˇ ˇ ˇ (3.A.10) = ψ f + ψ K1 ϕˇ , ψ P (x, D)M h h where j

K1r,C : H m (Rd+ ) ⊕ H m+3/2−k (Rd−1 ) → H m+1 (Rd+ ). With (f j , k hj ) as introduced in (3.3.2)–(3.3.3), we have obtained   j  j ∗ j C j f j j f Cj ˇ (3.A.11) = κ ψ P (x, D)M ψ PM k j h h   j ∗ j j  j ∗ j r,C j j f j ˇ ˇ = κ ψ f + κ ψ K1 ϕˇ k j h  f = ψ j f + K1r,j , h (m+3/2)

where K1r,j : H m (M) ⊕ HB

(∂M) → H m+1 (M) is bounded.

We now compute     j      j ∗ j C j f j j f Cj ˇ   ψ B (x, D)M (3.A.12) = κ ψ BM k j   h h k ∂M x

. + d =0

80

3. FREDHOLM PROPERTIES

Note that this function vanishes on the other connected components of ∂M. Arguing as above, we write j j j ψˇj B C (x , 0, D)M C = ψˇj B C (x, D)Mc Mb Ma ϕˇj ,

which we compute with the following lemma. Lemma 3.30. Let (f, h) ∈ H m (Rd+ ) ⊕ H m+3/2−k (Rd−1 ) and set  j f v = Mc Mb Ma ϕˇ ∈ H m+2 (Rd+ ). h We have

   D QD (ϕˇj f ) γ (v) r j f ˆ + K ϕˇ , = M∂ h ϕˇj h γ˜ N (v)

(3.A.13) where

ˆ r : H m (Rd+ ) ⊕ H m+3/2−k (Rd−1 ) → H m+5/2 (Rd−1 ) ⊕ H m+3/2 (Rd−1 ) K is a bounded operator. A proof of Lemma 3.30 is given below. With the notation B j (x , D ) = B j (x , xd = 0, D ), for j = k and j = k − 1, we have     j j γ D (v) f ψˇj B C (x, D)M C = ψˇj B k (x , D ) −iB k−1 (x , D ) Dd v|xd =0+ h  D γ (v) = ψˇj L∂,2 , γ˜ N (v) with v as in Lemma 3.30 and   L∂,2 = B k (x , D ) + iB k−1 (x , D )r0 (x , xd = 0) B k−1 (x , D ) . The matrix L∂,2 is the second row of the operator L∂ given in (3.2.20), with (3.A.13) and Lemma 3.11, we obtain   j j j f f r,C j C C j j j j = ψˇ χ∂ ϕˇ h + ψˇ K2 ϕˇ ψˇ B (x, D)M h h  j f , = ψˇj h + ψˇj K2r,C ϕˇj h using the support properties of χ∂ and ϕˇj and where j

K2r,C : H m (Rd+ ) ⊕ H m+3/2−k (Rd−1 ) → H m+5/2−k (Rd−1 ) is a bounded operator.

3.A. PROOF OF SOME TECHNICAL RESULTS

81

With (f j , k hj ) as introduced in (3.3.2)–(3.3.3) and with (3.A.12), we have obtained     j      j ∗ j C j f j j f Cj ˇ   ψ B (x, D)M (3.A.14) = κ ψ BM k j   h h k ∂M x =0+  jd  ∗  ∗ j f = κj ψˇj k hj + κj ψˇj K2r,C ϕˇj k j h  f = ψ j k h + K2r,j , h (m+3/2)

(∂M) → H m+5/2−k (k∂M) is bounded. where K2r,j : H m (M) ⊕ HB With K1r,j defined in (3.A.11) and K2r,j defined in (3.A.14), we define the (β + 2) × (β + 2) matrix operator ⎛ r,j ⎞ K1 ⎜ 0 ⎟ ⎟ ⎜ ⎜ .. ⎟ ⎜ . ⎟ ⎟ ⎜ ⎜ 0 ⎟ r,j ⎜ K = ⎜ r,j ⎟ ⎟, ⎜K2 ⎟ ⎜ 0 ⎟ ⎟ ⎜ ⎜ . ⎟ ⎝ .. ⎠ 0 where the 1 × (β + 2) matrix operator K2r,j is in the k + 2 line. Note that the zeros in the matrices stand for the zero 1 × 3 matrix operator. We recall that k is the order of the boundary operator on the only connected component of ∂M that meets the local chart C j . The operator K r,j is (m+3/2) (m+5/2) (∂M) into H m+1 (M) ⊕ HB (∂M). bounded from H m (M) ⊕ HB With (3.A.11) and (3.A.14), we have thus reached the conclusion of the first result of Lemma 3.14. For the second result, we write  j  j −1 ∗ P C (x, D) (κ ) j D C γ B (x, D)   C j (x, D)  j ∗ j  j −1 ∗ P j = κ ψˇ Mc Mb Ma ϕˇ . (κ ) j D C γ B (x, D)

 ∗ ψ M L = ψ κj ϕˇj Mc Mb Ma ϕˇj j

j



j

For u ∈ H m+2 (M), we set t (f, h) = Lu, that is, f = P u ∈ H m (M),

h = (0 h, . . . , β h)  (m+3/2) (∂M). = (Bu|0∂M , . . . , Bu|β∂M ∈ HB

82

3. FREDHOLM PROPERTIES

We then set ∗  uj = (κj )−1 u, u = ϕˇj uj ,

∗  f j = (κj )−1 f,

f = ϕˇj f j ,

h = ϕˇj|x

∗  h = (κj )−1 k h, k = 0, . . . , β,

k j k j

h .

+ d =0

We introduce ϕ˜j ∈ C ∞ (Rd+ ) be such that supp(ϕ˜j ) ⊂ κj (Oj ), ϕ˜j ≡ 1 on supp(ϕˇj ), and χ∂ ≡ 1 on supp(ϕ˜j|x =0+ ). We then have d

(3.A.15) P C (x, D)u = ϕˇj P C (x, D)uj + [P C (x, D), ϕˇj ]ϕ˜j uj = ϕˇj f j + K1 ϕ˜j uj j

j

j

= f + K1 ϕ˜j uj , and (3.A.16) B C (x, D)u|xd =0+ = ϕˇj B C (x, D)uj|x j

j

= =

ϕˇj|x =0+ k hj + d h + K2 ϕ˜j uj ,

+ [B C (x, D), ϕˇj ]ϕ˜j uj|x j

+ d =0

+ d =0

K2 ϕ˜j uj

where K1 is bounded from H m+2 (Rd+ ) into H m+1 (Rd+ ) and K2 is bounded from H m+2 (Rd+ ) into H m+5/2−k (Rd−1 ). With (3.2.8) and (3.A.15), we have u = Q(f ) − Q (0) (γ D (u)) + Q (−1) (˜ γ N (u)) + K3 ϕ˜j uj ,   with K3 w = QK1 w + Rw |Rd that is bounded from H m+2 (Rd+ ) into H m+3 (3.A.17)

+

(Rd+ ). Computing the Dirichlet trace of (3.A.17) (or equivalently, considering the first row in identity (3.2.14)), we have   γ D (u)  = QD (f ) + K4 ϕ˜j uj , Id −CD,D −CD,N γ˜ N (u) with K4 = γ D K3 . We see that K4 is bounded from H m+2 (Rd+ ) into H m+5/2 (Rd−1 ). We write (3.A.16) in the following form:  D γ (u) L∂,2 = h + K2 ϕ˜j uj , γ˜ N (u) where L∂,2 is the second row of the matrix L∂ defined in (3.2.20). We thus have    D  K4 γ (u) QD (f ) + ϕ˜j uj . L∂ = h K2 γ˜ N (u) With Lemma 3.11, we then have, using the support of u,  D  D  γ (u) γ (u) QD (f ) = χ = M + K5 ϕ˜j uj , ∂ ∂ h γ˜ N (u) γ˜ N (u)

3.A. PROOF OF SOME TECHNICAL RESULTS

83

where K5 is bounded from H m+2 (Rd+ ) into H m+5/2 (Rd−1 ) ⊕ H m+3/2 (Rd−1 ). With the operator Mc defined in (3.3.8) and (3.A.17), we write ⎞ ⎛ ⎛ ⎞ f f u = Mc ⎝γ D (u)⎠ + K3 ϕ˜j uj = Mc Mb ⎝QD (f )⎠ + K6 ϕ˜j uj h γ˜ N (u) where K6 is bounded from H m+2 (Rd+ ) into H m+3 (Rd+ ). We then obtain  j f j j Cj ˇj  ˜j uj , ˇ ˇ ψ u=ψ M k j + ψ K6 ϕ h that is

 f ψ u=ψ M + K ,j u, h j

j

j

where K ,j is bounded from H m+2 (M) into H m+3 (M).



Proof of Lemma 3.30. With the forms of the operators Ma , Mb , and Mc , we have    QD (ϕˇj f ) j (0) (−1) . v = Q(ϕˇ f ) + −Q M∂ Q ϕˇj h With (3.2.9) and (3.2.10) and the definition of the Calder´on projector in (3.2.13), computing γ D (v) and γ˜ N (v), we find  D   γ (v) QD (ϕˇj f ) QD (ϕˇj f ) + CM∂ . = QN (ϕˇj f ) ϕˇj h γ˜ N (v) We set

 D  D  r γ (v) QD (ϕˇj f ) , = − M∂ ϕˇj h rN γ˜ N (v)

and we have

  D   QD (ϕˇj f ) QD (ϕˇj f ) r + C − Id)M∂ . = QN (ϕˇj f ) ϕˇj h rN

We first compute t (rD , CD,N rN ). We have   D,D  Id 0 − Id CD,N C C − Id) = 0 CD,N CD,N CN,D CD,N (CN,N − Id)  CD,N CD,D − Id +R = −CD,D (CD,D − Id) −CD,D CD,N  D,D  Id 0 − Id CD,N C + R, = 0 −CD,D CD,D − Id CD,N with (3.2.16) and (3.2.17), where

 2 R : H m+3/2 (Rd−1 ) ⊕ H m+1/2 (Rd−1 ) → H −N (Rd−1 )

84

3. FREDHOLM PROPERTIES

is bounded for any N ∈ N. We thus find  D  QD (ϕˇj f ) r = CD,N QN (ϕˇj f ) CD,N rN   D,D  Id 0 − Id CD,N C QD (ϕˇj f ) + M∂ ϕˇj h 0 −CD,D CD,D − Id CD,N  j ϕˇ f + R ϕˇj h with

 2 R : H m (Rd+ ) ⊕ H m+3/2−k (Rd−1 ) → H −N (Rd−1 )   bounded for any N ∈ N. The matrix Id −CD,D −CD,N is the first row of the matrix L∂ in (3.2.20). With Lemma 3.11, we obtain  j  D  (1 − χ∂ )QD (ϕˇj f ) ˇf r r ϕ + Ka , = ϕˇj h CD,N QN (ϕˇj f ) + CD,D χ∂ QD (ϕˇj f ) CD,N rN  2 where Kar is bounded from H m (Rd+ )⊕H m+3/2−k (Rd−1 ) into H m+5/2 (Rd−1 ) . We introduce ϕ˜j , ϕ˜j ∈ C ∞ (Rd+ ) be such that (1) (2) (3) (4)

supp(ϕ˜j ) ∪ supp(ϕ˜j ) ⊂ κj (Oj ); ϕ˜j ≡ 1 on a neighborhood of supp(ϕˇj ); ϕ˜j ≡ 1 on a neighborhood of supp(ϕ˜j ); and χ∂ ≡ 1 on a neighborhood of supp(ϕ˜j ). |x =0+ d

= with Kbr : w → (1 − χ∂ )QD (ϕ˜j w). We write (1 − χ∂ )QD r We now prove that Kb is regularizing. We write

 , (1 − χ∂ )QD (ϕ˜j w) = (1 − χ∂ ) (1 − ϕ˜j )Q(ϕ˜j w) + (ϕˇj f )

Kbr (ϕˇj f ),

|xd =0

and we observe that the operator (1 − ϕ˜j )Qϕ˜j ∈ Ψ−N (Rd ) for any N ∈ N. We thus find that Kbr is bounded from H m (Rd+ ) into H m+5/2 (Rd−1 ). We now write CD,D χ∂ QD (ϕˇj f ) = CD,D QD (ϕˇj f ) − CD,D Kbr (ϕˇj f ). Since the first row of the map in (3.2.15) is precisely CD,N QN + CD,D QD , we find that CD,N QN (ϕˇj f ) + CD,D χ∂ QD (ϕˇj f ) = Kcr (ϕˇj f ), where Kcr is bounded from H m (Rd+ ) into H m+5/2 (Rd−1 ). Setting   r  Kb r f r f K2 f + Ka = , Kcr h h

3.A. PROOF OF SOME TECHNICAL RESULTS

85

we obtain that K r is bounded from H m (Rd+ ) ⊕ H m+3/2−k (Rd−1 ) into  m+5/2 d−1 2 2 H (R ) and  D  r r j f = K . ϕ ˇ 2 h CD,N rN Since CD,N ∈ Ψ−1 (Rd−1 ) with nonvanishing principal symbol −1   ρ|x =0+ (x , ξ )/2, we may apply a parametrix as given by Proposition 2.33 d of Volume 1 yielding   D r r j f ˆ = K ϕˇ , h rN ˆ r as in the statement of the lemma. with K



CHAPTER 4

Selfadjoint Operators Under General Boundary Conditions Contents 4.1. 4.2. 4.3. 4.3.1. 4.3.2. 4.3.3. 4.3.4. 4.4. 4.5. 4.6. 4.7. 4.7.1. 4.7.2.

Introduction and Setting Selfadjointness Preliminary Result on Symmetry Dimension d ≥ 4 Dimension d = 3 Dimension d = 2 Necessary Conditions for Symmetry to Hold Sufficient Conditions for Selfadjointness A Green Formula Spectral Properties ˇ Lopatinski˘ı–Sapiro Elliptic Problem The Nonhomogeneous Elliptic Problem A Boundary Lifting Map

87 88 90 90 90 93 95 96 98 100 110 110 112

4.1. Introduction and Setting Let (M, g) be a connected smooth compact Riemannian manifold with boundary. We consider a second-order differential operator P on M given by P = −Δg + R1 where R1 is a first-order differential operator. We also consider a smooth differential boundary operator B that takes the form B = B k + B k−1 ∂ν on each connected component N of ∂M with B k and B k−1 smooth differential operators on (a neighborhood of) N of order k and k − 1, respectively, with k ≤ β ∈ N. Thus, the operator B is of order β with differentiation transverse to ∂M of order less than or equal to one as in Chap. 2.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 4

87

88

4. SELFADJOINT OPERATORS UNDER GENERAL BOUNDARY CONDITIONS

We introduce the unbounded operator P on L2 (M) with domain (4.1.1)

D(P) = {u ∈ L2 (M); P u ∈ L2 (M) and Bu|∂M = 0},

and given by Pu = P u for u ∈ D(P). This definition is sensible: if u ∈ L2 (M) is such that P u ∈ L2 (M), then u ∈ WP (M), with this space defined in Sect. 18.6.2, and from Lemma 18.32 we know that both the Dirichlet and Neumann traces can be defined and γ D (u) = u|∂M ∈ H −1/2 (∂M) and (∂M); see (3.1.2) γ N (u) = ∂ν u|∂M ∈ H −3/2 (∂M); then Bu|∂M ∈ HB for the definition of this space. Without any further requirement one sees that the operator P is closed. (−1/2−k)

Lemma 4.1. The unbounded operator (P, D(P)) is closed. Proof. Let (un ) be a sequence in D(P) that converges to u in L2 (M) and such that fn = Pun = P un converges to some f in L2 (M). Observe that P u = f in 0 D  (M). Thus, u lies in the space WP (M). Moreover, (un ) ⊂ WP (M) and it converges to u in the topology of that space. The Dirichlet and Neumann trace operators γ D and γ N are defined on WP (M), and moreover continuous with values in H −1/2 (∂M) and H −3/2 (∂M), respectively. We thus have the convergence of γ D (un ) to γ D (u) in H −1/2 (∂M) and that of γ N (un ) to γ N (u) in H −3/2 (∂M). In particular Bun converges (−1/2−k) to Bu in HB (∂M). As Bun = 0 we conclude that Bu = 0.  ˇ In this chapter, we shall furthermore assume that the Lopatinski˘ı–Sapiro holds for (P, B) on the whole ∂M; see Definition 2.2. By Theorem 3.28 we then find (4.1.2)

D(P) = {u ∈ H 2 (M); Bu|∂M = 0}.

Our goal in this short chapter is to characterize the joint properties of the operators P and B for the unbounded operator (P, D(P)) to be selfadjoint. We choose to concentrate on the case β ≤ 1, that is, on boundary operators of order less than one. 4.2. Selfadjointness We note that 0 Dc∞ (M) ⊂ D(P) implying that D(P) is dense in L2 (M). We recall that 0 Dc∞ (M) is the space of smooth functions on M with support away from ∂M (see Section 8.1.3 of Volume 1). As a result the adjoint of P is well defined; see Section 11.4 (also in Volume 1). For (P, D(P)) to be selfadjoint it is first required to be a closed operator. ˇ We saw above that this holds without requiring the Lopatinski˘ı–Sapiro condition. A second requirement for (P, D(P)) to be selfadjoint is symmetry, that is, (P u, w)L2 (M) = (u, P w)L2 (M) ,

u, w ∈ D(P).

This implies constraints on both the structures of P and B.

4.2. SELFADJOINTNESS

89

Above all, if (P, D(P)) is symmetric, then P is itself symmetric on As P = −Δg + R1 and Δg are symmetric on 0 Dc∞ (M) this implies that R1 has to be symmetric on 0 Dc∞ (M). 0

Dc∞ (M).

Lemma 4.2. Let R1 be a first-order differential operator on M and be symmetric on 0 Dc∞ (M). Then R1 = iV +f , where V is a smooth real vector field and f is a smooth function such that Im f = divg V /2. Proof. The operator R1 is of the form R1 = X + f with X a complex smooth vector field. If R1 is symmetric on 0 Dc∞ (M), then R1 = t R1 = t X + f . Yet, by Proposition 17.7 we have t X = −X − div X. This yields g X = −X and f − f = − divg X. The result follows.  With the structure of R1 given by Lemma 4.2 we now consider u, w ∈ H 2 (M). With Proposition 18.29 and the Green formula given in Proposition 18.30 we then obtain (4.2.1) (P u, w)L2 (M) = (u, P w)L2 (M) − (γ N (u), γ D (w))L2 (∂M) + (γ D (u), γ N (w))L2 (∂M) + i(γ D (g(V, ν)u), γ D (w))L2 (∂M) . Symmetry of P is thus equivalent to having (4.2.2)

0 = −(γ N (u), γ D (w))L2 (∂M) + (γ D (u), γ N (w))L2 (∂M) + i(γ D (g(V, ν)u), γ D (w))L2 (∂M) ,

if u, w ∈ D(P). On each connected component of ∂M the differential boundary operator B is either of order k = 0 or 1. For k = 0 or 1, we denote by k∂M, the union of all the connected components of ∂M where the order of B is equal to k. On a connected component N of 0∂M under the Lopatinski˘ı– ˇ Sapiro condition it is equivalent to consider Bu|N = u|N , that is, a Dirichlet boundary operator. See Example 2.5-(1). In such case, that connected component yields a vanishing contribution to (4.2.2). On a connected component N of 1∂M, conditions on B are required for ˇ both the Lopatinski˘ı–Sapiro condition and the symmetry condition (4.2.2) to hold. In fact, as we shall see below, the boundary condition can then be reduced to the form B = ∂ν + B  , with B  a first-order tangential differential operator. Our first main result is the following one. Theorem 4.3. Let d ≥ 2 and let P = −Δg + R1 , with R1 a smooth firstorder differential operator on M, and B a boundary operator of order less ˇ than or equal to one such that (P, B) satisfies the Lopatinski˘ı–Sapiro condition on ∂M. The unbounded operator (P, D(P)) is selfadjoint on L2 (M) if and only if (1) R1 = iV + f with V is a smooth real vector field and f a smooth complex valued function on M such that Im f = divg V /2;

90

4. SELFADJOINT OPERATORS UNDER GENERAL BOUNDARY CONDITIONS

(2) On every connected component N of 1∂M then the condition Bu|N = 0 is equivalent (a) either to a homogeneous Dirichlet boundary condition u|N = 0; (b) or to the form ∂ν u|N + (iX  u + hu)|N = 0 with X  a real valued vector field and h a complex valued function on N with the condition D (g(V, ν)) = 0. 2 Im h − divg X  + γN

In the case d = 2, we moreover have |X  |g∂ = 1. In the case d ≥ 3, we moreover have |X  |g∂ < 1. Here and in what follows, we denote the Dirichlet and Neumann trace D and γ N . operators on a connected component N of ∂M by γN N Before proving this result we first show that the reduction of B into the form ∂ν + B  stated in item (2) can be carried out. This is based on both ˇ having P symmetric and the Lopatinski˘ı–Sapiro condition holding. 4.3. Preliminary Result on Symmetry ˇ Here, we first prove that if the Lopatinski˘ı–Sapiro condition holds and if the operator (P, D(P)) is symmetric, then on any connected component N of ∂M the boundary condition Bu|N = 0 is equivalent (1) either to a homogeneous Dirichlet boundary condition u|N = 0; (2) or to the form (∂ν u + B  u)|N = 0 with B  a first-order tangential differential operator. The case d ≥ 4 is the simplest to treat. The cases d = 3 and d = 2 require more analysis. After this reduction to simple forms for B on each connected component of ∂M we show that in the first-order case ∂ν +B  , the symmetry for the operator (P, D(P)) gives some constraints on the tangential operator B. ˇ 4.3.1. Dimension d ≥ 4. In dimension d ≥ 4, if the Lopatinski˘ı–Sapiro condition holds, then by Proposition 2.8 a boundary operator of order one on a connected component N of ∂M necessarily takes the form B = α∂ν + B  , where α does not vanish and B  is a first-order tangential operator (the argument based on (2.3.4) and (2.3.5) is given in Remark 2.9-(3)). Upon dividing by α we may thus assume that B = ∂ν + B  . Note that for the case d ≥ 4 we obtain the expected form without using the symmetry of P. The next two sections treat the cases d = 3 and d = 2. In these two cases we shall exploit the symmetry of P. 4.3.2. Dimension d = 3. We consider a connected component N of ∂M. There the boundary operator B takes the form

1

(4.3.1)

B = a∂ν + B  ,

4.3. PRELIMINARY RESULT ON SYMMETRY

91

where B  is a first-order tangential differential operator, that is, B  = X  + iY  +h where X  , Y  are two real valued vector fields on N and h is a complex valued function. The principal symbol of B is given by b(m, ω) = iaωn + iX  , ω   − Y  , ω  , ∗ N , ω ∈ R, and n with n = ν , that is, for ω = ω  + ωn nm , with ω  ∈ Tm n m the unitary outgoing cotangent vector. In the statement of Proposition 2.8 we have t = t + tν ν and v = v  + vν ν here given by tν = − Im a, t = −Y  , vν = Re a, v  = X  . By that proposition and by Proposition 2.10 two cases ˇ arise if the Lopatinski˘ı–Sapiro condition holds on N .

(1) For all m ∈ N we have either ν  | or |vm |g∂ < |tνm | |tm |g∂ < |vm

(4.3.2) or (4.3.3)

  2   2 ν 2 gm (tm , vm )2 > |tm |g∂ − (vm ) |vm |g∂ − (tνm )2 .

In particular, the complex valued function a does not vanish on N . Note that this condition coincides with the condition one finds in the case d ≥ 4 (see Proposition 2.8). (2) For all m ∈ N we have ν  | and |vm |g∂ > |tνm | |tm |g∂ > |vm

(4.3.4) and (4.3.5)

  2   2 ν 2 ) |vm |g∂ − (tνm )2 . gm (tm , vm )2 < |tm |g∂ − (vm

In this second case, the complex valued function a may vanish.  ) = span(X  , Y  ) = T N . Moreover, span(tm , vm m m m The following lemma shows that in the case of a symmetric operator the second case does not occur. ˇ Lemma 4.4. Let d = 3. Assume that the Lopatinski˘ı–Sapiro condition holds for (P, B) and that the unbounded operator (P, D(P)) is symmetric. Let N be a connected component of ∂M where B is given by (4.3.1). Then either the boundary condition on N is equivalent to having a homogeneous Dirichlet condition or conditions (4.3.2)–(4.3.3) hold. In the latter case, the function a does not vanish. If the boundary condition on N is equivalent to a homogeneous Dirichlet condition we shall rather consider it as a zero-order boundary condition. In the case of a true first-order boundary operator, with this lemma we see that if (P, D(P)) is symmetric the first case, that is (4.3.2) and (4.3.3), only occurs. Then the function a does not vanish on N . Upon dividing by a we may thus assume that B = ∂ν + B  . This reduction corresponds to item (2) of Theorem 4.3.

92

4. SELFADJOINT OPERATORS UNDER GENERAL BOUNDARY CONDITIONS

In this reduced form, since the principal symbol is now b(m, ω) = iωn + iX  , ω   − Y  , ω  , that is, tν = 0, vν = 1, t = −Y  , and v  = X  . Conditions (4.3.2) and (4.3.3) now read    2 2  2 (4.3.6) ) > |tm |g∂ − 1 |vm | g∂ . |tm |g∂ < 1 or (g∂ )m (tm , vm Proof. Let us assume that (4.3.4) and (4.3.5) hold. If so, we first prove that a = 0 meaning that B = B  and second we prove that the purely tangential boundary condition is equivalent to a homogeneous Dirichlet condition. We proceed by contradiction and we assume that a = 0. We may then choose an open set W in N where |a| ≥ C > 0 for some C > 0. Let u, w ∈ D(P) be such that both their Dirichlet and Neumann traces vanish on all the connected components of ∂M but N . Such function can be constructed by means of Theorem 18.25 once the two traces are also prescribed on N . D (u) and γ D (w) in H 3/2 (N ) with support in W , then we If we prescribe γN N N D (u) and γ N (w) = −a−1 B  γ D (w) and the functions have γN (u) = −a−1 B  γN N N u and w are in D(P) as Bu|N = Bw|N = 0. By (4.2.2) we have D D D D 0 = (a−1 B  γN (u), γN (w))L2 (N ) − (γN (u), a−1 B  γN (w))L2 (N ) D D + i(γN (g(V, ν)u), γN (w))L2 (N ) ,

which reads (4.3.7)

  D D (u), γN (w) L2 (N ) , 0 = QγN

D (g(V, ν)). It is a first-order differential with Q = a−1 B  − t B  a−1 + iγN operator on N . Thus Q = Z + k with Z a complex valued vector field and k a complex valued function. With B  = X  + iY  , by Proposition 17.7 we have

a−1 B  − t B  a−1 = a−1 (X  + iY  + h) + (X  − iY  − h)a−1 + (divg X  − i divg Y  )a−1 = 2 Re(a−1 )X  − 2 Im(a−1 )Y  + [X  − iY  , a−1 ] + a−1 h − ha−1 + (divg X  − i div Y  )a−1 . Thus Z = 2 Re a−1 X  − 2 Im a−1 Y  and D (g(V, ν))+[X  −iY  , a−1 ]+(divg X  −i div Y  )a−1 +2i Im(a−1 h). k = iγN D (u) and γ D (w) can be chosen arbitrary with support in W , from (4.3.7) As γN N we find that Q vanishes on any function supported in W . This gives k = 0 and Z = 0. Yet, rank(X  , Y  ) = 2 this implies that Re a−1 = Im a−1 = 0 in W ; a contradiction. We now know that a = 0. The operator B = B  . It thus only acts tangentially on N . We have B  = X  + iY  + h. It is a bounded operator from H 3/2 (N ) into H 1/2 (N ).

4.3. PRELIMINARY RESULT ON SYMMETRY

93

If B  is injective,1 then, the boundary condition Bu|N = B  u|N = 0 is equivalent to having u|N = 0, the homogeneous Dirichlet boundary condition. Let us assume now that B  is not injective and let us reach a contradiction. This will conclude the proof. In such case, there exists ϕ ∈ H 3/2 (N ) such that ϕ = 0 and B  ϕ = 0. We choose u, w ∈ H 2 (M) as above with vanishing traces on the other connected components of the boundary and on N we choose (4.3.8)

D (u) = ϕ, γN

N γN (u) = 0,

D γN (w) = 0,

N γN (w) = ϕ.

We have Bu|N = Bw|N = 0 and thus u, w ∈ D(P). However, with (4.2.2) that expresses the symmetry of the operator P we obtain |ϕ|2L2 (N ) = 0; a contradiction as ϕ = 0.  4.3.3. Dimension d = 2. Let N be a connected component of 1∂M. It is diffeomorphic to the unit circle S1 . There, the operator B takes the form (4.3.9)

B = a∂ν + bT  + c,

where T  is a real vector field on N such that |T  |g∂ = 1 and a, b, c are smooth complex valued functions. ˇ Lemma 4.5. Let d = 2. Assume that the Lopatinski˘ı–Sapiro condition holds for (P, B) and that the unbounded operator (P, D(P)) is symmetric. Let N be a connected component of 1∂M where B is as given in (4.3.9). Then the boundary condition on N is equivalent (1) either to the homogeneous Dirichlet condition u|N = 0; (2) or to ∂ν u|N + (iβT  + h)u|N = 0 with β, h smooth functions, β real valued, and T  a real vector field on N , such that (a) |T  |g∂ = 1; (b) |β| < 1 on N or |β| > 1 on N ; D (g(V, ν)) = 0 on N . (c) 2 Im h − divg (βT  ) + γN The proof shows in fact that either a ≡ 0 or a does not vanish on N . Case 1 corresponds to a ≡ 0; Case 2 to a = 0. Proof. We set B  = bT  + c, with b, c and T  as given in (4.3.9). As B has a nonvanishing principal symbol on N the open sets {a = 0} and {b = 0} form a cover of N . Set W to be an open set of N such that W  {a = 0}. For u, w ∈ D(P) such that their traces are supported in W , from (4.2.2), we have D D D D (u), γN (w))L2 (N ) − (γN (u), a−1 B  γN (w))L2 (N ) 0 = (a−1 B  γN D D + i(γN (g(V, ν)u), γN (w))L2 (N ) . 1In fact, by Lemma 3.19 it is Fredholm and its index is zero. If injective it is bijective with a bounded inverse by the closed graph theorem.

94

4. SELFADJOINT OPERATORS UNDER GENERAL BOUNDARY CONDITIONS

Such functions can be constructed by means of Theorem 18.25 by having both their Dirichlet and Neumann traces vanish on all the connected components of ∂M but N . On N their Dirichlet trace can be chosen arN (u) = bitrarily with support on W and the Neumann traces given by γN −1  D D D ∞ −a B γN (u). For simplicity we choose γN (u), γN (w) ∈ Cc (W ). We then obtain D (g(V, ν)) = 0 in W, a−1 B  − t B  a−1 + iγN

which reads, by Proposition 17.7, (4.3.10)

2 Re(a−1 b)T  + [T  , a−1 b] + 2i Im(a−1 c) + a−1 b divg T  D + iγN (g(V, ν)) = 0 in W.

ˇ This implies Re(a−1 b) = 0 in W . With the Lopatinski˘ı–Sapiro condition −1 holding, by Proposition 2.8, we also have a b = ±i. Assume that there is a point m0 ∈ W such that a−1 b(m0 ) ∈ i(−1, 1). Then, this remains true for all m in the same connected component in W . In fact, this excludes that a vanishes on W as |b| ≥ C > 0 in a neighborhood of {a = 0}. Thus W = W = N . We thus find that a does not vanish on N in this case and a−1 b(m) ∈ i(−1, 1) for all m ∈ N . Upon dividing B by a it reduces to the form ˜ |N = ∂ν u|N + (iβT  + h)u|N , Bu with β a real function such that |β| < 1. From (4.3.10) we also have D −[T  , β] + 2 Im h − β divg T  + γN (g(V, ν)) = 0 in N ,

which reads by (17.2.6) (4.3.11)

D 2 Im h − divg (βT  ) + γN (g(V, ν)) = 0 in N .

To the contrary, assume that there is a point m0 ∈ W such that a−1 b(m0 ) ∈ i(−∞, −1) ∪ i(1, +∞), the same argument as above shows that b does not vanish on N and a−1 b(m) ∈ i(−∞, −1) ∪ i(1, +∞) for all m ∈ N . Upon dividing B by the b it reduces to the form ˆ |N = α∂ν u|N + (iT  + ρ)u|N , Bu with α a real function such that |α| < 1. From (4.3.10) we also have D − [T  , α−1 ] + 2α−1 Im(ρ) − α−1 divg T  + γN (g(V, ν)) = 0

in {α = 0} = {a = 0}. This reads (4.3.12)

D T  (α) + (2 Im(ρ) − divg T  )α + γN (g(V, ν))α2 = 0,

in {α = 0}. By continuity observe in fact this first-order equation holds in {α = 0}. Observe also that (4.3.12) holds in int{α = 0}. Thus (4.3.12) everywhere on N . By the Cauchy–Lipschitz theorem then either α ≡ 0 or α does not vanish on N .

4.3. PRELIMINARY RESULT ON SYMMETRY

95

If α does not vanish, meaning that a does not vanish in the first place, upon dividing B by the a it reduces to the form ˜ |N = ∂ν u|N + (iβT  + h)u|N , Bu with β a real function such that |β| > 1. From (4.3.10) we also find (4.3.11). ˆ = iT  + ρ. Using the same Finally, if α ≡ 0, that is a ≡ 0, then B argument as in the proof of Lemma 4.4 at (4.3.8) we find that this firstˆ |N = 0 order operator is injective meaning that the boundary condition Bu is equivalent to a homogeneous Dirichlet boundary condition u|N = 0.  4.3.4. Necessary Conditions for Symmetry to Hold. With Lemmata 4.4 and 4.5, for any dimension d ≥ 2, we now know that if the Lopatinˇ ski˘ı–Sapiro condition holds and if the operator (P, D(P)) is symmetric, then on any connected component N of ∂M the boundary condition Bu|N = 0 is equivalent (1) either to a homogeneous Dirichlet boundary condition u|N = 0; (2) or to the form (∂ν u + B  u)|N = 0 with B  a first-order tangential differential operator. In the latter case, the following proposition gives some constraints on the tangential operator B  . We recall that P takes the form P = −Δg + iV + f with V a smooth real vector field and f a smooth complex function on M with the additional constraint Im f = divg V /2 by Lemma 4.2. Proposition 4.6. Let d ≥ 2 and let N be a connected component of ˇ ∂M where B = ∂ν + B  . If the Lopatinski˘ı–Sapiro condition holds for (P, B) on ∂M and (P, D(P)) is symmetric, then 1

(4.3.13)

D (g(V, ν)) = 0. B  − t B  + iγN

In particular, this means that B  = iX  + h with X  a real valued vector field and h a complex valued function on N with the condition (4.3.14)

D (g(V, ν)) = 0. 2 Im h − divg X  + γN

In the case d = 2, we moreover have |X  |g∂ = 1. In the case d ≥ 3, we moreover have |X  |g∂ < 1. Proof. Let u, w ∈ D(P) be such that both their Dirichlet and Neumann traces vanish on all the connected components of ∂M but N . Such function can be constructed by means of Theorem 18.25 once the two traces are also prescribed on N . We denote the Dirichlet and Neumann trace operators on D and γ N . N by γN N By (4.2.2) we have N D D N (u), γN (w))L2 (N ) + (γN (u), γN (w))L2 (N ) 0 = −(γN D D + i(γN (g(V, ν)u), γN (w))L2 (N ) .

96

4. SELFADJOINT OPERATORS UNDER GENERAL BOUNDARY CONDITIONS

N (u) = −B  γ D (u) and Because of the imposed boundary condition we have γN N N (w) = −B  γ D (w). Thus, upon prescribing γ D (u) and γ D (w) in H 3/2 (N ) γN N N N the functions u and w are uniquely defined. Moreover, we have D D D D (u), γN (w))L2 (N ) − (γN (u), B  γN (w))L2 (N ) 0 = (B  γN D D + i(γN (g(V, ν)u), γN (w))L2 (N ) ,

which reads 0=

  t  D  D D B − B  + iγN (g(V, ν)) γN (u), γN (w) L2 (N ) .

D (u) and γ D (w) can be arbitrarily chosen in H 3/2 (N ) we obtain (4.3.13). As γN N Let us write B  = Y  + h where Y  is a complex vector field and h is a smooth complex function on N . Then, by Proposition 17.7 we have

B  − t B  = Y  + Y  + divg Y  + h − h. From (4.3.13) we conclude that Y  = iX  with X  a real vector field on N . Moreover, we find B  − t B  = −i divg X  + 2i Im h, which with (4.3.13) yields (4.3.14). ˇ From the characterization of the Lopatinski˘ı–Sapiro condition in Propo sition 2.8, we find that |X |g∂ = 1 in the case d = 2 and |X  |g∂ < 1 in the case d ≥ 4. Finally, in the case d = 3 by Lemma 4.4 we have (4.3.2) and (4.3.3) that precisely coincides with the condition one finds in the case d ≥ 4 in Proposition 2.8. Thus, for the same reason we conclude that |X  |g∂ < 1.  The following lemma will be of use in what follows. ˇ Lemma 4.7. If the Lopatinski˘ı–Sapiro condition holds for (P, B) on ∂M and (P, D(P)) is symmetric, then (R1 u, v)L2 (M) + B  u|∂M , v¯|∂M H −1/2 (1∂M),H 1/2 (1∂M) = (u, R1 v)L2 (M) + u|∂M , B  v |∂M H 1/2 (1∂M),H −1/2 (1∂M) , for u, v ∈ H 1 (M) such that u|0∂M = v|0∂M = 0. The proof follows the arguments of the proofs of Lemma 4.2 and Proposition 4.6, noting that H 1 -regularity suffices for the performed computations. 4.4. Sufficient Conditions for Selfadjointness Here, we finish the proof of Theorem 4.3. Above in Lemma 4.2 and Proposition 4.6 we saw that under the Lopatinˇ ski˘ı–Sapiro condition if the unbounded operator (P, D(P)) is selfadjoint, then the two properties in Theorem 4.3 hold. We now prove that they are in fact sufficient for selfadjointness to hold. As explained above, in dimension d = 2 or d = 3, on a connected component N of 1∂M, if the boundary condition Bu|N = 0 is equivalent to

4.4. SUFFICIENT CONDITIONS FOR SELFADJOINTNESS

97

a homogeneous Dirichlet condition u|N = 0 we choose to use this latter and simpler form of the boundary condition. We thus consider N to be part of 0 ∂M. We refer to Lemmata 4.4 and 4.5 and their proofs for details. Proof of the sufficiency of the conditions in Theorem 4.3. The adjoint operator of P is well defined as D(P) is dense in L2 (M) as mentioned at the beginning of Sect. 4.2. Its domain is given by D(P∗ ) = {v ∈ L2 (M); ∃C > 0, ∀u ∈ D(P), |(v, P u)L2 (M) | ≤ CuL2 (M) }. Under the properties (1) and (2) in the statement of the theorem one finds that (4.2.2) holds meaning that P is symmetric. Hence, it only remains to prove that D(P∗ ) = D(P). If v ∈ D(P∗ ), one has P∗ v ∈ L2 (M) and (v, P u)L2 (M) = (P ∗ v, u)L2 (M) , for all u ∈ D(P) by the very definition of P∗ . We choose ϕ ∈ 0 Dc∞ (M) ⊂ D(P). We then have (P ∗ v, ϕ)L2 (M) = (v, P ϕ)L2 (M) = v, P ϕμg 0 D  (M),1 Dc∞ (M) = t P v, ϕμg 0 D  (M),1 Dc∞ (M) , where μg is the canonical positive density associated with the metric g (see Sect. 17.3). Now with the assumption made on X and f we have t P = P . We thus find P v, ϕμg 0 D  (M),1 Dc∞ (M) = (P ∗ v, ϕ)L2 (M) = P ∗ v, ϕμg 0 D  (M),1 Dc∞ (M) for all ϕ ∈ 0 Dc∞ (M) implying that P v = P ∗ v in 0 D  (M). As P ∗ v ∈ L2 (M) we thus have P v ∈ L2 (M). Consequently v lies in the space WP (M) defined in Sect. 18.6.2. By Lemma 18.32 the Dirichlet and Neumann traces γ D (v) and γ N (v) are well defined in H −1/2 (∂M) and H −3/2 (∂M), respectively. Moreover, by Lemma 18.33 we have the following Green-like formula (P v, w)L2 (M) + γ N (v), γ D (w)H −3/2 (∂M),H 3/2 (∂M) = (v, P w)L2 (M) + γ D (v), γ N (w)H −1/2 (∂M),H 1/2 (∂M) + γ D (g(X, ν)v), γ D (w)H −1/2 (∂M),H 1/2 (∂M) , for any w ∈ H 2 (M), as here t P = P . We now take w ∈ D(P). Since P v = P ∗ v and (P v, w)L2 (M) = (v, P w)L2 (M) we thus find 0 = −γ N (v), γ D (w)H −3/2 (∂M),H 3/2 (∂M) + γ D (v), γ N (w)H −1/2 (∂M),H 1/2 (∂M) + iγ D (g(V, ν)v), γ D (w)H −1/2 (∂M),H 1/2 (∂M) , using that X = iV here. First, if any, consider a connected component N of ∂M where B coincides with the Dirichlet boundary operator, that is, B is of order zero. By Theorem 18.25 pick w ∈ H 2 (M) such that γ D (w) = 0, that is, its

98

4. SELFADJOINT OPERATORS UNDER GENERAL BOUNDARY CONDITIONS

Dirichlet trace vanishes on the whole ∂M and ∂ν w|∂M\N = 0, that is, its Neumann trace vanishes also except on N . We then have Bw|∂M = 0 and thus w ∈ D(P) and we obtain D N γN (v), γN (w)H −1/2 (N ),H 1/2 (N ) = 0. N (w) can be chosen arbitrarily in H 1/2 (∂M), we obtain As the value of γN D that Bv|N = γN (v) = 0. Second, if any, consider a connected component N of ∂M where B is of order one, of the form B = ∂ν + B  . By Theorem 18.25 pick w such that both its Dirichlet and Neumann traces vanish on all connected components D (w) = ϕ ∈ H 3/2 (N ) be kept arbitrary and of ∂M but N . On N on let γN N (w) to be equal to −B  ϕ. Thus Bw on set γN |N = 0. With such choice we have

(4.4.1)

N D (v), ϕH −3/2 (N ),H 3/2 (N ) − γN (v), B  ϕH −1/2 (N ),H 1/2 (N ) 0 = −γN D + iγN (g(V, ν)v), ϕH −1/2 (N ),H 1/2 (N ) .

D (g(V, ν)) = 0 We write, using that B  − t B  + iγN D D γN (v), B  ϕH −1/2 (N ),H 1/2 (N ) = t B  γN (v), ϕH −3/2 (N ),H 3/2 (N ) D = B  γN (v) D + iγN (g(V, ν)v), ϕH −3/2 (N ),H 3/2 (N ) ,

which yields with (4.4.1) Bv|N , ϕH −3/2 (N ),H 3/2 (N ) = 0. As ϕ is arbitrary in H 3/2 (N ) we find that Bv|N = 0. We thus conclude that Bv|∂M = 0. As we saw above that P v ∈ L2 (M), we conclude that v ∈ D(P) from the definition of D(P) in (4.1.1).  4.5. A Green Formula The following subspace of L2 (M) associated with P is introduced in Sect. 18.6.2: (4.5.1)

WP (M) = {u ∈ L2 (M); P u ∈ L2 (M)}.

Note that the action of P on u is to be understood in the sense of distributions, as expressed in (18.3.1), and not as that of the unbounded operator (P, D(P)) on L2 (M). Properties of the space WP (M) are analyzed in Sect. 18.6.2. In particular, by Lemma 18.32, if u ∈ WP (M), then both its Dirichlet and Neumann traces, γ D (u) and γ N (u), are well defined and γ D (u) = u|∂M ∈ H −1/2 (∂M)

and γ N (u) = ∂ν u|∂M ∈ H −3/2 (∂M).

4.5. A GREEN FORMULA

99

If the unbounded operator (P, D(P)) is selfadjoint, meaning that the necessary of sufficient conditions given in Theorem 4.3 hold, one also has the following Green-like formula. Proposition 4.8. Let P and B be such that (P, D(P)) is selfadjoint. If u ∈ WP (M) and w ∈ H 2 (M) we have (P u, w)L2 (M) − (u, P w)L2 (M) + γ0N∂M (u), γ0D∂M (w)H −3/2 (0∂M),H 3/2 (0∂M) + Bu|1∂M , γ1D∂M (w)H −3/2 (1∂M),H 3/2 (1∂M) = γ0D∂M (u), γ0N∂M (w)H −1/2 (0∂M),H 1/2 (0∂M) + γ1D∂M (u), Bw|1∂M H −1/2 (1∂M),H 1/2 (1∂M) + iγ0D∂M (g(V, ν)u), γ0D∂M (w)H −1/2 (0∂M),H 1/2 (0∂M) . Recall that 0∂M is the union of all the connected components of ∂M where B is the Dirichlet boundary operator and 1∂M is the union of all the connected components of ∂M where B is of order one with the properties given in Theorem 4.3. For both u, w ∈ H 2 (M), with Proposition 4.8 seen in Proposition 4.6 and its proof. we find (P u, w)L2 (M) + (γ0N∂M (u), γ0D∂M (w))L2 (0∂M) + (Bu|1∂M , γ1D∂M (w))L2 (1∂M) = (u, P w)L2 (M) + (γ0D∂M (u), γ0N∂M (w))L2 (0∂M) + (γ1D∂M (u), Bw|1∂M )L2 (1∂M) + i(γ0D∂M (g(V, ν)u), γ0D∂M (w))L2 (0∂M) . Note also that in the case where both u and w have vanishing Dirichlet traces on 0∂M, we have the simple form, for u ∈ WP (M) and w ∈ H 2 (M), (4.5.2) (P u, w)L2 (M) + Bu|1∂M , γ1D∂M (w)H −3/2 (1∂M),H 3/2 (1∂M) = (u, P w)L2 (M) + γ1D∂M (u), Bw|1∂M H −1/2 (1∂M),H 1/2 (1∂M) , which in the case where u and w are moreover both in H 2 (M) gives (4.5.3)

(P u, w)L2 (M) + (Bu|1∂M , γ1D∂M (w))L2 (1∂M) = (u, P w)L2 (M) + (γ1D∂M (u), Bw|1∂M )L2 (1∂M) .

Note that this latter formula is also a consequence of Lemma 4.7. Proof. We start from the result of Lemma 18.33 that reads here (P u, w)L2 (M) + γ N (u), γ D (w)H −3/2 (∂M),H 3/2 (∂M) = (u, P w)L2 (M) + γ D (u), γ N (w)H −1/2 (∂M),H 1/2 (∂M) + iγ D (g(V, ν)u), γ D (w)H −1/2 (∂M),H 1/2 (∂M) .

100 4. SELFADJOINT OPERATORS UNDER GENERAL BOUNDARY CONDITIONS

On 1∂M with B = ∂ν + B  , with B  a first-order tangential differential operator we have Bu|1∂M = (∂ν u + B  u)|1∂M ∈ H −3/2 (1∂M), Bw|1∂M = (∂ν w + B  w)|1∂M ∈ H 1/2 (1∂M). The condition in Theorem 4.3 precisely mean t B  = B  + iγ D (g(V, ν) (as seen in Proposition 4.6 and its proof). It follows that γ1N∂M (u), γ1D∂M (w)H −3/2 (1∂M),H 3/2 (1∂M) = Bu|1∂M , γ1D∂M (w)H −3/2 (1∂M),H 3/2 (1∂M) − B  u|1∂M , γ1D∂M (w)H −3/2 (1∂M),H 3/2 (1∂M) , and B  u|1∂M , γ1D∂M (w)H −3/2 (1∂M),H 3/2 (1∂M) = γ1D∂M (u), t B  w|1∂M H −1/2 (1∂M),H 1/2 (1∂M) = γ1D∂M (u), B  w|1∂M H −1/2 (1∂M),H 1/2 (1∂M) − iγ1D∂M (u), γ1D∂M (g(V, ν)w)H −1/2 (1∂M),H 1/2 (1∂M) . We thus find γ1N∂M (u), γ1D∂M (w)H −3/2 (1∂M),H 3/2 (1∂M) − γ1D∂M (u), γ1N∂M (w)H −1/2 (∂M),H 1/2 (∂M) = Bu|1∂M , γ1D∂M (w)H −3/2 (1∂M),H 3/2 (1∂M) − γ1D∂M (u), Bw|1∂M H −1/2 (1∂M),H 1/2 (1∂M) + iγ1D∂M (u), γ1D∂M (g(V, ν)w)H −1/2 (1∂M),H 1/2 (1∂M) , 

which yields the result. 4.6. Spectral Properties

In the case where (P, D(P)) is selfadjoint, meaning that the assumptions of Theorem 4.3 are fulfilled, we have following spectral decomposition. ˇ Theorem 4.9. Let P and B be such that the Lopatinski˘ı–Sapiro condition holds on ∂M and (P, D(P)) is selfadjoint. Then, the spectrum of (P, D(P)) is a sequence of real eigenvalues (μj )j whose modulus goes to +∞. Each eigenvalue is of finite multiplity. Moreover, there exists a Hilbert basis (φj )j∈N of L2 (M) made of eigenfunctions of (P, D(P)), each associated with the eigenvalues μj . Proof. For c ∈ C, we define the bounded operator (4.6.1)

Lc : H 2 (M) → L2 (M) ⊕ H 3/2,1/2 (∂M) u → (P u + cu, Bu|0∂M , Bu|1∂M ),

4.6. SPECTRAL PROPERTIES

101

with the notation of (3.1.3) in Chap. 3 with (3/2)

H 3/2,1/2 (∂M) = HB

(∂M) = H 3/2 (0∂M) ⊕ H 1/2 (1∂M).

By Theorem 3.1 the Fredholm properties of this operator Lc is equivalent to ˇ having the Lopatinski˘ı–Sapiro condition. Moreover, the index of Lc is zero: in the case d ≥ 4, this follows from Theorem 3.15; in the case d = 3, this follows from Theorem 3.16; in the case d = 2, we find that the form of B given in Theorem 4.3 allows one to apply the first point of Theorem 3.25. First, we consider the case c = i. Since (P, D(P)) is selfadjoint, we have ker(i Id +P) = {0} by Theorem 11.18 of Volume 1. Hence, the operator : L2 (M) ⊕ Li is injective and consequently surjective. The inverse L−1 i 3/2,1/2 2 H (∂M) → H (M) is thus well defined and bounded by the open 2 mapping theorem. Observe then that L−1 i restricted to L (M)⊕{0} provides the inverse of i Id +P. We obtain that (i Id +P)−1 : L2 (M) → H 2 (M) is a bounded operator. As the injection ι : H 2 (M) → L2 (M) is compact by the Rellich–Kondrachov theorem (Theorem 18.7), the resolvent map (i Id +P)−1 maps L2 (M) into itself in a compact way. In particular, from the spectral properties of compact operators, it has a bounded and at most countable spectrum. Second, we claim that Lc is injective for some c ∈ R. In fact, if 0 = u ∈ H 2 (M) is such that Lc u = 0, then Bu|∂M = 0 and P u + iu = (i − c)u. As i − c = 0, this means that 1/(i − c) is an eigenvalue of (i Id +P)−1 . The set {1/(i − c); c ∈ R} is the image of the horizontal line i + R by the inversion map z → 1/z. It is the circle centerer at −i/2 with radius 1/2. Above, we saw that the spectrum of (i Id +P)−1 is at most countable. We may thus find c ∈ R such that 1/(i − c) is not one of its eigenvalues. For such a well-chosen value of c, the map Lc is injective. With the analysis used above for Li transposed to Lc , we find that (c Id +P)−1 is a well defined compact map on L2 (M). Since c Id +P is selfadjoint, then so is (c Id +P)−1 . From the spectral properties of selfadjoint compact operators, we find that its spectrum is formed by a sequence of real eigenvalues of finite multiplicity that accumulate to zero and that zero is not an eigenvalue. Moreover, there exists a Hilbert basis of L2 (M) made of eigenfunctions (φj )j∈N associated with the sequence of eigenvalues. We conclude that the spectrum of (P, D(P)) is formed of a sequence of real eigenvalues whose modulus go to ∞ and any eigenvalue μ is of finite  multiplicity, as (μ + c)−1 is then an eigenvalue of (c Id +P)−1 . As stated in Theorem 4.3 on any connected component N of ∂M the boundary operator B is (1) either (equivalent to) the Dirichlet operator and the part of the boundary where this occurs is denoted by 0∂M; (2) or of the form Bu|N = ∂ν u|N + (iX  + h)u|N where with X  a real valued vector field on N and h a complex valued function on

102 4. SELFADJOINT OPERATORS UNDER GENERAL BOUNDARY CONDITIONS

N with some joint condition on P , X  , and h. The part of the boundary where this occurs is denoted by 1∂M. To understand the basic behavior of the sequence of eigenvalues we shall need the following lemma. Lemma 4.10. Let X  be such that |X  |g∂ < 1 on 1∂M. Set B  = iX  + h. There exists 0 < C0 < 1 and C > 0 such that |(B  γ D (w), γ D (w))L2 (1∂M) | ≤ C0 ∇g w2L2 V (M) + Cw2L2 (M) , for all w in H 1 (M). Proof. Let us first focus on one of the connected components N of ∂M. In a neighborhood of N we use normal geodesic coordinates (m , z) ∈ O = N × [0, Z0 ) as given by Theorem 17.22. In such coordinates the metric g takes the form 1

 g(m ,z) = gm  (z) ⊗ 1z + dz ⊗ dz .

Since N is compact we have   |Xm  |g∂ = |Xm |g  (0) ≤ C1 < 1,

for some C1 > 0 and for all m ∈ N . Let Y  (z) be a smooth real vector field on N that depends smoothly on the parameter z ∈ [0, Z0 ) and such that Y  (0) = X  and satisfies |Y  (z)m |g (z)m ≤ C2 for some C1 ≤ C2 < 1 and for all (m , z) ∈ O. Using (17.2.1), this implies that |Y  w(m , z)| = |g(m ,z) (Y  (z)m , ∇g w(m ,z) )| (4.6.2)

  = |gm  (z)(Y (z)m , ∇g  (z) w(m ,z) )|

≤ C2 |∇g (z) w(m , z)|g (z)m . We first choose w ∈ H 2 (O) supported near z = 0, say in N × [0, Z0 /2), and we compute i(X  w|z=0+ , w|z=0+ ))L2 (N ) = (Dz Y  (z)w, w)L2 (O) − (Y  (z)w, Dz w)L2 (O)  1 + Y  (z)w, wDz log det g L2 (O) 2 = (Y  (z)Dz w, w)L2 (O) − (Y  (z)w, Dz w)L2 (O)

   1 + [Dz , Y  (z)] − (Dz log det g)Y  (z) w, w 2 . 2 L (O) By Proposition 17.7 giving the transpose of a vector field and using that N has no boundary we find (Y  (z)Dz w, w)L2 (O) = −(Dz w, Y  (z)w)L2 (O) − (Dz w, divg Y  (z)w)L2 (O) .

4.6. SPECTRAL PROPERTIES

103

We thus obtain i(X  w|z=0+ , w|z=0+ ))L2 (N ) = −2 Re(Dz w, Y  (z)w)L2 (O) + (Rw, w)L2 (O) , where R is a first-order differential operator. We now have, for ε > 0 to be fixed below, 2|(Dz w, Y  (z)w)L2 (O) | ≤ (1 + ε)−1 Dz w2L2 (O) + (1 + ε)Y  (z)wL2 (O) 2

≤ (1 + ε)−1 Dz w2L2 (O) + C22 (1 + ε) ∫

[0,Z0 )

∇g (z) w2L2 V (N ) dz,

  by (4.6.2). We choose ε > 0 so that C3 = max (1 + ε)−1 , C22 (1 + ε) < 1. We then have   2|(Dz w, Y  (z)w)L2 (O) | ≤ C3 Dz w2L2 (O) + ∫ ∇g (z) w2L2 V (N ,g (z)) dz [0,Z0 )

=

C3 ∇g w2L2 V (O) .

We also write, for any η > 0, |(Rw, w)L2 (O) | ≤ η∇g w2L2 V (O) + Cη w2L2 (O) . If we choose η > 0 such that CN = C3 + η < 1 we then obtain (4.6.3)

|(X  w|z=0+ , w|z=0+ ))L2 (N ) | ≤ CN ∇g w2L2 V (O) + Cw2L2 (O) .

This estimate extends to any function w ∈ H 1 (O) by density. Denote by N 1 , . . . , N n the connected components of 1∂M and by O1 , . . . , n O the associated open neighborhoods where we use normal geodesic coordinates as above. These neighborhoods are chosen disjoint. Let ψ 1 , . . . , ψ n ∈ C ∞ (M) be such that ψ j ≡ 1 in a neighborhood of N j , 0 ≤ ψ j ≤ 1, and supp(ψ j ) ⊂ Oj , j = 1, . . . n. Let now w ∈ H 1 (M). Estimate (4.6.3) applies to ψ j w, j = 1, . . . , n. We then write  (B  γ D (ψ j w), γ D (ψ j w))L2 (N j ) (B  γ D (w), γ D (w))L2 (1∂M) = 1≤j≤n

≤



1≤j≤n

2

CN j ∇g (ψ j w)L2 V (O) + Cw2L2 (O) ,

j with CN j < 1. We set C0 = max1≤j≤n  CN j . jSince the supports of ψ , j = 1, . . . , n, are disjoint if we set ψ = 1≤j≤n ψ we obtain

(B  γ D (w), γ D (w))L2 (1∂M) ≤ C0 ∇g (ψw)2L2 V (O) + Cw2L2 (O) ≤ C0 ψ∇g (w)2L2 V (O) + C  w2L2 (O) , using that [∇g , ψ] is a bounded function. As ψ ≤ 1 the result follows. From Lemma 4.10 we deduce the following norm equivalence.



104 4. SELFADJOINT OPERATORS UNDER GENERAL BOUNDARY CONDITIONS

Corollary 4.11. Let (P, D(P)) be selfadjoint. In the case d = 2 assume furthermore that X  is such that |X  |g∂ < 1 on 1∂M. Then, there exists C∗ > 0 and C > 0 such that C −1 u2H 1 (M) ≤ (Pu, u)L2 (M) +C∗ u2L2 (M) ≤ Cu2H 1 (M) ,

u ∈ D(P).

Proof. Let u ∈ D(P). With (18.5.5) consequence of the divergence formula of Proposition 18.28, we write (P u, u)L2 (M) = ∇g u2L2 V (M) − (γ N (u), γ D (u))L2 (∂M) + (R1 u, u)L2 (M) . On 0∂M, the part of ∂M where B is of order zero, we have γ D (u) = 0. On 1 ∂M, the part of ∂M where B is of order one, we have γ N (u) = −B  γ D (u) leading to (P u, u)L2 (M) = ∇g u2L2 V (M) + (B  γ D (u), γ D (u))L2 (1∂M) + (R1 u, u)L2 (M) . In the case d ≥ 3 we have |X  |g∂ < 1 on 1∂M by Proposition 4.6. In the case d = 2 this property is assumed here. By Lemma 4.10 we have for some 0 < C0 < 1 and C1 > 0 (P u, u)L2 (M) ≥ (1 − C0 )∇g u2L2 V (M) − C1 u2L2 (M) − |(R1 u, u)L2 (M) |. For any 0 < η < (1 − C0 ), we have |(R1 u, u)L2 (M) | ≤ η∇g u2L2 V (M) + Cη u2L2 (M) , yielding (P u, u)L2 (M) ≥ (1 − C0 − η)∇g u2L2 V (M) − (C1 + Cη )u2L2 (M) . The result follows.  From Corollary 4.11 we obtain the following result similar to the classical case of Dirichlet boundary condition as presented in Sect. 10.1 of Volume 1. Theorem 4.12. Let (P, D(P)) be selfadjoint and the (P, B) fulfills the ˇ Lopatinski˘ı–Sapiro condition. In the case d = 2 assume furthermore that X   is such that |X |g∂ < 1 on 1∂M. The eigenvalues counted with multiplicities can be sorted in a nondecreasing sequence μ0 ≤ μ1 ≤ · · · ≤ μ n ≤ · · · that goes to +∞. Proof. It suffices to prove that there exists M ∈ R such that μ ≥ M for any eigenvalue as we know that the sequence of the modulus of the eigenvalues goes to +∞. Let thus μ be an eigenvalue and let φ be a unitary eigenfunction. We have μ = (P φ, φ)L2 (M) . By Corollary 4.11 we have  μ ≥ Cφ2H 1 (M) − C∗ ≥ −C∗ . We show through an example that having B = ∂ν +iX  +h with |X  |g∂ > 1 in a connected component of 1∂M may have a dramatic effect on the spectrum of (P, D(P)).

4.6. SPECTRAL PROPERTIES

105

Proposition 4.13. On the unit disc D consider P = −Δ = −∂x2 − ∂y2 and the boundary operator B = ∂ν +2i∂θ . The associated operator (P, D(P)) is selfadjoint and its spectrum is given the union of two real sequences, one going to +∞ and the other one going to −∞. Observe that with such a spectral behavior the operator (P, D(P)) cannot be the generator of a C0 -semigroup. Proof. As one can readily check, from Theorem 4.3 and Proposition 2.8 the unbounded operator (P, D(P)) is selfadjoint. We use polar coordinates (r, θ) for which the Laplace operator reads −Δ = −r−1 ∂r (r∂r ) − r−2 ∂θ2 . Note that B reads Bu|∂D = ∂r u|r=1 + 2i∂θ u|r=1 . First, set wn = r2n einθ . We have Bwn |∂D = 0 yielding wn ∈ D(P). We have wn 2L2 (D) = 2π/(4n + 1). We compute P wn = −3n2 r2n−2 einθ , which leads to 2π 1 3 (P wn , wn )L2 (D) = ∫ ∫ (P wn )wn rdrdθ = − πn, 2 0 0

implying the existence of a sequence of eigenvalues that goes to −∞. Second, set wn = ψ(r)einθ , with ψ ∈ Cc∞ (1/4, 3/4; R). We have Bwn |∂D = 0 yielding wn ∈ D(P). We have wn L2 (D) = Cst. We compute P wn = (−ψ  − r−1 ψ  + r−2 n2 ψ)einθ , which leads to 1

2

(P wn , wn )L2 (D) = 2π ∫ (−ψ  − r−1 ψ  )ψ rdr + 2πn2 r−1/2 ψL2 (0,1) , 0

implying the existence of a sequence of eigenvalues that goes to +∞.



In the case (P, D(P)) is selfadjoint as above, we now provide some norm equivalence on H 1 (M) and on its linear subspace formed by functions with vanishing Dirichlet trace on 0∂M and some density result of D(P) in that space, Consider u ∈ H 2 (M) and v ∈ H 1 (M) with v|0∂M = 0. One has (4.6.4)

(P u, v)L2 (M) = (∇g u, ∇g v)L2 V (M) + (R1 u, v)L2 (M) − (γ N (u), γ D (v))L2 (1∂M) .

If moreover Bu|1∂M = 0 one finds (Pu, v)L2 (M) = (∇g u, ∇g v)L2 V (M) + (R1 u, v)L2 (M) + (B  γ D (u), γ D (v))L2 (1∂M) .

106 4. SELFADJOINT OPERATORS UNDER GENERAL BOUNDARY CONDITIONS

Motivated by this computation, for u, v ∈ H 1 (M), we define (4.6.5)

˜ (u, v) = (∇g u, ∇g v)L2 V (M) + (R1 u, v)L2 (M) N + B  γ D (u), γ D (¯ v )H −1/2 (1∂M),H 1/2 (1∂M) .

˜ (., .) is Hermitian symmetric. One sets By Lemma 4.7 the bilinear form N (4.6.6)

˜ (u, u) = ∇g u2 2 N (u) = N L V (M) + (R1 u, u)L2 (M) u)H −1/2 (1∂M),H 1/2 (1∂M) , + B  γ D (u), γ D (¯

and Nλ (u) = N (u) + λu2L2 (M) . We introduce the following space HD1 (M) = {u ∈ H 1 (M); u|0∂M = 0}, equipped with the usual H 1 -norm, yielding a Hilbert space structure. Above we saw that (4.6.7)

˜ (u, v), (Pu, v)L2 (M) = N

for u ∈ D(P) and v ∈ HD1 (M). Finally, for λ ∈ R, we set Pλ = P + λ IdL2 (M) with D(Pλ ) = D(P). Proposition 4.14. Let (P, D(P)) be selfadjoint and the (P, B) fulfills ˇ the Lopatinski˘ı–Sapiro condition. In the case d = 2 assume furthermore  that X is such that |X  |g∂ < 1 on 1∂M. Let μ0 be the lowest eigenvalue of (P, D(P)). Then, Nλ (.)1/2 is a norm on HD1 (M) for λ > −μ0 that is equivalent to the usual H 1 -norm. Moreover, D(P) is dense in HD1 (M). Consequently, the inner product ˜ (u, v) + λ(u, v)L2 (M) , (u, v) → N is an inner product that yields the Hilbert space structure on HD1 (M). Proof. By Theorem 4.12, with λ > −μ0 , the operator (Pλ , D(P)) is selfadjoint and positive. For u ∈ D(P), with the computation carried out above, one has  (μj + λ)|uj |2 ≥ 0, Nλ (u) = (Pλ u, u)L2 (M) = j∈N

with uj = (u, φj )L2 (M) , j ∈ N, with (φj )j∈N a Hilbert basis of eigenfunctions of (Pλ , D(P)), each associated with the eigenvalues μj + λ; see Theorem 4.9. This implies that Nλ (.)1/2 is a norm on D(P) for λ > −μ0 . Observe that two values of λ > −μ0 yield two associated norms Nλ (.)1/2 that are equivalent. One observes that Nλ (u)  u2H 1 (M) for u ∈ H 1 (M). By Lemma 4.10 and the Young inequality, there exists 0 < C1 < 1 and C2 > 0 such that Nλ (u) ≥ (1 − C1 )∇g u2L2 V (M) + (λ − C2 )u2L2 (M) .

4.6. SPECTRAL PROPERTIES

107

Thus, for λ > max(C2 , −μ0 ) one has Nλ (.)1/2 equivalent to .H 1 (M) on H 1 (M). The above two observations, put together, show that Nλ (.)1/2 and .H 1 (M) are equivalent on D(P) if λ > −μ0 . Note that this equivalence may not hold on H 1 (M) if λ ≤ max(C2 , −μ0 ). The norm Nλ (.)1/2 is associated with the ˜λ (u, v) = N ˜ (u, v) + λ(u, v)L2 (M) . inner product N The space HD1 (M) is closed in H 1 (M) and one has D(P) ⊂ HD1 (M). Below we show that the orthogonal of D(P) in HD1 (M) with respect to the ˜λ (., .) for λ > max(C2 , −μ0 ) is the trivial set. Consequently, inner product N D(P) is dense in HD1 (M) for the norm Nλ (.)1/2 (with λ > max(C2 , −μ0 )) and thus also for the norm .H 1 (M) . Let now λ > −μ0 . As explained above, there exists C > c > 0 such that cuH 1 (M) ≤ Nλ (u)1/2 ≤ CuH 1 (M) ,

u ∈ D(P).

If v ∈ HD1 (M) and (vn )n ⊂ D(P) converges to v for the norm .H 1 (M) one has cvn H 1 (M) ≤ Nλ (vn )1/2 ≤ Cvn H 1 (M) . Observing that u → Nλ (u)1/2 is continuous on H 1 (M) one finds, passing to the limit, cvH 1 (M) ≤ Nλ (v)1/2 ≤ CvH 1 (M) , which proves that Nλ (.)1/2 is a norm of HD1 (M), that is moreover equivalent to the norm .H 1 (M) . Consider v ∈ HD1 (M) that is in the orthogonal of D(P) with respect to ˜λ (., .) in the case λ > max(C2 , −μ0 ). For any u ∈ D(P) the inner product N one has, with Lemma 4.7, ˜λ (u, v) 0=N = λ(u, v)L2 (M) + (∇g u, ∇g v)L2 V (M) + (R1 u, v)L2 (M) + (B  γ D (u), γ D (v))L2 (1∂M) = λ(u, v)L2 (M) + (−Δg u, v)L2 (M) + (R1 u, v)L2 (M) + (γ N (u) + B  γ D (u), γ D (v))L2 (1∂M) = (Pλ u, v)L2 (M) . Consider the operator Lλ : H 2 (M) → L2 (M) ⊕ H 3/2,1/2 (∂M) u → (P u + λu, u|0∂M , Bu|1∂M ), with the notation of (3.1.3) in Chap. 3 with (3/2)

H 3/2,1/2 (∂M) = HB

(∂M) = H 3/2 (0∂M) ⊕ H 1/2 (1∂M).

This operator is Fredholm with a zero index as explained in the beginning of the proof of Theorem 4.9. With λ > −μ0 , this operator is injective and

108 4. SELFADJOINT OPERATORS UNDER GENERAL BOUNDARY CONDITIONS

thus surjective. This implies in particular that Ran(Pλ ) = L2 (M). Hence,  having (Pλ u, v)L2 (M) = 0 for all u ∈ D(P) gives v = 0. For any λ ∈ R, on D(Pλ ) = D(P) the graph norm u2D(Pλ ) = u2L2 (M) + Pλ u2L2 (M) is a norm equivalent to the graph norm u2D(P) = u2L2 (M) + Pu2L2 (M) . Since (P, D(P)) is a closed operator then the inner product (u, v) → (u, v)L2 (M) + (Pu, Pv)L2 (M) yields a Hilbert space structure on D(P). Lemma 4.15. The norms u → uD(P) , u → u → Pλ uL2 (M) for λ > −μ0 are equivalent on D(P).

uH 2 (M) and

Proof. Naturally, one has Pλ uL2 (M)  uD(P)  uH 2 (M) . For λ > −μ0 one has ker(Pλ ) = {0}. Thus the Fredholm map (4.6.1) with c = λ is injective, and thus bijective as its index is zero and the inverse 2 3/2,1/2 (∂M) → H 2 (M) is bounded as recalled at map L−1 λ : L (M) ⊕ H the beginning of the proof of Theorem 4.9. Hence, for u ∈ D(P) one has  Bu|∂M = 0 implying that uH 2 (M)  Pλ uL2 (M) . The Hilbert basis (φj )j∈N of eigenfunctions of the unbounded selfadjoint operator (P, D(P)) on L2 (M) allows one to give a description of the spaces HD1 (M) and D(P), as is done in Section 10.1.2 of Volume 1 in the case of Dirichlet boundary conditions. Naturally, if u ∈ L2 (M) one has u =  j∈N uj φj , with uj = (u, φj )L2 (M) , and where convergence takes place in L2 (M), meaning that (uj )j∈N ∈ 2 (C). Let λ > −μ0 and set λj = μj + λ > 0. Proposition 4.16. Let u ∈ L2 (M) and uj = (u, φj )L2 (M) , j ∈ N. First, we have the following equivalences: u ∈ D(P) ⇔ (μj uj )j∈N ∈ 2 (C) ⇔ (λj uj )j∈N ∈ 2 (C), 1/2

u ∈ HD1 (M) ⇔ (|μj |1/2 uj )j∈N ∈ 2 (C) ⇔ (λj uj )j∈N ∈ 2 (C). Second, the bilinear map (u, v) →



λ2j uj v¯j

j∈N

is an inner product on D(P) that yields the same Hilbert space structure on D(P) as that given by the H 2 -inner product and  2 λj |uj |2 , u2H 2 (M)  u2D(P)  Pλ u2L2 (M) = j∈N

4.6. SPECTRAL PROPERTIES

for u ∈ D(P). Third, the bilinear map (u, v) →



109

λj uj v¯j

j∈N

is an inner product on HD1 that yields the same Hilbert space structure on HD1 as that given by the H 1 -inner product and  λj |uj |2 , u2H 1 (M)  Nλ (u)2 = j∈N

for u ∈

HD1 .

Proof. Arguing as in the proof of Theorem 4.9 one sees that P−1 λ is a −1 −1 (φ ) = λ φ . Since Ran(P compact operator on L2 (M). One has P−1 j j j λ λ )= D(P) this gives the characterization of D(P), as u ∈ D(P) reads u =  λ−1 v φ for some (vj )j∈N ∈ 2 (C). Moreover one finds that if u =  j∈N j j j (λj uj )j∈N ∈ 2 (C) and Pλ u = j∈N λj uj φj . Thus, j∈N uj φj ∈ D(P), then we have Pλ uL2 (M) = j∈N λ2j |uj |2 . By Lemma 4.15, u → Pλ uL2 (M) is a norm equivalent to the H 2 -norm on D(P) and  2 (u, v) → λj uj v¯j j∈N

is the inner product on D(P) associated with this norm. This concludes the second point of the proposition. From Proposition 4.14 u → Nλ (u) is a norm on HD1 (M) and D(P) is ˜ (u, v) + λ(u, v)L2 (M) , is an inner product dense in HD1 (M)and (u, v) → N that yields the Hilbert space structure on HD1 (M). If u, v ∈ D(P), by (4.6.7) one has ˜ (u, v) + λ(u, v)L2 (M) = (Pu, v)L2 (M) =  λj uj v¯j . N j∈N

Then, arguing by density as in the proof of Proposition 10.5 of Volume 1 one obtains the characterization of HD1 (M) and one proves the third point of the proposition.  With Proposition 4.16 one finds  ˜ (u, v) =  μj uj v¯j , μj |uj |2 , and N N (u) = j∈N

j∈N

for u, v ∈ HD1 (M). If μ0 ≥ 0 one has N (u) ≥ 0 but N (.)1/2 may not be a norm on HD1 (M) unless μ0 > 0, that is, if ker(P) = {0}. However, in the case μ0 ≥ 0 one has the following Cauchy–Schwarz like inequality ˜ (u, v)| ≤ N (u)1/2 N (v)1/2 , (4.6.8) u, v ∈ HD1 (M). |N  With Proposition 4.16 if u ∈ D(P) and u = j∈N uj φj one has Pu =  j∈N μj uj φj . A simple consequence is the following result.

110 4. SELFADJOINT OPERATORS UNDER GENERAL BOUNDARY CONDITIONS

Proposition 4.17. The range of P is closed in L2 (M) and Ran(P) = ker(P)⊥ . ˇ 4.7. Lopatinski˘ı–Sapiro Elliptic Problem In this section we carry on with the analysis of the operator P in the case ˇ (P, B) fulfills the Lopatinski˘ı–Sapiro condition and (P, D(P)) is selfadjoint. As above if d = 2 we further assume that |X  |g∂ < 1 on 1∂M to have the conclusion of Theorem 4.12. 4.7.1. The Nonhomogeneous Elliptic Problem. Let k ∈ N. Denote by L the map (4.7.1)

L : H k+2 (M) → H k (M) ⊕ H k+3/2,k+1/2 (∂M) u → (P u, u|0∂M , Bu|1∂M ).

We set n = dim ker(P), meaning that {φ0 , . . . , φn−1 } is a basis of E0 = ker(P). Recall that nul L, def L, and ind(L) are independent of the chosen value k ∈ N; see Proposition 3.4. With the property of L recalled after (4.7.10), the range of L, Ran(L), is closed and of codimension equal to n (see Definition 11.6 of Volume 1) since its index is zero. For u ∈ L2 (M) we set u = (Id −ΠE0 )u, that is, the projection of u onto E0⊥ , yielding (4.7.2)

u=u−

n−1  j=0

uj φj =



uj φj ,

uj = (u, φj )L2 (M) .

j≥n

Let k ∈ N. Naturally, if u ∈ H k (M) one has u ∈ H k (M) because of the H k -regularity of the eigenfunctions φ0 , . . . , φn−1 . Setting (4.7.3)

H k (M) = {u; u ∈ H k (M)} = (IdL2 (M) −ΠE0 )(H k (M)),

one sees that H k (M) is a closed linear subspace of H k (M), thus a Hilbert space. Consider the nonhomogeneous elliptic problem (4.7.4) P u = f, in M,

u|0∂M = g0 in 0∂M,

Bu|1∂M = g1 in 1∂M.

If u ∈ H k+2 (M), then f ∈ H k (M), g0 ∈ H k+3/2 (0∂M), and g1 ∈ H k+1/2 (1∂M). For j = 0, . . . , n − 1, by Proposition 4.8 one computes (4.7.5) (f, φj )L2 (M) + (g1 , γ1D∂M (φj ))L2 (1∂M) − (g0 , γ0N∂M (φj ))L2 (0∂M) = 0. Since codim Ran(L) = n then one sees that conditions (4.7.5), for j = 0, . . . , n−1, characterize Ran(L) as a subset of H k (M)⊕H k+3/2,k+1/2 (∂M): they are necessary and sufficient conditions for the resolution of (4.7.4). Theorem 4.18 (Elliptic Problem—Strong Solutions). Let k ∈ N. Let f ∈ H k (M), g0 ∈ H k+3/2 (0∂M), and g1 ∈ H k+1/2 (1∂M) be such that (4.7.5) hold, for j = 0, . . . , n − 1. Then, there exists a unique v ∈ H k+2 (M) such that (4.7.4) holds. Moreover, any other solution u ∈ WP (M) of (4.7.4)

ˇ 4.7. LOPATINSKI˘I–SAPIRO ELLIPTIC PROBLEM

111

is such that u − v ∈ E0 = ker(P), meaning in particular that u ∈ H k+2 (M). Finally, there exists C > 0 such that   (4.7.6) vH k+2 (M) ≤ C f H k (M) + |g0 |H k+3/2 (0∂M) + |g1 |H k+1/2 (1∂M) . Proof. Consider L : H 2+k (M) → Ran(L) given by Lu = Lu. Since Ran(L) is closed it is complete with the norm of H k (M)⊕H k+3/2,k+1/2 (∂M). Hence, L is bijective and bounded. By the open mapping theorem L−1 is bounded and v = L−1 (f, g0 , g1 ) ∈ H 2+k (M) is a solution of (4.7.4) and estimate (4.7.6) holds. Consider now u ∈ WP (M). Then, by Lemma 18.32 one has P u ∈ L2 (M) and the traces u|0∂M and Bu|1∂M make sense in H −1/2 (0∂M) and H −3/2 (0∂M), respectively. Assume that P u = f , u|0∂M = g0 , and Bu|1∂M = g1 . By Theorem 3.28, one has u ∈ H k+2 (M). Observe that w = u − v ∈ H k+2 (M) is such that P w = 0, w|0∂M = 0, Bw|1∂M = 0 meaning that w ∈ ker(P).  Consider now the space H 1D (M) = {u; u ∈ HD1 (M)} = (IdL2 (M) −ΠE0 )(HD1 (M)). It is a closed linear subspace of HD1 (M). On the space H 1D (M), one can introduce a variational form of the elliptic problem (4.7.4) in the case u|0∂M = g0 = 0. ˜ (u, v)+λ(u, v)L2 (M) On HD1 (M), for λ > −μ0 , the bilinear map (u, v) → N is an inner product that yields the Hilbert structure of the space by Proposition 4.14. Moreover, one has  λj |uj |2 , u2H 1 (M)  Nλ (u) = j∈N

by Proposition 4.16. Observe that if u ∈ H 1D (M), then   λj |uj |2  μj |uj |2 = N (u). Nλ (u) = j≥n

j≥n

˜ (u, v) yields the Hilbert structure of the Thus the bilinear map (u, v) → N 1 space H D (M). Theorem 4.19 ( Elliptic Problem—Variational Form). Let f ∈ L2 (M) and g1 ∈ H −1/2 (1∂M) be such that (4.7.7) (f, φj )L2 (M) +g1 , γ1D∂M (φj )H −1/2 (1∂M),H 1/2 (1∂M) = 0,

j = 0, . . . , n − 1.

Then, there exists a unique v ∈ H 1D (M) such that (4.7.8) ˜ (v, u) = (f, u)L2 (M) +g1 , u ¯|1∂M H −1/2 (1∂M),H 1/2 (1∂M) , N

u ∈ HD1 (M).

Moreover, if g1 ∈ H 1/2 (1∂M), then v ∈ H 2 (M) and v is the unique solution to the elliptic problem (4.7.4) provided by Theorem 4.18 in the case g0 = 0.

112 4. SELFADJOINT OPERATORS UNDER GENERAL BOUNDARY CONDITIONS

Remark 4.20. Let v ∈ H 1D (M) and j = 0, . . . , n − 1. Then by (4.6.7) we have (4.7.9)

˜ (v, φj ) = (v, Pφj )L2 (M) = 0, N

as φj ∈ ker(P). One thus finds that the conditions (4.7.7) are necessary for (4.7.8) to hold. Proof. As u → (u, f )L2 (M) +u|1∂M , g1 H 1/2 (1∂M),H −1/2 (1∂M) is bounded on H 1D (M), and as the Hilbert structure of H 1D (M) is given by the inner ˜ (., .), there exists v ∈ H 1 (M) such that product N D ˜ (v, u) = (f, u)L2 (M) +g1 , u N ¯|1∂M H −1/2 (1∂M),H 1/2 (1∂M) ,

u ∈ H 1D (M).

by the Riesz theorem. With conditions (4.7.7) and (4.7.9) one sees that this identity also holds for u = φj , j = 0, . . . , n − 1. Thus (4.7.8) holds for all u ∈ HD1 (M). Consider now g1 ∈ H 1/2 (1∂M) and the unique solution w ∈ H 2 (M) to (4.7.5) provided by Theorem 4.18 in the case g0 = 0. Let u ∈ HD1 (M). Since w ∈ H 2 (M) and w|0∂M = 0, with (4.6.5) one finds ˜ (w, u) = (∇g w, ∇g u)L2 V (M) +(R1 w, u)L2 (M) +(B  w 1 , u 1 ) 2 1 N | ∂M | ∂M L ( ∂M) = (∇g w, ∇g u)L2 V (M) +(R1 w, u)L2 (M) +(∂ν w|1∂M , u|1∂M )L2 (1∂M) + (g1 , u|1∂M )L2 (1∂M) = (P w, u)L2 (M) + (g1 , u|1∂M )L2 (1∂M) = (f, u)L2 (M) + (g1 , u|1∂M )L2 (1∂M) . Consequently, since w ∈ H 2 (M) ⊂ H 1 (M), with the uniqueness of v proven above we have w = v.  4.7.2. A Boundary Lifting Map. For λ ∈ R, we set Pλ = P + λ IdL2 (M) and (4.7.10)

Lλ : H 2 (M) → L2 (M) ⊕ H 3/2,1/2 (∂M) u → (Pλ u, u|0∂M , Bu|1∂M ),

as in (3.1.3) in Chap. 3. This map is Fredholm of index zero as seen in the beginning of the proof of Theorem 4.9. Below we shall consider λ > −μ0 . Then, the unbounded operator Pλ = P + λ IdL2 (M) with domain D(P) is such that ker(Pλ ) = {0}. We introduce here a map that is the counterpart of the Dirichlet lifting introduced in Section 10.5 in Volume 1 and in Sect. 18.6.3 in the present volume or the Neumann lifting map introduced in Sect. 18.7.1 or the mixed Dirichlet-Neumann lifting map introduced in Sect. 18.8.2. The setting of the latter map is quite close to that we face here. We shall thus closely follow this section.

ˇ 4.7. LOPATINSKI˘I–SAPIRO ELLIPTIC PROBLEM

113

For r, s ∈ R we set (4.7.11)

H r,s (∂M) = H r (0∂M) ⊕ H s (1∂M).

For a function w ∈ H 2 (M), we set (4.7.12)

γ(w) = (w|0∂M , Bw|1∂M ),

and we have γ(w) ∈ H 3/2,1/2 (∂M). By Lemma 18.32, for a function w ∈ WP (M) we have γ(w) ∈ H −1/2,−3/2 (∂M). Proposition 4.21. Let h ∈ H −1/2,−3/2 (∂M). There exists a unique u ∈ WP (M) such that Pλ u = 0 and γ(u) = h. Moreover, the map H −1/2,−3/2 (∂M) → WP (M) h → u is bounded. Definition 4.22 (Boundary Lifting Map). We call the bounded map MPλ ,B : H −1/2,−3/2 (∂M) → WP (M) given by Proposition 4.21 the boundary lifting map adapted to (Pλ , B). Proof of Proposition 4.21. Recalling that ker(Pλ ) = {0}, we denote by Rλ : L2 (M) → H 2 (M) the resolvent map associated with the homogeneous boundary problem (4.7.13)

Pλ u = f in M,

Bu|∂M = 0 on ∂M.

First, we address uniqueness. By linearity, we consider u ∈ WP (M) such that Pλ u = 0 and γ(u) = 0. Let f ∈ L2 (M) and choose w = Rλ f . We have w ∈ H 2 (M) and γ(w) = 0. By Proposition 4.8, or rather (4.5.3), we obtain (u, f )L2 (M) = (u, Pλ w)L2 (M) = 0. As f ∈ L2 (M) is arbitrary we conclude that u = 0. Second, we address existence. To ease −1/2,−3/2 (∂M). For a function w ∈ H 2 (M) set H=H

notation

we

set

γ  (w) = (∂ν w|0∂M , w|1∂M ) ∈ H  = H 1/2,3/2 (∂M). For h = (0h, 1h) ∈ H and k = (0k, 1k) ∈ H  , we set k, hH  ,H = −0k, 0hH 1/2 (0∂M),H −1/2 (0∂M) + 1k, 1hH 3/2 (1∂M),H −3/2 (1∂M) . For h ∈ H, consider then the map U : L2 (M) → C f → γ  Rλ f, hH  ,H . By the trace formula of Theorem 18.25, this form is bounded and we have |U (f )| ≤ |h|H Rλ f H 2 (M)  |h|H f L2 (M) .

114 4. SELFADJOINT OPERATORS UNDER GENERAL BOUNDARY CONDITIONS

By the Riesz theorem, there exists u ∈ L2 (M) such that U (f ) = (f, u)L2 (M) for all f ∈ L2 (M). Moreover, uL2 (M)  |h|H . If ϕ ∈ 0 Dc∞ (M) and f = Pλ ϕ, then ϕ = Rλ f and we find U (f ) = γ  (ϕ), hH  ,H = 0. Consequently Pλ u, ϕμg 0 D  (M),1 Dc∞ (M) = u, Pλ ϕμg 0 D  (M),1 Dc∞ (M) = (Pλ ϕ, u)L2 (M) = U (f ) = 0. Thus, in the sense of distributions we have Pλ u = 0. Thus u ∈ WP (M) and uWP (M) = uL2 (M) + P uL2 (M)  |h|H . Lemma 4.23. There exists a bounded map M0 : H  → H 2 (M) such that γ  ◦ M0 = IdH  and γ ◦ M0 = 0. This is a particular case of Theorem 18.25. A proof follows from Lemmata 18.26 and 18.27 by working locally at the boundary. Let k ∈ H  . For v = M0 k ∈ H 2 (M) as given by Lemma 4.23, we have (Pλ v, u)L2 (M) = k, γ(u)H  ,H , by the Green formula of Proposition 4.8. We also have (Pλ v, u)L2 (M) = U (Pλ v) = γ  Rλ (Pλ v), hH  ,H = γ  (v), hH  ,H = k, hH  ,H . For all k ∈ H  we thus find k, h − γ(u)H  ,H = 0, implying γ(u) = h, which concludes the existence part of the proof.  Proposition 4.24. Let k ∈ N. If h ∈ H k−1/2,k−3/2 (∂M), then MP,B (h) ∈ H k (M). Moreover, for some C > 0, we have MP,B (h)H k (M) ≤ C|h|H k−1/2,k−3/2 (∂M) . Proof. The case k = 0 is treated above. The proof can be adapted from that of Proposition 18.39 for k ≥ 2. The case k = 1 follows by an interpolation argument [74, 236].  Set W P (M) = {u; u ∈ WP (M)} = (IdL2 (M) −ΠE0 )(WP (M)). The following theorem extends Theorem 4.18 to lower regularity for the boundary data. Theorem 4.25. Let f ∈ L2 (M), g0 ∈ H −1/2 (0∂M), and g1 ∈ H −3/2 ( ∂M) be such that 1

(4.7.14)

0 = (f, φj )L2 (M) + g1 , γ1D∂M (φj )H −3/2 (1∂M),H 3/2 (1∂M) − g0 , γ0D∂M (φj )H −1/2 (0∂M),H 1/2 (0∂M) ,

ˇ 4.7. LOPATINSKI˘I–SAPIRO ELLIPTIC PROBLEM

115

for j = 0, . . . , n − 1. Then, there exists a unique v ∈ W P (M) such that (4.7.4) holds. Any other solution u ∈ WP (M) of (4.7.4) is such that u − v ∈ E0 = ker(P). Moreover, there exists C > 0 such that   vL2 (M) ≤ C f L2 (M) + |g0 |H −1/2 (0∂M) + |g1 |H −3/2 (1∂M) . (4.7.15) If one has g0 ∈ H 1/2 (0∂M) and g1 ∈ H −1/2 (1∂M), then v ∈ W P (M) ∩ H 1 (M) and there exists C > 0 such that   vH 1 (M) ≤ C f L2 (M) + |g0 |H 1/2 (0∂M) + |g1 |H −1/2 (1∂M) . (4.7.16) This theorem generalizes what is obtained in Sect. 18.6 for the DirichletLaplace problem, in Sect. 18.7 for the Neumann-Laplace problem, and in Sect. 18.8 for the mixed Dirichlet-Neumann-Laplace problem. Note that the cases 0∂M = ∅ or 1∂M = ∅ are contained in the present result. Note, however, that in the case 1∂M = ∅, that is, the Dirichlet-Laplace problem, weaker regularity for f , namely H −1 (M), can be considered; see Theorem 18.40. The last result in the theorem makes sense since u|0∂M ∈ H 1/2 (0∂M) and Bu|1∂M ∈ H −1/2 (1∂M) by Lemma 18.42 since WP (M) ∩ H 1 (M) = Wg (M) ∩ H 1 (M). Proof. Let u, v ∈ WP (M) be two solutions of (4.7.4). Then P (u−v) = 0, (u − v)|0∂M = 0 and B(u − v)|1∂M = 0. Thus, by Theorem 3.28 one has u − v ∈ H 2 (M) and thus u − v ∈ ker(P). This implies the uniqueness of a solution in W P (M). Let λ > −μ0 and w = MPλ ,B (g0 , g1 ) meaning that w ∈ WP (M) with P w = −λw, w|0∂M = g0 and Bw|1∂M = g1 by Proposition 4.21 and Definition 4.22. For j = 0, . . . , n − 1, with the Green formula of Proposition 4.8 we compute (P w, φj )L2 (M) + g1 , γ1D∂M (φj )H −1/2 (1∂M),H 1/2 (1∂M) = g0 , γ0N∂M (φj )(L2 (0∂M) , using that P φj = 0, γ0D∂M (φj ) = 0, and Bφj |1∂M = 0. We thus have (f − P w, φj )L2 (M) = 0. Thus by Theorem 4.18 there exists u ∈ H 2 (M) such that P u = f − P w, u|0∂M = 0, and Bu|1∂M = 0. We set v = u + w ∈ WP (M). We have P v = f , v|0∂M = g0 and Bv|1∂M = g1 . We then find that v ∈ W P (M) and P v = f , v |0∂M = g0 and Bv |1∂M = g1 . The case g0 ∈ H 1/2 (0∂M) and g1 ∈ H −1/2 (1∂M) follows similarly using Proposition 4.24.  Finally we have the following result.

116 4. SELFADJOINT OPERATORS UNDER GENERAL BOUNDARY CONDITIONS

Proposition 4.26. Let f ∈ L2 (M) and g1 ∈ H −1/2 (1∂M) be such that (4.7.17) (f, φj )L2 (M) + g1 , γ1D∂M (φj )H −1/2 (1∂M),H 1/2 (1∂M) = 0,

j = 0, . . . , n − 1.

The unique solution in H 1D (M) to (4.7.8) coincides with the unique solution to (4.7.4) provided by Theorem 4.25 in the case g0 = 0. Proof. Let v ∈ W P (M) ∩ HD1 (M) be the unique solution to (4.7.5) provided by Theorem 4.25, that is P v = f and Bv|1∂M = g1 . Note that Bv|1∂M makes sense by Lemma 18.42 since WP (M) ∩ H 1 (M) = Wg (M) ∩ H 1 (M). Let u ∈ D(P). With Lemma 4.7 we have ˜ (v, u) = (∇g v, ∇g u)L2 V (M) + (v, R1 u)L2 (M) N + v|1∂M , B¯ u|1∂M H −1/2 (1∂M),H 1/2 (1∂M) = (v, P u)L2 (M) + v|1∂M , ∂ν u ¯|1∂M H −1/2 (1∂M),H 1/2 (1∂M) + v|1∂M , B  u|1∂M H −1/2 (1∂M),H 1/2 (1∂M) = (v, P u)L2 (M) + v|1∂M , Bu|1∂M H −1/2 (1∂M),H 1/2 (1∂M) which with (4.5.2) gives ˜ (v, u) = (P v, u)L2 (M) + Bv 1 N

¯|1∂M H −3/2 (1∂M),H 3/2 (1∂M) | ∂M , u

= (f, u)L2 (M) + g1 , u ¯|1∂M H −1/2 (1∂M),H 1/2 (1∂M) . Since D(P) is dense in HD1 (M) one obtains (4.7.8).



Part 2

Carleman Estimates on Riemannian Manifolds

CHAPTER 5

Estimates on Riemannian Manifolds for Dirichlet Boundary Conditions Contents 5.1. 5.2. 5.3. 5.4. 5.4.1. 5.4.2.

Setting Estimates Away from the Boundary Estimates at the Boundary Global Estimations A Global Estimate with an Inner Observation A Global Estimate with a Boundary Observation

119 121 123 126 126 127

This chapter aims to transpose the results obtained in Chapter 3 of Volume 1 to the framework of Riemannian manifolds. Using local coordinates one can use directly the local results obtained in that chapter. Passing from regular open sets to Riemannian manifolds is thus fairly straightforward, up to reproducing some of the patching arguments of Chapter 3. 5.1. Setting We consider a smooth Riemannian manifold (M, g). Elementary facts on manifolds and Riemannian manifolds are presented in Chap. 15. We assume that M is σ-compact, that is, M admits an exhaustion by compact sets (K n )n . A subset L of M is then said to be bounded if there exists n ∈ N such that L ⊂ K n . In local coordinates the Laplace–Beltrami operator reads    ∂i (det g)1/2 g ij ∂j f , Δg f = (det g)−1/2 1≤i,j≤d

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 5

119

120

5. ESTIMATES ON MANIFOLDS FOR DIRICHLET CONDITIONS

where, at each point of M, (g ij ) is the inverse matrix of the metric g = (gij ) and det g = det(gij ). We set P0 = −Δg and we shall consider more general operators P of the form P = P0 + R1 where R1 is a first-order differential operator with bounded coefficients on M. Those operators are the sum of a bounded function and a locally finite sum of vectors fields with bounded coefficients (See Remark 16.9). For ϕ ∈ C ∞ (M) we introduce the conjugated operator, for τ > 0, Pϕ = eτ ϕ P0 e−τ ϕ . In each local chart, its representative has the form    (Di + iτ (dϕ)i ) (det g)1/2 g ij (Dj + iτ (dϕ)j ) , (det g)−1/2 1≤i,j≤d



where dϕ has 1≤i≤d (dϕ)i dxi for representative. With such a local representative we may see Pϕ as a second-order differential operator with a large parameter on M (see Sect. 16.3.4). Its principal symbol is the function on the cotangent bundle T ∗ M with τ as a parameter, given by pϕ (m, ω, τ ) = (ω + iτ dϕ(m), ω + iτ dϕ(m))gm , with (., .)gm defined in (17.1.4). In local coordinates, this reads  (ξi + iτ (dϕ(x))i )g ij (x)(ξj + iτ (dϕ(x))j ). pϕ (x, ξ, τ ) = 1≤i,j≤d

Following Section 3.2.1 we set 1 1 P2 = (Pϕ + Pϕ∗ ), P1 = (Pϕ − Pϕ∗ ), 2 2i that are formally selfadjoint, and we have Pϕ = P2 + iP1 . Adjoint operators are defined similarly to transpose operators with an additional complex conjugation (see Sect. 16.3.3). They are differential operators of order two with a large parameter. Their principal symbols are given by p2 (m, ω, τ ) = |ω|2gm − τ 2 |dϕ(m)|2gm , p1 (m, ω, τ ) = 2τ (ω, dϕ(m))gm , with |.|gm defined in (17.1.4). In local coordinates, they read    g ij (x) ξi ξj − τ 2 (dϕ(x))i (dϕ(x))j , p2 (x, ξ, τ ) = 1≤i,j≤d

p1 (x, ξ, τ ) = 2τ



g ij (x) ξi (dϕ(x))j .

1≤i,j≤d

The sub-ellipticity property of Definition 3.2 of Volume 1 can be extended to the case of a manifold as follows. The Poisson bracket of two functions on a manifold is recalled in Sect. 15.7.2.

5.2. ESTIMATES AWAY FROM THE BOUNDARY

121

Definition 5.1 (Sub-ellipticity). Let V be a bounded open set in M. We say that the weight function ϕ ∈ C ∞ (M; R) and P have the subellipticity condition in V if dϕ = 0 in V and if (5.1.1) ∀(m, ω) ∈ T ∗ V , ∀τ > 0, pϕ (m, ω, τ ) = 0 ⇒ {p2 , p1 }(m, ω, τ ) > 0.

Having ϕ and P satisfying the sub-ellipticity condition of Definition 5.1 in V precisely means that in each local chart (O, κ) the representatives of ϕ and P satisfy the sub-ellipticity condition of Definition 3.2 in κ(V ∩ O). 5.2. Estimates Away from the Boundary The counterpart of Theorem 3.11 of Volume 1 is the following result. Theorem 5.2. Let P = P0 +R1 with P0 = −Δg where R1 is a first-order differential operator with bounded coefficients. Let V be a bounded open set in M such that V ∩ ∂M = ∅, and let ϕ and P have the sub-ellipticity property of Definition 5.1 in V ; then, there exist τ∗ > 0 and C > 0 such that τ 3 eτ ϕ u2L2 (M) + τ eτ ϕ D u2L2 Λ1 (M) + τ −1 eτ ϕ H u2L2 Λ2 (M) (5.2.1)

≤ Ceτ ϕ P u2L2 (M) ,

for u ∈ Cc∞ (V ) and τ ≥ τ∗ . Here, D stands for the covariant derivative on M and H = D2 is the Hessian defined by means of the Levi-Civita connection. For a function such as u we simply have D u = du. These notions are recalled in Sects. 17.4 and 17.7. In particular, the gradient of u on (M, g), given in local coordinates by,  ij g ∂j u, i = 1, . . . , d, (∇g u)i = 1≤j≤d

is related to D u by the relation ∇g u = (D u) (see Sect. 17.1.3 for the musical isomorphisms on a Riemannian manifold). The space L2 Λ(M) (resp. L2 Λ2 (M)) is the Hilbert space of L2 one-forms (resp. 2-covariant tensors) on M. These spaces and the associated norms .L2 Λ(M) and .L2 Λ2 (M) are given in Sect. 18.2. Note that eτ ϕ D uL2 Λ1 (M) = eτ ϕ ∇g uL2 V (M) , with the space L2 V (M) of L2 -vector fields and its associated norm given in Sect. 18.1. Proof. We first observe that it suffices to prove the estimate for P0 in place of P by adapting Remark 3.13 to the manifold setting. Let the local charts C i = (Oi , κi ), i ∈ I, form an atlas of M. As V is bounded, that is, contained in a compact set of M as recalled above, there exists J ⊂ I with #J < ∞ such that V ⊂ ∪i∈J Oi . Let ( χi )i∈J

122

5. ESTIMATES ON MANIFOLDS FOR DIRICHLET CONDITIONS

be a C ∞ -partition of unity of V subordinated to this open covering (see Definition 15.4 and Theorem 15.14). i i i i Let f = P0 u. For each i ∈ J , we let uC (resp. P0C , f C , ϕ, χC ) be the i i i representative of u (resp. P0 , f , ϕC , χi ) in the chart C i . We have P0C uC = i i f C . The operator P0C is an elliptic second-order differential operator on ˜ i = κi (Oi ) with smooth real principal part. Setting v i = χC i uC i ∈ Cc∞ (O ˜ i) O i i i i i i i we have P0C v i = χC f C + [P0 , χC ]uC . As ϕC and P0C satisfy the subellipticity condition in κi (V ∩ Oi ) that is a neighborhood of supp(v i ), the Carleman estimate of Theorem 3.11 applies. We can then obtain Ci



2

τ 3 eτ ϕ v i L2 (O˜ i ) + τ

1≤j≤d



+ τ −1

2

Ci

eτ ϕ Dj v i L2 (O˜ i ) Ci

eτ ϕ (∂j ∂k −

1≤j,k≤d Ci

 1≤≤d

2

Ci

2

 eτ ϕ P0C v i L2 (O˜ i ) + eτ ϕ uC L2 (O˜ i ) + i

2

Γjk ∂ ) v i 

i

˜i) L 2 (O

 1≤j≤d

Ci

2

eτ ϕ Dj uC L2 (O˜ i ) , i

for τ ≥ τ i for some τ i > 0 chosen sufficiently large, using that [P0 , χC ] is a first-order differential operator. The symbol Γjk stands for the Christoffel symbol associated with the metric g and the Levi-Civita connection in the ˜ i in the local chart C i (see (17.4.11) in Sect. 17.4). The L2 -norms on O above inequality are for the Lebesgue measure dx in Rd . However, the i same inequality holds for dx replaced by μCg , the local representative of the i Riemannian canonical density μg (see Sect. 17.3), that is, μCg = T(det gCi )1/2 , i

using the notation of Section 8.1.2 of Volume 1, precisely meaning μCg = i (det g C )1/2 dx. We refer the reader to Sect. 17.1.3 for the musical isomorphisms on a Riemannian manifold and to Sects. 17.4 and 17.7 for the definition of the covariant derivative D. Then, using that, for a function w on M, i

• The covariant derivative D w = dw is a one-form, with moreover (D w) = ∇g w, with ∇g w the Riemannian gradient, and that both i their representatives in C i have norms equivalent to that of DwC ; • the Hessian H w = D2 w is a 2-covariant tensor and its represen i tative in C i is given by (H w)Cjk = (∂j ∂k − 1≤≤d Γjk ∂ )wC , see (17.7.2); on the manifold, we thus obtain 2

2

2

τ 3 eτ ϕ χi uL2 (M) + τ eτ ϕ D( χi u)L2 Λ1 (M) + τ −1 eτ ϕ H( χi u)L2 Λ2 (M) 2

 eτ ϕ P0 χi uL2 (M) + eτ ϕ u2L2 (M) + eτ ϕ D u2L2 Λ1 (M) .

5.3. ESTIMATES AT THE BOUNDARY

123

Commuting χi with D, H = D2 and P0 yields the estimation 2

2

2

τ 3 eτ ϕ χi uL2 (M) + τ eτ ϕ χi D uL2 Λ1 (M) + τ −1 eτ ϕ χi H uL2 Λ2 (M)  eτ ϕ P0 u2L2 (M) + τ eτ ϕ u2L2 (M) + eτ ϕ D u2L2 Λ1 (M) , for τ ≥ τ i . Summing over i ∈ J , for τ ≥ max τ i , recalling that #J < ∞ and using the properties of the partition of unity, we obtain with the triangular inequality, τ 3 eτ ϕ u2L2 (M) + τ eτ ϕ D u2L2 Λ1 (M) + τ −1 eτ ϕ H u2L2 Λ2 (M)  τϕ i 2  τϕ i 2  τ3 e χ uL2 (M) + τ e χ D uL2 Λ1 (M) i∈J

+ τ −1

 i∈J

i∈J 2 eτ ϕ χi H uL2 Λ2 (M)

 e P0 u2L2 (M) + τ eτ ϕ u2L2 (M) + eτ ϕ D u2L2 Λ1 (M) . τϕ

The result follows by choosing τ > 0 sufficiently large.



5.3. Estimates at the Boundary Let ν be the unique outward pointing vector field along ∂M such that, for all m ∈ ∂M, gm (νm , νm ) = 1 and gm (νm , u) = 0 for all u ∈ Tm ∂M, that is, ν is unitary and orthogonal to Tm ∂M in the sense of g. For a smooth function f on M, we define its normal derivative at m ∈ ∂M by (5.3.1)

∂ν f (m) = νm (f ) = df (m)(νm ).

The counterparts of the local estimations of Lemmata 3.15 and 3.16 are given by the following results. Lemma 5.3. Let P = P0 + R1 with P0 = −Δg where R1 is a first-order differential operator with bounded coefficients. Let m0 ∈ ∂M. Let V 0 be a bounded open set in M such that m0 ∈ V 0 , and let ϕ ∈ C ∞ (V 0 ) and P have the sub-ellipticity property of Definition 5.1 in V 0 . Then, there exits an open neighborhood V 1 of m0 in M such that V 1 ⊂ V 0 and there exist τ∗ > 0 and C > 0 such that  (5.3.2) τ 3 eτ ϕ u2L2 (M) + τ eτ ϕ D u2L2 Λ1 (M) ≤ C eτ ϕ P u2L2 (M)  2 + τ 3 |eτ ϕ u|∂M |2L2 (∂M) + τ |eτ ϕ D u|∂M |L2 Λ1 (∂M) + τ |eτ ϕ ∂ν u|∂M |2L2 (∂M) , for u ∈ C ∞ (M), with supp(u) ⊂ V 1 and τ ≥ τ∗ . Here, D denotes the covariant derivative on ∂M and the norms |.|L2 (∂M) and |.|L2 Λ(∂M) are defined by means of the metric g∂ inherited on ∂M from that on M. On the boundary ∂M, the H 1 -norm is given by |w|2H 1 (∂M) = |w|2L2 (∂M) + | D w|L2 Λ1 (∂M) , 2

124

5. ESTIMATES ON MANIFOLDS FOR DIRICHLET CONDITIONS

(see Sects. 18.1 and 18.2). We then have |eτ ϕ D u|L2 Λ1 (∂M)  τ 2 |eτ ϕ u|2L2 (∂M) + |eτ ϕ u|∂M |2H 1 (∂M) , 2

implying that the Carleman estimate of the previous theorem can thus be written in the same form as that of Lemma 3.15 of Volume 1:  (5.3.3) τ 3 eτ ϕ u2L2 (M) + τ eτ ϕ D u2L2 Λ1 (M) ≤ C eτ ϕ P u2L2 (M)  + τ 3 |eτ ϕ u|∂M |2L2 (∂M) + τ |eτ ϕ u|∂M |2H 1 (∂M) + τ |eτ ϕ ∂ν u|∂M |2L2 (∂M) . Lemma 5.4. Let P = P0 + R1 with P0 = −Δg where R1 is a first-order differential operator with bounded coefficients. Let m0 ∈ ∂M. Let V 0 be a bounded open set in M such that m0 ∈ V 0 , and let ϕ ∈ C ∞ (V 0 ) and P have the sub-ellipticity property of Definition 5.1 in V 0 and ∂ν ϕ(m0 ) < 0. Then, there exits an open neighborhood V 1 of m0 in M such that V 1 ⊂ V 0 and there exist τ∗ > 0 and C > 0 such that (5.3.4) τ 3 eτ ϕ u2L2 (M) + τ eτ ϕ D u2L2 Λ1 (M) + τ |eτ ϕ ∂ν u|∂M |2L2 (∂M)   2 ≤ C eτ ϕ P u2L2 (M) + τ 3 |eτ ϕ u|∂M |2L2 (∂M) + τ |eτ ϕ D u|∂M |L2 Λ1 (∂M) , for u ∈ C ∞ (M), with supp(u) ⊂ V 1 and τ ≥ τ∗ . Proof of Lemma 5.3 (resp. 5.4). We first observe that it suffices to prove the estimate for P0 in place of P by adapting Remark 3.13 to the manifold setting. Choose an open neighborhood V 1 ⊂ V 0 of m0 in M that is contained in a local chart C = (O, κ) at the boundary. In this local chart M ∩ V 1 is given by {xd ≥ 0} ∩ κ(O) and ∂M ∩ V 1 = {xd = 0} ∩ κ(O). In κ(V 1 ), the elliptic problem, through the representatives of the operator and the involved functions, takes precisely the form given in Sections 3.4 and 3.5. The result of Theorem 3.28 of Volume 1 (resp. 3.29) applies to the ˜ representative P0C of the operator P0 . Under the assumptions of Lemma 5.3 we then have, for τ∗ > 0, C˜

2

C˜

2

τ 3 eτ ϕ wL2 (Rd ) + τ eτ ϕ DwL2 (Rd ) +

 e

+

˜

τ ϕC

2 ˜ P0C wL2 (Rd ) +

C˜

2

+ τ 3 |eτ ϕ w|xd =0+ |L2 (Rd−1 ) 2

C˜

C˜

2

+ τ |eτ ϕ D w|xd =0+ |L2 (Rd−1 ) + τ |eτ ϕ Dd w|xd =0+ |L2 (Rd−1 ) , ∞

for τ ≥ τ∗ and w ∈ C c (U+1 ), with U+1 = κ ˜ (V 1 ) ∩ Rd+ . ∞ d For some open set U+ of R+ the space C c (U+ ) is defined in (3.4.11) as (5.3.5)

∞

C c (U+ ) = {u = v|Rd ; v ∈ Cc∞ (Rd ) and supp v ⊂ U }, +

where U is some open set of Rd such that U+ = U ∩ Rd+ . Note that the ∞ space C c (U+ ) is independent of the choice of U .

5.3. ESTIMATES AT THE BOUNDARY

125

As in the proof of Theorem 5.2, we change the Lebesgue measure for the ˜ (V 1 ) and local representative of the Riemannian canonical density μg in κ similarly we change the Lebesgue measure on κ(∂M ∩ V 1 ) by the canonical density μg∂ , which only affects constants in the estimation. We refer the reader to Sect. 17.1.3 for the musical isomorphisms on a Riemannian manifold and to Sects. 17.4 and 17.7 for the definition of the covariant derivative D and its connection with the Riemannian gradient in the case of functions. Using that (D u) = ∇g u, (D u|∂M ) = ∇g∂ u|∂M and that ˜

˜

their representatives have norms equivalent to that of DuC and D uC|xd =0+ , ˜

respectively, and using that the representative ∂ν u|∂M is −∂d uC|xd =0+ = ˜

−iDd uC|xd =0+ , we then have,

τ 3 eτ ϕ w2L2 (M) +τ eτ ϕ D u2L2 Λ1 (M)  eτ ϕ P0 u2L2 (M) +τ 3 |eτ ϕ u|∂M |2L2 (∂M) + τ |eτ ϕ D u|∂M |L2 Λ1 (∂M) + τ |eτ ϕ ∂ν u|∂M |2L2 (∂M) , 2

for τ ≥ τ∗ and u ∈ C ∞ (M) with supp(u) ⊂ V 1 . Here, because of the chosen neighborhood V 1 , there is, however, no need for the introduction of partition of unity as opposed to the proof of Theorem 5.2. We leave to the reader the adaptation of the estimate of Theorem 3.29 under the assumptions of Lemma 5.4.  Patching estimates together, as is done in Section 3.5, we need not reduce the size of the considered open set where the sub-ellipticity condition hold. The following theorems are the counterparts of Theorems 3.28 and 3.29. Theorem 5.5. Let P = P0 + R1 with P0 = −Δg where R1 is a firstorder differential operator with bounded coefficients. Let V be a bounded open set in M and let ϕ ∈ C ∞ (V ) and P have the sub-ellipticity property of Definition 5.1 in V . Then, there exist τ∗ > 0 and C > 0 such that  (5.3.6) τ 3 eτ ϕ u2L2 (M) + τ eτ ϕ D u2L2 Λ1 (M) ≤ C eτ ϕ P u2L2 (M)

 2 + τ 3 |eτ ϕ u|∂M |2L2 (∂M) + τ |eτ ϕ D u|∂M |L2 Λ1 (∂M) + τ |eτ ϕ ∂ν u|∂M |2L2 (∂M) ,

for u ∈ C ∞ (M), with supp(u) ⊂ V and τ ≥ τ∗ . Note that V ∩ ∂M may be empty in the previous statement. Theorem 5.6. Let P = P0 + R1 with P0 = −Δg where R1 is a firstorder differential operator with bounded coefficients. Let V be a bounded open set in M and set V∂ = V ∩ ∂M, and let ϕ ∈ C ∞ (V ) and P have the sub-ellipticity property of Definition 5.1 in V and ∂ν ϕ < 0 in V∂ in the case

126

5. ESTIMATES ON MANIFOLDS FOR DIRICHLET CONDITIONS

V∂ = ∅. Then, there exist τ∗ > 0 and C > 0 such that (5.3.7) τ 3 eτ ϕ u2L2 (M) + τ eτ ϕ D u2L2 Λ1 (M) + τ |eτ ϕ ∂ν u|∂M |2L2 (∂M)   2 ≤ C eτ ϕ P u2L2 (M) + τ 3 |eτ ϕ u|∂M |2L2 (∂M) + τ |eτ ϕ D u|∂M |L2 Λ1 (∂M) , for u ∈ C ∞ (M), with supp(u) ⊂ V and τ ≥ τ∗ . Using Lemmata 5.3 and 5.4, the proofs of Theorems 5.5 and 5.6 can be readily adapted from that of Theorem 5.2 and Theorems 3.28 and 3.29, using a partition of unity. 5.4. Global Estimations We can adapt the results of Sections 3.6.1 and 3.6.2 to the case of a Riemannian manifold. ˜ g) 5.4.1. A Global Estimate with an Inner Observation. Let (M, be a smooth σ-compact Riemannian manifold (possibly not compact) with ˜ or without boundary and let M be a bounded connected open set of M. Let Γ0 be an open set of ∂M such that ∂M is smooth in a neighborhood of ˜ is smooth, we need not assume Γ0 . Note that, whereas the boundary of M ∂M to be smooth everywhere. Let also ω0 be an open subset of M. As ˜ and P = P0 + R1 above P0 denotes the Laplace–Beltrami operator on M where R1 is a first-order differential operator with bounded coefficients. Definition 5.7. A real valued function ϕ ∈ C ∞ (M) is said to be a global Carleman weight function on M adapted to Γ0 and ω0 if it satisfies ∂ν ϕ|∂M (m) < 0,

for m ∈ Γ0 ,

and if the sub-ellipticity property of Definition 5.1 for (P, ϕ) is fulfilled in M \ ω0 . The construction of a global weight function ϕ with these properties can be done by adapting Proposition 3.31 of volume 1 and its proof to the manifold case. Theorem 5.8 (Global Carleman Estimate—Inner Observation). Let P = P0 + R1 with P0 = −Δg where R1 is a first-order differential operator with bounded coefficients. Let also W0 be a neighborhood of ∂M \ Γ0 ˜ Let ω be an open set of M such that ω0  ω. Let ϕ ∈ C ∞ (M) be a in M. global weight function adapted to Γ0 and ω0 in the sense of Definition 5.7. Then, there exist τ∗ > 0 and C ≥ 0 such that τ 3 eτ ϕ u2L2 (M) + τ eτ ϕ D u2L2 Λ1 (M) + τ |eτ ϕ ∂ν u|Γ0 |2L2 (Γ ) 0  ≤ C eτ ϕ P u2L2 (M) + τ 3 eτ ϕ u2L2 (ω) + τ 3 |eτ ϕ u|Γ0 |2L2 (Γ ) + τ |eτ ϕ D u|Γ0 |L2 Λ1 (Γ 2

0

0

 , )

5.4. GLOBAL ESTIMATIONS

127

for τ ≥ τ∗ and u ∈ C ∞ (M) vanishing in W0 ∩ M. The case where M is itself a smooth connected compact Riemannian manifold and Γ0 = ∂M yields the following corollary. Corollary 5.9. Let M be a smooth bounded connected Riemannian manifold. Let P = P0 + R1 with P0 = −Δg where R1 is a first-order differential operator with bounded coefficients. Let ω, ω0 be two nonempty open subsets of M such that ω0  ω. Let ϕ ∈ C ∞ (M) be a global weight function adapted to Γ0 = ∂M and ω0 in the sense of Definition 5.7. Then, there exist τ∗ > 0 and C ≥ 0 such that τ 3 eτ ϕ u2L2 (M) + τ eτ ϕ D u2L2 Λ1 (M) + τ |eτ ϕ ∂ν u|∂M |2L2 (∂M)  ≤ C eτ ϕ P u2L2 (M) + τ 3 eτ ϕ u2L2 (ω)

 2 + τ 3 |eτ ϕ u|∂M |2L2 (∂M) + τ |eτ ϕ D u|∂M |L2 Λ1 (∂M) ,

for u ∈ C ∞ (M) and τ ≥ τ∗ . 5.4.2. A Global Estimate with a Boundary Observation. We consider the same setting as in Sect. 5.4.1. Here, We consider an open set Γ0 of ∂M such that Γ0  ∂M. Definition 5.10. A real valued function ϕ ∈ C ∞ (M) is said to be a global Carleman weight function on M adapted to Γ0 if it satisfies ∂ν ϕ|Γ0 (m) < 0,

for m ∈ Γ0 ,

and if the sub-ellipticity property of Definition 5.1 for (P, ϕ) is fulfilled in M. Theorem 5.11 (Global Carleman Estimate—Boundary Observation). Let P = P0 + R1 with P0 = −Δg where R1 is a first-order differential operator with bounded coefficients. Let Γ0 and Γobs be two nonempty open sets of ∂M such that Γobs \ Γ0 = ∅, and such that ∂M is smooth in a neighborhood of Γ0 ∪ Γobs . Let also W0 be a neighborhood of ∂M \ (Γ0 ∪ Γobs ) ˜ in M. Let ϕ ∈ C ∞ (M) be a global weight function adapted to Γ0 in the sense of Definition 5.10. Then, there exist τ∗ > 0 and C ≥ 0 such that τ 3 eτ ϕ u2L2 (M) + τ eτ ϕ D u2L2 Λ1 (M) + τ |eτ ϕ ∂ν u|Γ0 |2L2 (Γ ) 0  τϕ 2 2 τϕ ≤ C e P uL2 (M) + τ |e ∂ν u|Γobs |L2 (Γ ) obs

+ τ |e 3

τϕ

u|∂M |2L2 (∂M)

+ τ |e

for τ ≥ τ∗ and u ∈ C ∞ (M) vanishing in W0 ∩ M.

τϕ

 2 D u|∂M |L2 Λ1 (∂M) , 

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5. ESTIMATES ON MANIFOLDS FOR DIRICHLET CONDITIONS

For the construction of the global weight function ϕ one can pick an open set Γ1 of ∂M such that Γ1  Γobs \ Γ0 and then adapt Proposition 3.39 of volume 1 and its proof to the manifold case. The case where M is itself a smooth connected compact Riemannian manifold yields the following corollary. Corollary 5.12. Let P = P0 + R1 with P0 = −Δg where R1 is a firstorder differential operator with bounded coefficients. Let Γ0 , Γobs be two nonempty open subsets of ∂M such that Γ0 ∪ Γobs = ∂M and Γobs \ Γ0 = ∅. Let ϕ ∈ C ∞ (M) be a global weight function adapted to Γ0 in the sense of Definition 5.10. Then, there exist τ∗ > 0 and C ≥ 0 such that τ 3 eτ ϕ u2L2 (M) + τ eτ ϕ D u2L2 Λ1 (M) + τ |eτ ϕ ∂ν u|∂M |2L2 (∂M)  ≤ C eτ ϕ P u2L2 (Ω) + τ |eτ ϕ ∂ν u|Γobs |2L2 (Γ ) obs

+ τ |e 3

for u ∈ C ∞ (M) and τ ≥ τ∗ .

τϕ

u|∂M |2L2 (∂M)

+ τ |e

τϕ

 2 D u|∂M |L2 Λ1 (∂M) ,

CHAPTER 6

Pseudo-Differential Operators on a Half-Space Contents 6.1. More on Tangential Symbols and Operators 6.1.1. Additional Classes of Symbols 6.1.2. Corresponding Classes of Operators 6.2. Adapted Sobolev Norms and Continuity Results 6.2.1. A Microlocal Norm Equivalence 6.3. Quadratic Forms in a Half-Space and G˚ arding Inequality 6.4. Estimates for First-Order Operators 6.4.1. A Perfectly Elliptic Estimate 6.4.2. An Elliptic Estimate with a Boundary Observation 6.4.3. Estimate Under Sub-ellipticity 6.5. Notes Appendix 6.A. Smooth Factorisation of Polynomials 6.A.1. Some Results from Complex Analysis 6.A.2. An Algebraic Identity 6.A.3. Roots Regularity

130 130 131 133 135 136 143 144 145 146 147 148 148 148 148 149

In Chapter 2 of Volume 1 we presented the basic results that allowed us to derive estimates away from boundaries and at boundaries in Chapter 3 also in Volume 1, in particular in the case of Dirichlet boundary conditions. In the present chapter, we wish to present additional material on pseudo-differential operators with a large parameter, focusing on the case of operators defined on the half-space Rd+ . This will allow us to introduce the basic tools for the derivation of Carleman estimates at a boundary in the ˇ case of general boundary conditions, namely the Lopatinski˘ı–Sapiro condition, that we shall present in Chap. 8.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 6

129

130

6. PSEUDO-DIFFERENTIAL OPERATORS ON A HALF-SPACE

6.1. More on Tangential Symbols and Operators As in Chapter 2 of Volume 1, we consider symbols that depend on a parameter τ ≥ 1 meant to be large. For x ∈ Rd or Rd+ and ξ ∈ Rd we shall often write  = (x, ξ, τ ). Similarly, for ξ  ∈ Rd−1 we shall write  = (x, ξ  , τ ). As introduced in Section 2.10, for X = Rd or X = Rd+ , we recall that a( ) ∈ STm,τ (X × Rd−1 ) if a( ) ∈ C ∞ (X × Rd−1 ) and (6.1.1) |∂xα ∂ξβ a(x, ξ  , τ )| ≤ Cα,β λT,τ , x ∈ X, ξ  ∈ Rd−1 , τ ∈ [1, +∞),  1 where λT,τ = |(ξ  , τ )| = |ξ  |2 + τ 2 2 . Note that we make clear that τ is a parameter rather than a variable in the notation of the symbol space. Indeed smoothness with respect to τ is not an issue here. More generally, if U is a conic open set of X × Rd−1 × [1, +∞) we say that a ∈ STm,τ microlocally in U if (6.1.1) holds in for  = (x, ξ  , τ ) ∈ U . m−|β|

6.1.1. Additional Classes of Symbols. We consider symbols that behave polynomially in the ξd variable, as already presented in Remark 2.46(2) in Chapter 2. Definition 6.1. Let a() ∈ C ∞ (X × Rd ), with τ as a parameter in [1, +∞), and m ∈ N and r ∈ R. We say that a ∈ Sτm,r (X × Rd ) if m  aj ( )ξdj , aj ∈ STm−j+r (X × Rd−1 ), a() = ,τ j=0

for x ∈ X, ξ ∈

Rd ,

τ ∈ [1, +∞), and ξd ∈ R.

We also simply write a ∈ Sτm,r . If U is conic open set of X × Rd−1 × [1, +∞) we say that a ∈ Sτm,r microlocally for  ∈ U if each aj is in STm−j+r ,τ microlocally in U , j = 0, . . . , m. Note that we have (6.1.2)





Sτm,r ⊂ Sτm+m ,r−m ,

m, m ∈ N, r ∈ R.

We call the principal symbol of a the symbol m  σ(aj )( )ξdj , σ(a)() = j=0

which is a representative of the class of a in Sτm,r /Sτm,r−1 . Note that Sτm,r ⊂ Sτm+r . For example, consider a(x, ξ, τ ) = λT,τ ξd for λT,τ ≥ 1. We have a ∈ Sτ1,1 ⊂ Sτ2,0 and yet a ∈ / Sτ2 . In fact observe that  differentiating with respect to ξ yields 1−|α|

|∂ξα a(x, ξ, τ )| ≤ Cα λT,τ |ξd |. An estimate of the form of (2.2.1) is, however, not achieved for |α| ≥ 2. In Definition 2.6, polyhomogeneous symbols were introduced: we recall  m−j m if a that a ∼ j∈N am−j ∈ Sτ,ph is homogeneous of degree m − j m−j ∈ Sτ

6.1. MORE ON TANGENTIAL SYMBOLS AND OPERATORS

131

with respect to (ξ, τ ) for each j ∈ N. Additionally, we define tangential polyhomogeneous symbols. They are characterized by an asymptotic expansion where each term is positively homogeneous (ξ  , τ ). m (X × Rd−1 ) or simply Definition 6.2. We shall say that a ∈ ST,τ,ph

m  if there exists a(j) ∈ STm−j ST,τ,ph ,τ , homogeneous of degree m − j in (ξ , τ ) for |(ξ  , τ )| ≥ r0 , with r0 ≥ 0, such that

(6.1.3)

a∼



a(j) ,

in the sense that

a−

N  j=0

j≥0

−1 a(j) ∈ STm−N . ,τ

A representative of the principal symbol is then given by the first term in the expansion. We denote it by σ(a). We have m ⊂ Sτm , Sτ,ph

m ST,τ,ph ⊂ STm,τ .

m,r (X × Rd ) or Then, for m ∈ N and r ∈ R, we shall say that a() ∈ Sτ,ph m,r simply Sτ,ph , if

a() =

m  j=0

aj ( )ξdj ,

m−j+r with aj ∈ ST,τ,ph ,  = ( , ξd ).

A representative of the principal symbol is given by is homogeneous of degree m in (ξ, τ ). We have

m

j=0 σ(aj )(

 )ξ j d

and

m,r ⊂ Sτm,r . Sτ,ph m ⊂ 6.1.2. Corresponding Classes of Operators. For m ∈ R, as ST,τ,ph  the operator OpT (a) = a(x, D , τ ) is given in Definition 2.42 of Volm ume 1 and we write OpT (a) ∈ Ψm T,τ,ph (X) or simply ΨT,τ,ph . m−1 m The principal symbol of A is σ(A) = σ(a) in ST,τ,ph /ST,τ,ph .

STm,τ ,

Definition 6.3. For m ∈ N, r ∈ R, and a ∈ Sτm,r with a() =

m  j=0

aj ( )ξdj ,

aj ∈ STm−j+r (X × Rd−1 ), ,τ

we set a(x, D, τ ) = Op(a) =

m  j=0

aj (x, D , τ )Ddj ,

m,r and we write A = Op(a) ∈ Ψm,r τ (X) or simply Ψτ . The principal symbol of A is σ(A) = σ(a) in Sτm,r /Sτm,r−1 . We denote by Ψm,r τ,ph (X) the subclass of these operators associated with m,r symbols in Sτ,ph . The principal symbol of A is then σ(A) = σ(a) in m,r m,r−1 /Sτ,ph . Sτ,ph

We provide a notion of formal adjoint.

132

6. PSEUDO-DIFFERENTIAL OPERATORS ON A HALF-SPACE

Definition 6.4 (Formal Adjoint). Let b ∈ Sτm,r , with Op(b) =

m  j=0

OpT (bj )Ddj ,

bj ∈ STm+r−j . ,τ

We set Op(b)∗ =

m  j=0

Ddj OpT (bj )∗ .

In other words, in this definition we ignore the possible occurrence of boundary terms when performing the operator transposition. Note that here m+r−j OpT (bj )∗ ∈ Ψm+r−j . Observe also that OpT (bj )∗ ∈ Ψm+r−j T,τ T,τ,ph if bj ∈ ST,τ,ph . m For a ∈ STm,τ (resp. ST,τ,ph ) we have [Dd , OpT (a)] = OpT (Dd a) ∈ Ψm T,τ m (resp. ΨT,τ,ph ) and more generally, for j ≥ 1, we have

[Ddj , Op(a)] =

j−1 

αk ∈ STm,τ

OpT (αk )Ddk ,

m (resp. ST,τ,ph ),

k=0

where the symbols αk involve various derivatives of a in the xd -direction. m−j+r As an application we see that if we consider aj ∈ STm−j+r (resp. ST,τ,ph ), ,τ then we have m m   Ddj OpT (aj ) = OpT (˜ aj )Ddj , j=0

j=0

m−j+r (resp. ST,τ,ph ) and its principal part satisfies σ(˜ aj ) ≡ aj where a ˜j ∈ STm−j+r ,τ m−j+r m−j+r−1 in STm−j+r /STm−j+r−1 (resp. ST,τ,ph /ST,τ,ph ). Hence ,τ ,τ  

 m m m,r−1 Ddj OpT (aj ) = aj ( )ξdj mod Sτm,r−1 (resp. Sτ,ph ). σ j=0

j=0

From the calculus rules given in Section 2.10 of Volume 1 for tangential operators and the above observations we have the following results on principal symbols. 





m,r ) and b ∈ Sτm ,r (resp. Proposition 6.5. Let a ∈ Sτm,r (resp. Sτ,ph 

m ,r ) with Sτ,ph

a() =

m  j=0

aj ( )ξdj ,

b() =

m  j=0

bj ( )ξdj .

(resp. Ψm,r (1) We have Op(a)∗ ∈ Ψm,r τ τ,ph ) and m    σ Op(a)∗ () ≡ aj ( )ξdj ∈ Sτm,r /Sτm,r−1 j=0

m,r m,r−1 (resp. Sτ,ph /Sτ,ph ).

Moreover, we have Op(a)∗ − Op(a) ∈ Ψm,r−1 (resp. Ψm,r−1 τ τ,ph ).

6.2. ADAPTED SOBOLEV NORMS AND CONTINUITY RESULTS 





133



,r+r ,r+r (2) We have Op(a)Op(b) ∈ Ψm+m (resp. Ψm+m ) and τ τ,ph

  σ Op(a)Op(b) ≡

 0≤j≤m 0≤k≤m









(aj bk )( )ξdj+k ∈ Sτm+m ,r+r /Sτm+m ,r+r −1 







m+m ,r+r m+m ,r+r −1 (resp. Sτ,ph /Sτ,ph ).

We have 







,r+r −1 ,r+r −1 Op(a)Op(b)u − Op(ab)u ∈ Ψm+m (resp.Ψm+m ). τ τ,ph

6.2. Adapted Sobolev Norms and Continuity Results As in Chapters 2 and 3 of Volume 1, for u and v defined in Rd+ = {xd > 0}, we often use the notation (u, v)+ = ∫ u(x)v(x)dx, Rd+

u+ = uL2 (Rd ) . +

For u and v defined on {xd = 0} we use v (x )dx , (u, v)∂ = ∫ u(x )¯ Rd−1

|u|∂ = |u|L2 (Rd−1 ) ,

We recall that ΛsT,τ := Op(λsT,τ ) and we define the following norm for functions in Rd+ , for m ∈ N and s ∈ R, u2τ,m,s = ΛsT,τ u2τ,m ,

u ∈ S (Rd+ ),

with v2τ,m = v2τ,m,0 = τ 2m v2+ + v2H m (Rd ) +



m  j=0

τ

2j

v2H m−j (Rd ) , +

v ∈ S (Rd+ ).

From pseudo-differential symbol calculus we obtain the following inequality. Proposition 6.6. If a ∈ Sτm,r , with m ∈ N and r ∈ R, then for m ∈ N and r ∈ R there exists C > 0 such that Op(a)uτ,m ,r ≤ Cuτ,m+m ,r+r ,

u ∈ S (Rd+ ).

A consequence of this results is the following property. Corollary 6.7. Let m, m ∈ N and r ∈ R. There exists C > 0 such that uτ,m,r ≤ Cuτ,m+m ,r−m ,

u ∈ S (Rd+ ).

134

6. PSEUDO-DIFFERENTIAL OPERATORS ON A HALF-SPACE 



−m Proof. We write u = Λm T,τ v with v = ΛT,τ u. As we have by (6.1.2) 







m 0,m ⊂ Sτm ,0 , λm T,τ ∈ ST,τ = Sτ

we obtain 



−m uτ,m,r = Λm T,τ vτ,m,r  vτ,m+m ,r = ΛT,τ uτ,m+m ,r

= uτ,m+m ,r−m , 

by Proposition 6.6. The counterpart of Remark 2.27 in Chapter 2. Remark 6.8. We observe that we have, for some C > 0, uτ,m,s ≤ Cτ − uτ,m,s+ ,

u ∈ S (Rd+ ),

for m ∈ N and s ∈ R and  ≥ 0. This implies that uτ,m,s  uτ,m,s+ for τ sufficiently large. For a sufficiently smooth function u defined in Rd+ we set, for m ∈ N, trm (u) = (u|xd =0+ , Dd u|xd =0+ , . . . , Ddm u|xd =0+ ) on {xd = 0} and we define the following norm (6.2.1)

| trm (u)|2τ,m,s =

m  j=0

2

|Λm+s−j Ddj u|xd =0+ |∂ . T,τ

We see that without any ambiguity we may write | tr(u)|2τ,m,s in place of | trm (u)|2τ,m,s . Let u ∈ S (Rd+ ) and set Su = {v ∈ S (Rd ); u = v|Rd }. +

We define, for r ∈ R, (6.2.2)

uτ,r = inf vτ,r , v∈Su

with vτ,r = Λrτ vL2 (Rd ) .

For r ∈ N, this norm can be proven equivalent to the norm uτ,r = uτ,r,0 defined above, by using a properly designed extension operator [2, Theorem 5.21]. We have the following trace inequality. Proposition 6.9 (Trace Inequality). Let s > 0. There exists C > 0 such that |u|xd =0+ |τ,s ≤ Cuτ,s+1/2 ,

u ∈ S (Rd+ ).

6.2. ADAPTED SOBOLEV NORMS AND CONTINUITY RESULTS

135

Proof. For v ∈ Su , denote by F v(ξ) the Fourier transform of v with respect to x and by F  v(ξ  , xd ) its Fourier transform with respect to x . We have |u|xd =0+ |2τ,s = |v|xd =0 |2τ,s 2  = (2π)1−d |λsT,τ F  v(ξ  , 0)|Rd−1 = (2π)−d ∫ λ2s T,τ | ∫ F v(ξ)dξd | dξ 2

≤ (2π)

−d

∫

Rd−1

λ2s T,τ (∫ R

λ−2s−1 dξd )(∫ τ R

R Rd−1 2s+1 2 λτ |F v(ξ)| dξd )dξ  ,

by the Cauchy–Schwarz inequality. We compute −2s−1 2 2 −s−1/2 λ2s dξd = λ2s dξd = ∫ (1 + σ 2 )−s−1/2 dσ < ∞, T,τ ∫ λτ T,τ ∫ (λT,τ + ξd ) R

R

R

which gives |u|xd =0+ |τ,s  vτ,s+1/2 . We conclude with (6.2.2).



Corollary 6.10 (Trace Inequality). Let m ∈ N and s ∈ R. For some C > 0, we have | tr(u)|τ,m,s ≤ Cuτ,m+1,s−1/2 ,

u ∈ S (Rd+ ).

Proof. With (6.2.1) we write m m 2   2 m−j+s−1/2 j j |Λm−j+s D u | = |ΛT,τ Dd u|xd =0+ |τ,1/2 . | tr(u)|2τ,m,s = + T,τ d |xd =0 ∂ j=0

j=0

From the trace inequality of Proposition 6.9 we obtain | tr(u)|τ,m,s 

m  j=0

m−j+s−1/2

ΛT,τ

Ddj uτ,1 

m+1  j=0

m+1−j+s−1/2

ΛT,τ

Ddj u+

 uτ,m+1,s−1/2 .  6.2.1. A Microlocal Norm Equivalence. One has τ ∈ ST1,τ yielding, for s ∈ R, τ u0,s  u0,s+1 . However, in a microlocal region where τ  λT,τ , that is, if |ξ  |  τ one can expect a norm equivalence. The following proposition states that it is indeed the case, yet up to a remainder term. Proposition 6.11. Let U be a bounded open set of Rd+ and U be conic open sets of U × Rd−1 × R+ with C0 > 0 such that λT,τ ≤ C0 τ in U . Let s, s , s ∈ R with s = s + s . There exist C, C  > 0 such that for χ ∈ ST0,τ , homogeneous of degree zero with supp(χ) ⊂ U , and N ∈ N, there exists CN such that 

Cτ s OpT (χ)uτ,0,s − CN uτ,0,−N 

≤ OpT (χ)uτ,0,s ≤ C  τ s OpT (χ)uτ,0,s + CN uτ,0,−N , for all u ∈ S (Rd+ ). The result of Proposition 6.11 can be generalized.

136

6. PSEUDO-DIFFERENTIAL OPERATORS ON A HALF-SPACE

Proposition 6.12. Let U be a bounded open set of Rd+ and U be conic open sets of U × Rd−1 × R+ with C0 > 0 such that λT,τ ≤ C0 τ in U . Let L ∈ Ψm,r , with m ∈ N and r ∈ R, and let s, s , s ∈ R with s = s + s . There exist C, C  > 0 such that for χ ∈ ST0,τ , homogeneous of degree zero with supp(χ) ⊂ U , and N ∈ N, there exists CN such that 

Cτ s LOpT (χ)vτ,0,s − CN vτ,m,−N 

≤ LOpT (χ)vτ,0,s ≤ C  τ s LOpT (χ)vτ,0,s + CN vτ,m,−N , for all v ∈ S (Rd+ ). Proof of Proposition 6.12. Note that SU is compact. Thus, there exists V a conic open set of U ×Rd−1 ×R+ such that U ⊂ V and λT,τ ≤ C1 τ , for some C1 > 0 by homogeneity. ˆ ⊂V Let χ ˆ ∈ ST0,τ , homogeneous of degree zero, be such that supp(χ) and χ ˆ = 1 on U . One has ˆ sT,τ LOpT (χ) ΛsT,τ LOpT (χ) = OpT (χ)Λ

mod Ψm,−∞ , τ

yielding ˆ sT,τ LOpT (χ)v+ + CN vτ,m,−N , LOpT (χ)vτ,0,s ≤ OpT (χ)Λ for some CN > 0.   ˆ sT,τ ∈ ΨsT,τ giving We then observe that τ −s OpT (χ)Λ 

ˆ sT,τ LOpT (χ)v+  τ s LOpT (χ)vτ,0,s . OpT (χ)Λ We have thus obtained 

LOpT (χ)vτ,0,s ≤ Cτ s LOpT (χ)vτ,0,s + CN vτ,m,−N , for some C > 0 independent of N and χ. With this inequality, exchanging the rˆ oles of s and s and changing s into −s one obtains the second sought inequality.  6.3. Quadratic Forms in a Half-Space and G˚ arding Inequality Definition 6.13. Let u ∈ S (Rd+ ). We say that (6.3.1)

Q(u) =

N 

(As u, B s u)+ ,

As = Op(as ), B s = Op(bs ),

s=1

is an interior quadratic form of type (m, σ) with smooth coefficients, if for m,σ  m,σ  each s = 1, . . . N , we have as () ∈ Sτ,ph (Rd+ × Rd ) and bs () ∈ Sτ,ph (Rd+ × Rd ), with σ  + σ  = 2σ,  = (x, ξ, τ ). The principal symbol of the quadratic form Q is defined as the class of (6.3.2)

q() =

N 

as ()bs ()

s=1

in

2m,2σ (Rd+ Sτ,ph

×

2m,2σ−1 Rd )/Sτ,ph (Rd+

× Rd ).

6.3. QUADRATIC FORMS IN A HALF-SPACE AND G˚ ARDING INEQUALITY

137

Remark 6.14. Note that σ  and σ  can vary with s ∈ {1, . . . , N }. Their sum yet remains constant equal to 2σ. In what follows we shall not write this dependency explicitly for concision. Observe that several quadratic forms may share the same principal symbol. As an example, in one dimension, for N = 1 we can choose A = Dd2 ∈ Ψτ2,0 and B = Λ2T,τ ∈ Ψτ0,2 ⊂ Ψτ2,0 yielding to λ2T,τ ξd2 for principal symbol. The choice A = B = ΛT,τ Dd ∈ Ψτ1,1 ⊂ Ψτ2,0 leads to the same principal symbol. In fact, if u ∈ Cc∞ (Rd+ ), then Q(u) =

N 

((B s )∗ ◦ As u, u)+ .

s=1

The symbol of Q thus coincides with the principal symbol of N principal s )∗ ◦ As . Note that considering test functions with nonvanishing (B s=1 traces at the boundary xd = 0+ will naturally generate boundary terms when performing such operator transpositions. Such questions will be dealt with below. For s = 1, . . . , N , we have as () =

m  j=0

asj ( )ξdj ,

bs () =

m  j=0



bsj ( )ξdj ,

 = ( , ξd ),  = (x, ξ  , τ ),



m−j+σ m−j+σ with asj ∈ Sτ,ph and bsj ∈ Sτ,ph , and

As = Op(as ) = B s = Op(bs ) =

m  j=0 m  j=0

Asj Ddj ,

Asj = OpT (asj ),

Bjs Ddj ,

Bjs = OpT (bsj ).

Then, for u ∈ S (Rd+ ), the quadratic form given by (6.3.1) reads m  m    Cj,k Ddj u, Ddk u + ,

Q(u) =

j=0 k=0

where Cj,k are tangential operators given by Cj,k =

N  s=1

(Bks )∗ Asj ,

with symbols cjk ( ) =

N 

(bsk )∗ ◦ asj ( ) ∈ ST,τ,ph

2(m+σ)−(j+k)

.

s=1

We have the following lemma that follows directly from Proposition 6.6.

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6. PSEUDO-DIFFERENTIAL OPERATORS ON A HALF-SPACE

Lemma 6.15. We consider the interior quadratic form of type (m, σ), as above, m  m    Q(u) = Cj,k Ddj u, Ddk u + , Cj,k = OpT (cj,k ), j=0 k=0

2(m+σ)−(j+k)

cj,k ∈ ST,τ,ph

.

We have, for some C > 0, |Q(u)| ≤ Cu2τ,m,σ ,

u ∈ S (Rd+ ).

Next, we consider the case of a quadratic form with a vanishing principal symbol. Such a result is useful when comparing quadratic forms associated with the same principal symbol. Lemma 6.16. We consider the interior quadratic form of type (m, σ), as above, m  m    Q(u) = Cj,k Ddj u, Ddk u + , Cj,k = OpT (cj,k ), j=0 k=0

2(m+σ)−(j+k)

cj,k ∈ ST,τ,ph

,

and we further assume that its principal symbol vanishes, that is,  2(m+σ)−−1 cj,k ≡ 0 mod ST,τ,ph , ∀ ∈ {0, . . . , 2m}. 0≤j,k≤m j+k=

Then the following estimate holds   |Q(u)| ≤ C u2τ,m,σ−1/2 + | tr(u)|2τ,m−1,σ+1/2 ,

u ∈ S (Rd+ ).

Proof. Let  ∈ {0, . . . , 2m}. We introduce α = max(0,  − m) and β = min(m, ). Note that α + β = . We set I =

 0≤j,k≤m j+k=

β      Cj,k Ddj u, Ddk u + = C−k,k Dd−k u, Ddk u + . k=α

First, we consider 0 <  < 2m. For k > α we write     C−k,k Dd−k u, Ddk u + = C−k,k Dd−k+1 u, Ddk−1 u +   + Op(Dd c−k,k )Dd−k u, Ddk−1 u +   − i C−k,k Dd−k u|xd =0+ , Ddk−1 u|xd =0+ ∂ , which by induction yields,     C−k,k Dd−k u, Ddk u + = C−k,k Ddβ u, Ddα u + +

k−α  

Op(Dd c−k,k )Dd−k+s−1 u, Ddk−s u

s=1 k−α  

−i

s=1

 +

C−k,k Dd−k+s−1 u|xd =0+ , Ddk−s u|xd =0+

 ∂

.

6.3. QUADRATIC FORMS IN A HALF-SPACE AND G˚ ARDING INEQUALITY

139

2(m+σ)−

As Dd c−k,k ∈ ST,τ,ph we note that     Op(Dd c−k,k )D−k+s−1 u, Dk−s u  (6.3.3) d d + m+σ+k−−s+ 21

 ΛT,τ

m+σ+k−−s+ 21

 ΛT,τ

m+σ−k+s− 21

Dd−k+s−1 u+ ΛT,τ

m+σ−k+s− 21

uτ,−k+s−1 ΛT,τ

Ddk−s u+

uτ,k−s

 uτ,−k+s−1,m+σ+k−−s+ 1 uτ,k−s,m+σ−k+s− 1 2



2

u2τ,m−1,σ+ 1 , 2

by Corollary 6.7 as m + k −  − s ≥ 0 and m − 1 − k + s ≥ 0. Similarly we write   (C−k,k D−k+s−1 u|x =0+ , Dk−s u|x =0+ )∂   | tr(u)|2 d

d

d

τ,m−1,σ+ 12 .

d

We thus obtain β    (C−k,k Ddβ u, Ddα u)+   u2τ,m−1,σ+ 1 + | tr(u)|2τ,m−1,σ+ 1 . |I | −  2

k=α

2

As by assumption we have β  k=α



C−k,k =

0≤j,k≤m j+k=

2(m+σ)−−1

Cj,k ∈ ΨT,τ,ph

,

we find, as α + β = , β       m+σ−β − 12 β   m+σ−α − 21 α   Dd u+ ΛT,τ D d u + C−k,k Ddβ u, Ddα u +   ΛT,τ k=α

 uτ,β ,m+σ−β − 1 uτ,α ,m+σ−α − 1 2



2

u2τ,m,σ− 1 , 2

by Corollary 6.7 since m − α ≥ 0 and m − β ≥ 0. In the case 0 <  < 2m we have thus obtained (6.3.4)

|I |  u2τ,m,σ− 1 + | tr(u)|2τ,m−1,σ+ 1 . 2

2

Second, we consider  = 0. Then I0 = (C0,0 u, u)+ and as C0,0 ∈ 2(m+σ)−1 we find |I0 |  u2τ,m,σ− 1 . ΨT,τ,ph 2 Third, we consider  = 2m. Then we have I2m = (Cm,m Ddm u, Ddm u)+ 2 with Cm,m ∈ Ψ2σ−1 T,τ,ph yielding |I2m |  uτ,m,σ− 12 . This concludes the proof.  We have the following microlocal G˚ arding inequality for the quadratic forms we have introduced. Theorem 6.17 (Microlocal G˚ arding Inequality). Let K be a compact d set of R+ and let U be a conic open set of Rd+ × Rd−1 × R+ contained in K × Rd−1 × R+ . Let also χ ∈ ST0,τ be homogeneous of degree 0, be such

140

6. PSEUDO-DIFFERENTIAL OPERATORS ON A HALF-SPACE

that supp(χ) ⊂ U . Let Q be an interior quadratic form of type (m, σ) with 2m,2σ satisfying, for (one of the representatives of ) its principal symbol q ∈ Sτ,ph some C0 > 0 and r0 > 0, 2σ Re q() ≥ C0 λ2m τ λT,τ ,

for τ ≥ r0 ,  = ( , ξd ),  = (x, ξ  , τ ) ∈ U , ξd ∈ R.

For 0 < C1 < C0 and N ∈ N there exist τ∗ , C > 0, and CN > 0 such that Re Q(OpT (χ)u) ≥ C1 OpT (χ)u2τ,m,σ − C| tr(OpT (χ)u)|2τ,m−1,σ+1/2 − CN u2τ,m,−N , for u ∈ S (Rd+ ) and τ ≥ τ∗ . The important feature of this version of the G˚ arding inequality is that it concerns functions defined on a half-space. Compare with Theorem 2.29 of Volume 1. Remark 6.18. In the case U = U0 × Rd−1 × R+ , with U0 a bounded open subset of Rd+ , by continuity, there exists a U1 bounded open subset of Rd+ such that U1 is a neighborhood of U0 , with dist(U0 , ∂U 1 ) ≥ δ > 0 and 2σ Re q() ≥ C0 λ2m τ λT,τ ,

for τ ≥ r0 ,

for C1 < C0 < C0 , where  = ( , ξd ) with  = (x, ξ  , τ ) ∈ U1 × Rd−1 × R+ and ξd ∈ R. (See the beginning of the proof of Theorem 2.28 in Section 2.A.6.1 of Volume 1 for a detailed argumentation.) Then, there exist C and τ∗ > 0 such that (6.3.5)

Re Q(u) ≥ C1 u2τ,m,σ − C| tr(u)|2τ,m−1,σ+1/2 ,

for u ∈ S (Rd+ ) with supp(u) ⊂ U0 . This is obtained from Theorem 6.17 by choosing χ = χ(x) ∈ C ∞ (Rd ) with supp(χ|xd >0 ) ⊂ U1 and χ ≡ 1 on U0 and by taking τ sufficiently large. We thus obtain the counterpart of Theorem 2.28 for quadratic forms in a half-space. Observe that the assumption that U0 is bounded is not necessary. One could combine the arguments of the proofs of Theorems 2.28 and 6.17 to obtain a direct proof of (6.3.5). Yet, we shall not need such a result here and we leave this proof to the reader. Proof of Theorem 6.17. We introduce the interior quadratic form N   1  ˜ (As u, B s u)+ + (B s u, As u)+ , Q(u) = Re Q(u) = 2 s=1

that we may write in the form of (6.3.1) with 2N terms in the sum. Its principal symbol as given by (6.3.2) is then in the class N  s N   1  2m,2σ 2m,2σ−1 b as + as bs = Re as bs ∈ Sτ,ph /Sτ,ph . 2 s=1 s=1

6.3. QUADRATIC FORMS IN A HALF-SPACE AND G˚ ARDING INEQUALITY

141

Without any loss of generality we may thus assume that the interior quadratic form Q has a real principal symbol q(). 2m,2σ / We first pick a representative of the principal symbol q() in Sτ,ph 2m,2σ−1 that reads Sτ,ph

q() =

2m  j=0

qj ( )ξdj ,

2m+2σ−j 2m+2σ−j−1 qj ∈ ST,τ,ph /ST,τ,ph ,

( , ξ

  0 for  = d ), and  = (x, ξ , τ ). We choose qj a representative of qj that is homogeneous of degree 2m + 2σ − j in (ξ  , τ ) for τ ≥ r0 with r0 > 0 (see Definition 6.2). Then

q 0 () =

(6.3.6)

2m  j=0

qj0 ( )ξdj

is a real and homogeneous representative of the principal symbol q, and we have for C1 < C0 < C0 , 2σ q 0 () ≥ C0 λ2m τ λT,τ ,

 ∈ U ,

ξd ∈ R,

for τ ≥ r0 ,

with r0 > 0 chosen sufficiently large. Arguing as in the proof of Theorem 2.29 in Section 2.A.6.2 of Volume 1, ˜ ⊂ U and there exists χ ˜ ∈ ST0,τ , homogeneous of degree 0, such that supp(χ) χ ˜ ≡ 1 on supp(χ). 2σ With C1 < L < C0 , we see that q 0 ()−Lλ2m τ λT,τ is a polynomial function with real coefficients in the variable ξd of order 2m, that takes positive values for ξd ∈ R,  ∈ U and τ ≥ r0 . The leading coefficient a0 ( ) ∈ Sτ2σ is homogeneous of degree 2σ in (ξ  , τ ), for τ ≥ r0 , and a0 ( ) ≥ Cλ2σ T,τ for √  σ C > 0 and  ∈ U . This gives a0 ∈ Sτ microlocally in U . The roots of the polynomial come into conjugated pairs and are functions of the other variables  = (x, ξ  , τ ) ∈ U . We may thus write 2σ  q 0 () − Lλ2m τ λT,τ = a0 ( )f ()f (), τ ≥ r0 ,

 = ( , ξd ),  ∈ U , ξd ∈ R,

with f () =

m 

i=1

where

ρ+ i ,

  ξd − ρ+ i ( ) ,

i = 1, . . . , m, denote the roots with positive imaginary parts.

m,0 microlocally for  ∈ U and Lemma 6.19. We have f () is in Sτ,ph τ ≥ r0 > 0.

A proof is given below. Note in particular that this uses the homogeneity of the functions qj,0 in (6.3.6). We have   2σ ˜2 ( )a0 ( )|f |2 () ∈ Sτ2m,2σ , (6.3.7) χ ˜2 ( ) q 0 () − Lλ2m τ λT,τ = χ

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6. PSEUDO-DIFFERENTIAL OPERATORS ON A HALF-SPACE

for  = ( , ξd ) with  ∈ Rd+ × Rd−1 × R+ , ξd ∈ R for τ ≥ r0 . Observe that ˜ 2τ,m,σ is an interior quadratic form of type (m, σ) with prinv → OpT (χ)v 2σ ˜ − LOpT (χ)v ˜ 2τ,m − cipal symbol χ ˜2 λ2m τ λT,τ . We thus see that Q(OpT (χ)v) √ Op(χ ˜ a0 f )v2+ is an interior quadratic form of type (m, σ) with a vanish2m,2σ 2m,2σ−1 /Sτ,ph . Lemma 6.16 then yields ing principal symbol in Sτ,ph   √ ˜ − LOpT (χ)v ˜ 2 − Op(χ ˜ a0 f )v2  (6.3.8)  Re Q(OpT (χ)v) 

τ,m,σ v2τ,m,σ−1/2

+

+ 2 | tr(v)|τ,m−1,σ+1/2 ,

for v ∈ S (Rd+ ). The triangular inequality then yields   ˜ ≥ LOpT (χ)v ˜ 2τ,m,σ − C v2τ,m,σ−1/2 + | tr(v)|2τ,m−1,σ+1/2 . Re Q(OpT (χ)v) ˜ = OpT (χ)u + Ru with We now set v = OpT (χ)u. We have OpT (χ)v R ∈ ∩N ∈N Ψ−N by pseudo-differential calculus. We then obtain the sought T,τ estimate by taking τ > 0 sufficiently large.  Proof of Lemma 6.19. We write 2σ  p( , ξd ) = q 0 (ρ) − Lλ2m τ λT,τ = a0 ( )

 0≤j≤2m

b2m−j ( )ξdj ,

( )

the coefficient b2m−j are, respectively, homogeneous of degree 2m − j  with respect to (ξ , τ ). We have b0 ≡ 1 and we recall that a0 ( ) ≥ Cλ2σ T,τ with C > 0 for  ∈ U . If  lie in the compact set1 K × Sd−1 , then by +  Lemma 6.29, the roots of ξd → p( , ξd ) are contained in a bounded open ball B(0, R0 ) with R0 > 0. Let γ 0 be the closed path in C, oriented counter-clockwise, composed with (1) the real interval [−R0 , R0 ]; (2) the upper half circle centered at 0 of radius R0 . Im(z) γ0

D0 −R0

0

R0

Re(z)

As the polynomial z → p( , z) does not have any real root if  ∈ U , with what precedes, for  ∈ SU , with the notation introduced in (1.7.2), the roots of the polynomial that have a positive imaginary part all lie in the 1Observe that if  were to lie in Rd × Sd−1 instead, then one could simply argue that +

2m,2σ . One can then the coefficients b2m−j ( ) remain in a compact set since q 0 (ρ) ∈ Sτ,ph still apply Lemma 6.29 and prove that the roots are contained in a bounded set.

6.4. ESTIMATES FOR FIRST-ORDER OPERATORS

143

interior D0 of γ 0 . In particular, p( , z) does not vanish if z ∈ γ 0 . As  lie in the compact set SU we find that (6.3.9)

|p( , z)| ≥ C > 0,

  ∈ SU , z ∈ γ 0 ,

for some C > 0. By Proposition 6.24 we then obtain that, for  ∈ N, s ( ) :=

 1≤i≤m

  ρ+ i ( ) =

1 z d ∫ p( , z) dz 2πi γ 0 dz p( , z)

is a C ∞ -function of (x, ξ  ) for  ∈ SU , from the regularity of the integrand on the right-hand-side. By (6.3.9) and using again that  lie in the compact set SU we find that s ( ), and all its derivatives with respect to (x, ξ  ), are . bounded functions of  ∈ SU

  We recall that f ( , z) = 1≤i≤m (z − ρ+ i ( )), that we write  fm−j ( )z j , f ( , z) = 0≤j≤m

with f0 ≡ 1. By Lemma 6.26 in the appendix below, we have f1 ( ) = −s1 ( ),   2f2 ( ) = − s2 ( ) + f1 ( ) s1 ( ) ,   3f3 ( ) = − s3 ( ) + f1 ( ) s2 ( ) + f2 ( ) s1 ( ) , .. .    m fm ( ) = − sm ( ) + f1 ( ) sm−1 ( ) + · · · + fm−1 ( ) s1 ( ) . By a finite induction we find that the coefficients fj ( ) of f ( , z) and all their derivatives with respect to (x, ξ  ) are bounded functions of  ∈ SU . According to Lemma 6.30 in the appendix below, the set of the roots  ) is homogeneous of degree one in the variables (ξ  , τ ). This implies ( ρ+ i that s ( ) is homogeneous of degree  in (ξ  , τ ) for  ∈ U . Consequently, the formulae above yield the function fj ( ) to be homogeneous of degree j in (ξ  , τ ). Hence, we find that fj ∈ Sτj microlocally in U , j = 1, . . . , m. This concludes the proof.  6.4. Estimates for First-Order Operators In this section we derive estimates for first-order operators in a half-space of the form L = Dd − OpT (b) with b( ) ∈ ST1,τ . Depending on the sign of the imaginary part of b, inequalities of different natures are obtained. They are to be used in other chapters below. There, first-order operators will appear through the factorization of a conjugated operator; however, there, the sign of the imaginary part of b will vary. In such occurrence, microlocal estimations can be obtained with proper cutoffs in phase space. We have thus chosen to derive such microlocal estimates here.

144

6. PSEUDO-DIFFERENTIAL OPERATORS ON A HALF-SPACE

6.4.1. A Perfectly Elliptic Estimate. Let U be a conic open set of and b( ) ∈ ST1,τ , with principal symbol b1 ( ) homogeneous of degree one, such that, for C > 0, Rd+ × Rd−1 × R+

Im b1 ( ) ≤ −CλT,τ ,

 ∈ U .

We set L = Dd − OpT (b) and we have the following microlocal elliptic estimate. Lemma 6.20. Let χ ∈ ST0,τ , homogeneous of degree 0, be such that supp(χ) ⊂ U . Let s ∈ R. There exist C > 0 and τ∗ > 0 such that for any N ∈ N, there exists CN > 0 such that (6.4.1)

OpT (χ)uτ,1,s + |OpT (χ)u|xd =0+ |τ,s+1/2 ≤ CLOpT (χ)uτ,0,s + CN uτ,0,−N ,

for τ ≥ τ∗ and u ∈ S (Rd+ ). The proof is based on a multiplier method. We set w = OpT (χ)u. Proof. With an integration by parts, with r = 2s + 1, we compute   2 Re(Lw, iΛrT,τ w)+ = 2 Re( Dd − OpT (b) w, iΛrT,τ w)+ = Re(i(ΛrT,τ OpT (b) − OpT (b)∗ ΛrT,τ )w, w)+ + Re(ΛrT,τ w|xd =0+ , w|xd =0+ )∂ . We have Re(ΛrT,τ w|xd =0+ , w|xd =0+ )∂ = |w|xd =0+ |2τ,s+1/2 and the principal symbol σ of the operator i(ΛrT,τ OpT (b) − OpT (b)∗ ΛrT,τ ) is real and satisfies σ( ) = −2λrT,τ Im b1 ( )  λ2s+2 T,τ

in a neighborhood of supp(χ).

Then, the microlocal G˚ arding inequality of Theorem 2.50 of Volume 1 yields 2

2

−N Re(Lw, iΛrT,τ w)+ ≥ |w|xd =0+ |2τ,s+1/2 + CΛs+1 T,τ w+ − CN ΛT,τ u+ .

The Young inequality gives 2

|(Lw, iΛrT,τ w)+ |  ε−1 ΛsT,τ Lw2τ,+ + εΛs+1 T,τ w+ , which yields, for ε chosen sufficiently small, |w|xd =0+ |τ,s+1/2 + ΛT,τ wτ,0,s+1  Lwτ,0,s + uτ,0,−N . Finally, we write

  Dd wτ,0,s   Dd − OpT (b) wτ,0,s + wτ,0,s+1 = Lwτ,0,s + wτ,0,s+1 .

Together, these last two inequalities yield the result.



6.4. ESTIMATES FOR FIRST-ORDER OPERATORS

145

6.4.2. An Elliptic Estimate with a Boundary Observation. Let U be a conic open set of Rd+ × Rd−1 × R+ and b( ) ∈ ST1,τ , with principal symbol b1 ( ) homogeneous of degree one, such that, for C > 0, Im b1 ( ) ≥ CλT,τ ,

 ∈ U .

We set L = Dd − OpT (b) and we have the following microlocal elliptic estimate. Lemma 6.21. Let χ ∈ ST0,τ , homogeneous of degree 0, be such that supp(χ) ⊂ U . Let s ∈ R. There exist C > 0 and τ∗ > 0 such that for any N ∈ N, there exists CN > 0 such that

 OpT (χ)uτ,1,s ≤ C LOpT (χ)uτ,0,s + |OpT (χ)u|xd =0+ |τ,s+1/2 + CN uτ,0,−N ,

(6.4.2)

for τ ≥ τ∗ , u ∈ S (Rd+ ). The proof is very similar to that of Lemma 6.20. Proof. With an integration by parts, with r = 2s + 1, we compute   2 Re(Lw, −iΛrT,τ w)+ = 2 Re( Dd − OpT (b) w, −iΛrT,τ w)+ = Re(i(OpT (b)∗ ΛrT,τ − ΛrT,τ OpT (b))w, w)+ − Re(ΛrT,τ w|xd =0+ , w|xd =0+ )∂ . We have Re(ΛrT,τ w|xd =0+ , w|xd =0+ )∂ = |w|xd =0+ |2τ,s+1/2 and the principal symbol σ of the operator i(OpT (b)∗ ΛrT,τ − ΛrT,τ OpT (b)) is real and satisfies σ( ) = 2λrT,τ Im b1 ( )  λ2s+2 T,τ

in a neighborhood of supp(χ).

Then, the microlocal G˚ arding inequality of Theorem 2.50 of Volume 1 yields 2

2

−N Re(Lw, −iΛrT,τ w)+ + |w|xd =0+ |2τ,s+1/2 ≥ CΛs+1 T,τ w+ − CN ΛT,τ u+ .

The Young inequality gives 2

|(Lw, −iΛrT,τ w)+ |  ε−1 ΛsT,τ Lw2τ,+ + εΛs+1 T,τ w+ , which yields, for ε chosen sufficiently small, ΛT,τ wτ,0,s+1  Lwτ,0,s + |w|xd =0+ |τ,s+1/2 + uτ,0,−N . Finally, we write

  Dd wτ,0,s   Dd − OpT (b) wτ,0,s + wτ,0,s+1 = Lwτ,0,s + wτ,0,s+1 .

Together, these last two inequalities yield the result.



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6.4.3. Estimate Under Sub-ellipticity. Let b( ) ∈ ST1,τ , with principal symbol b1 ( ) homogeneous of degree one, L = Dd − OpT (b) with principal symbol () = ξd − b1 ( ). Introduce   1 1 L2 = L + L∗ = Dd − OpT (b) + (OpT (b)∗ , 2 2   1 1 ∗ L−L =− OpT (b) − OpT (b)∗ . L1 = 2i 2i Observe that L2 ∈ Ψτ1,0 and L1 ∈ Ψ1T,τ are both formally selfadjoint. Their respective principal symbols are 2 () = Re () = ξd − Re b1 ( ) ∈ Sτ1,0 , 1 ( ) = Im () = − Im b1 ( ) ∈ ST1,τ . 1 {, } = {2 , 1 } is a tangential symbol. We observe that 2i Let U be a conic open set of Rd+ × Rd−1 × R+ be such that the following sub-ellipticity property holds

(6.4.3) 1 {, }( ) = {2 , 1 }( ) > 0,  ∈ U ,  = ( , ξd ). 2i Here, the sub-ellipticity property is written in the same form as for a secondorder operator (see Definition 3.2 of Volume 1). For the present first-order operator, there is a more natural form. In fact, consider  ∈ U such that 1 ( ) = 0. Then, setting  = ( , ξd ) with ξd = Re b1 ( ) we have () = 0 thus implying {2 , 1 }( ) > 0 by (6.4.3). The sub-ellipticity property thus reads () = 0 ⇒

(6.4.4)

1 ( ) = 0 ⇒ {2 , 1 }( ) > 0,

 ∈ U .

Lemma 6.22. Assume that (6.4.3) holds and SU is compact. Let χ ∈ ST0,τ , homogeneous of degree 0, be such that supp(χ) ⊂ U . There exist C > 0 and τ∗ > 0 such that for any N ∈ N, there exists CN > 0 such that   τ −1/2 OpT (χ)vτ,1 ≤ C LOpT (χ)v+ + |OpT (χ)v|xd =0+ |τ,1/2 + CN vτ,0,−N , for τ ≥ τ∗ , and v ∈ S (Rd+ ). Proof. We set w = OpT (χ)v. We compute, with an integration by parts, Lw2+ = L2 w2+ + L1 w2+ + 2 Re(L2 w, iL1 w)+ = L2 w2+ + L1 w2+ + i([L2 , L1 ]w, w)+ + (L1 w|xd =0+ , w|xd =0+ )∂ . We then obtain, Lw2+ + |w|xd =0+ |2τ,1/2  L1 w2+ + i([L2 , L1 ]w, w)+ .

6.5. NOTES

147

Then, for μ > 0 to be chosen below and for τ chosen sufficiently large so that τ −1 μ ≤ 1, we obtain   (6.4.5) Lw2+ + |w|xd =0+ |2τ,1/2  τ −1 μL1 w2+ + iτ ([L2 , L1 ]w, w)+     τ −1 μL21 + iτ [L2 , L1 ] w, w + . Next, we observe that the principal symbol of the operator μL21 +iτ [L2 , L1 ] ∈ Ψ2T,τ is μ|1 |2 + τ {2 , 1 } ∈ ST2,τ . As we have assumed that SU is compact, from the sub-ellipticity property (6.4.4), Lemma 3.56 of Volume 1 and the homogeneity of the symbol μ|1 |2 +τ {2 , 1 }, we deduce the following lemma. Lemma 6.23. There exist μ0 > 0, C > 0 such that μ|1 |2 ( ) + τ {2 , 1 }( ) ≥ Cλ2T,τ ,

 = (x, ξ  , τ ) ∈ U ,

for μ ≥ μ0 . arding Let μ = μ0 and N ∈ N. As supp(χ) ⊂ U , the microlocal G˚ inequality of Theorem 2.50 of Volume 1 yields for some C > 0 and C  > 0,    Re μL21 + iτ [L2 , L1 ] w, w + ≥ C  w2τ,0,1 − CN v2τ,0,−N , for τ > 0 chosen sufficiently large. From (6.4.5) we thus obtain Lw+ + |w|xd =0+ |τ,1/2 + vτ,0,−N  τ −1/2 wτ,0,1 . From the form of L we observe that Dd w+  Lw+ + wτ,0,1 , 

yielding the result. 6.5. Notes

Here, in a half-space, the operators and symbols we treat are differential in the direction normal to the boundary and pseudo-differential in the tangential direction. Symbols of this form already appear in the proof of Carleman estimates at the boundary in Section 3.4.2. With these operators we consider quadratic forms. Away from boundaries, such differential quadratic forms are studied in the work of L. H¨ ormander [172, Chapter 8]. Quadratic forms in a half-space associated with the symbols we define here can be found in the work of D. Tataru [314] and some joint work with M. Bellassoued [70]. One of the main goals of this chapter is the derivation of the G˚ arding inequality of Theorem 6.17 that applied to quadratic forms in a half space. It can be found in the works of D. Tataru [314], M. Eller [131] and in [70]. Here we give a microlocal version of the inequality. Note that for pseudo-differential operators obtaining a G˚ arding inequality on a half-space is not straight-forward. Some sufficient conditions are given in the work of N. Lerner and X. Saint-Raymond [231], with some improvement given by F. Herau [168].

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In Sect. 6.4, in preparation for later chapters, we also derive microlocal estimates for first-order operators at a boundary. Decomposition in firstorder factors is an approach that goes back to the work of A.P. Calder´ on [100]. The estimates we write and their proofs can be found in a joint work with N. Lerner [211]. Appendix 6.A. Smooth Factorisation of Polynomials 6.A.1. Some Results from Complex Analysis. Let Ω be an open set of C and γ be a closed continuous path in Ω such that (1) indγ (z) = 0 if z ∈ C \ Ω. (2) indγ (z) = 0 or 1 for z ∈ Ω \ γ. Set Oγ = {z ∈ Ω \ γ; indγ (z) = 1}. The following proposition is a consequence of the residue formula. For a holomorphic function in Ω that does not vanish on γ, we denote by (ζi )i∈Iγ the zeros of f in Oγ (counted with multiplicities). Proposition 6.24. Let f be a holomorphic function in Ω that does not vanish on γ. Let  ∈ N. We have   f  (z)z  1 ∫ dz = ζi . 2πi γ f (z) i∈Iγ In particular, the case  = 0 yields (6.A.1)

Nγ (f ) = #Iγ =

1 f  (z) ∫ dz, 2πi γ f (z)

as the number of zeros of f in Oγ (counted with multiplicities). We also recall the Rouch´e theorem (see for instance [294]). Theorem 6.25 (Rouch´e Theorem). Under the same setting as above for γ, if f and g are two holomorphic functions in Ω that satisfy |f (z) − g(z)| < |f (z)|,

for all z ∈ γ,

then Nγ (f ) = Nγ (g). 6.A.2. An Algebraic Identity. The following result is classical and known as the Newton identities.  Lemma 6.26 (Newton Identities). Let q(z) = 0≤k≤n qn−k z k be a monic and let (ri )1≤i≤n be its roots (counted with polynomial, that is, q0 = 1,  multiplicities). If we set s = 1≤i≤n ri ,  ∈ N, we have  kqk + s qk− = 0, k ∈ {1, . . . , n}. 1≤≤k

6.A. SMOOTH FACTORISATION OF POLYNOMIALS

149

Proof. We have q(z) = 1≤k≤n (z − rk ) and we thus find 

qk z k = z n q(1/z) = (1 − rk z). 0≤k≤n

1≤k≤n

Differentiating both sides with respect to z we obtain, after multiplication by z,    

rj z kqk z k = − (1 − rk z) . 1≤j≤n

1≤k≤n

1≤k≤n k=j

Choosing |z| sufficiently small so as to have |rj z| < 1 for all j we write, with a Neumann series,

  rj z 

kqk z k = − (1 − rk z) 1≤j≤n 1 − rj z 1≤k≤n 1≤k≤n

    rj z  qm z m =− 1≤j≤n ≥1

=−



s z 

≥1





0≤m≤n

qm z m .

0≤m≤n

The result then follows by identifying the coefficients of z k on the righthand-side.  Setting qk = 0 for k ≥ n + 1, observe that the proof yields moreover that  s qk− = 0 if k ≥ n + 1. 1≤≤k

6.A.3. Roots Regularity. For n ∈ N we shall use the notation Jn = {1, . . . , n}. Let N ∈ N and p(t, z) be a monic polynomial function in z ∈ C of degree N with coefficients that continuously depend on t ∈ U , an open set of a d-dimensional smooth manifold M, p(t, z) = z N + a1 (t)z N −1 + · · · + aN −1 (t)z + aN (t). Let t0 ∈ U and denote by αj (t0 ), j ∈ Jn , with n ≤ N , the different roots of z → p(t0 , z) with respective multiplicities mj , meaning that m1 + · · · + mn = N , that is,

(z − αj (t0 ))mj . p(t0 , z) = j∈Jn

An important consequence of the Rouch´e theorem is the following regularity result for the roots of a polynomial function. Lemma 6.27. There exist U 0 ⊂ U an open neighborhood of t0 and N functions t → β j,k (t), j ∈ Jn , k = 1, . . . , mj , that enumerate the roots of p(t, z) for t ∈ U 0 , that is,



(z − β j,k (t)), t ∈ U 0 , z ∈ C. p(t, z) = j∈Jn 1≤k≤mj

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and such that lim β j,k (t) = αj (t0 ),

t→t0

j ∈ Jn , k = 1, . . . , mj .

Observe that the smoothness of the coefficients ak , k = 1, . . . , N does not necessarily imply the smoothness of the roots t → β j,k (t), j ∈ Jn , k = 1, . . . , mj . This is well illustrated in the following example. Let p(t, z) = z 2 − t. Then, a pair of continuous roots is given by, for j = 1, 2, √ t if t ≥ 0, j j α (t) = (−1) × √ i −t if t < 0. One sees that these two roots are not smooth at t = 0. Any other choice of functions, say √ t if t ≥ 0, √ αj (t) = (−1)j × −i −t if t < 0. does not yield smooth functions at t = 0 either. The crossing of the roots is responsible for the lack of smoothness of the roots. Observe that the argument of the proof of Lemma 6.19 can be adapted to prove the following result. Proposition 6.28. Let γ be a closed continuous path in C such that indγ (z) = 0 or 1 for z ∈ C \ γ. Assume that z → p(t0 , z) does not vanish on γ. Set J(γ) = {j ∈ Jn ; indγ (αj (t0 )) = 1}. For U 0 as given by Lemma 6.27 there exists δ > 0 such that B(t0 , δ) ⊂ U 0 and indγ (β j,k (t)) = indγ (αj (t0 )),

j ∈ Jn , k = 1, . . . , mj , t ∈ B(t0 , δ).

Moreover, assume that the coefficients of z → p(t, z) are of class C  (resp. analytic) in t ∈ U . Then, the coefficients of the polynomial function



(z − β j,k (t)) z → j∈J(γ) 1≤k≤mj

are of class C  (resp. analytic) in t ∈ B(t0 , δ). In particular, isolated roots (of constant multiplicity) are of class C  (resp. analytic) in t ∈ B(t0 , δ). This result shows that the lack of smoothness of the roots is precisely due to roots crossing. Away from change of multiplicities, roots are smooth functions of the parameter t. If the coefficients of p are moreover analytic in t, then, away from change of multiplicities, the roots are analytic in t. Note that the existence of δ > 0 in the statement of this proposition is a simple consequence of the limit of the roots β j,k (t) as t → t0 , as stated in Lemma 6.27. The following lemmata are used in the proof of Lemma 6.19. Lemma 6.29. Let K ⊂ U be a compact set of M. There exists a bounded set BK where all the roots of the polynomial z → p(t, z) lie if t ∈ K.

6.A. SMOOTH FACTORISATION OF POLYNOMIALS

151

Lemma 6.30. Assume that M = M × M where M is a manifold and  is a conic submanifold (with or without boundary) of Rd for some d ∈ N. Accordingly for t ∈ M we write t = (t , t ) with t ∈ M and t ∈ M . Let then U be a conic open set of M (with respect to the variable t ). In addition to the hypotheses of Lemma 6.27, assume that each coefficient ak (t) is homogeneous of degree k in t for |t | ≥ r0 > 0. Let t0 = (t0 , t0 ) ∈ U with t0 = 0. Then the neighborhood U 0 of Lemma 6.27 can be chosen conic and the set M

R(t) = {β j,k (t); j ∈ Jn , k = 1, . . . , mj }, is homogeneous of degree one, meaning R(t , λt ) = λR(t , t ) for λ > 0 and (t , t ) ∈ U 0 . Proof of Lemma 6.27. Let j ∈ Jn . We consider a closed circular curve γ j : [0, 1] → C with center αj (t0 ), with radius chosen sufficiently small for αj (t0 ) to be the only root of z → p(t0 , z) in Dj , the interior disk of C j = γ j ([0, 1]). We set ε = minz∈C j |p(t0 , z)|/2 > 0 (we shall omit some cumbersome superscript j in what follows). Let z ∈ C j . By continuity of p, there exists a neighborhood Vz ⊂ C of z and a neighborhood Uz ⊂ U of t0 such that t ∈ U z , ζ ∈ Vz . |p(t, ζ) − p(t0 , z)| < ε, Since C j ⊂ ∪z∈C Vz , and C j is compact, we can extract a finite covering with such neighborhoods, viz., there exists z 1 , . . . , z  ∈ C j such that C j ⊂ ∪k=1 Vz k . Then, U j = ∩k=1 Uz k ⊂ U defines a neighborhood of t0 such that for all z ∈ C j and all t ∈ U j we have |p(t, z) − p(t0 , z)| < 2ε ≤ |p(t0 , z)|. By the Rouch´e theorem, for each t ∈ U j the polynomial z → p(t, z) has mj roots in the disc Dj , that we denote by β j,k (t), k = 1, . . . , mj . Arguing the same for each j ∈ Jn , the neighborhood U 0 as in the statement of the lemma is simply obtained by setting U 0 = ∩j∈Jn U j . For each j ∈ Jn , since we can let the radius of the circle C j go to zero, meaning that the disk Dj shrinks to the point αj (t0 ), we obtain lim β j,k (t) = αj (t0 ),

t→t0

k = 1, . . . , mj ,

which concludes the proof.



Proof of Lemma 6.29. Because of the continuity of the coefficients with respect to the parameter t, we have |p(t, z) − z N | ≤ CK (1 + |z|)N −1 , for some CK > 0 if t ∈ K. Then, as there exists RK > 0 such that |z|N > CK (1 + |z|)N −1 for |z| ≥ RK , we see that the roots of z → p(t, z) are  confined in the ball B(0, RK ) uniformly with respect to t ∈ K.

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6. PSEUDO-DIFFERENTIAL OPERATORS ON A HALF-SPACE 

Proof of Lemma 6.30. Set M = M ∩Sd −1 and U = U ∩(M ×M ). We then consider t1 = (t0 , t1 ) ∈ U such that t1 = νt0 for ν > 0. By the homogeneity of the coefficients ak (t) we have

−1 (6.A.2) (ν z − αj (t0 , t0 ))mj p(t1 , z) = ν N p(t0 , ν −1 z) = ν N =



j∈Jn

(z − ναj (t0 , t0 ))mj .

j∈Jn

We thus set α ˜ j = ναj (t0 , t0 ), j ∈ Jn , that enumerate the different roots of 1 z → p(t , z), with multiplicity mj . Lemma 6.27 yields an open set U0 ⊂ U and N functions (t , t ) → β˜j,k (t , t ) that enumerate the roots for (t , t ) ∈ U0 with continuity at t1 :



(z − β˜j,k (t , t )), p(t , t , z) = j∈Jn 1≤k≤mj

and lim

(t ,t )→t1

β˜j,k (t , t ) = α ˜j ,

j ∈ Jn .

We then introduce the open set U 0 = {(t , νt ); (t , t ) ∈ U0 and ν > 0} in M × M and we define the functions β j,k (t , t ) to coincide with β˜j,k (t , t ) on U0 and to be homogeneous of degree one in t . Arguing as in (6.A.2) we see that we obtain the sought result. 

CHAPTER 7

Sobolev Norms with a Large Parameter on a Manifold Contents 7.1. 7.2. 7.3. 7.4. 7.5.

Nonnegative Sobolev Orders on a Manifold Manifold Without Boundary Trace Norms Carleman Weight Function and Sobolev-Norm Estimation Notes

153 154 156 156 162

In Sect. 18.2, Sobolev spaces of order k, k ∈ N, are defined on a Riemannian manifold. In Sect. 18.5, Sobolev spaces of order s, s ∈ R, are defined on a compact Riemannian manifold without boundary. Here, we define their associated norms with a large parameter τ . We also provide an estimation of the bound of the map u → eτ ϕ u with respect to these norms. 7.1. Nonnegative Sobolev Orders on a Manifold First, we consider a smooth Riemannian manifold M, possibly with a boundary. For k ∈ N, we define the following norm: u2τ,k = τ 2k u2L2 (M) + u2H k (M) 

k  j=0

τ 2j u2H k−j (M) .

It is the counterpart of the norm defined in (2.6.1) in Rd .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 7

153

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7. PSEUDO-DIFFERENTIAL OPERATORS ON A HALF-SPACE

Second, we consider a smooth Riemannian manifold N that is compact and without boundary. Then, for s ∈ R+ , we define (7.1.1)

u2τ,s = τ 2s u2L2 (N ) + u2H s (N ) ,

with the norm .H s (N ) as in Sect. 18.5. In application we have in mind, N will be equal to ∂M or a connected component of ∂M, where M is a smooth compact Riemannian manifold. As in other sections, we denote norms on boundaries by |.| rather than . for an easier reading of the statements of the results we prove. In place of (7.1.1), we write (7.1.2)

|u|2τ,s = τ 2s |u|2L2 (∂M) + |u|2H s (∂M) .

Note that for 0 ≤ s < r, we have (7.1.3)

|u|τ,s  τ s−r |u|τ,r ,

implying |u|τ,s  |u|τ,r for τ > 0 chosen sufficiently large. 7.2. Manifold Without Boundary On a smooth Riemannian manifold without boundary N , the previous section applies. We are now interested in the extension of the above Sobolev norms to the case s < 0. We proceed similarly to the definition of H s (N ) in Sect. 18.5. On N , with g its Riemannian metric, we define P0 has the unbounded operator on L2 (N ) with domain D(P0 ) = {u ∈ H 1 (N ); Δg u ∈ L2 (N )}, given by P0 u = −Δg u. In Sect. 18.4, we find that D(P0 ) = H 2 (N ) and that P0 is selfadjoint with spectrum made of eigenvalues with finite multiplicities, given by 0 = μ0 < μ1 ≤ μ2 ≤ · · · ≤ μ n ≤ · · · We denote an associated Hilbert basis of eigenfunctions by (φj )j . If s ∈ R,  −s/2   then u ∈ H s (N ) if u = j uj φj , with (uj )j ⊂ C such that μj uj j ∈ 2 (C); see the beginning of Sect. 18.5 and Proposition 18.16 for the case s ≥ 0 and Proposition 18.19 for the case s < 0. Then, one has  u2H s (N )  (1 + μj )s |uj |2 . j

For any s ∈ R, we now introduce the norm  u2τ,H s (N )  (1 + τ 2 + μj )s |uj |2 , j

where τ ≥ 0. Note that this corresponds to considering Pτ = 1 + τ 2 + P0 in place of P0 above. For τ ≥ 0 kept fixed, the norms .τ,H s (N ) and .H s (N ) are equivalent. For s ≥ 0, we have (1 + τ 2 + μj )s  τ 2s + (1 + μj )s ,

7.2. MANIFOLD WITHOUT BOUNDARY

155

and we recover (7.2.1)

uτ,H s (N )  τ s uL2 (N ) + uH s (N ) ,

as in (7.1.1). This equivalence in the case s ≥ 0 allows us to write uτ,s in place of uτ,H s (N ) . We use this notation also in the case s < 0 as no risk of confusion may arise: if we exclude the case of Rd , we have not introduced any norm in the case of a large parameter for s < 0 up to now. Note that we have (7.2.2)

uτ,s ≤ uH s (N ) , for s ≤ 0,

uH s (N ) ≤ uτ,s , for s ≥ 0.

In what follows, for s ∈ R, we shall denote by Hτs (N ) the space H s (N ) if equipped with the norm .τ,s . With the definition of these norms for both s ≥ 0 and s < 0, one can identify Hτ−s (N ) as the dual space of Hτs (N ) as is done for H −s (N ) and H s (N ) in Sect. 18.5. We have the following result. Proposition 7.1. Let s ∈ R and A = (C i )1≤i≤N be a finite atlas of N with C i = (Oi , κi ), and let (ψi )1≤i≤N be a smooth partition of unity subordinated to the open covering (Oi )1≤i≤N . There exists K = KA > 0 such that  i Λsτ (ψi u)C L2 (Rd ) ≤ Kuτ,s , K −1 uτ,s ≤ 1≤i≤N

for u ∈ C ∞ (N ). Moreover, if i ∈ {1, . . . , N }, for some K = Ki > 0, we have K −1 uτ,s ≤ Λsτ uC L2 (Rd ) ≤ Kuτ,s , i

if u ∈ C ∞ (N ) and supp(u) ⊂ Oi . We recall that Λsτ = Op(λsτ ) with λ2τ = τ 2 + |ξ|2 . Proof. In the case s ≥ 0, with (7.2.1), one sees that the proof follows from the counterpart result in Proposition 18.21. Then, the case s < 0 is obtained as in the proof of Proposition 18.22.  Naturally, with the norms introduced above for s ≥ 0 and s < 0, one has the following continuity result. Proposition 7.2. Let Qτ be a differential operator with a large parameter with smooth coefficients of order k ∈ N as in Sect. 16.3.4. Then, for s ∈ R, there exists C > 0 such that Qτ uτ,s ≤ Cuτ,s+k ,

u ∈ C ∞ (N ), τ > 0.

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7. PSEUDO-DIFFERENTIAL OPERATORS ON A HALF-SPACE

7.3. Trace Norms As in the above sections, consider ν to be the unique outward pointing vector field on M along ∂M such that, for all m ∈ ∂M, gm (νm , νm ) = 1 and gm (νm , v) = 0 for all v ∈ Tm ∂M. For a function u defined in M, we set trk (u) = (u|∂M , ∂ν u|∂M , . . . , ∂νk u|∂M ), where ∂νj u|∂M is defined in (17.7.3). As shown in Proposition 17.25, in normal geodesic coordinates (to be used locally below), we have ∂νj u|∂M = (−∂xd )j u|xd =0+ . With the norms introduced in the previous section, we then define the following norm, for k ∈ N and r ∈ R, k 

2

| trk (u)|τ,k,r =

2

j=0

|∂νj u|∂M |τ,k−j+r .

As there shall be no ambiguity, we shall write | tr(u)|τ,k,r instead of | trk (u)|τ,k,r . For k,  ∈ N and r, s ≥ 0, if k +r < +s and k ≤ , we have | tr(u)|τ,k,r  | tr(u)|τ,,s for τ > 0 chosen sufficiently large. For a smooth function f defined on ∂M, we shall write |f tr(u)|2τ,k,r =

k  j=0

2

|f ∂νj u|∂M |τ,k−j+r .

7.4. Carleman Weight Function and Sobolev-Norm Estimation Our goal in this section is the proof of the following result. Proposition 7.3. Let s ∈ R, and let N be a smooth compact Riemannian manifold without boundary, Ω be an open set of N , and ϕ ∈ C ∞ (Ω). Let also K be a compact set of Ω, then there exists C > 0 such that for any τ ≥ 1, we have (7.4.1)

eτ ϕ uτ,s ≤ Ceτ maxK ϕ uτ,s ,

for all u ∈ Hτs (N ) supported in K. Note that the result of this proposition is clear in the case s ∈ N. The proof we provide is thus of interest in the case s is not an integer. As we shall see, the proof can be reduced to a similar estimate on Rd by means of Proposition 7.1. On the flat space, the argument relies on the introduction of a classical Littlewood–Paley decomposition yet here adapted to the Sobolev norms with the large parameter τ > 0. Then, the product au with a = eτ ϕ is split into different terms in the spirit of the paraproduct. We pick φ ∈ Cc∞ (R), 0 ≤ φ ≤ 1, such that φ(t) = 1 for |t| ≤ 1 and φ(t) = 0 for |t| ≥ 3/2. We also set ϕ(t) = φ(t/2) − φ(t); note that ϕ is supported on {t ∈ R; 1 ≤ |t| ≤ 3} and 0 ≤ ϕ ≤ 1. We further set

7.4. CARLEMAN WEIGHT FUNCTION AND SOBOLEV-NORM ESTIMATION

157

ϕ0 = φ and, for k ≥ 1, ϕk (t) = ϕ(21−k t), the latter being supported in {t ∈ R; 2k−1 ≤ |t| ≤ 3 · 2k−1 }. We have ∞ 

(7.4.2)

ϕk = 1,

k=0

with convergence in C ∞ (R) as the sum is locally finite. Note that any t ∈ R lies in the support of at most two functions in the summation. For u ∈ S  (Rd ), and τ ≥ 1, we define Δτ,k u for k ∈ N through their Fourier transform: u(ξ). F Δτ,k u(ξ) = ϕk (|ξ|/τ )ˆ

 One thus has u = k∈N Δτ,k u with convergence in S  (Rd ). Observe that F Δτ,k u is supported in {ξ ∈ Rd ; 2k−1 τ ≤ |ξ| ≤ 3τ 2k−1 } for k ≥ 1. For k ≥ 3, we define also Sτ,k u =

k−3 

Δτ,k u.

j=0

u(ξ). Note that Sτ,k u is supported in One has F Sτ,k u(ξ) = φ(23−k ξ/τ )ˆ d k−4 {ξ ∈ R , |ξ| ≤ 3τ 2 }. With their forms in the Fourier domain, one sees that the operators Δτ,k and Sτ,k are uniformly bounded on L2 (Rd ) since the functions φ and ϕ are bounded.  Lemma 7.4. Let s ∈ R. One has u2τ,s  k∈N (τ 2k )2s Δτ,k u2L2 (Rd ) . Proof. Since ϕk ≥ 0 and since any point meets at most two such functions, for all t ∈ R, there exists k ∈ N such that ϕk (t) ≥ 1/2. One thus finds  2 1/4 ≤ ϕk (t) ≤ 2, t ∈ R. k∈N

With this property, we first prove that one has  u2τ,s  (7.4.3) Δτ,k u2τ,s . k∈N

In fact, one writes ˆ, u ˆ)L2 (Rd )  u2τ,s = (λ2s τ u =

 k∈N

   2s 2 λτ ϕk (|ξ|/τ )ˆ u, u ˆ L2 (Rd )

k∈N s λτ ϕk (|ξ|/τ )ˆ u2L2 (Rd ) ,

which is (7.4.3). We then write (7.4.4)

2

Δτ,k u2τ,s = λsτ F Δτ,k u2L2 (Rd )  (τ 2k )s F Δτ,k uL2 (Rd ) ,

since λτ  τ 2k on supp(F Δτ,k u). We shall need the following estimates in what follows.



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7. PSEUDO-DIFFERENTIAL OPERATORS ON A HALF-SPACE

Lemma 7.5. There exists C > 0 such that a ∈ L∞ (Rd ), k ∈ N.

Δτ,k aL∞ (Rd ) + Sτ,k aL∞ (Rd ) ≤ CaL∞ (Rd ) , One also has Δτ,k aL∞ (Rd ) ≤ Cτ −1 2−k aW 1,∞ (Rd ) ,

a ∈ W 1,∞ (Rd ), k ≥ 1.

A proof is given below. With Lemma 7.5, for two functions a and u in S (Rd ), one may write  (7.4.5) au = Δτ,j a Δτ,k u k,j∈N

=



k≥j+3

=

∞  k=3



Δτ,j a Δτ,k u +

Δτ,j a Δτ,k u +

j≥k+3

Sτ,k a Δτ,k u +

∞ 

Sτ,k u Δτ,k a +

k=3

 |j−k|≤2

 |j−k|≤2

Δτ,j a Δτ,k u

Δτ,j a Δτ,k u,

where convergence holds in L∞ (Rd ). With the support theorem (see Proposition 8.42 of Volume 1), we find (7.4.6)

  supp F Sτ,k aΔτ,k u ⊂ {ξ ∈ Rd ; 5τ 2k−4 ≤ |ξ| ≤ 27τ 2k−4 },   and Δτ,j Sτ,k aΔτ,k u = 0 if |j − k| ≥ 3.

k≥3

One also finds (7.4.7)

  supp F Δτ,j aΔτ,k u ⊂ {ξ ∈ Rd ; |ξ| ≤ 3τ 2max(k,j) },   and Δτ,n Δτ,j aΔτ,k u = 0 if n ≥ 3 + max(k, j).

k, j ∈ N.

Proposition 7.6. Let s ∈ R. There exists C > 0 such that 

 τ −|α| ∂ α aL∞ (Rd ) uτ,s , (7.4.8) auτ,s ≤ C |α|≤|s|+1

for a ∈ W |s|+1,∞ (Rd ) and u ∈ Hτs (Rd ). For s ∈ N, a direct proof is obtained by computing τ s−|α| ∂ α (au). In what follows, we shall assume s ∈ R \ N. To estimate the Hτs -norm of the product au, we need the following lemma, whose proof is given below. Lemma 7.7. We have the following three estimations: (1) There exists C0 > 0 such that (7.4.9)

Sτ,k aΔτ,k uL2 (Rd ) ≤ C0 aL∞ (Rd ) Δτ,k uL2 (Rd ) , for a ∈ L∞ (Rd ) and u ∈ Hτs (Rd ).

k ≥ 3,

7.4. CARLEMAN WEIGHT FUNCTION AND SOBOLEV-NORM ESTIMATION

(2) Let s > 0. There exists C1 such that if     n s Δτ,j aΔτ,k u 2 δn = (τ 2 ) 

L (Rd )

|j−k|≤2, k≥n

159

n ∈ N,

,

then δ = (δn )n∈N ∈ 2 (N) and δ2 ≤ C1 aL∞ (Rd ) uτ,s ,

(7.4.10)

for a ∈ L∞ (Rd ) and u ∈ Hτs (Rd ). (3) Let s ∈ (0, 1). There exists C2 such that (τ 2k )s Sτ,k uΔτ,k aL2 (Rd ) ≤ Cτ −1 aW 1,∞ (Rd ) uτ,s 2−(1−s)k ,

(7.4.11)

for a ∈ W 1,∞ (Rd ) and u ∈ Hτs (Rd ). Proof of Proposition 7.6. We prove that each term in the r.h.s. of (7.4.5) fulfills the estimation in (7.4.8). With the support of the Fourier transform of Sτ,k aΔτ,k u given in (7.4.6), we have (7.4.12)  ∞      Sτ,k aΔτ,k u   (τ 2k )s Sτ,k aΔτ,k uL2 (Rd ) . (τ 2n )s Δτ,n k=3

|n−k|≤2

L2 (Rd )

With Lemma 7.4 and (7.4.9), this implies  uτ,s .

∞

k=3 Sτ,k aΔτ,k uτ,s

 aL∞ (Rd )

With the support of the Fourier transform of Δτ,j aΔτ,k u given in (7.4.7), we have       Δτ,j aΔτ,k u  (τ 2n )s Δτ,n    (τ 2n )s 

|j−k|≤2

 |j−k|≤2 max(j,k)+2≥n

L2 (Rd )

  Δτ,j aΔτ,k u

L2 (Rd )

= δn .

If s > 0 with (7.4.10), one has (δn )n∈N ∈ 2 (N) and      2  (τ 2n )2s Δτ,n Δτ,j aΔτ,k u   a2L∞ (Rd ) u2τ,s . |j−k|≤2

n∈N

This implies that  Lemma 7.4.



L2 (Rd )

|j−k|≤2 Δτ,j (a)Δτ,k (u)τ,s

 aL∞ (Rd ) uτ,s by

For s ∈ (0, 1), with the support of the Fourier transform of Sτ,k uΔτ,k a given in (7.4.6) and (7.4.11), we have  ∞      Sτ,k uΔτ,k a   (τ 2k )s Sτ,k uΔτ,k aL2 (Rd ) (τ 2n )s Δτ,n k=3

L2 (Rd )

|n−k|≤2

 τ −1 aW 1,∞ (Rd ) uτ,s 2−(1−s)n .

160

7. PSEUDO-DIFFERENTIAL OPERATORS ON A HALF-SPACE

With Lemma 7.4, we find  s ∈ (0, 1).

∞

 τ −1 aW 1,∞ (Rd ) uτ,s for

k=3 Sτ,k uΔτ,k aτ,s

With the above estimations, the result of the proposition is proven for s ∈ (0, 1). We now prove the result for other values of s. For s > 0, s ∈ / N, we have s = p + σ with σ ∈ (0, 1) and p = s!. Applying the result of the proposition proven for σ ∈ (0, 1), we have  p−|α| α  τ ∂ (au)τ,σ  τ p−|α|−|β| ∂ α a∂ β u)τ,σ auτ,s  |α|≤p

 

|α|+|β|≤p



τ

|α|+|β|≤p



|α|≤p+1

p−|α|−|β|−1

∂ aW 1,∞ (Rd ) ∂ β uτ,σ α

 τ −|α| ∂ α aL∞ (Rd ) uτ,s ,

as τ p−|β| ∂ β uτ,σ  uτ,s . We thus obtain the result of the proposition for s ≥ 0, recalling that the case s ∈ N follows from explicit computations. To obtain the result for s < 0, we argue by duality. Let σ = −s, for u ∈ Hτ−σ (Rd ), v ∈ Hτσ (Rd ), and a ∈ W σ+1 (Rd ). First note that au ∈ Hτ−σ (Rd ). Second, we write |au, vHτ−σ (Rd ),H σ (Rd ) | = |u, avHτ−σ (Rd ),H σ (Rd ) | ≤ uτ,−σ avτ,σ τ

 τ   uτ,−σ τ −|α| ∂ α aL∞ (Rd ) vτ,σ , |α|≤σ+1



which implies the result for s < 0. We may now proceed with the proof of Proposition 7.3.

Proof of Proposition 7.3. Below we prove that (7.4.1) holds in the case of Rd . This suffices for the conclusion in the case N is a smooth Riemannian manifold without boundary. In fact, with a finite atlas A = (C i )1≤i≤N , with C i = (κi , Oi ), and a subordinated partition of unity (ψi )1≤i≤N , by applying Proposition 7.1 twice, one has   Ci i eτ ϕ uτ,s  Λsτ ψi eτ ϕ uC L2 (Rd ) 1≤i≤N





i

e

τ supx∈κ(K∩Oi ) ϕC (x)

1≤i≤N

 eτ maxK ϕ

 1≤i≤N

e

τ maxK ϕ

 i Λsτ ψi uC L2 (Rd )

 i Λsτ ψi uC L2 (Rd )

uτ,s .

In what follows, we can assume that Ω is an open set in Rd and we use the Euclidean distance. There exists χτ ∈ Cc∞ (Ω) such that χτ ≡ 1 in a neighborhood of K, χτ (x) = 0 if dist(x, K) ≥ τ −1 , and for any multiindex α, one has |∂ α χτ |  τ |α| . As eτ ϕ u = eτ ϕ χτ u, by Proposition 7.6

7.4. CARLEMAN WEIGHT FUNCTION AND SOBOLEV-NORM ESTIMATION

161

with a = eτ ϕ χτ therein, we only need to estimate the L∞ -norm on Rd of τ −α ∂ α (eτ ϕ χτ ) for |α| = |s|! + 1. This term is a linear combination of terms of the form 

τ −|α|+k eτ ϕ (∂ β1 ϕ) · · · (∂ βk ϕ)∂ β χτ ,

(7.4.13)

with 1 ≤ k ≤ |α|, 1 ≤ |βi |, i = 1, . . . , k, and β1 + · · · + βk + β  = α. In particular, k + |β  | ≤ |α|. Hence, each term given by (7.4.13) has its L∞ norm estimated by Ceτ ϕ L∞ (Kτ ) , for some C > 0 independent of τ , and where Kτ = {x ∈ Rd ; dist(x, K) ≤ τ −1 }. The conclusion follows if we prove that eτ ϕ(y)  eτ maxK ϕ , for all y ∈ Kτ . We note that if y ∈ Kτ \ K, there exists x ∈ K such that |x − y| ≤ τ −1 . We then write ϕ(y) ≤ ϕ(x) + |ϕ(x) − ϕ(y)| ≤ ϕ(x) + C|x − y| ≤ max ϕ + Cτ −1 . K

We deduce that eτ ϕ(y) ≤ eτ maxK ϕ+C , which gives the result.



Proof of Lemma 7.5. Set φ˜ and ϕ˜ as the inverse Fourier transforms of φ(|ξ|) and ϕ(|ξ|), respectively. One has ˜ x) ∗ a and Δτ,k a = (τ 2k−1 )d ϕ(τ ˜ 2k−1 x) ∗ a, k ≥ 1, Δτ,0 a = τ d φ(τ and ˜ 2k−3 x) ∗ a, k ≥ 3. Sτ,k a = (τ 2k−3 )d φ(τ ˜ n τ x) and (τ 2n )d ϕ(2 ˜ n τ x) are bounded in L1 (Rd ) uniformly with As (τ 2n )d φ(2 respect to n and τ , we obtain the first inequality of the lemma. For the second inequality, for k ≥ 1, we write   Δτ,k a(x) = (τ 2k−1 )d ∫ ϕ˜ τ 2k−1 (x − y) a(y) dy Rd

   = (τ 2k−1 )d ∫ ϕ˜ τ 2k−1 (x − y) a(y) − a(x) dy, Rd

as ϕ˜ has a zero mean since ϕ(0) = 0. We deduce that    Δτ,k aL∞ (Rd )  (τ 2k−1 )d aW 1,∞ (Rd ) ∫ ϕ˜ τ 2k−1 (x − y) |x − y| dy Rd

2

−k −1

τ

aW 1,∞ (Rd ) ,

by a change of variables and using that |x|ϕ(x) ˜ ∈ L1 (Rd ). Proof of Lemma 7.7. Let a ∈ L∞ (Rd ) and u ∈ Hτs (Rd ). Lemma 7.5, we have

 From

Sk (a)Δk (u)L2 (Rd )  Sk (a)L∞ Δk (u)L2 (Rd )  aL∞ Δk (u)L2 (Rd ) . This gives (7.4.9).

162

7. PSEUDO-DIFFERENTIAL OPERATORS ON A HALF-SPACE

Let now s > 0. We have by Lemma 7.5     −s(k−n) ks s   Δτ,j aΔτ,k u 2 d  aL∞ 2 (2 τ Δk (u)L2 (Rd ) ). τ s 2sn  |j−k|≤2, k≥n

L (R )

k≥n

As (2−sn )n is in 1 (N) since s > 0 and (2ks τ s Δk (u)L2 (Rd ) )k is in 2 (N) with norm estimated by uτ,s by Lemma 7.4, by discrete convolution, we obtain that δ ∈ 2 with δ as in the statement and (7.4.10) follows. Let now s ∈ (0, 1) and k ≥ 3. As Sτ,k is uniformly bounded on L2 (Rd ) and from Lemma 7.5, we have (τ 2k )s Sτ,k uΔτ,k aL2 (Rd )  (τ 2k )s uL2 (Rd ) Δτ,k aL∞ (Rd )  2−k(1−s) τ s−1 uL2 (Rd ) aW 1,∞ (Rd )  2−k(1−s) τ −1 uτ,s aW 1,∞ (Rd ) , 

which proves (7.4.11) 7.5. Notes

The contents we present in this chapter are very classical and somewhat already used in the previous chapters, yet not for norms of orders that are integers. The present chapter is thus mainly written to make precise Sobolev norms with a large parameter at boundaries, in particular in the case of noninteger Sobolev orders (positive or negative). Trace inequalities are also written. This prepares for Chap. 8 where these norms are needed to derive more precise Carleman estimates at boundaries. The basic material for (usual) Sobolev spaces on a Riemannian manifold can be found in Chapter 18. The Sobolev-norm estimation of the map u → eτ ϕ u given in Proposition 7.3 is based on the analysis tools developed for the so-called paraproduct with extensions to paradifferential calculus. The reader is referred to the original article by J.-M. Bony [76] and the articles by Y. Meyer [248, 249]. See also the books of Y. Meyer and R.R. Coifman [250], M.E. Taylor [318], and H. Bahouri et al. [50].

CHAPTER 8

Estimates for General Boundary Conditions Contents 8.1. 8.2. 8.3.

Introduction ˇ Lopatinski˘ı–Sapiro Condition After Conjugation ˇ Carleman Estimate Under the Lopatinski˘ı–Sapiro Condition 8.3.1. Statements 8.3.2. Choice of Local Coordinates 8.3.3. A Basic Estimate ˇ 8.3.4. A Microlocal Estimate Under the Lopatinski˘ı–Sapiro Condition 8.3.5. Patching Microlocal Estimates Together 8.3.6. A Shifted Estimate 8.3.7. A Basic Microlocal Elliptic Estimate 8.4. Estimates Without Any Prescribed Boundary Condition 8.4.1. A First Estimate 8.4.2. A Refined Estimate 8.4.3. A Shifted Refined Estimate 8.5. Global Estimates 8.6. Notes Appendix 8.A. Some Technical Proofs 8.A.1. A Norm Computation 8.A.2. The Classical Carleman Argument Revisited 8.A.3. Proof of Lemma 8.19 8.A.4. Proof of Lemma 8.20 8.A.5. Proof of Theorem 8.24 8.A.6. Proof of Theorem 8.30

163 164 176 176 178 180 180 185 187 188 188 189 189 191 192 193 195 195 195 195 196 199 200 201

8.1. Introduction ˇ In Chap. 2, Lopatinski˘ı–Sapiro boundary conditions are introduced. Our goal is now to use these conditions and to generalize the Carleman estimates obtained in Chap. 5 in the case of Dirichlet boundary conditions on © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 8

163

164

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

ˇ a Riemannian manifold. The Lopatinski˘ı–Sapiro framework for the study of a boundary value problem is natural as it means that the elliptic system formed by the elliptic operator and the boundary operator is Fredholm; see Chap. 3. We thus consider (M, g) a d-dimensional smooth compact Riemannian manifold. We refer to Chap. 15 for the notions of differential geometry that are needed in what follows. The elliptic operator under consideration is P = −Δg + R1 with R1 a first-order differential operator on M with bounded coefficients. We also consider a differential operator B defined at (or in a neighborhood of) the boundary ∂M. The elliptic problem under consideration is thus the following: (8.1.1)

P u = f ∈ L2 (M),

Bu|∂M = f∂ ,

with f∂ in a prescribed space depending on the nature of the operator B. For a reader interested in a second-order elliptic operator in an open set of Rd , we refer to Sect. 18.9 that exposes how the Riemannian point of view can be used. As in Chap. 2, the boundary operator B is chosen of the form (8.1.2)

B = B k + B k−1 ∂ν ,

one every connected component N of ∂M, with B k and B k−1 smooth differential operators on N of order k and k − 1, respectively, with k ≤ β. Naturally, if B is of order zero on N , one has B k−1 = 0. ˇ In Sect. 8.2, we show that the Lopatinski˘ı–Sapiro condition is robust under conjugation by a weight function, a key ingredient in the derivation of a Carleman estimate. In Sect. 8.3, we show how Carleman estimates ˇ can be obtained under the Lopatinski˘ı–Sapiro boundary condition. Finally, in Sect. 8.4, we take advantage of the analysis tools we develop to derive estimates in the case no boundary condition is imposed; yet the Dirichlet trace and a first-order trace need to be known. This allows us to refine the counterpart estimates derived in Chap. 5 (see Lemma 5.3 and Theorem 5.5). ˇ 8.2. Lopatinski˘ı–Sapiro Condition After Conjugation We naturally use the same notation as in Sect. 2.2. Let ϕ be a smooth function of M. By Proposition 16.17, the operator Pϕ = eτ ϕ P e−τ ϕ is a second-order differential operator with a large parameter on M, and its principal symbol is given by σ(Pϕ )(m, ω, τ ) = p(m, ω + iτ dϕ(m)) = (ω + iτ dϕ(m), ω + iτ dϕ(m))g , with (., .)g the bilinear form for cotangent vectors associated with the metric at m, see (17.1.4) in Sect. 17.1. Following Sect. 2.2, for (m, ω  ) ∈ T ∗ ∂M, z ∈ R, and τ ≥ 0, we set ω = ω  − znm in the above symbol and define pˇϕ (m, ω  , z, τ ) = p(m, ω  − znm + iτ dϕ(m)).

ˇ 8.2. LOPATINSKI˘I–SAPIRO CONDITION AFTER CONJUGATION

165

We recall that n is the outward pointing unitary one-form on M along ∂M, that is, n = ν , with ν the unique outward pointing unitary vector field that is orthogonal to Tm ∂M in the sense of g. Let m ∈ ∂M. If v ∈ Tm M, we have v = tνm + v  with v  ∈ Tm ∂M. Going back to the definition of a tangent vector (see Sect. 15.3), we have v  (ϕ) = v  (ϕ|∂M ) = dϕ|∂M (m)(v  ), where dϕ|∂M = d(ϕ|∂M ) is the differential on the submanifold ∂M of the restriction ϕ|∂M . In particular, ∗ ∂M. We thus have d(ϕ|∂M )(m) ∈ Tm dϕ(m)(v) = tdϕ(m)(νm ) + dϕ(m)(v  ) = t∂ν ϕ(m) + dϕ|∂M (m)(v  ). We thus write dϕ(m) = ∂ν ϕ(m)nm + dϕ|∂M (m), which is the decomposition ∗ ∂M ⊕ T ∗ ∂M = T ∗ M of dϕ(m) according to the orthogonal direct sum Nm m m (orthogonality to be understood with respect to gm ). Decomposing ω + iτ dϕ(m) accordingly, that is,   ω + iτ dϕ(m) = ω  + iτ dϕ|∂M (m) + − z + iτ ∂ν ϕ(m) nm , yields (8.2.1)

 2 pˇϕ (m, ω  , z, τ ) = z − iτ ∂ν ϕ(m) + α2 ,

with α = α(m, ω  , τ ) such that Re α ≥ 0 and (8.2.2)

α2 = p(m, ω  + iτ dϕ|∂M (m)) = |ω  |g∂ − τ 2 |dϕ|∂M (m)|2g + 2iτ (ω  , dϕ|∂M (m))g∂ 2

∂

=

2 |ω  |g∂

−τ

2

|dϕ|∂M (m)|2g ∂

+ 2iτ ω  (ϕ|∂M ),

as we have, noticing that ω  ∈ Tm ∂M, (ω  , dϕ(m))g = ω  (ϕ) = ω  (ϕ|∂M ) = dϕ|∂M (m)(ω  ) = (ω  , dϕ|∂M (m))g∂ , where g∂ is the induced metric on ∂M. We thus obtain pˇϕ (m, ω  , z, τ ) = (z − γ2 )(z − γ1 ), with (8.2.3)

γj = γj (m, ω  , τ ) = iτ ∂ν ϕ(m) + i(−1)j α(m, ω  , τ ),

j = 1, 2.

As in Definition 2.2 (see also Remark 2.2), we define the following polynomial:

 (z − γj ), pˇ+ ϕ (m, ω , z, τ ) = Im γj ≥0

 ˇϕ have negawith the convention that pˇ+ ϕ (m, ω , z, τ ) = 1 if all the roots of p tive imaginary parts. Depending on the sign of these roots, the polynomial pˇ+ ϕ is thus of degree 0, 1, or 2.

We also define a conjugated version of the boundary operator B. Assume that m is in a connected component N of ∂M where B is of order k ≤ β. We have B = B k + B k−1 ∂ν , where B k and B k−1 are smooth differential

166

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

operators on N of order k and k − 1, respectively. We denote by b(m, ω), bk (m, ω  ), and bk−1 (m, ω  ) their respective principal symbols. We set Bϕ = eτ ϕ Be−τ ϕ , which is a differential operator with a large parameter of order k on a neighborhood of ∂M, and its principal symbol is given by   bϕ (m, ω, τ ) = b m, ω + iτ dϕ(m)   = bk m, ω  + iτ dϕ|∂M (m)    + ibk−1 m, ω  + iτ dϕ|∂M (m) ωn + iτ ∂ν ϕ(m) , ∗ ∂M. For (m, ω  ) ∈ T ∗ ∂M, and τ ≥ 0, where ω = ω  + ωn nm with ω  ∈ Tm we set ˇbϕ (m, ω  , z, τ ) = b(m, ω  − znm + iτ dϕ(m))   = bk m, ω  + iτ dϕ|∂M (m)    + ibk−1 m, ω  + iτ dϕ|∂M (m) iτ ∂ν ϕ(m) − z .     Note that bk m, ω  + iτ dϕ|∂M (m) and bk−1 m, ω  + iτ dϕ|∂M (m) are polynomial functions in τ of degree less than or equal to k and k −1, respectively, by Proposition 16.13.

ˇ We may now state the Lopatinski˘ı–Sapiro condition for the conjugated operators, first microlocally, second at one point, and third locally. ˇ Definition 8.1 (Lopatinski˘ı–Sapiro Condition for Conjugated Opera ∗ tors). Let (m, ω ) ∈ T ∂M and τ ≥ 0 with (ω  , τ ) = 0. We say that the ˇ Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) at (m, ω  , τ ) if for any polynomial function f (z) with complex coefficients, there exist c ∈ C and a polynomial function (z) with complex coefficients such that, for all z ∈ C, (8.2.4) p+ (m, ω  , z, τ ). f (z) = cˇbϕ (m, ω  , z, τ ) + (z)ˇ ϕ

ˇ We say that the Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) at m ∈ ∂M  if it holds at (m, ω , τ ) for all ω  ∈ T ∗ ∂M and τ ≥ 0 such that (ω  , τ ) = 0. ˇ If Γ ⊂ ∂M, we say that the Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) in Γ if it holds at m for all m ∈ Γ. Remark 8.2. In Definition 8.1, we have τ ≥ 0. Note that the case τ = 0 ˇ coincides with the Lopatinski˘ı–Sapiro condition for (P, B); see Definition 2.2. Remark 8.3. With the Euclidean division of polynomials, we see that it suffices to consider the polynomial function f (z) to be of degree less than  that of pˇ+ ϕ (m, ω , z, τ ) in (8.2.4). In any case, the degree of f (z) can be chosen less than or equal to one. Remark 8.4. Similarly to what was observed in Remark 2.4, note that ˇ the Lopatinski˘ı–Sapiro property is written here without any use of local coordinates. It is thus a geometrical condition. As the principal parts of the conjugated operators Pϕ and Bϕ are geometrical objects (see Sect. 16.3.4),

ˇ 8.2. LOPATINSKI˘I–SAPIRO CONDITION AFTER CONJUGATION

167

ˇ if a particular local chart is chosen, then the Lopatinski˘ı–Sapiro condition for (P, B, ϕ) can be equivalently expressed in this chart. Lemma 8.5. Let (m, ω  ) ∈ T ∗ ∂M and τ ≥ 0 with (ω  , τ ) = 0. The ˇ Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) at (m, ω  , τ ) if and only if  (1) either pˇ+ ϕ (m, ω , z, τ ) = 1 +  (2) or pˇϕ (m, ω , z, τ ) = z − γ and ˇbϕ (m, ω  , γ, τ ) = 0.  ˇϕ (m, ω  , z, τ ) = (z − γ1 )(z − γ2 ), that is, Proof. If pˇ+ ϕ (m, ω , z, τ ) = p both roots γ1 and γ2 have nonnegative imaginary parts, then condition (8.2.4) cannot hold, as by Remark 8.3, it means that the vector space of polynomials of degree less than or equal to one would be generated by the single polynomial ˇbϕ (m, ω  , z, τ ).  If pˇ+ ϕ (m, ω , z, τ ) is of degree one, then one of the roots, say γ1 , has a negative imaginary part and the second one, γ2 , has a nonnegative imag ı– inary part. We have pˇ+ ϕ (m, ω , z, τ ) = z − γ2 . Then, the Lopatinski˘  ˇ Sapiro condition holds at (m, ω , τ ) if for any f (z), the polynomial function z → f (z) − cˇb(m, ω  , z, τ ) admits γ2 for a root for some c ∈ C. A necessary and sufficient condition is then ˇb(m, ω  , γ2 , τ ) = 0. Observe that in this case, the root configuration is precisely that of the non-conjugated operators (compare with Proposition 2.3).  If finally pˇ+ ϕ (m, ω , z, τ ) = 1, that is, both roots have negative imaginary parts, then the condition (8.2.4) holds trivially. 

Proposition 8.6. Let (m0 , ω 0 ) ∈ T ∗ ∂M and τ 0 ≥ 0 with (ω 0 , τ 0 ) = 0 ˇ such that the Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) at (m0 , ω 0 , τ 0 ). Then, there exists a conic neighborhood of (m, ω 0 , τ 0 ) ∈ T ∗ ∂M × R+ where ˇ the Lopatinski˘ı–Sapiro condition holds also. 0 ˇ condition holds for Let m ∈ ∂M be such that Lopatinski˘ı–Sapiro (P, B, ϕ) at m. Then, there exists a neighborhood of m0 in ∂M where the ˇ Lopatinski˘ı–Sapiro condition holds also. Proof. The second part of the proof follows from the first part, using homogeneity and compacity. ˇ If the Lopatinski˘ı–Sapiro condition holds at (m0 , ω 0 , τ 0 ), by Lemma 8.5, one faces the following two cases. 0 0 0 Case 1: pˇ+ ϕ (m , ω , z, τ ) = 1, meaning that both roots γ1 and γ2 are such that Im γ1 < 0 and Im γ2 < 0. With the continuity of the roots as given by Lemma 6.27, one sees that pˇ+ ϕ remains equal to one in a conic neighborhood. Case 2: one root, say γ1 (m0 , ω 0 , τ 0 ), has a nonnegative imaginary part. Then, 0 0 0 0 0 0 pˇ+ ϕ (m , ω , z, τ ) = z − γ1 (m , ω , τ ),

and   ˇbϕ m0 , ω 0 , γ1 (m0 , ω 0 , τ 0 ), τ 0 = 0.

Im γ2 (m0 , ω 0 , τ 0 ) < 0

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8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

Again, with the continuity of the roots, one has Im γ2 (m, ω  , τ ) < 0 and   ˇbϕ m, ω  , γ1 (m, ω  , τ ), τ = 0 (8.2.5) for (m, ω  , τ ) in a conic neighborhood U of (m0 , ω 0 , τ 0 ).  For a point (m, ω  , τ ) ∈ U , either Im γ1 (m, ω  , τ ) < 0 yielding pˇ+ ϕ (m, ω ,  +   z, τ ) = 1 or Im γ1 (m, ω , τ ) ≥ 0 yielding pˇϕ (m, ω , z, τ ) = z − γ1 (m, ω , τ ). In ˇ either case, Lemma 8.5 implies that the Lopatinski˘ı–Sapiro condition holds   at (m, ω , τ ). ˇ The following theorem states how the Lopatinski˘ı–Sapiro condition before conjugation of Definition 2.2 for (P, B) may or may not imply the ˇ Lopatinski˘ı–Sapiro condition after conjugation of Definition 8.1 for (P, B, ϕ) for some choices of ϕ. Theorem 8.7. Let P , B, and ϕ be as above. Let m ∈ ∂M. ˇ condition for (P, B, ϕ) (1) If ∂ν ϕ(m) ≥ 0, then the Lopatinski˘ı–Sapiro does not hold at m. ˇ (2) Let the Lopatinski˘ı–Sapiro condition hold for (P, B) at m, and let ϕ be such that ∂ν ϕ(m) < 0. We have the following results: (a) If B is of order zero with nonvanishing principal symbol (Dirichˇ let boundary conditions), then the Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) at m. (b) Let B be of order 1 ≤ k ≤ β on N the connected component of ∂M that contains m. There exists μ0 > 0, depending only on ˇ the principal symbols p and b, such that the Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) at m if (8.2.6)

|dϕ|∂M (m)|g∂ ≤ μ0 |∂ν ϕ(m)|. ˇ By continuity, if the Lopatinski˘ı–Sapiro condition holds for (P, B) in a compact subset Γ of ∂M, then there exists μ0 > 0 ˇ such that if (8.2.6) holds in Γ, then the Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) in Γ. (c) Assume that B is of order k = 1 on N the connected component of ∂M that contains m with principal symbol of the ∗ M, ω = ω n + ω  form b(m, ω) = ωn + ω  , tm  for ω ∈ Tm n m  ∗  with ω ∈ Tm ∂M, and where t is a real vector field on ∂M ˇ (see Example 2.5-(3)). If d = 2, the Lopatinski˘ı–Sapiro condiˇ tion holds for (P, B, ϕ) at m. If d ≥ 3, the Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) at m if and only if we have

(8.2.7)

|∂ν ϕ(m)| > dϕ|∂M (m), tm . (d) Assume that B is of order k = 1 on N the connected component of ∂M that contains m with principal symbol of the form   for ω ∈ T ∗ M, ω = ω n + ω  with b(m, ω) = ωn + iω  , vm n m m  ∗ ω ∈ Tm ∂M, and where v  is a real vector field on ∂M. If

ˇ 8.2. LOPATINSKI˘I–SAPIRO CONDITION AFTER CONJUGATION

169

 | = 1 and the Lopatinski˘ ˇ d = 2, necessarily |vm ı–Sapiro condi | tion holds for (P, B, ϕ) at m. If d ≥ 3, necessarily |vm g∂ < 1 ˇ and the Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) at m if and only if  2  2  2 (8.2.8) (1 − |vm |g∂ )(∂ν ϕ(m))2 > |vm |g∂ |dϕ|∂M (m)|2g∂ − dϕ|∂M (m), vm  .

The remainder of this section is devoted to the proof of this theorem. Remark 8.8. Observe that Item (2b) states that if the Lopatinski˘ı– ˇ Sapiro condition holds for (P, B) at m, if ∂ν ϕ(m) < 0, and if ϕ is constant ˇ on ∂M, then the Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) at m. By Item (2c), observe also that in the case of the Neumann boundary ˇ condition, we see that the Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) at m if and only if ∂ν ϕ(m) < 0. Observe also that Item (2c) covers the case a boundary operator with principal symbol of the form b(m, ω) = cω, vm  where c ∈ C∗ and v is vector ν = n , v  = 0. field in M along ∂M with vm m m Proof of Theorem 8.7. ˇϕ occurs for some choices of ω  Item (1). We prove that the equality pˇ+ ϕ =p ˇ and τ . Then by Lemma 8.5, the Lopatinski˘ı–Sapiro condition cannot hold for (P, B, ϕ). First, we assume that ∂ν ϕ(m) = 0, and then for ω  = 0 and τ > 0, we have α2 ≤ 0 for α defined in (8.2.2) implying that Re α = 0. Consequently, for the roots of z → pˇϕ (m, ω  , z, τ ) given in (8.2.3), we have Im γj = 0 for ˇϕ . j = 1, 2. Hence, pˇ+ ϕ =p ∗ ∂M. Second, we assume that ∂ν ϕ(m) > 0, and we pick some ω  ∈ Tm Then, we write   2 α(m, ω  , τ )2 = τ 2 − |dϕ|∂M (m)|2g + |ω  /τ |g + 2iτ −1 ω  (ϕ|∂M ) . As τ → ∞, we have (α/τ )2 → c ∈ C with Re c ≤ 0, implying that Re α/τ → 0. As the roots of z → pˇϕ (m, ω  , z, τ ) read   γj = iτ ∂ν ϕ(m) + (−1)j α(m, ω  , τ )/τ , j = 1, 2, we see that Im γj > 0 for τ > 0 chosen sufficiently large for both j = 1, 2. ˇϕ . Hence, as above, pˇ+ ϕ =p ˇ Item (2). As the Lopatinski˘ı–Sapiro condition holds for (P, B), it is sufficient to only consider τ > 0 in the proof by Remark 8.2. The condition ∂ν ϕ(m) < 0 implies Im γ1 < 0 with γ1 given in (8.2.3). If 0 ≤ Re α < τ |∂ν ϕ|, we have Im γ2 < 0 and thus pˇ+ ϕ = 1; then the ˇ Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) at m by Lemma 8.5. If Re α ≥ τ |∂ν ϕ|, we have Im γ2 ≥ 0 and thus pˇ+ ϕ = z − γ2 ; then by Lemma 8.5, ˇ the Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) at m if and only if ˇbϕ (m, ω  , γ2 , τ ) = 0. The Lopatinski˘ı–Sapiro ˇ condition for (P, B, ϕ) at m

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8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

∗ ∂M and τ > 0, we have thus holds if and only if for any ω  ∈ Tm Re α(m, ω  , τ ) ≥ τ |∂ν ϕ(m)| ⇒ ˇbϕ (m, ω  , γ2 , τ ) = 0. (8.2.9)

Item (2a). If B is of order zero, the principal symbol is such that b(m, ω) = 0 is independent of ω. The condition ˇbϕ (m, ω  , γ2 , τ ) = 0 thus holds. Item (2b). We consider the case Re α ≥ τ |∂ν ϕ(m)| > 0, that is, Im γ2 ≥ 0. Lemma 8.11 below yields the equivalent condition 4τ 2 (∂ν ϕ(m))2 Re α2 − 4τ 4 (∂ν ϕ(m))4 + (Im α2 )2 ≥ 0. In turn, the definition of α2 in (8.2.2) gives the equivalent condition  2  (∂ν ϕ(m))2 |ω  |g∂ − τ 2 |d(ϕ|∂M )(m)|2g − τ 2 (∂ν ϕ(m))4 + (ω  (ϕ|∂M ))2 ≥ 0, which reads (8.2.10)

(τ ∂ν ϕ(m))2 |dϕ(m)|2g ≤ (∂ν ϕ(m))2 |ω  |g∂ + (ω  (ϕ|∂M ))2 . 2

In particular, we see that ω  = 0 if Im γ2 ≥ 0. With (8.2.3), and writing dϕ(m) = dϕ|∂M (m) + ∂ν ϕ(m)nm , we compute   ˇbϕ (m, ω  , z = γ2 , τ ) = b m, ω  − γ2 nm + iτ dϕ(m)  = b m, ω  − i|ω  |g∂ nm − i(α − |ω  |g∂ )nm  + iτ dϕ|∂M (m) . We now claim that for ε > 0, there exists μ0 > 0 such that (8.2.11) |dϕ|∂M (m)|g ≤ μ0 |∂ν ϕ(m)| and Im γ2 ≥ 0 ∂

⇒ |α − |ω  |g∂ | + τ |(dϕ|∂M )(m)|g ≤ ε|ω  |g∂ . ∂

|b(m, ω 

i|ω  |

C0 |ω  |kg∂ ,

We then use that − for some C0 > 0 as g∂ nm )| ≥ ˇ the Lopatinski˘ı–Sapiro condition holds for (P, B) at m, for ω  = 0, and as b is homogeneous of degree k in ω here. With the mean value theorem and using that dω b is homogeneous of degree k − 1 in ω, we then find that |ˇbϕ (m, ω  , z = γ2 , τ )| ≥ C|ω  |kg∂ > 0 for μ0 > 0 chosen sufficiently small and ϕ satisfying the condition (8.2.6), using that in the case Im γ2 ≥ 0, we have ω  = 0, as seen above. This concludes the proof of the item (2b) of the theorem. Proof of the Claim (8.2.11). Condition (8.2.10) and |(dϕ|∂M )(m)|g ≤ ∂ μ0 |∂ν ϕ(m)| imply, with the Cauchy–Schwarz inequality, τ 2 (∂ν ϕ(m))4 ≤ (∂ν ϕ(m))2 |ω  |g∂ + |ω  |g∂ |(dϕ|∂M )(m)|2g 2

2

∂

≤ (1 +

2 μ20 )(∂ν ϕ(m))2 |ω  |g∂ ,

yielding τ |∂ν ϕ(m)| ≤ (1 + μ0 )|ω  |g∂ . We thus obtain (8.2.12)

τ |(dϕ|∂M )(m)|g ≤ μ0 (1 + μ0 )|ω  |g∂ . ∂

ˇ 8.2. LOPATINSKI˘I–SAPIRO CONDITION AFTER CONJUGATION

171

The form of α2 given in (8.2.2) and inequality (8.2.12) give α2 = |ω  |2g∂ (1 + O(μ0 )). We thus obtain α = |ω  |g∂ |(1 + O(μ0 )).

(8.2.13)

Inequalities (8.2.12) and (8.2.13) yield the claim (8.2.11). ∗ ∂M is of dimension Item (2c). Firstly, we consider the case d = 2. Then Tm ∗ one. We pick em such that span(em ) = Tm ∂M and |em |g∂ = 1. Let s ∈ R be such that τ dϕ|∂M (m) = sem . For ω  = s em , we have

α2 = p(m, ω  + iτ dϕ|∂M (m)) = (s + is)2 |em |2g∂ = (s + is)2 , which gives α = sgn(s )(s + is). We thus find, by (8.2.3), ˇbϕ (m, ω  , z = γ2 , τ ) = ω  + iτ dϕ|∂M (m), t  − iα m     = (s + is) em , tm  − i sgn(s ) . ∗ ∂M and τ > 0, we have Assume that for some ω  ∈ Tm

|s | = Re α(m, ω  , τ ) ≥ τ |∂ν ϕ(m)| and ˇbϕ (m, ω  , z = γ2 , τ ) = 0. Then, s = 0, and thus 0 = em , tm  − i sgn(s ); we obtain a contradiction. ˇ The Lopatinski˘ı–Sapiro condition for (P, B, ϕ) thus holds at m. Secondly, we treat the case d ≥ 3. Assume that (8.2.7) holds. We have ˇbϕ (m, ω  , z, τ ) = ω  + iτ dϕ|∂M (m), t  + iτ ∂ν ϕ(m) − z, m yielding, by (8.2.3), ˇbϕ (m, ω  , z = γ2 , τ ) = ω  + iτ dϕ|∂M (m), t  − iα. m

(8.2.14)

∗ ∂M and τ > 0 we have Assume that for some ω  ∈ Tm

(8.2.15)

Re α(m, ω  , τ ) ≥ τ |∂ν ϕ(m)| and ˇbϕ (m, ω  , z = γ2 , τ ) = 0.

We then have τ dϕ|∂M (m), tm  = Re α ≥ τ |∂ν ϕ(m)|. We have thus reached a contradiction with (8.2.7). Let us now assume that (8.2.7) does not hold at m, that is, (8.2.16)

dϕ|∂M (m), tm  ≥ |∂ν ϕ(m)| > 0.

∗ ∂M and τ > 0 such that (8.2.15) holds. By (8.2.14), We wish to find ω  ∈ Tm  ˇ the condition bϕ (m, ω , z = γ2 , τ ) = 0 reads

(8.2.17)

α(m, ω  , τ ) = τ dϕ|∂M (m) − iω  , tm ,

yielding α(m, ω  , τ )2 = τ dϕ|∂M (m), tm 2 − ω  , tm 2 − 2iτ dϕ|∂M (m), tm ω  , tm .

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8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

With α2 given by (8.2.2), considering the real and imaginary parts separately, we find   2 (8.2.18) τ 2 dϕ|∂M (m), tm 2 + |dϕ|∂M (m)|2g = ω  , tm 2 + |ω  |g∂ , ∂

(8.2.19)

dϕ|∂M (m), tm ω  , tm 



= −ω (ϕ|∂M ) = −ω  , dϕ|∂M (m) .

∗ ∂M of dimenThe solutions ω  of (8.2.19) form a linear subspace of Tm sion d − 2 ≥ 1. If we pick one of them different from zero, there exists τ > 0 such that (8.2.18) holds also, as here dϕ|∂M (m), tm  = 0. We thus have

α(m, ω  , τ )2 = τ dϕ|∂M (m) − iω  , tm 2 . We may thus choose α as in (8.2.17), since our requirement is Re α ≥ 0 (see above (8.2.2)) and here we have dϕ|∂M (m), tm  ≥ 0 by (8.2.16). We thus obtain Re α(m, ω  , τ ) = τ dϕ|∂M (m), tm . By (8.2.16), we thus obtain that (8.2.15) holds for this choice of ω  ∈ Tm ∂M ˇ and τ > 0, meaning that the Lopatinski˘ı–Sapiro condition does not hold for (P, B, ϕ) at m. This concludes the proof of the item (2c) of the theorem. Item (2d). Firstly, we consider the case d = 2. By Proposition 2.8, if  | ˇ the Lopatinski˘ı–Sapiro condition holds for (P, B), then |vm g∂ = 1. Here ∗ ∗ ∂M and Tm ∂M is of dimension one. We pick em such that span(em ) = Tm |em |g∂ = 1. Let s ∈ R be such that τ dϕ|∂M (m) = sem . For ω  = s em , we have α2 = p(m, ω  + iτ dϕ|∂M (m)) = (s + is)2 |em |2g∂ = (s + is)2 , which gives α = sgn(s )(s + is). We thus find, by (8.2.3), ˇbϕ (m, ω  , z = γ2 , τ ) = iω  + iτ dϕ|∂M (m), v   − iα m     = i(s + is) em , vm  − sgn(s ) . ∗ ∂M and τ > 0, we have Assume that for some ω  ∈ Tm

|s | = Re α(m, ω  , τ ) ≥ τ |∂ν ϕ(m)| and ˇbϕ (m, ω  , z = γ2 , τ ) = 0.   − sgn(s ); we obtain a contradiction as Then, s = 0, and thus 0 = em , vm  ˇ condition for (P, B, ϕ) thus holds at m. |vm |g∂ = 1. The Lopatinski˘ı–Sapiro ˇ Secondly, we treat the case d ≥ 3. Since the Lopatinski˘ı–Sapiro condition  holds for (P, B), then |vm |g∂ < 1 by Proposition 2.8. Assume that (8.2.8) ˇ condition holds for holds. If dϕ|∂M (m) = 0, then the Lopatinski˘ı–Sapiro  = 0, (P, B, ϕ) at m by Item (2b) of the theorem that is proven above. If vm ˇ then the Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) at m by Item (2c) of the theorem that is proven above.  = 0 and dϕ We may thus assume that vm |∂M (m) = 0. We have

ˇbϕ (m, ω  , z, τ ) = iω  + iτ dϕ(m), v   + iτ ∂ν ϕ(m) − z. m

ˇ 8.2. LOPATINSKI˘I–SAPIRO CONDITION AFTER CONJUGATION

173

By (8.2.3), it yields (8.2.20)

ˇbϕ (m, ω  , z = γ2 , τ ) = iω  + iτ dϕ|∂M (m), v   − iα(m, ω  , τ ). m

∗ ∂M and τ > 0, we have Assume that for some ω  ∈ Tm

(8.2.21)

Re α(m, ω  , τ ) ≥ τ |∂ν ϕ(m)| and ˇbϕ (m, ω  , z = γ2 , τ ) = 0.

We then have (8.2.22)

 ω  , vm  = Re α ≥ τ |∂ν ϕ(m)|.

  Since ˇbϕ (m, ω  , z = γ2 , τ ) = 0, we have α(m, ω  , τ ) = ω  + iτ dϕ|∂M (m), vm implying  2  2  − τ 2 dϕ|∂M (m), vm  α(m, ω  , τ )2 = ω  , vm   + 2iτ dϕ|∂M (m), vm ω  , vm .

With α2 given by (8.2.2), considering the real and imaginary parts separately, we find

  2  2 (8.2.23)  = |ω  |2g∂ − ω  , vm  τ 2 |dϕ|∂M (m)|2g∂ − dϕ|∂M (m), vm and (8.2.24)

  ω  , vm . (ω  , dϕ|∂M (m))g∂ = dϕ|∂M (m), vm

 ) ≤ 1, then dϕ   If rank(dϕ|∂M (m) , vm |∂M (m) = avm with a = 0 since  |2 )ω  , v   = 0. dϕ|∂M (m) = 0 here. Condition (8.2.24) leads to a(1 − |vm g∂ m    Since |vm |g∂ < 1, we obtain ω , vm  = 0, which contradicts (8.2.22).  ) = 2 in addition to condition Assume now that rank(dϕ|∂M (m) , vm (8.2.8). Introduce

(8.2.25)

  vm . u = dϕ|∂M (m) − dϕ|∂M (m), vm

 ) = 2. Condition (8.2.24) reads ω  , u  = 0. Note that rank(u , vm 





Set Q = span((vm ) , dϕ|∂M (m)) = span((vm ) , (u ) ) and by ω0 the orthogonal projection of ω  onto Q. Introduce also p the orthogonal pro onto {u }⊥ . Since rank(u , v  ) = 2, we have p = 0 and jection of vm m    , u ) u and (p ) ∈ Q. We have Q = span((p ) , (u ) ). p = vm − |u |−2 (v g g∂ m ∂  |p |−2 (p ) . Lemma 8.9. We have ω0 = ω  , vm g∂  |p |−2 (p ) . We have Proof. Set ω ˜  = ω  , vm g∂      ˜  , p  = ω  , p  − ω  , vm  = −|u |−2 ω  − ω g∂ (vm , u )g∂ ω , u  = 0,

˜  , u  = 0, as ω ˜  is also orthogonal since ω  , u  = 0. We also have ω  − ω      ˜ is orthogonal to Q meaning that to u , since p , u  = 0. Thus, ω − ω ˜ .  ω0 = ω

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8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

  = aτ |∂ ϕ(m)|. We write ω  = ω  + ζ  , Set a ≥ 1 to be such that ω  , vm ν 0  where ζ is orthogonal to Q by the definition of ω0 . Note that we have ζ  = 0 ∗ ∂M = Q. in the case d = 3 since Tm We shall need the following lemma whose proof is given in Appendix 8.A.1.

Lemma 8.10. We have    2   |dϕ|∂M (m)|2g∂ − dϕ|∂M (m), vm  −2  2 . |p |g∂ − 1 = 1 − |vm |g∂  2  2 |vm |g∂ |dϕ|∂M (m)|2g∂ − dϕ|∂M (m), vm From condition (8.2.23), we have  2

|ζ | = |ω  |2g∂ − |ω0 |2g∂

  2  2 = ω  , vm  − |ω0 |2g∂ + τ 2 |dϕ|∂M (m)|2g∂ − dϕ|∂M (m), vm 

   2 2  2 = τ 2 a2 1 − |p |−2 g∂ (∂ν ϕ(m)) + |dϕ|∂M (m)|g∂ − dϕ|∂M (m), vm  .

With Lemma 8.10, we obtain

   2  2 |ζ  |2 = M a2 |vm |g∂ − 1 (∂ν ϕ(m))2 + |vm |g∂ |dϕ|∂M (m)|2g∂   2 − dϕ|∂M (m), vm  , with M = τ2

 2 |dϕ|∂M (m)|2g∂ − dϕ|∂M (m), vm > 0,  |2 |dϕ 2  2 |vm |∂M (m)|g∂ − dϕ|∂M (m), vm  g∂

 | since |vm g∂ < 1. Necessarily, we have    2  2  2 |g∂ |dϕ|∂M (m)|2g∂ − dϕ|∂M (m), vm  |vm |g∂ − 1 (∂ν ϕ(m))2 + |vm   2  2 2  2 2 ≥ a |vm |g∂ − 1 (∂ν ϕ(m)) + |vm |g∂ |dϕ|∂M (m)|g∂  2 − dϕ|∂M (m), vm 

= |ζ  |2 /M ≥ 0. This gives a contradiction with (8.2.8). Having condition (8.2.8) thus implies ˇ that the Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) at m. Let us now assume that (8.2.8) does not hold at m, that is,    2  2  2 (8.2.26) |vm |g∂ |dϕ|∂M (m)|2g∂ − dϕ|∂M (m), vm  ≥ 1 − |vm |g∂ (∂ν ϕ(m))2 .  |   Since |vm g∂ < 1 here, observe that this implies that rank(dϕ|∂M (m) , vm ) = ∗ ∂M and τ > 0 such that (8.2.21) holds. By 2. We wish to find ω  ∈ Tm ˇ (8.2.20), the condition bϕ (m, ω  , z = γ2 , τ ) = 0 reads

(8.2.27)

 α(m, ω  , τ ) = ω  + iτ dϕ|∂M (m), vm .

 2 and If we find ω  and τ > 0 such that α(m, ω  , τ )2 = ω  + iτ dϕ|∂M (m), vm   ≥ τ |∂ ϕ(m)|, then (8.2.27) holds since Re α ≥ 0, and thus (8.2.21) ω  , vm ν ˇ is fulfilled meaning that the Lopatinski˘ı–Sapiro condition does not hold for

ˇ 8.2. LOPATINSKI˘I–SAPIRO CONDITION AFTER CONJUGATION

175

(P, B, ϕ) at m. We then conclude that condition (8.2.8) is necessary for the ˇ Lopatinski˘ı–Sapiro condition to hold for (P, B, ϕ) at m. The remainder of the proof thus concerns the construction of (ω  , τ ).  2 is equivalent We recall that having α(m, ω  , τ )2 = ω  +iτ dϕ|∂M (m), vm  to (8.2.23)–(8.2.24). With u as given by (8.2.25), the second condition reads ω  , u  = 0. We shall also use the notation introduced in the proof of the sufficiency of condition (8.2.8) given above. Let τ > 0 to be kept fixed and a ≥ 1 to be determined below. Set   ω0 = a|p |−2 g∂ τ |∂ν ϕ(m)|(p ) ∈ Q,

and set ω  = ω0 +ζ  with ζ  orthogonal to Q. Note that in the case d = 3, we ∗ ∂M meaning that ζ  = 0. We also have ω  , v   = aτ |∂ ϕ(m)|. have Q = Tm ν m We have ω  , u  = 0 since (u ) ∈ Q and since p is orthogonal to u . Hence condition (8.2.24) is fulfilled. Now condition (8.2.23) reads

  2  = |ω0 |2g∂ + |ζ  |2g∂ − (aτ ∂ν ϕ(m))2 . τ 2 |dϕ|∂M (m)|2g∂ − dϕ|∂M (m), vm As |ω0 |2g∂ = (aτ ∂ν ϕ(m))2 /|p |2g∂ , we find

  2 2 |dϕ|∂M (m)|2g∂ ϕ(m)) + τ |ζ  |2g∂ = 1 − |p |−2 (aτ ∂ ν g∂   2 − dϕ|∂M (m), vm 

   2  2 |g∂ − 1 (∂ν ϕ(m))2 + |vm |g∂ |dϕ|∂M (m)|2g∂ = M a2 |vm   2 − dϕ|∂M (m), vm  . with the computations made above. By (8.2.26), we have    2  2  2 |g∂ |dϕ|∂M (m)|2g∂ − dϕ|∂M (m), vm  ≥ 0. |vm |g∂ − 1 (∂ν ϕ(m))2 + |vm  | Since |vm g∂ < 1 for some a ≥ 1, we obtain   2   2  2 a |vm |g∂ − 1 (∂ν ϕ(m))2 + |vm |g∂ |dϕ|∂M (m)|2g∂ − dϕ|∂M (m), vm  = 0.

This means that for this value of a ≥ 1, if we choose ω  = ω0 , that is, ζ  = 0, condition (8.2.23) is also fulfilled. Note that with the choice of ζ  = 0, we cover the cases d = 3 and d ≥ 4 together. We have thus obtained ∗ ∂M and τ > 0 such that α(m, ω  , τ )2 = ω  + iτ dϕ  2 ω  ∈ Tm |∂M (m), vm    ≥ τ |∂ ϕ(m)|. This concludes the proof of the Item (2d) of the and ω  , vm ν theorem.  Lemma 8.11. Let t ∈ C and s = t2 . We then have, for r0 > 0, | Re t|  r0

⇔

4r02 Re s − 4r04 + (Im s)2  0.

Proof. Let t = x + iy. We have Re s = x2 − y 2 and Im s = 2xy, and we observe that 4r02 Re s − 4r04 + (Im s)2 = 4(r02 + y 2 )(x2 − r02 ), which gives the result.



176

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

ˇ 8.3. Carleman Estimate Under the Lopatinski˘ı–Sapiro Condition The Sobolev norms used in the statements of the Carleman estimates are introduced in Chap. 7. 8.3.1. Statements. In the neighborhood of a point m0 of the boundary ∂M, we prove the following estimate. Proposition 8.12. Let (M, g) be a smooth compact Riemannian manifold with boundary, and let P = −Δg + R1 with R1 a first-order differential operator with bounded coefficients on M. Let m0 ∈ ∂M and V 0 be an open neighborhood of m0 in M that meets one connected component of ∂M. Let ϕ ∈ C ∞ (M) be such that the pair (P, ϕ) has the sub-ellipticity property of Definition 5.1 in V 0 . Consider B a differential operator of order k in V 0 of the form of (8.1.2), Moreover, assume that (P, B, ϕ) satisfies ˇ the Lopatinski˘ı–Sapiro condition of Definition 8.1 at m0 . Then there exist 1 a neighborhood V of m0 in M and two constants C and τ∗ > 0 such that (8.3.1)

τ −1/2 eτ ϕ uτ,2 + |eτ ϕ|∂M tr(u)|τ,1,1/2 

≤ C eτ ϕ P uL2 (M) + |eτ ϕ Bu|∂M |τ,3/2−k ,

for all u ∈ C ∞ (M), with supp(u) ⊂ V 1 , and τ ≥ τ∗ . ˇ Remark 8.13. The Lopatinski˘ı–Sapiro condition for (P, B, ϕ) is only used in the estimate of Lemma 8.19 that represents a key point of the proof of Proposition 8.12. As pointed out in Remark 8.37, from the proof of Lemma 8.19 in Appendix 8.A.3, one can allow the boundary operator B to continuously depend on a parameter s that lies in a compact set S and such ˇ that the Lopatinski˘ı–Sapiro condition holds for (P, B, ϕ) for all values of s ∈ S. Note that the order of the operator should however remain constant as s varies. In such case, all the constants obtained in the Carleman estimate of Proposition 8.12 can be chosen uniform with respect to s ∈ S. This property can then be extended to all resulting Carleman estimates under ˇ Lopatinski˘ı–Sapiro conditions derived in this chapter. Arguing as in Sect. 5.3, for Theorems 5.5 and 5.6, patching together estimates as above, we obtain the following result. Theorem 8.14. Let (M, g) be a smooth compact Riemannian manifold with boundary, and let P = −Δg + R1 with R1 a first-order differential operator with bounded coefficients on M. Let V be an open set of M and set V∂ = V ∩ ∂M. Let ϕ ∈ C ∞ (M) be such that the pair (P, ϕ) has the subellipticity property of Definition 5.1 in V . If V∂ = ∅, consider B a differential operator of order β in V . For 0 ≤ k ≤ β, denote by k∂M the union of the connected components of ∂M where B is of order k. Moreover, assume that ˇ (P, B, ϕ) satisfies the Lopatinski˘ı–Sapiro condition of Definition 8.1 at all

ˇ 8.3. CARLEMAN ESTIMATE UNDER THE LOPATINSKI˘I–SAPIRO CONDITION 177

points m ∈ V∂ . Then, there exist C and τ∗ > 0 such that (8.3.2)

τ −1/2 eτ ϕ uτ,2 + |eτ ϕ|∂M tr(u)|τ,1,1/2

 |eτ ϕ Bu|k∂M | ≤ C eτ ϕ P uL2 (M) + 1≤k≤β

 τ,3/2−k

,

for all u ∈ C ∞ (M), with supp(u) ⊂ V , and τ ≥ τ∗ . ˇ Note that the Lopatinski˘ı–Sapiro condition for (P, B, ϕ) is assumed on a ˇ closed set. By Proposition 8.6, the Lopatinski˘ı–Sapiro condition holds also in a neighborhood of V∂ . This is needed to have all the proper estimations in the patching procedure. Remark 8.15. With commutator arguments, as in Chapter 3 of Volume 1, we see that estimate (8.3.2) reads τ 3/2 eτ ϕ uL2 (M) + τ 1/2 eτ ϕ D uL2 Λ1 (M) + τ −1/2 eτ ϕ H uL2 Λ2 (M)

 + |eτ ϕ|∂M tr(u)|τ,1,1/2 ≤ C eτ ϕ P uL2 (M) + |eτ ϕ Bu|k∂M | 1≤k≤β



τ,3/2−k

where D u stands for the covariant derivative on M of u and H u = D2 u is the Hessian of u, defined by means of the Levi-Civita connection. These notions are recalled in Sects. 17.4 and 17.7. We prove the result of Proposition 8.12 by means of local coordinates at the point of interest at the boundary. In a local chart, C = (O, κ) such that m0 ∈ O, the boundary ∂M is given by {xd = 0} and M = {xd ≥ 0}. More precisely, we have (see Sect. 15.1) (8.3.3) κ(∂M ∩ O) = {xd = 0} ∩ κ(O),

κ(M ∩ O) = {xd ≥ 0} ∩ κ(O).

In this chart, the principal part of the Laplace-Beltrami operator takes the form  (8.3.4) g C,ij (x)ξi ξj . pC (x, ξ) = 1≤i,j≤d

We denote the principal symbol of the operator B by b(m, ω) and by bC (x, ξ) its local representative. In this local chart, the operator B is of order 0 ≤ k ≤ β. The form of the Carleman estimate of Proposition 8.12 is insensitive to lower order terms in the operator P following Remark 3.13. The same holds for the boundary operator B. With the local setting introduced above, proving Proposition 8.12 is thus equivalent to proving the following proposition. Proposition 8.16. Let P0 = Op(pC ), B0 = Op(bC ). Let m0 ∈ ∂M and V 0 be an open neighborhood of m0 in M that meets one connected component of ∂M. Let x0 = κ(m0 ). Assume that (P0 , ϕC ) has the subellipticity property of Definition 3.2 of Volume 1 in U 0 with U 0 = κ(V 0 ∩O). ˇ Moreover, assume that (P0 , B0 , ϕC ) satisfies the Lopatinski˘ı–Sapiro condition

178

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

of Definition 8.1 at x0 . Then, there exist a bounded neighborhood U+ of x0 in Rd+ such that U+ ⊂ U 0 and two constants C and τ∗ > 0 such that C

(8.3.5) τ −1/2 eτ ϕ uτ,2 + |e

τ ϕC

|xd =0+

tr(u)|τ,1,1/2

 C C ≤ C eτ ϕ P0 u+ + |eτ ϕ B0 u|xd =0+ |τ,3/2−k ,

∞

for all u ∈ C c (U+ ) and τ ≥ τ∗ . For some open set W+ = W ∩ Rd+ with W an open set of Rd , the space is defined in (3.4.11). By extension for U+ an open set of Rd+ , we

∞ C c (W+ )

set (8.3.6)

∞

C c (U+ ) = {u = v|Rd ; v ∈ Cc∞ (Rd ) and supp v ⊂ U }, +

where U is some open set of Rd such that U+ = U ∩ Rd+ . Note that the ∞ space C c (U+ ) is independent of the choice of U . The following sections are devoted to the proof of Proposition 8.16. 8.3.2. Choice of Local Coordinates. Choosing normal geodesic coordinates, near m0 , by possibly reducing the size of the open set O and changing the coordinate map κ (see Sects. 9.4 and 17.6), yields a local representative of the metric that satisfies C C (x) = 1, and gdj (x) = 0 for j = 1, . . . , d − 1, gdd

for x ∈ κ(O). We moreover choose the map κ such that 0 = κ(m0 ). We set x = (x , xd ) with x = (x1 , . . . , xd−1 ) and accordingly ξ = (ξ  , ξd ) with ξ  = (ξ1 , . . . , ξd−1 ). The principal part of the Laplace–Beltrami operator then reads  r(x, ξ  ) = g C,ij (x)ξi ξj  |ξ  |2 . pC (x, ξ) = ξd2 + r(x, ξ  ), 1≤i,j≤d−1 ∗ M, we write Let m ∈ ∂M and x = κ(m). In this chart if ω ∈ Tm ∗ ∂M if and only if ξ = ω C = ξ = t (T κ(m))−1 ω (see Sect. 15.4). Then, ω ∈ Tm d d 0. An outward pointing vector field is simply given by its representative νxC = (0, . . . , 0, −1). It is the unique outward pointing vector field on O such that, for all m ∈ ∂M ∩ O, gm (νm , νm ) = 1 and gm (νm , u) = 0 for all

as in Sect. 2.2, we have nC = (0, . . . , 0, −1) u ∈ Tm ∂M. Setting nm = νm x because of the form of the metric in the chosen normal geodesic coordinates. We also have ∂ν ϕ(m) = −∂d ϕC (x) for x = κ(m). In the chosen normal geodesic coordinates, the local representative of ∂ν is −∂d . Hence, the representative of the boundary operator B of order k takes the form

B C (x, D) = B k,C (x, D ) − iB k−1,C (x, D )Dd ,

ˇ 8.3. CARLEMAN ESTIMATE UNDER THE LOPATINSKI˘I–SAPIRO CONDITION 179

where B k,C (x, D ) and B k−1,C (x, D ) are the local representatives of B k and B k−1 . We denote by bk,C (x, ξ  ) and bk−1,C (x, ξ  ) their respective principal symbols. With bC (x, ξ) defined above, we have bC (x, ξ) = bk,C (x, ξ  ) − ibk−1,C (x, ξ  )ξd and B0 = Op(bC ). In the local chart, we define the conjugated operators of P0 and B0 : C

C

C

P0,ϕ = eτ ϕ P0 e−τ ϕ ,

C

B0,ϕ = eτ ϕ B0 e−τ ϕ .

The principal symbol of P0,ϕ and B0,ϕ is, respectively, p0,ϕ (x, ξ, τ ) = pC (x, ξ + iτ dϕC (x)), and (8.3.7) with

  b0,ϕ (x, ξ, τ ) = bCϕ (x, ξ, τ ) = bC x, ξ + iτ dϕC (x)    k−1,C = bk,C (x, ξ  , τ ) ξd + iτ ∂d ϕC , ϕ (x, ξ , τ ) − ibϕ    j,C x, ξ  + iτ dϕC|xd =0+ (x) , bj,C ϕ (x, ξ , τ ) = b

By (8.2.1)–(8.2.2), for m ∈ ∂M,

ω

∈

∗ ∂M, Tm

j = k, k − 1. and z ∈ R, we obtain

pˇϕ (m, ω  , z, τ ) = p0,ϕ (x, ξ  , z, τ ) = pC (x, ξ  + iτ dx ϕC (x), z + iτ ∂d ϕC (x)) = (z + iτ ∂d ϕC (x))2 + α(x, ξ  , τ )2 , for x = (x , 0) = κ(m), with (ξ  , 0) = (ω  )C , and where (8.3.8)

α(x, ξ  , τ )2 = r(x, ξ  + iτ dx ϕC (x)) = r(x, ξ  ) − τ 2 r(x, dx ϕC (x)) + 2iτ r˜(x, ξ  , dx ϕC (x)),

with r˜(x, ., .) the bilinear form associated with the quadratic form r(x, .). As in Sect. 8.2, α is chosen such that Re α ≥ 0. We thus have pˇϕ (m, ω  , z, τ ) = p0,ϕ (x, ξ  , z, τ ) = (z − γ1 )(z − γ2 ), with (8.3.9)

γj (x, ξ  , τ ) = −iτ ∂d ϕC (x) + i(−1)j α(x, ξ  , τ ).

ˇ By the first result of Theorem 8.7, if the Lopatinski˘ı–Sapiro condition holds 0 C 0 for (P, B, ϕ) at m , then ∂d ϕ (x ) > 0, and locally we have ∂d ϕC (x) > 0, implying  1 if Re α(x, ξ  , τ ) < τ ∂d ϕC (x), +  (8.3.10) pˇϕ (m, ω , z, τ ) = z − γ2 (x, ξ  , τ ) if Re α(x, ξ  , τ ) ≥ τ ∂d ϕC (x). ˇ Moreover, by Lemma 8.5, if the Lopatinski˘ı–Sapiro condition holds at 0   ∗  (m , ω , τ ) with ω ∈ Tm0 ∂M, τ ≥ 0 and (ω , τ ) = 0, we have (8.3.11)

Re α(x0 , ξ  , τ ) ≥ τ ∂d ϕC (x0 ) ⇒ bC (x0 , ξ  + iτ dx ϕC (x0 ), γ2 + iτ ∂xd ϕC (x0 )) = 0,

180

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

as ˇbϕ (m0 , ω  , z) = bC (x0 , ξ  + iτ dx ϕC (x0 ), z + iτ ∂xd ϕC (x0 )); see the arguˇ mentation above (8.2.9). Here, as the Lopatinski˘ı–Sapiro condition holds at 0  ∗ m , property (8.3.11) is valid for any ω ∈ Tm0 ∂M and τ ≥ 0 such that (ω  , τ ) = 0. Observe that if we have 0 ≤ Re α < τ ∂d ϕC , by Lemma 8.11, this reads 4(τ ∂d ϕC )2 Re α2 − 4(τ ∂d ϕC )4 + (Im α2 )2 < 0, which, with (8.3.8), gives (8.3.12)   (∂d ϕC )2 r(x, ξ  ) + r˜2 (x, ξ  , dx ϕC ) < (τ ∂d ϕC )2 r(x, dx ϕC ) + (∂d ϕC )2 . In particular, this yields |ξ  |  τ , or rather λT,τ  τ.

(8.3.13)

8.3.3. A Basic Estimate. An argument based on the G˚ arding inequality for interior quadratic forms gives the following basic estimation. Note that the argument is in fact quite close to that used in Chapter 3 of Volume 1 and only relies on the sub-ellipticity property. A proof is given in Appendix 8.A.2. Lemma 8.17. Let P0 = Op(pC ). Assume that (P0 , ϕC ) has the subellipticity property of Definition 3.2 of Volume 1 in U 0 with U 0 = κ(V 0 ∩O). Then, there exist two constants C and τ∗ > 0 such that   τ −1 w2τ,2 ≤ C P0,ϕ w2+ + | tr(w)|2τ,1,1/2 , (8.3.14) ∞

for τ ≥ τ∗ and w ∈ C c (U 0 ). ∞

The space C c (U 0 ) is defined in (8.3.6). In this lemma, all traces appear on the right-hand side of the estimation: no assumption is made on the weight function at the boundary. This basic estimate is used in the following section as well as in Sect. 8.4. ˇ 8.3.4. A Microlocal Estimate Under the Lopatinski˘ı–Sapiro Condition. In the local setting described above, we prove the following result. Proposition 8.18. Let x0 = κ(m0 ) be such that x0d = 0. Assume that (P0 , ϕC ) has the sub-ellipticity property of Definition 3.2 of Volume 1 in ∗ ∂M, with local representative U 0 with U 0 = κ(V 0 ∩ O). Let ω 0 ∈ Tm 0 0 C 0 d−1 0 (ω ) = ξ ∈ R , and τ ≥ 0 such that (ξ 0 , τ 0 ) = 0, and assume that the ˇ Lopatinski˘ı–Sapiro condition of Definition 8.1 holds at (m0 , ω 0 , τ 0 ). Then, there exists U a conic open neighborhood of (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+

ˇ 8.3. CARLEMAN ESTIMATE UNDER THE LOPATINSKI˘I–SAPIRO CONDITION 181

such that for χ ∈ ST0,τ , homogeneous of degree 0, with supp(χ) ⊂ U , there exist C > 0 and τ∗ > 0 such that (8.3.15) τ −1/2 OpT (χ)vτ,2 + | tr(OpT (χ)v)|τ,1,1/2 

≤ C P0,ϕ v+ + |B0,ϕ v|xd =0+ |τ,3/2−k + vτ,2,−1 , for τ ≥ τ∗ , v ∈ S (Rd+ ). For the proof of the proposition, we use some of the notation and techniques introduced in Chapter 3 of Volume 1. In what follows, to ease the notation, we shall write ϕ, p, b, and bϕ in place of ϕC , pC , bC , and bCϕ = b0,ϕ , respectively. With (8.3.16)

P2 =

 1 ∗ ∈ Dτ2 , P0,ϕ + P0,ϕ 2

P1 =

 1 ∗ ∈ τ Dτ1 P0,ϕ − P0,ϕ 2i ∞

see Sections 3.2.1 and 3.4.2), we write, for some w ∈ C c (U 0 ), (8.3.17) P0,ϕ w2+ = Q(w) + 2 Re(P2 w, iP1 w)+ ,

with Q(w) = P2 w2+ + P1 w2+ .

The computations of Section 3.4.2 give, see (3.4.20)–(3.4.21),   ˜ (8.3.18) 2 Re(P2 w, iP1 w)+ = Re i[P2 , P1 ]w, w + + τ Re B(w), with (8.3.19)

˜ B(w) = 2(∂d ϕDd w|xd =0+ , Dd w|xd =0+ )∂ + 2(˜ r(x, D , dx ϕ)w|xd =0+ , Dd w|xd =0+ )∂   + 2 r˜(x, D , dx ϕ)Dd w|xd =0+ , w|xd =0+ ∂     − 2 ∂d ϕ OpT (r) − p(x, τ dϕ) w|xd =0+ , w|xd =0+ ∂ + (Op(c0 )w|xd =0+ , Dd w|xd =0+ )∂    + Op(˜ c0 )Dd + Op(c1 ) w|xd =0+ , w|xd =0+ ∂ ,

c0 ) ∈ D 0 and Op(c1 ) ∈ DT1,τ . We have this estimation for with Op(c0 ), Op(˜ ˜ B(w): (8.3.20)

˜ |B(w)|  | tr(w)|2τ,1,0 .

According to (8.3.10), we face two cases in the proof of Proposition 8.18: either one root is the upper complex half-plane, that is, Im γ2 ≥ 0, implying 0 0 pˇ+ ϕ (m , ω , z) = z − γ2 , or both roots γ1 and γ2 are in the lower complex 0 0 half-plane, implying pˇ+ ϕ (m , ω , z) = 1.

182

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

8.3.4.1. Case 1: One Root in the Upper Complex Half-Plane. Here, we have 0 0 0 0 0 0 pˇ+ ϕ (m , ω , z, τ ) = z − γ2 (x , ξ , τ ),

that is, Im γ1 (x0 , ξ 0 , τ 0 ) < 0 and Im γ2 (x0 , ξ 0 , τ 0 ) ≥ 0. With the Lopatinˇ ski˘ı–Sapiro condition holding at the considered point, by (8.3.11), we have moreover bϕ (x0 , ξ 0 , ξd = γ2 , τ 0 ) = b(x0 , ξ 0 + iτ 0 dx ϕ(x0 ), γ2 + iτ 0 ∂xd ϕ(x0 )) = 0. As the roots γ1 and γ2 are locally smooth with respect to (x, ξ  , τ ) and homogeneous of degree one in (ξ  , τ ) by Proposition 6.28, there exist U a conic open neighborhood of (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ and C, C  > 0 such that SU is compact and γ1 ( ) = γ2 (x, ξ  , τ ), Im γ2 ( ) ≥ −CλT,τ , and Im γ1 ( ) ≤ −C  λT,τ ,

and bϕ (x, ξ  , ξd = γ2 ( ), τ ) = 0,

(8.3.21)

if  = (x, ξ  , τ ) ∈ U . ˜ ∈ ST0,τ be homogeneous of We let χ ∈ ST0,τ be as in the statement and χ degree zero and be such that supp(χ) ˜ ⊂ U and χ ˜ ≡ 1 on supp(χ). From the smoothness and the homogeneity of the roots, we have χγ ˜ j ∈ ST1,τ , j = 1, 2. We set ˜ 2 ) and P − = Dd − OpT (χγ ˜ 1 ). P + = Dd − OpT (χγ Lemma 8.19. There exist C > 0 and τ∗ > 0 such that for any N ∈ N, there exists CN > 0 such that | tr(OpT (χ)v)|τ,1,1/2   ≤ C |B0,ϕ OpT (χ)v|xd =0+ |τ,3/2−k + |P + OpT (χ)v|xd =0+ |τ,1/2 + CN | tr(v)|τ,1,−N , for τ ≥ τ∗ , v ∈ S (Rd+ ). ˇ This lemma is the key point where the Lopatinski˘ı–Sapiro condition for (P, B, ϕ) is used. A proof is given in Appendix 8.A.3. From Lemma 6.20 (with s = 0), we have for τ chosen sufficiently large, (8.3.22) OpT (χ)uτ,1 + |OpT (χ)u|xd =0+ |τ,1/2  P − OpT (χ)u+ + uτ,0,−N , for u ∈ S (Rd+ ). Let now v ∈ S (Rd+ ). We apply estimate (8.3.22) for u = P + v, yielding |OpT (χ)P + v|xd =0+ |τ,1/2  P − OpT (χ)P + v+ + vτ,1,−N  P0,ϕ v+ + vτ,1 ,

ˇ 8.3. CARLEMAN ESTIMATE UNDER THE LOPATINSKI˘I–SAPIRO CONDITION 183 1,0 using that P − OpT (χ)P + = OpT (χ)P − P + mod Ψτ,ph = OpT (χ)P0,ϕ 1,0 mod Ψτ,ph . We set w = OpT (χ)v. We observe that we have

|P + w|xd =0+ |τ,1/2  |OpT (χ)P + v|xd =0+ |τ,1/2 + |v|xd =0+ |τ,1/2  |OpT (χ)P + v|xd =0+ |τ,1/2 + vτ,1 , using the trace inequality of Proposition 6.9. We thus obtain |P + w|xd =0+ |τ,1/2  P0,ϕ v+ + vτ,1 . Together with Lemma 8.19, this yields the estimate (8.3.23)

| tr(w)|τ,1,1/2  |B0,ϕ w|xd =0+ |τ,3/2−k + P0,ϕ v+ + vτ,2,−1 ,

using the trace inequality of Corollary 6.10. By Lemma 8.17, we have τ −1/2 wτ,2  P0,ϕ w+ + | tr(w)|τ,1,1/2 , for τ > 0 chosen sufficiently large. As [P0,ϕ , OpT (χ)] ∈ Ψτ1,0 , we obtain (8.3.24)

τ −1/2 wτ,2  P0,ϕ v+ + v1 + | tr(w)|τ,1,1/2 .

Finally, with (8.3.23) and (8.3.24), we obtain τ

−1/2

wτ,2 + | tr(w)|τ,1,1/2  P0,ϕ v+ + | tr(w)|τ,1,1/2 + v1  P0,ϕ v+ + |B0,ϕ w|xd =0+ |τ,3/2−k + vτ,2,−1 .

As w = OpT (χ)v, with a commutator argument, we have |B0,ϕ w|xd =0+ |τ,3/2−k  |B0,ϕ v|xd =0+ |τ,3/2−k + | tr(v)|τ,1,−1/2  |B0,ϕ v|xd =0+ |τ,3/2−k + vτ,2,−1 , using the trace inequality of Corollary 6.10. We then obtain (8.3.15), which concludes the proof for Case 1.  8.3.4.2. Case 2: Both Roots in the Lower Complex Half-Plane. In this case, we have 0 0 0 ˇϕ (m0 , ω 0 , z, τ 0 ) pˇ− ϕ (m , ω , z, τ ) = p

= pC (x0 , ξ 0 + iτ 0 dx ϕC (x0 ), z + iτ 0 ∂d ϕC (x0 )) = (z − γ1 (x0 , ξ 0 , τ 0 ))(z − γ2 (x0 , ξ 0 , τ 0 )). As the roots γ1 and γ2 depend continuously on the parameters (x, ξ  , τ ), there exists U a conic open neighborhood of (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ such that both Im γ1 < 0 and Im γ2 < 0 if (x, ξ  , τ ) ∈ U . In particular, ∂d ϕC|xd =0+ > 0 in U . Letting χ ∈ ST0,τ be as in the statement, we set w = OpT (χ)v. For ε ∈ (0, 1) to be set below, as Q(w) ≥ 0, from (8.3.17), we write in fact (8.3.25)

P0,ϕ w2+ ≥ εQ(w) + 2 Re(P2 w, iP1 w)+ .

184

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

First, the set K = {(x, ξ, τ ); (x, ξ  , τ ) ∈ U , ξd ∈ R, |ξ|2 + τ 2 = 1} is a compact set recalling that U 0 = κ(V 0 ∩O) is bounded since the manifold is compact here. On K, we have |p(x, ξ + iτ dϕ(x))| ≥ C0 > 0, for (x, ξ, τ ) ∈ K. By homogeneity, we obtain (8.3.26)

|p(x, ξ + iτ dϕ(x))| ≥ C0 λ2τ ,

(x, ξ  , τ ) ∈ U ,

ξd ∈ R.

Denoting by p2 and p1 the (real) principal symbols of P2 and P1 , respectively, we have (8.3.27) (x, ξ  , τ ) ∈ U , ξd ∈ R. q(x, ξ, τ ) = p22 (x, ξ, τ ) + p21 (x, ξ, τ )  λ4τ , As the interior quadratic form Q has q for its principal symbol in the sense of Definition 6.13, by the G˚ arding inequality of Theorem 6.17, we obtain (8.3.28)

Re Q(w) ≥ Cw2τ,2 − C  | tr(w)|2τ,1,1/2 − CN v2τ,2,−N ,

for τ > 0 chosen sufficiently large. Second, as we have [P2 , P1 ] ∈ τ Dτ2 , we have   (8.3.29) | Re i[P2 , P1 ]w, w + |  τ w2τ,2,−1  τ −1 w2τ,2 (see Remark 6.8). Third, we have the following lemma. Lemma 8.20. There exist C, CN > 0 and τ∗ > 0 such that 2 2 ˜ τ Re B(Op T (χ)v) ≥ C| tr(OpT (χ)v)|τ,1,1/2 − CN | tr(v)|τ,1,−N , for τ ≥ τ∗ and v ∈ S (Rd+ ). A proof is given in Appendix 8.A.4. With this lemma, from (8.3.18) and (8.3.29), we obtain (8.3.30) 2 Re(P2 w, iP1 w)+ ≥ C| tr(w)|2τ,1,1/2 − C  τ w2τ,2,−1 − CN | tr(v)|2τ,1,−N  v2τ,2,−N , ≥ C| tr(w)|2τ,1,1/2 − C  τ w2τ,2,−1 − CN

using the trace inequality of Corollary 6.10. With (8.3.25), (8.3.28), and (8.3.30), we obtain   Cεw2τ,2 + (C − C  ε)| tr(w)|2τ,1,1/2 ≤ C  P0,ϕ w2+ + τ w2τ,2,−1 + CN v2τ,2,−N . Choosing τ > 0 sufficiently large and ε > 0 sufficiently small allows one to obtain, for any N ∈ N, w2τ,2 + | tr(w)|2τ,1,1/2  P0,ϕ w2+ + CN v2τ,2,−N . As w = OpT (χ)v, with a commutator argument, we then write P0,ϕ w+  P0,ϕ v+ + vτ,2,−1 ,

ˇ 8.3. CARLEMAN ESTIMATE UNDER THE LOPATINSKI˘I–SAPIRO CONDITION 185

yielding wτ,2 + | tr(w)|τ,1,1/2  P0,ϕ v+ + vτ,2,−1 ,

(8.3.31)



which concludes the proof for Case 2.

Remark 8.21. This is in fact a much better microlocal estimate than the sought estimate of Proposition 8.18. Observe, for instance, that this estimate is elliptic as the OpT (χ)vτ,2 is recovered from P0,ϕ OpT (χ)v+ and that no trace information is needed. Traces are in fact estimated from P0,ϕ OpT (χ)v+ also. The microlocal nature of this estimate is crucial with that respect. A similar estimate without the microlocal cutoff OpT (χ) cannot hold. 8.3.5. Patching Microlocal Estimates Together. In the framework of the normal geodesic coordinates introduced in Sect. 8.3.2, we now deduce from the microlocal estimate of Proposition 8.18 the following local estimate. Proposition 8.22. Under the assumption of Proposition 8.16, there exist a bounded neighborhood U+ of x0 in Rd+ such that U+ ⊂ U 0 and two constants C and τ∗ > 0 such that

 τ −1/2 vτ,2 + | tr(v)|τ,1,1/2 ≤ C P0,ϕ v+ + |B0,ϕ v|xd =0+ |τ,3/2−k , ∞

for all v ∈ C c (U+ ) and τ ≥ τ∗ . With the experience of Chapter 3 of Volume 1, it is now classical to deduce the result of Proposition 8.16 by setting v = eτ ϕ u, with a commutator argument between eτ ϕ and Dd , to prove (8.3.32)

| tr(v)|τ,1,1/2  |e

τ ϕ|x

d =0

+

tr(u)|τ,1,1/2 .

Proof of Proposition 8.22. As was done in Sect. 8.3.4 above, in what follows, to ease the notation, we shall write ϕ and p in place of ϕC and pC , respectively. ˇ With x0 as in the statement of Proposition 8.16, the Lopatinski˘ı–Sapiro 0 0 0 0 ∗ d−1 ∼ condition for (P0 , B0 , ϕ) holds at (x , ξ , τ ) for all ξ ∈ Tx0 R = Rd−1 0 0 0 and τ ≥ 0 such that (ξ , τ ) = 0. It is in fact sufficient to consider (ξ 0 , τ 0 ) in the half-unit sphere Sd−1 + , using the notation introduced in (1.7.3). By Proposition 8.18 for all (ξ 0 , τ 0 ) ∈ Sd−1 + , there exists a conic open 0 0 0 0 neighborhood Uy0 of  = (x , ξ , τ ) in U 0 × Rd−1 × R+ such that the estimate (8.3.15) holds. In fact, by reducing U0 , we can choose U0 = O0 × Γ0 , where O0 is an open set in U 0 and Γ0 is a conic open set in Rd−1 × R+ . With the compactness of Sd−1 + , we can thus find finitely many ⊂ ∪j∈J Γj . We then such open sets Uj = Oj × Γj , j ∈ J, such that Sd−1 + set O = ∩j∈J Oj that is an open neighborhood of x0 in U 0 , and we set Vj = O × Γj ⊂ Uj . We also choose an open neighborhood U of x0 in Rd such that U+ = U ∩ U 0  O.

186

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

We then choose a smooth partition of unity, χj , j ∈ J, of the closed set F = U+ × Sd−1 in the manifold R = Rd × Sd−1 subordinated to the covering + + by the open sets Vj ∩ R according to Theorem 15.14. We then extend each χj smoothly to Rd × Rd−1 × R+ by homogeneity of degree 0 for |(ξ  , τ )| ≥ 1. We have supp(χj ) ⊂ Vj and  χj ( ) = 1, j∈J

conic neighborhood of U+ ×Rd−1 ×R+ and |(ξ  , τ )| ≥ 10. for  = (x, ξ  , τ ) in a  We also set χ = 1 − j∈J χj As supp(χj ) ⊂ Uj , we can apply the microlocal estimate of Proposition 8.18 (8.3.33)

τ −1/2 Op(χj )vτ,2 + | tr(Op(χj )v)|τ,1,1/2  P0,ϕ v+ + |B0,ϕ v|xd =0+ |τ,3/2−k + vτ,2,−1 ,

for τ > 0 chosen sufficiently large and for v = w|Rd with w ∈ Cc∞ (U ). + Observe then that, for any N ∈ N, using the support of v,  OpT (χj )vτ,2 + OpT (χ)vτ,2 vτ,2 ≤ j∈J





j∈J

and | tr(v)|τ,1,1/2 ≤ 

 j∈J



j∈J





j∈J

OpT (χj )vτ,2 + vτ,2,−N ,

| tr(OpT (χj )v)|τ,1,1/2 + | tr(OpT (χ)v)|τ,1,1/2 | tr(OpT (χj )v)|τ,1,1/2 + | tr(v)|τ,1,−N | tr(OpT (χj )v)|τ,1,1/2 + vτ,2,−N .

Summing estimates (8.3.33) for each j ∈ J, we thus obtain τ −1/2 vτ,2 + | tr(v)|τ,1,1/2  P0,ϕ v+ + |B0,ϕ v|xd =0+ |τ,3/2−k + vτ,2,−1 . Choosing now τ > 0 sufficiently large, we obtain the sought estimate.



We observe that the same proof allows us to deduce the following microlocal “semi-global” estimate. Proposition 8.23. Let P0 = Op(pC ), B0 = Op(bC ). Assume that (P0 , ϕC ) has the sub-ellipticity property of Definition 3.2 of Volume 1 in U 0 with U 0 = κ(V 0 ∩ O). Let K be a compact set of U 0 . Let V be a conic set of K × Rd−1 × R+ . Assume that (P0 , B0 , ϕC ) satisfies the Lopatinski˘ı– ˇ Sapiro condition of Definition 8.1 at all (x0 , ξ 0 , τ 0 ) ∈ V ∩ {xd = 0}. Then, there exists U a conic open set of Rd+ × Rd−1 × R+ such that V ⊂ U , and if

ˇ 8.3. CARLEMAN ESTIMATE UNDER THE LOPATINSKI˘I–SAPIRO CONDITION 187

χ ∈ Sτ0 is homogeneous of degree 0 and such that supp(χ) ⊂ U , there exist two constants C and τ∗ > 0 such that (8.3.34) τ −1/2 OpT (χ)vτ,2 + | tr(OpT (χ)v)|τ,1,1/2 

≤ C P0,ϕ v+ + |B0,ϕ v|xd =0+ |τ,3/2−k + vτ,2,−1 , for τ ≥ τ∗ , v ∈ S (Rd+ ). In particular, one can choose χ ≡ 1 in a neighborhood of V . 8.3.6. A Shifted Estimate. With the estimations of Proposition 8.22, we can prove the following estimate, that is closer in form to those proven in Chapter 3 of Volume 1 at boundaries. Theorem 8.24. Under the assumptions of Theorem 8.14, there exist C and τ∗ > 0 such that τ 1/2 eτ ϕ uτ,1 + τ 1/2 |eτ ϕ|∂M tr(u)|τ,1,0

 ≤ C eτ ϕ P uL2 (M) + τ 1/2 |eτ ϕ Bu|k∂M | 1≤k≤β

 τ,1−k

,

for all u ∈ C ∞ (M), with supp(u) ⊂ V , and τ ≥ τ∗ . A proof is given in Appendix 8.A.5. Weaker norms both in the interior of M or at the boundary ∂M are obtained as compared to the estimation in Theorem 8.14. However, the right-hand side of the estimate also shows weaker norms for the boundary operator. Note that τ 1/2 |eτ ϕ|∂M tr(u)|τ,1,0  τ 3/2 |eτ ϕ|∂M u|∂M |L2 (∂M) + τ 1/2 |eτ ϕ|∂M D u|∂M |L2 Λ1 (∂M) + τ 1/2 |eτ ϕ|∂M ∂ν u|∂M |L2 (∂M) , where D denotes the covariant derivative on ∂M. As we also have τ 1/2 eτ ϕ uτ,1  τ 3/2 eτ ϕ uL2 (M) + τ 1/2 eτ ϕ D uL2 Λ1 (M) , we obtain the following estimation: τ 3/2 eτ ϕ uL2 (M) + τ 1/2 eτ ϕ D uL2 Λ1 (M) + τ 1/2 |eτ ϕ|∂M ∂ν u|∂M |L2 (∂M)   eτ ϕ P uL2 (M) + τ 1/2 |eτ ϕ Bu|k∂M | . 1≤k≤β

τ,1−k

Observe that in the case β = 0, that is, B is the Dirichlet boundary operator, we recover precisely the statement of Theorem 5.6.

188

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

8.3.7. A Basic Microlocal Elliptic Estimate. With a microlocalization procedure as in Sect. 8.3.4, we can improve the result of Lemma 8.17 in regions where roots are not real. This result is used in Chapter 14; the reader may thus skip this section. We use the local notation of Sect. 8.3.2. Lemma 8.25. Assume that (P0 , ϕC ) has the sub-ellipticity property of Definition 3.2 of Volume 1 in U 0 with U 0 = κ(V 0 ∩ O). Let χ ∈ ST0,τ be homogeneous of degree 0 and such that Im γ1 and Im γ2 do not vanish in supp(χ). Then, there exist C > 0 and τ∗ > 0 such that (8.3.35)

OpT (χ)vτ,2  P0,ϕ v+ + | tr(OpT (χ)v)|τ,1,1/2 + vτ,2,−1 ,

for τ ≥ τ∗ , v ∈ S (Rd+ ). Proof. We use the notations P1 and P2 as defined in (8.3.16) and denote by p2 and p1 their respective (real) principal symbols. Observe that w → P0,ϕ w2+ is an interior quadratic form in the sense of Definition 6.13, with principal symbol q = p22 + p21 . Having Im γ1 = 0 and Im γ2 = 0 in supp(χ), one obtains (8.3.36)

q(x, ξ, τ )  λ4τ ,

(x, ξ  , τ ) ∈ U ,

for U a neighborhood of supp(χ). Then, with the microlocal G˚ arding inequality of Theorem 6.17, one finds OpT (χ)vτ,2  P0,ϕ OpT (χ)v+ + | tr(OpT (χ)v)|τ,1,1/2 + vτ,2,−N . Finally, with [P0,ϕ , OpT (χ)] ∈ Ψτ1,1 , we obtain the result.



8.4. Estimates Without Any Prescribed Boundary Condition We prove now estimates without assuming any boundary condition. Such a case is considered in Theorem 3.28 of Volume 1 in the case of an open set of Rd and in Theorem 5.5 in the case of a manifold with boundary. In those results, both the Dirichlet and Neumann traces are assumed known and allow one to obtain an estimation. In that case, no particular assumption is made on the weight function at the boundary. Here, we wish to obtain bounds by means of norms on the Dirichlet trace u|∂M and a first-order trace Bu|∂M , that is, where B = ∂ν + B  is a first-order boundary operator. For this to make sense, one assumes ˇ B fulfills the Lopatinski˘ı–Sapiro condition of Definition 2.2 along with P . Such boundary operators are characterized in Proposition 2.8. However, no particular assumption on the weight function is made at the boundary. Naturally, as in the rest of this book, the sub-ellipticity property of the pair (P, ϕ) is assumed. We first prove such an estimate with Sobolev norms as in Theorem 8.14 with an observation of all traces. Second, we improve upon this estimate, removing some the trace observations. Third, we provide a shifted estimate, similar to Theorems 3.28 and 5.5. The improvement made in the second

8.4. ESTIMATES WITHOUT ANY PRESCRIBED BOUNDARY CONDITION

189

step can be important in applications. The microlocal nature of the proof of the estimates in the present chapter is crucial with regard to this refinement. 8.4.1. A First Estimate. Proposition 8.26. Let (M, g) be a smooth compact Riemannian manifold with boundary, and let P = −Δg + R1 with R1 a first-order differential operator with bounded coefficients on M. Let V be an open set of M such that V ∩ ∂M = ∅. Let ϕ ∈ C ∞ (M) be such that the pair (P, ϕ) has the subellipticity property of Definition 5.1 in V . Then, there exist C and τ∗ > 0 such that 

(8.4.1) τ −1/2 eτ ϕ uτ,2 ≤ C eτ ϕ P uL2 (M) + |eτ ϕ tr(u)|τ,1,1/2 , for all u ∈ C ∞ (M), with supp(u) ⊂ V , and τ ≥ τ∗ . This result simply follows from patching estimates in local charts as in Lemma 8.17 together. This estimation is different from its counterparts in Chapter 3 of Volume 1, namely Theorem 3.28 in an open set and Theorem 5.5 on a manifold by the norms that appears on both the left-hand side and the right-hand side of the estimation. There is a shift by a half-tangential derivative. We refer to Sect. 8.3.6 where this issue is discussed. 8.4.2. A Refined Estimate. Let B = ∂ν + B  be a first-order boundˇ ary operator such that (P, B) satisfies the Lopatinski˘ı–Sapiro condition. We prove an estimate with terms involving both the Dirichlet trace u|∂M and the trace Bu|∂M . Theorem 8.27. Let (M, g) be a smooth compact Riemannian manifold with boundary, and let P = −Δg + R1 with R1 a first-order differential operator with bounded coefficients on M. Let V be an open set of M such that V∂ = V ∩∂M = ∅. Let B  be a differential operator of order one on ∂M ˇ such that (P, B) fulfills the Lopatinski˘ı–Sapiro condition of Definition 2.2 on  ∞ V ∩ ∂M for B = ∂ν + B . Let ϕ ∈ C (M) be such that the pair (P, ϕ) has the sub-ellipticity property of Definition 5.1 in V∂ . Then, there exist C and τ∗ > 0 such that (8.4.2) τ −1/2 eτ ϕ uτ,2 + |eτ ϕ tr(u)|τ,1,1/2

 ≤ C eτ ϕ P uL2 (M) + τ 3/2 |eτ ϕ u|∂M |L2 (∂M) + |eτ ϕ Bu|∂M |τ,1/2 , for all u ∈ C ∞ (M), with supp(u) ⊂ V , and τ ≥ τ∗ . Remark 8.28. (1) Note that one does not assume here that (P, B, ϕ) fulfills the Lopatinˇ ski˘ı–Sapiro condition of Definition 8.1, as opposed to Theorem 8.14.

190

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

(2) An important case is naturally B = ∂ν since this operator fulfills ˇ the Lopatinski˘ı–Sapiro condition along with P ; see example 2.5(2). Note that the estimate one obtains in this case is stronger than that of Proposition 8.26 by replacing |eτ ϕ u|∂M |τ,3/2 by τ 3/2 |eτ ϕ u|∂M |L2 (∂M) on the right-hand side. In the framework of the normal geodesic coordinates introduced in Sect. 8.3.2, we prove in fact the following proposition, that is, the counterpart of Proposition 8.22 Proposition 8.29. Let P0 = Op(pC ) and B0 = Op(bC ). Let x0 = κ(m0 ) ˇ be such that x0d = 0. Assume that (P0 , B0 ) satisfies the Lopatinski˘ı–Sapiro condition of Definition 2.2 at x0 . Assume that (P0 , ϕC ) has the sub-ellipticity property of Definition 3.2 of Volume 1 in U 0 with U 0 = κ(V 0 ∩ O). There exist a bounded neighborhood U+ of x0 in Rd+ such that U+ ⊂ U 0 and two constants C and τ∗ > 0 such that τ −1/2 vτ,2 + | tr(v)|τ,1,1/2 

≤ C P0,ϕ v+ + τ 3/2 |v|xd =0+ |∂ + |B0,ϕ v|xd =0+ |τ,1/2 , ∞

for all v ∈ C c (U+ ) and τ ≥ τ∗ . With the experience of Chapter 3 of Volume 1, we deduce, with a commutator argument, a local version of Theorem 8.27 by setting v = eτ ϕ u. Arguing as in Sect. 5.3, for Theorems 5.5 and 5.6, patching together such local estimates, we then obtain the result of Theorem 8.27. ˇ conProof. Let ξ 0 ∈ Rd−1 \ {0}. Observe that the Lopatinski˘ı–Sapiro 0 0 0 dition for (P0 , B0 , ϕ) holds at (x , ξ , τ = 0) for any weight function ϕ, ˇ since the Lopatinski˘ı–Sapiro condition of Definition 8.1 coincides with the ˇ Lopatinski˘ı–Sapiro condition of Definition 2.2 at such a point. The Lopatinˇ ski˘ı–Sapiro condition thus holds on the conic set V = {x0 }×Rd−1 ×{τ = 0}. By Proposition 8.23, there exists U a conic open set of Rd+ × Rd−1 × R+ such that V ⊂ U and for χ ∈ Sτ0 , homogeneous of degree 0 and such that supp(χ) ⊂ U and χ ≡ 1 in a neighborhood of V , we have (8.4.3) τ −1/2 OpT (χ)vτ,2 + | tr(OpT (χ)v)|τ,1,1/2  P0,ϕ v+ + |B0,ϕ v|xd =0+ |τ,1/2 + vτ,2,−1 , for τ > 0 chosen sufficiently large. We set χ ˜ = 1 − χ ∈ Sτ0 . By Lemma 8.17, we have (8.4.4)

˜ τ,2  P0,ϕ OpT (χ)v ˜ + + | tr(OpT (χ)v)| ˜ τ −1/2 OpT (χ)v τ,1,1/2 ,

for τ > 0 chosen sufficiently large.

8.4. ESTIMATES WITHOUT ANY PRESCRIBED BOUNDARY CONDITION

191

Observe that τ −3/2 ΛT,τ OpT (χ) ˜ ∈ Ψ0T since τ  λT,τ in supp(χ). ˜ This yields 3/2

|OpT (χ)v ˜ |xd =0+ |τ,3/2  τ 3/2 |v|xd =0+ |∂ . We thus find | tr(OpT (χ)v)| ˜ ˜ |xd =0+ |τ,3/2 + |Dd OpT (χ)v ˜ |xd =0+ |τ,1/2 τ,1,1/2  |OpT (χ)v  |OpT (χ)v ˜ |xd =0+ |τ,3/2 + |B0,ϕ OpT (χ)v ˜ |xd =0+ |τ,1/2  τ 3/2 |v|xd =0+ |∂ + |B0,ϕ v|xd =0+ |τ,1/2 + |[B0,ϕ , OpT (χ)]v ˜ |xd =0+ |τ,1/2 , ˜ ∈ Ψ0 with the using that B0,ϕ − Dd ∈ Ψ1T,τ . Since τ −1/2 ΛT,τ [B0,ϕ , OpT (χ)] same support argument as above, we obtain 1/2

3/2 | tr(OpT (χ)v)| ˜ |v|xd =0+ |∂ + |B0,ϕ v|xd =0+ |τ,1/2 . τ,1,1/2  τ

From (8.4.4), we find (8.4.5) τ −1/2 OpT (χ)v ˜ τ,2 + | tr(OpT (χ)v)| ˜ τ,1,1/2 ˜ + + τ 3/2 |v|xd =0+ |∂ + |B0,ϕ v|xd =0+ |τ,1/2  P0,ϕ OpT (χ)v  P0,ϕ v+ + τ 3/2 |v|xd =0+ |∂ + |B0,ϕ v|xd =0+ |τ,1/2 + vτ,1 , ˜ ∈ Ψτ1,0 . With (8.4.3) and (8.4.5), one obtains as [P0,ϕ , OpT (χ)] τ −1/2 vτ,2 + | tr(v)|τ,1,1/2  τ −1/2 OpT (χ)vτ,2 + τ −1/2 OpT (χ)v ˜ τ,2 + | tr(OpT (χ)v)|τ,1,1/2 + | tr(OpT (χ)v)| ˜ τ,1,1/2  P0,ϕ v+ + τ 3/2 |v|xd =0+ |∂ + |B0,ϕ v|xd =0+ |τ,1/2 + vτ,2,−1 . We then conclude the proof by taking τ > 0 sufficiently large.



8.4.3. A Shifted Refined Estimate. Theorem 8.30. Under the assumptions of Theorem 8.27, there exist C and τ∗ > 0 such that (8.4.6) τ 1/2 eτ ϕ uτ,1 + τ 1/2 |eτ ϕ tr(u)|τ,1,0 

≤ C eτ ϕ P uL2 (M) + τ 3/2 |eτ ϕ u|∂M |L2 (∂M) + τ 1/2 |eτ ϕ Bu|∂M |L2 (∂M) , for all u ∈ C ∞ (M), with supp(u) ⊂ V , and τ ≥ τ∗ . A comparison with the statement of Theorem 5.5 shows that the trace norms on the right-hand side of the above estimate are weaker. No tangential derivative of the trace is needed here. In fact, this tangential derivative actually appears on the left-hand side of the estimate. A proof of Theorem 8.30 is given in Appendix 8.A.6.

192

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

Remark 8.31. As pointed out in Remark 8.28, an important case is the Neumann boundary operator B = ∂ν . 8.5. Global Estimates ˜ g) be a σ-compact We consider the setting of Sect. 5.4, and we let (M, Riemannian manifold (possibly not compact) with or without boundary and ˜ Let also Γ0 be an open set of M be a bounded connected open set of M. ∂M such that ∂M is smooth in a neighborhood of Γ0 . Note that, whereas ˜ is smooth, we need not assume ∂M to be smooth evthe boundary of M erywhere. Let also ω0 be an open subset of M. As for Theorem 5.8 using Theorem 8.14 (respectively, Theorem 8.24), one can adapt Section 3.6.1 and one obtains the following global estimates. Theorem 8.32 (Global Carleman Estimate—Inner Observation). Let P = P0 + R1 with P0 = −Δg , where R1 is a first-order differential operator ˜ with bounded coefficients. Let also W0 be a neighborhood of ∂M \ Γ0 in M. ∞ Let ω be an open set of M such that ω0  ω. Let ϕ ∈ C (M) be a global weight function adapted to Γ0 and ω0 in the sense of Definition 5.7. Consider B a differential operator of order β in Γ0 . For 0 ≤ k ≤ β, denote by k∂M the union of the connected components of ∂M where B is ˇ of order k. Moreover, assume that (P, B, ϕ) satisfies the Lopatinski˘ı–Sapiro condition of Definition 8.1 at all points m ∈ Γ0 . Then, there exist τ∗ > 0 and C ≥ 0 such that τ −1/2 eτ ϕ uτ,2 + |eτ ϕ|∂M tr(u)|τ,1,1/2   ≤ C eτ ϕ P uL2 (M) + τ 3/2 eτ ϕ uL2 (ω) + |eτ ϕ Bu|Γ0 ∩k∂M | 1≤k≤β

τ,3/2−k

 ,

for τ ≥ τ∗ and u ∈ C ∞ (M) vanishing in W0 ∩M. One also has the estimate τ 1/2 eτ ϕ uτ,1 + τ 1/2 |eτ ϕ|∂M tr(u)|τ,1,0   ≤ C eτ ϕ P uL2 (M) + τ 3/2 eτ ϕ uL2 (ω) + τ 1/2 |eτ ϕ Bu|Γ0 ∩k∂M | 1≤k≤β

τ,1−k

 .

If Γ0 = ∂M, then one can write a corollary in the form of Corollary 5.9 As for Theorem 5.11 using Theorem 8.14 (respectively, 8.24) and Theorem 8.27 (respectively, Theorem 8.30), one can adapt Section 3.6.1 and one obtains the following global estimates. Theorem 8.33 (Global Carleman Estimate—Boundary Observation). Let P = P0 +R1 with P0 = −Δg where R1 is a first-order differential operator with bounded coefficients. Let Γ0 and Γobs be two nonempty open sets of ∂M such that Γobs \ Γ0 = ∅ and such that ∂M is smooth in a neighborhood of ˜ Γ0 ∪ Γobs . Let also W0 be a neighborhood of ∂M \ (Γ0 ∪ Γobs ) in M. ∞ Let ϕ ∈ C (M) be a global weight function adapted to Γ0 in the sense of Definition 5.10.

8.6. NOTES

193

Consider B a differential operator of order β in Γ0 . For 0 ≤ k ≤ β, denote by k∂M the union of the connected components of ∂M where B is ˇ of order k. Moreover, assume that (P, B, ϕ) satisfies the Lopatinski˘ı–Sapiro condition of Definition 8.1 at all points m ∈ Γ0 . ˜ = Consider also B  a first-order differential operator in Γobs and B  ˜ ˇ ∂ν + B , and assume that (P, B) fulfills the Lopatinski˘ı–Sapiro condition of Definition 2.2 on Γobs . Let also ψ0 , ψobs ∈ C ∞ (M) be such that ψ0 |∂M and ψobs |∂M form a partition of unity of F = Γ0 ∪ Γobs \ W 0 associated with the covering by Γ0 and Γobs , that is, supp(ψ0 |∂M ) ⊂ Γ0 , supp(ψobs |∂M ) ⊂ Γobs , and ψ0 |∂M + ψobs |∂M ≡ 1 in a neighborhood of F . Then, there exist τ∗ > 0 and C ≥ 0 such that τ −1/2 eτ ϕ uτ,2 + |eτ ϕ|∂M tr(u)|τ,1,1/2  ˜ |Γ | ≤ C eτ ϕ P uL2 (M) + τ 3/2 |eτ ϕ u|Γobs |L2 (Γ ) + |eτ ϕ ψobs Bu obs τ,1/2 obs   + |eτ ϕ ψ0 Bu|Γ0 ∩k∂M | , τ,3/2−k

1≤k≤β

for τ ≥ τ∗ and u ∈ C ∞ (M) vanishing in W0 ∩ M. One also has the estimate τ 1/2 eτ ϕ uτ,1 + τ 1/2 |eτ ϕ|∂M tr(u)|τ,1,0  ≤ C eτ ϕ P u2L2 (M) + τ 3/2 |eτ ϕ u|Γobs |L2 (Γ

obs )

+ τ 1/2

˜ |Γ | 2 + τ 1/2 |eτ ϕ Bu obs L (Γobs )   , |eτ ϕ Bu|Γ0 ∩k∂M |

1≤k≤β

τ,1−k

for τ ≥ τ∗ and u ∈ C ∞ (M) vanishing in W0 ∩ M. Remark 8.34. The cutoff functions ψ0 and ψobs in the two estimates of Theorem 8.33 are useful for Sobolev norms of noninteger order, as we have not introduced properly Sobolev norms of fractional orders on a manifold with boundary like Γ0 and Γobs . If M is itself a smooth connected compact Riemannian manifold and Γ0 ∪Γobs = ∂M, then one can write a corollary in the form of Corollary 5.12. 8.6. Notes ˇ In Sect. 8.2, we present how Lopatinski˘ı–Sapiro conditions need to be adapted to cover the case of a conjugated operator for the purpose of the derivation of a Carleman estimate. Such extended conditions can be found in the work of D. Tataru [314] and joint work with M. Bellassoued [70]. The proof of the Carleman estimate of Theorem 8.12 under such extended

194

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

ˇ Lopatinski˘ı–Sapiro conditions follows also from these two references. Here, we however make use of estimates for first-order factors as derived in Sect. 6.4. Note that the positivity argument of Sect. 8.3.4.2 exploits the fact that the roots of the symbol, viewed as polynomials in ξd , are microlocally located in the lower complex half-plane. This is generalized to operators of arbitrary order in [70]. The techniques we use yield estimates with optimal spaces for trace terms, for example, a Sobolev norm on ∂M of order 3/2 for the trace u|∂M . Compare with the results in Chapter 3 of Volume 1: there we obtained a Sobolev norm on ∂M of order 1 for the trace u|∂M is obtained. We show in Sect. 8.3.6 that the estimates we find here imply estimates of the form of those obtained in Chapter 3. In Sect. 8.4, with the techniques introduced for the treatment of the ˇ Lopatinski˘ı–Sapiro conditions, we revisit the case where no boundary condition is imposed as we done for instance1 in Theorem 5.5, and we obtain a substantial improvement by removing the term |eτ ϕ D u|∂M |L2 Λ1 (∂M) from the right-hand side of the inequality. This improvement is crucial for the application to the logarithmic stabilization of the wave equation through a boundary damping treated in Chap. 10. As mentioned above, Neumann conditions are a particular cases of Lopatinˇ ski˘ı–Sapiro type conditions. For the derivation of a Carleman estimate for a second-order elliptic operator, their treatment can be found in [219]. The Neumann and Robin conditions for the associated parabolic operator are treated in the work of A. Fursikov and O. Yu. Imanuvilov [156]. Mixed Zaremba type conditions are treated in a joint work with P. Cornilleau [108]. Ventcel boundary condition are treated by R. Buffe [92]. The Lopatinski˘ı– ˇ Sapiro conditions for higher order operators are treated in a joint work with M. Bellassoued [70]. Clamped boundary conditions for the bi-Laplace operator are treated in [215]. For the Lam´e system, O. Yu. Imanuvilov and M. Yamamoto consider in [182] boundary conditions related to the stress associated with the solution. Here, we only treat boundary conditions as given by differential operators, thus local conditions. Nonlocal boundary conditions are also of interest. The case of an integral condition for the Neumann trace can be found in the work of Q. Lu and Z. Yin [240]. The proofs of Carleman estimates for various boundary conditions were followed by the studies of interfaces associated with transmission conditions as in the works of A. Doubova et al. [127], M. Bellassoued [68] extended by Le Rousseau and Robbiano [213, 214], and some joint works with N. Lerner [211] and M. L´eautaud [210]. For transmission problems for higher order

1See Theorem 3.28 of Volume 1 for the counterpart result in an open set of the Euclidean space.

8.A. SOME TECHNICAL PROOFS

195

elliptic operators, we refer to joint work with M. Bellassoued [71]. For transmission problems with regularity as low as Lipschitz in the principal part, we refer to the work of M. Di Cristo et al. [118] in the elliptic case and E. Francini and S. Vessella [151] in the parabolic case. The content of the present book and most of the above references concern scalar operators; estimates for elliptic systems like the Lam´e system for elasticity are derived in the work of M. Bellassoued [67, 69].

Appendix 8.A. Some Technical Proofs 8.A.1. A Norm Computation. Here we prove Lemma 8.10.     We have v  = p + x with x = |u |−2 g∂ (vm , u )g∂ u . As p is orthogonal to   2  2  2 x , we obtain |p |g∂ = |vm |g∂ − |x |g∂ , yielding  2  |u |2g∂ |p |2g∂ = |vm |g∂ |u |2g∂ − (vm , u )2g∂ .

Recalling the definition of u in (8.2.25), we compute  2  2  2  |vm |g∂ − 2dϕ|∂M (m), vm  , |u |2g∂ = |dϕ|∂M (m)|2g∂ + dϕ|∂M (m), vm

and      2 , u )g∂ = dϕ|∂M (m), vm  1 − |vm | g∂ , (vm yielding, after algebraic simplifications,  2   2  2 |g∂ |u |2g∂ − (vm , u )2g∂ = |vm |g∂ |dϕ|∂M (m)|2g∂ − dϕ|∂M (m), vm  . |vm

We thus find |p |2g∂ =

 |2 |dϕ 2  2 |vm |∂M (m)|g∂ − dϕ|∂M (m), vm  g∂ ,  2 |v  |2 − 2dϕ  2 |dϕ|∂M (m)|2g∂ + dϕ|∂M (m), vm |∂M (m), vm  m g∂

and the result follows.



8.A.2. The Classical Carleman Argument Revisited. To prove Lemma 8.17, we start with the following lemma whose proof is given below. Lemma 8.35. Set Q(w) = P2 w2+ + P1 w2+ . There exist C0 , C > 0, μ > 0, and τ∗ > 0 such that   μQ(w) + τ Re i[P2 , P1 ]w, w + ≥ C0 w2τ,2 − C| tr(w)|2τ,1,1/2 , ∞

for τ ≥ τ∗ and w ∈ C c (U 0 ).

196

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

Proof of Lemma 8.17. Let μ > 0 be as given by Lemma 8.35, and let τ ≥ μ. With this lemma, the identities (8.3.17)–(8.3.18), and the estimation (8.3.20), we then write P0,ϕ w2+ ≥ μτ −1 Q(w) + 2 Re(P2 w, iP1 w)+   ˜ = μτ −1 Q(w) + Re i[P2 , P1 ]w, w + + τ Re B(w)   ≥ C0 τ −1 w2τ,2 − C  τ −1 | tr(w)|2τ,1,1/2 + τ | tr(w)|2τ,1,0 ≥ C0 τ −1 w2τ,2 − C  | tr(w)|2τ,1,1/2 , 

which gives (8.3.14). Proof of Lemma 8.35. We have [P2 , P1 ] ∈ τ Dτ2 . Writing     τ Re i[P2 , P1 ]w, w + = Re iτ −1 [P2 , P1 ]w, τ 2 w + ,

we see that this is an interior quadratic form of type (2, 0) (see   Definition 6.13). We thus see that K(w) = μQ(w) + τ Re i[P2 , P1 ]w, w + is also such an interior quadratic form with principal symbol k() = μ|p0,ϕ ()|2 + τ {p2 , p1 }(),

 = (x, ξ, τ ).

The sub-ellipticity property of (P0 , ϕ) yields by Lemma 3.8 of Volume 1 k()  λ4τ ,

 ∈ U 0 × Rd × [1, +∞),

for μ > 0 chosen sufficiently large. The G˚ arding inequality of Theorem 6.17 yields K(w) ≥ Cw2τ,2 − C  | tr(w)|2τ,1,1/2 , for some C, C  > 0 and for τ > 0 chosen sufficiently large (no microlocalization is needed here and the estimation follows from (6.3.5) in Remark 6.18).  8.A.3. Proof of Lemma 8.19. The principal symbol of B0,ϕ (x, D, τ ) is given in (8.3.7) and is homogeneous of degree k in λτ . One sets Bϕj (x, D , τ ) = OpT (bkϕ ), j = k, k − 1, that is,   B0,ϕ (x, D, τ ) = Bϕk (x, D , τ ) − iBϕk−1 (x, D , τ ) Dd + iτ ∂d ϕ(x) ˜ k (x, D , τ ) − iB k−1 (x, D , τ )Dd , =B ϕ

ϕ

with ˜ϕk (x, D , τ ) = Bϕk (x, D , τ ) + τ Bϕk−1 (x, D , τ )∂d ϕ(x) ∈ Dτk . B ˜ϕk is given by ˜bkϕ (x, ξ  , τ ) = bkϕ (x, ξ  , τ ) + τ bk−1 The principal symbol of B ϕ (x, ξ  , τ )∂d ϕ(x) and is homogeneous of degree k in λT,τ . For σ ∈ R, we introduce the operator 

3/2−σ−k k−1 k   1 ˜ Bσϕ = Λ3/2−σ−k Bϕ (x, D , τ )ΛT,τ , Bϕ (x, D , τ ) −iΛT,τ T,τ

8.A. SOME TECHNICAL PROOFS

197

with principal symbol 

˜bk (x, ξ  , τ ) −iλ5/2−σ−k bk−1 (x, ξ  , τ ) . bσϕ (x, ξ  , τ ) = λ3/2−σ−k ϕ ϕ T,τ T,τ We also introduce

 3/2−σ Pσ,+ = −Λ1/2−σ , Op (˜ γ ) Λ 2 T T,τ T,τ

with principal symbol

 3/2−σ . pσ,+ = −λ1/2−σ χγ ˜ λ 2 T,τ T,τ

We set Mσ = (Bσϕ )∗ Bσϕ + (Pσ,+ )∗ Pσ,+ . It is 2 × 2 matrix operator of order  2 3 − 2σ. If V = t (v 0 , v 1 ) ∈ S (Rd−1 ) , we have 2

(Mσ V, V )(L2 (Rd−1 ))2 = |Bσϕ V |2∂ + |Pσ,+ V |∂ . We have the following lemma. SW

Lemma 8.36. Let W be an open conic set of Rd+ × Rd−1 × R+ such that is compact and bϕ (x, ξ  , ξd = γ2 (x, ξ  , τ ), τ ) = 0,

(8.A.1)

(x, ξ  , τ ) ∈ W ,

and such that χ ˜ ≡ 1 on W . Let χ ˆ ∈ ST0,τ be homogeneous of degree zero and be supported in W . There exists C > 0 such that for N ∈ N, there exist CN > 0 and τ∗ > 0 such that ˆ OpT (χ)V ˆ )(L2 (Rd−1 ))2 ≥ C|OpT (χ)V ˆ |2τ,3/2−σ − CN |V |2τ,−N , (Mσ OpT (χ)V,  2 for V = t (v 0 , v 1 ) ∈ S (Rd−1 ) and τ ≥ τ∗ . 2

2

Here |V |2τ,s = |v 0 |τ,s + |v 1 |τ,s . The proof of Lemma 8.36 can be found below. We shall use this lemma in the case σ = 0 with W ⊂ U an open conic neighborhood of supp(χ) where χ ˜ ≡ 1. We choose χ ˆ ∈ ST0,τ satisfying moreover χ ˆ ≡ 1 on supp(χ). Condition (8.A.1) holds by (8.3.21) as W ⊂ U . ˇ Recall that this condition is a direct consequence of the Lopatinski˘ı–Sapiro condition. For u ∈ S (Rd+ ), we have 3/2−k

|B0,ϕ u|xd =0+ |2τ,3/2−k = |ΛT,τ

2

2

B0,ϕ u|xd =0+ |∂ = |B0ϕ U |∂ ,

+ = D − Op (˜ where U = t (u|xd =0+ , Λ−1 d T γ2 ) T,τ Dd u|xd =0+ ). Recalling that P ˜ 2 , we have with γ˜2 = χγ 2

2

1/2

2

|P + u|xd =0+ |τ,1/2 = |ΛT,τ P + u|xd =0+ |∂ = |P0,+ U |∂ . We thus obtain (8.A.2)

2

|B0,ϕ u|xd =0+ |2τ,3/2−k + |P + u|xd =0+ |τ,1/2 = (M0 U, U )L2 (Rd−1 )2 .

198

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

Lemma 8.36, for σ = 0 here, gives ˆ OpT (χ)U ˆ )(L2 (Rd−1 ))2 ≥ C|OpT (χ)U ˆ |2τ,3/2 − CN |U |2τ,−N , (M0 OpT (χ)U, for N ∈ N and τ ≥ 1 chosen sufficiently large. We now set u = OpT (χ)v, that is, U = (OpT (χ)v|xd =0+ , Λ−1 T,τ Dd OpT (χ)v|xd =0+ ). Because of the support conditions of χ ˆ and χ and from pseudo-differential calculus, we find, for any N ∈ N, ˆ OpT (χ)U ˆ )(L2 (Rd−1 ))2 (M0 U, U )(L2 (Rd−1 ))2 ≥ Re(M0 OpT (χ)U, − CN | tr(v)|2τ,1,−N , and ˆ |2τ,3/2 ≥ |U |2τ,3/2 − CN | tr(v)|2τ,1,−N , |OpT (χ)U yielding (M0 U, U )(L2 (Rd−1 ))2 ≥ C|U |2τ,3/2 − CN | tr(v)|2τ,1,−N = C| tr(OpT (χ)v)|2τ,1,1/2 − CN | tr(v)|2τ,1,−N . By (8.A.2), this concludes the proof of Lemma 8.19.



Proof of Lemma 8.36. The matrix operator Mσ has the following 2× 2 matrix principal symbol:    = (x, ξ  , τ ). ( ) = (bσϕ )∗ bσϕ + (pσ,+ )∗ pσ,+ ( ), For z ∈ C2 , we have (( )z, z)C2 = |bσϕ ( )z|2 + |pσ,+ ( )z|2 . Having bσϕ ( )z = pσ,+ ( )z = 0 for z = 0 precisely means  σ  bϕ ( ) det σ,+ = 0, p ( )  k   ˜b ( ) − ibk−1 ( )˜ γ2 . In W , we have χ that is, 0 = λ3−2σ−k ˜ ≡ 1 meaning ϕ ϕ T,τ γ˜2 = γ2 and then bϕ (x, ξ  , ξd = γ2 , τ ) = 0, which is precisely excluded in W by (8.A.1). As a result, by homogeneity and the compactness of SW , we have (8.A.3)

2 (( )z, z)C2 ≥ Cλ3−2σ T,τ |z|C2 ,

for some C > 0 for  ∈ W . The microlocal G˚ arding inequality for systems of Theorem 2.32 of Volume 1 then yields the result.  Remark 8.37. Assume that the boundary operator B continuously depends on a parameter s that lies in a compact set S and such that the ˇ Lopatinski˘ı–Sapiro condition holds for all values of s ∈ S. Then because of the compactness of S and thus of SW × S, the estimate in (8.A.3) holds uniformly with respect to s ∈ S, meaning that the result of Lemma 8.19 can

8.A. SOME TECHNICAL PROOFS

199

be written uniformly with respect to s ∈ S. An inspection of the remainder of the proofs of Propositions 8.18 and 8.12 shows that their statements then follow with constants that are uniform with respect to s ∈ S. 8.A.4. Proof of Lemma 8.20. Setting W = (Λ1T,τ w|xd =0+ , Dd w|xd =0+ ), with (8.3.19), we write (8.A.4)

˜ Re B(w) ≥ 2 Re(OpT (a)W, W )∂ − C| tr(w)|xd =0+ |2τ,1,−1/2

with a = (aij )1≤i,j≤2 given by

  OpT (a11 ) = −Λ−1 T,τ ∂d ϕ|xd =0+ OpT (r) − p(x, τ dϕ) |x

+ d =0

Λ−1 T,τ ,

˜(x, D , dx ϕ)|xd =0+ , OpT (a12 ) = Λ−1 T,τ r OpT (a21 ) = r˜(x, D , dx ϕ)|xd =0+ Λ−1 T,τ , OpT (a22 ) = ∂d ϕ|xd =0+ , with principal parts in ST0,τ given by   −2  a11 0 = −λT,τ ∂d ϕ|xd =0+ r(x, ξ ) − p(x, τ dϕ) |x

+ d =0

,

−1 21 ˜(x, ξ  , dx ϕ)|xd =0+ , a12 0 = a0 = λT,τ r

a22 0 = ∂d ϕ|xd =0+ . We set a0 = (aij 0 )1≤i,j≤2 . It is symmetric. As we have ∂d ϕ|xd =0+ > 0 in U , we thus obtain, for Z ∈ R2 ,   (8.A.5) a0 (x, ξ  , τ )Z, Z R2 ≥ CZ2R2 , (x, ξ  , τ ) ∈ U , τ ∈ [1, +∞), |(ξ  , τ )| ≥ R,

for some C and R positive if and only if det a0 (x, ξ  , τ ) > 0 in U , by compactness and homogeneity. We compute   λ2T,τ det a0 (x, ξ  , τ ) = −(∂d ϕ)2 r(x, ξ  ) − p(x, τ dϕ) |x =0+ d

− r˜2 (x, ξ  , dx ϕ)|xd =0+ . By (8.3.10), we have 0 ≤ Re α < τ ∂d ϕ. By (8.3.12), this reads   (∂d ϕ)2 r(x, ξ  ) − τ 2 r(x, dx ϕ) − τ 2 (∂d ϕ)4 + r˜2 (x, ξ  , dx ϕ) < 0, which is precisely det a0 (x, ξ  , τ ) > 0. By (8.A.4), (8.A.5), and Theorem 2.32, for any N ∈ N, we have ˜ Re B(w) ≥ C| tr(w)|2τ,1,0 − C  | tr(w)|2τ,1,−1/2 − CN | tr(v)|2τ,1,−N . For τ > 0 chosen sufficiently large, we obtain ˜ Re B(w) ≥ C| tr(w)|2 − CN | tr(v)|2 τ,1

τ,1,−N .

Observe that we have, for any N ∈ N,  | tr(v)|τ,1,−N , τ s | tr(w)|τ,1,s ≥ C| tr(w)|τ,1,s+s − CN

200

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS

by the G˚ arding inequality of Theorem 2.29 of Volume 1, since, by (8.3.13),  we have λT,τ  τ in U . We thus obtain the result. 8.A.5. Proof of Theorem 8.24. In the framework of the normal geodesic coordinates introduced in Sect. 8.3.2, we prove in fact the following proposition. Proposition 8.38. Under the assumption of Proposition 8.16, there exist a bounded neighborhood U+ of x0 in Rd+ such that U+ ⊂ U 0 and two constants C and τ∗ > 0 such that 

τ v2τ,1 + τ | tr(v)|2τ,1,0 ≤ C P0,ϕ v2+ + τ |B0,ϕ v|xd =0+ |2τ,1−k , ∞

for all v ∈ C c (U+ ) and τ ≥ τ∗ . With the experience of Chapter 3 of Volume 1, it is now classical to deduce, with a commutator argument, a local version of Theorem 8.24 by setting v = eτ ϕ u. Arguing as in Sect. 5.3, for Theorems 5.5 and 5.6, patching together such local estimates, we then obtain the result of Theorem 8.24. ˜+ be the open set of U 0 given by Proof of Proposition 8.38. Let U Proposition 8.22, and let U+ be a second open set of U 0 such that U+  ˜+ with x0 ⊂ U+ . We choose χ ∈ C ∞ ˜ U c (U+ ) be such that χ ≡ 1 in a neighborhood of U+ . −1/2 We then set w = τ 1/2 χΛT,τ v, and we apply Proposition 8.22 to w: −1/2

2

−1/2

2

(8.A.6) χΛT,τ vτ,2 + τ | tr(χΛT,τ v)|τ,1,1/2 −1/2

2

−1/2

2

 τ P0,ϕ χΛT,τ v+ + τ |B0,ϕ χΛT,τ v|xd =0+ |τ,3/2−k . We then write, with Corollary 6.7, −1/2

vτ,2,−1/2 = χvτ,2,−1/2 = ΛT,τ χvτ,2 −1/2

−1/2

 χΛT,τ vτ,2 + [ΛT,τ , χ]vτ,2 −1/2

 χΛT,τ vτ,2 + vτ,2,−3/2 . Thus, for τ > 0 chosen sufficiently large, we obtain −1/2

vτ,2,−1/2  χΛT,τ vτ,2 , which yields (8.A.7)

−1/2

τ 1/2 vτ,1  vτ,1,1/2  vτ,2,−1/2  χΛT,τ vτ,2 .

8.A. SOME TECHNICAL PROOFS −1/2

k,−3/2

We now observe that [Ddk χ, ΛT,τ ] ∈ Ψτ

201

, for k = 0, 1, which gives

−1/2

| tr(v)|τ,1,0 = |ΛT,τ tr(χv)|τ,1,1/2 −1/2

−1/2

 |ΛT,τ χv|xd =0+ |τ,3/2 + |ΛT,τ Dxd (χv)|xd =0+ |τ,1/2 −1/2

−1/2

 |χΛT,τ v|xd =0+ |τ,3/2 + |Dxd (χΛT,τ v)|xd =0+ |τ,1/2 −1/2

−1/2

+ |[ΛT,τ , χ]v|xd =0+ |τ,3/2 + |[ΛT,τ , Dxd χ]v|xd =0+ |τ,1/2 −1/2

 | tr(χΛT,τ v)|τ,1,1/2 + | tr(v)|τ,1,−1 . Thus, for τ > 0 chosen sufficiently large, we obtain −1/2

| tr(v)|τ,1,0  | tr(χΛT,τ v)|τ,1,1/2 .

(8.A.8) −1/2

1,−1/2

As τ 1/2 [P0,ϕ , χΛT,τ ] ∈ τ 1/2 Ψτ

⊂ Ψτ1,0 , we also have

(8.A.9) −1/2 −1/2 −1/2 τ 1/2 P0,ϕ χΛT,τ v+  τ 1/2 χΛT,τ P0,ϕ v+ + τ 1/2 [P0,ϕ , χΛT,τ ]v+  P0,ϕ v+ + vτ,1 . −1/2

−3/2

Finally, we observe that [B0,ϕ χ, ΛT,τ ] ∈ ΨT,τ −1/2

1,k−5/2

−1/2

if k = 0 and [B0,ϕ χ, ΛT,τ ] ∈

1,k−5/2

Ψτ if k ≥ 1; thus, [B0,ϕ χ, ΛT,τ ] ∈ Ψτ for any value of k. We then find (8.A.10) −1/2 |B0,ϕ χΛT,τ v|xd =0+ |τ,3/2−k  |B0,ϕ v|xd =0+ |τ,1−k −1/2

+ |[B0,ϕ χ, ΛT,τ ]v|xd =0+ |τ,3/2−k  |B0,ϕ v|xd =0+ |τ,1−k + | tr(v)|xd =0+ |τ,1,−1 . With (8.A.6) and (8.A.7)–(8.A.10), we obtain τ v2τ,1 + τ | tr(v)|2τ,1,0  P0,ϕ v2+ + τ |B0,ϕ v|xd =0+ |2τ,1−k + v2τ,1 + τ | tr(v)|xd =0+ |2τ,1,−1 . We then obtain the result by choosing τ > 0 sufficiently large.



8.A.6. Proof of Theorem 8.30. In the framework of the normal geodesic coordinates introduced in Sect. 8.3.2, we prove in fact the following proposition. Proposition 8.39. Under the assumption of Proposition 8.29, there exist a bounded neighborhood U+ of x0 in Rd+ such that U+ ⊂ U 0 and two constants C and τ∗ > 0 such that

τ 1/2 vτ,1 + τ 1/2 | tr(v)|τ,1,0 ≤ C P0,ϕ v+ + τ 3/2 |v|xd =0+ |∂  + τ 1/2 |B0,ϕ v|xd =0+ |∂ ,

202

8. ESTIMATES FOR GENERAL BOUNDARY CONDITIONS ∞

for all v ∈ C c (U+ ) and τ ≥ τ∗ . With the experience of Chapter 3 of Volume 1 we deduce, with a commutator argument, a local version of Theorem 8.30 by setting v = eτ ϕ u. Arguing as in Sect. 5.3, for Theorems 5.5 and 5.6, patching together such local estimates, we then obtain the result of Theorem 8.30. ˜+ be the open set of U 0 given by Proof of Proposition 8.39. Let U Proposition 8.29, and let U+ be a second open set of U 0 such that U+  ˜ ˜+ with x0 ⊂ U+ . We choose χ ∈ C ∞ U c (U+ ) be such that χ ≡ 1 in a neighborhood of U+ . −1/2 We then set w = τ 1/2 χΛT,τ v, and we apply Proposition 8.29 to w: (8.A.11)

−1/2

−1/2

χΛT,τ vτ,2 + τ 1/2 |χΛT,τ v|xd =0+ |τ,3/2 −1/2

−1/2

 τ 1/2 P0,ϕ χΛT,τ v+ + τ 2 |χΛT,τ v|xd =0+ |∂ −1/2

+ τ 1/2 |B0,ϕ (χΛT,τ v)|xd =0+ |τ,1/2 . By (8.A.7), we have, for τ > 0 sufficiently large, −1/2

τ 1/2 vτ,1  χΛT,τ vτ,2 .

(8.A.12)

−1/2

−3/2

We now observe that [χ, ΛT,τ ] ∈ ΨT,τ , which gives −1/2

|v|xd =0+ |τ,1 = |ΛT,τ χv|xd =0+ |τ,3/2 −1/2

−1/2

 |χΛT,τ v|xd =0+ |τ,3/2 + |[ΛT,τ , χ]v|xd =0+ |τ,3/2 −1/2

 |χΛT,τ v|xd =0+ |τ,3/2 + |v|xd =0+ |∂ . Thus, for τ > 0 chosen sufficiently large, we obtain −1/2

|v|xd =0+ |τ,1  |χΛT,τ v|xd =0+ |τ,3/2 .

(8.A.13) By (8.A.9), we have (8.A.14)

−1/2

τ 1/2 P0,ϕ χΛT,τ v+  P0,ϕ v+ + vτ,1 . −1/2

Finally, as τ 1/2 χΛT,τ

−1/2

1,−3/2

∈ Ψ0T,τ and [B0,ϕ , χΛT,τ ] ∈ Ψτ

, we have

(8.A.15) −1/2 −1/2 τ 2 |χΛT,τ v|xd =0+ |∂ + τ 1/2 |B0,ϕ (χΛT,τ v)|xd =0+ |τ,1/2  τ 3/2 |v|xd =0+ |∂ + τ 1/2 |B0,ϕ v|xd =0+ |∂ + τ 1/2 |Dd v|xd =0+ |τ,−1  τ 3/2 |v|xd =0+ |∂ + τ 1/2 |B0,ϕ v|xd =0+ |∂ ,

8.A. SOME TECHNICAL PROOFS

203

using that B0,ϕ − Dd ∈ Ψ1T,τ . With (8.A.11) and (8.A.12)–(8.A.15), we obtain τ 1/2 vτ,1 + τ 1/2 |v|xd =0+ |τ,1  P0,ϕ v+ + τ 3/2 |v|xd =0+ |∂ + τ 1/2 |B0,ϕ v|xd =0+ |∂ + v2τ,1 . By choosing τ > 0 sufficiently large, we find τ 1/2 vτ,1 + τ 1/2 |v|xd =0+ |τ,1  P0,ϕ v+ + τ 3/2 |v|xd =0+ |∂ + τ 1/2 |B0,ϕ v|xd =0+ |∂ . Finally, we write |Dd v|xd =0+ |∂ ≤ |v|xd =0+ |τ,1 + |B0,ϕ v|xd =0+ |∂ , and we obtain the sought estimate.



Part 3

Applications

CHAPTER 9

Quantified Unique Continuation on a Riemannian Manifold Contents 9.1. 9.2. 9.3. 9.4.

Unique Continuation Estimate Away from Boundaries Unique Continuation Estimates Up to Boundaries Boundary Initiated Unique Continuation: Improved Estimates Notes

208 210 218 219

In Chapter 5 of Volume 1, we saw how the Carleman estimates proven in Chapter 3 (also in Volume 1) for a second-order elliptic operator on an open set Ω of Rd could be used to provide unique continuation results. Moreover, a quantification of the unique continuation property could be expressed through an interpolation inequality of the form, for some δ ∈ (0, 1),  δ uH 1 (U ) ≤ Cu1−δ H 1 (Ω) f L2 (Ω) + uL2 (ω) , where U and ω are open subsets of an open set Ω of Rd and where u ∈ H 2 (Ω) satisfies P u = f + g, with f ∈ L2 (Ω),   |g(x)| ≤ C0 |u(x)| + |Du(x)| a.e. in Ω,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 9

207

208

9. QUANTIFIED UNIQUE CONTINUATION ON A RIEMANNIAN MANIFOLD

along with homogeneous Dirichlet boundary conditions. With the Carleman estimates of Chap. 5 obtained for a Riemannian manifold, away from and also at boundaries, the result of Chapter 5 can be extended to the manifold case along with homogeneous Dirichlet boundary conditions. The purpose of the present chapter is to generalize these results to the case of the general boundary conditions treated in Chaps. 2, 3, and 8, namely Lopatinˇ ski˘ı–Sapiro type boundary conditions. We shall consider (M, g) a d-dimensional smooth Riemannian manifold, possibly noncompact. On M, we consider the operator P = −Δg + R1 with R1 a first-order differential operator on M. When needed, we shall also consider a differential operator B of order β in a neighborhood of part of ∂M in the form introduced in Chap. 2. 9.1. Unique Continuation Estimate Away from Boundaries We consider U and V two open sets of M such that U is connected, U ∩ ∂M = ∅, and U  V  M; see Fig. 9.1. The following result, away from boundaries, is deduced from the results of Section 5.3.1 of Volume 1. Theorem 9.1 (Propagation of “smallness” Away from Boundaries). Let ω be an open subset of U . Let C0 > 0. There exist C > 0 and δ ∈ (0, 1) such that  δ uH 1 (U ) ≤ Cu1−δ (9.1.1) + u , f  2 2 1 L (V) L (ω) H (V) for u ∈ H 2 (V) satisfying P u = f +  with f ∈ L2 (V) and   |(m)| ≤ C0 |u(m)| + | D u(m)|g a.e. in V. ∂M M

ω m

∂M

0

γ m∗

O1

O0

On−1 U

On

V

Figure 9.1. Propagation of smallness on a Riemannian manifold away from the boundary

9.1. UNIQUE CONTINUATION ESTIMATE AWAY FROM BOUNDARIES

209

Here, D denotes the Levi-Civita connection; see Sect. 17.4. Proof. We may assume that f L2 (V) ≤ uH 1 (V) , since otherwise the estimate we wish to prove is obvious. Because of the compactness of U , it suffices to prove that for any m∗ ∈ U , there exist a neighborhood W of m∗ and δ ∈ (0, 1) such that  δ uH 1 (W )  u1−δ H 1 (V) f L2 (V) + uL2 (ω) . As U is connected, there exists a continuous path γ : [0, 1] → U such that γ(0) = m0 ∈ ω and γ(1) = m∗ . Because of the compactness of γ([0, 1]), we ˜ j , κj ), such that can pick a finite number of local charts (C j )0≤j≤n , C j = (O j j j ˜ ˜ ˜ γ([0, 1]) ⊂ ∪0≤j≤n O . Replacing O by O ∩ V if needed, we may assume ˜ j ⊂ V. With Lemma 15.13, there exists an open set Oj such that that O j ˜ j , j = 0, . . . , n, and γ([0, 1]) ⊂ ∪0≤j≤n Oj . We order the local charts O O to have m0 ∈ O0 , m∗ ∈ On , and Oj ∩ Oj−1 = ∅, j = 1, . . . , n. See Fig. 9.1 for an illustration of this setting. ˜ 0 ) and ω 0 = In the local chart C 0 , we set V 0 = κ0 (O0 ), V˜ 0 = κ0 (O 0 ˜ 0 ). In V˜ 0 , the local representative uC of u is in H 2 and satisfies κ0 (ω ∩ O P C uC = f C + C , 0

0

0

0

where P C , f C , and C denote the local representatives of P , f , and , respectively, and we have  0  0 0 0 f C ∈ L2 (V˜ 0 ), |C (x)|  |uC (x)| + |duC (x)| a.e. in V˜ 0 . 0

0

0

Then, by Theorem 5.6 of Volume 1, we have, for some δ0 ∈ (0, 1),  δ 0 0 1−δ0 0 0 uC H 1 (V 0 )  uC H 1 (V˜ 0 ) f C L2 (V˜ 0 ) + uC L2 (ω0 ) 0 , which gives, using the assumption made at the beginning of the proof,  δ0 0 uH 1 (O0 ) + f L2 (V)  u1−δ . H 1 (V) f L2 (V) + uL2 (O0 ∩ω) Similarly, for j = 1, . . . , n, we have, for some δj ∈ (0, 1), δ 1−δj  f L2 (V) + uL2 (Oj ∩Oj−1 ) j . uH 1 (Oj ) + f L2 (V)  uH 1 (V) By induction, we then obtain  δ uH 1 (On ) + f L2 (V)  u1−δ H 1 (V) f L2 (V) + uL2 (ω) ,

with δ = 0≤j≤n δj .



210

9. QUANTIFIED UNIQUE CONTINUATION ON A RIEMANNIAN MANIFOLD

9.2. Unique Continuation Estimates Up to Boundaries We now consider U and V two open sets of M such that U is connected, and U  V  M (if M is connected and compact, we can choose U = V = M). Here, we require U ∩ ∂M = ∅ as opposed to what was assumed in Sect. 9.1. We first place ourselves at a point m0 of ∂M ∩ U , and we prove how smallness can be propagated from regions of V away from the boundary up to a neighborhood of m0 . At the boundary, we consider Lopatinski˘ı– ˇ Sapiro type boundary conditions and the result we obtain follows from the Carleman estimate proven in Chap. 8. The boundary operator B we consider is of order β ∈ N, of the form introduced in Chap. 2, that is, B = B k + B k−1 ∂ν ,

0 ≤ k ≤ β,

on every connected component N of ∂M, with B k and B k−1 smooth differential operators on N of order k and k − 1, respectively. If B is of order zero on N , one has B k−1 = 0. For each k ∈ {0, . . . , β}, we denote by k∂M the union of all the connected components of ∂M where the order of B is exactly k. For Uˆ an open set of M, we define the following open set: (9.2.1) Uˆε = {m ∈ Uˆ; distg (m, ∂M) > ε}, which is not empty for ε > 0 chosen sufficiently small. Here, distg (., .) denotes the Riemannian distance (see Sect. 17.1). Lemma 9.2 (Local Interpolation Inequality at a Boundary). Let k ∈ {0, . . . , β}, and assume that V ∩ ∂M ⊂ k∂M. Let Uˆ be an open set of M ˇ such that U  Uˆ  V. The Lopatinski˘ı–Sapiro condition of Definition 2.2 is assumed to hold for (P, B) in V ∩ ∂M. Let m0 ∈ U ∩ ∂M, C0 > 0, and χU ∈ C ∞ (V ∩ ∂M) such that χU ≡ 1 in a neighborhood of U ∩ ∂M. There exist W an open neighborhood of m0 in V, ε ∈ (0, 1), δ ∈ (0, 1), and C > 0, such that we have

δ (9.2.2) uH 1 (W ) ≤ Cu1−δ + |χ h| + u f  k 2 1−k 1 1 U ˆ L (V) H ( ∂M) H (V) H (U ε ) , for u ∈ H 2 (V) satisfying P u = f +  with f ∈ L2 (V) and   |(m)| ≤ C0 |u(m)| + | D u(m)|g a.e. in V, and Bu|V∩∂M = h|V∩∂M + ∂ with h ∈ H 3/2−k (k∂M) and ⎧ ⎪ ⎨∂ ≡ 0 a.e. in V ∩ ∂M |∂ (m)| ≤ C0 |u(m)| a.e. in V ∩ ∂M   ⎪ ⎩ |∂ (m)| ≤ C0 |u(m)| + | D u(m)|g a.e. in V ∩ ∂M

if k = 0, if k = 1, if k ≥ 2.

The proof is evidently very similar to that of Lemma 5.10 of Volume 1.

9.2. UNIQUE CONTINUATION ESTIMATES UP TO BOUNDARIES

211

Remark 9.3. The proof is based on the Carleman estimate of Theorem 8.24 itself a consequence of Proposition 8.12 and Theorem 8.14. By Remark 8.13, if the boundary operator B continuously depends on a paˇ rameter s that lies in a compact set S and such that the Lopatinski˘ı–Sapiro condition holds for all values of s ∈ S, then all the Carleman estimates derived in Chap. 8 under these boundary conditions hold with constants that are uniform with respect to s ∈ S. Note that the order of the operator should however remain constant as s varies. Consequently, the quantified unique continuation result of Lemma 9.2 also follows with constants that uniform with respect to s ∈ S. The same holds for the results in Theorems 9.4 and 9.6 below. Proof. With the same argument as in the beginning of the proof of Proposition 5.1 of Volume 1, we may simply assume P = P0 = −Δg . We use a local chart C = (O, κ) such that m0 ∈ O ⊂ Uˆ, χU ≡ 1 in a neighborhood of O and such that the associated local coordinates given by κ are normal geodesic coordinates; see Sections 9.4. With the coordinates x = (x , xd ) = κ(m), the boundary κ(O ∩ ∂M) is given by κ(O) ∩ {xd = 0} and κ(O) ⊂ {xd ≥ 0}. We assume moreover that κ(m0 ) = 0. In this chart, a local representative of the metric that satisfies C C (x) = 1, and gdj (x) = 0 for j = 1, . . . , d − 1, gdd

for x ∈ κ(O). In what follows, to ease the notation, we shall write ϕ, p, and g in place of ϕC , pC , and g C , respectively. Let r > 0 be chosen such that B(x(1) , 4r)∩Rd+  κ(O) with x(1) = (0, 2r). We then set ϕ(x) = e−γ|x−x

(1) |2

.

By Lemma 3.5 of Volume 1, for γ > 0 chosen sufficiently large, the subellipticity condition of Definition 3.2 of Volume 1 holds also in V 0 , open set in O, given by κ(V 0 ) = {x ∈ Rd+ ; r/2 < |x − x(1) | < 4r}. We fix γ equal to this value. Observe that ∂ν ϕ(0) = −∂d ϕ(0) = −4γrϕ(0) < 0. If B is of order k = 0, ˇ then the Lopatinski˘ı–Sapiro condition of Definition 8.1 holds for (P, B, ϕ) in a neighborhood of x = κ(m0 ) = 0 by Theorem 8.7-(2a). If k ≥ 1, let μ0 be as given by Theorem 8.7-(2b). Observe that ∇x ϕ(0) = 0 implying that condition (8.2.6) holds in a neighborhood of x = κ(m0 ) = 0. In both cases, k = 0 or k ≥ 1, there exists an open neighborhood V 1 ⊂ V 0 of m0 such that ˇ the Lopatinski˘ı–Sapiro condition of Definition 8.1 holds for (P, B, ϕ) at all 1 1 points of V∂ = V ∩ ∂M. In V 1 , we may thus apply the Carleman estimate of Theorem 8.24. Let r0 > 0 such that r0 < r/4 to be chosen below, and let χ0 ∈ Cc∞ (R) be such that  1 if |s| < r0 , χ0 (s) = 0 if 2r0 < |s|.

212

9. QUANTIFIED UNIQUE CONTINUATION ON A RIEMANNIAN MANIFOLD

Let also χ1 ∈ Cc∞ (B(x(1) , 4r)) be such that  1 if 3r/4 < |x − x(1) | < r1 , χ1 (x) = 0 if |x − x(1) | < 5r/8 or r1 < |x − x(1) |, where r1 and r1 are such that 2r < r1 < r1 < 3r and will be chosen below. We set χ(x) = χ0 (xd )χ1 (x). As we have supp(χ) ⊂ {x ∈ Rd+ ; xd ≤ 2r0 } ∩ {x ∈ Rd+ ; |x − x(1) | ≤ r1 }. Upon choosing r0 > 0 and r1 −2r > 0 sufficiently small, we obtain supp(χ) ⊂ κ(V 1 ). Below, we shall also write χ in place of χ ◦ κ−1 ∈ C ∞ (M) with support in V 1 as no confusion should arise. The geometry associated with the functions we have just introduced is illustrated in Fig. 9.2. Note that we have chosen the notation to match that of the proof of Lemma 5.10 of Volume 1. Figure 9.2 is naturally a small variation of Figure 5.5. x ∈ Rd−1

κ(O) κ(V 0 )

0

κ(m ) = 0

r0 2r0 κ(V 1 )

3r/4 5r/8 r/2 x(1)

xd

2r r1

r1 3r 4r

κ(∂M ∩ O)

Figure 9.2. Geometry near the boundary for the application of the local Carleman estimate of Theorem 8.24 in the local chart C = (O, κ)

9.2. UNIQUE CONTINUATION ESTIMATES UP TO BOUNDARIES

213

Below we shall apply the Carleman estimate of Theorem 8.24 in V 1 neighborhood of m0 as given above to the function v = χu. In both inequalities, we find the term eτ ϕ P vL2 (M) that we estimate as follows: eτ ϕ P vL2 (M)  eτ ϕ χf L2 (V) + eτ ϕ χL2 (V) + eτ ϕ [P, χ]uL2 (V) . By the assumption made on , we obtain eτ ϕ χL2 (V)  eτ ϕ χuL2 (V) + eτ ϕ χ D uL2 Λ1 (V)  eτ ϕ vL2 (V) + eτ ϕ D vL2 Λ1 (V) + eτ ϕ [D, χ]uL2 (V) . The operators [P, χ] and [D, χ] are differential operators of order less than or equal to 1 with bounded coefficients and supported in supp(dχ) ⊂ A0 ∪ A1 with κ(A0 ) = {x ∈ Rd+ ; r0 ≤ xd ≤ 2r0 and |x − x(1) | ≤ r1 }, κ(A1 ) = {x ∈ Rd+ ; 0 ≤ xd ≤ 2r0 and r1 ≤ |x − x(1) | ≤ r1 }. We thus obtain (9.2.3)

eτ ϕ P vL2 (V)  eτ ϕ χf L2 (V) + eτ ϕ vL2 (V) + eτ ϕ D vL2 Λ1 (V) + eτ ϕ uL2 (A0 ∪A1 ) + eτ ϕ D uL2 Λ1 (A0 ∪A1 ) .

We set (9.2.4)

C1 = e−γr1 ,

C2 = e−γ(r+r1 /2) ,

2

C3 = e−γ(2r−2r0 ) .

2

2

We have 0 < C1 < C2 < C3 . We also pick C1 such that C1 < C1 < C2 . We have sup ϕ ≤ C1 ,

(9.2.5)

A1

C2 ≤ inf ϕ, W

sup ϕ ≤ C3 , supp(χ)

where the open set W in O is given by (9.2.6) κ(W ) = {x ∈ Rd+ ; 0 ≤ xd ≤ r0 } ∩ {x ∈ Rd+ ; |x − x(1) | < r + r1 /2}. This is an open neighborhood of m0 = κ−1 (0) in U as 2r < r + r1 /2. Since r + r1 /2 < r1 and xd < r0 if x ∈ κ(W ), we have χ = χ0 χ1 ≡ 1, and thus u ≡ v in W . We apply Theorem 8.24 to v = χu in V 1 , that is, (9.2.7)

τ 1/2 eτ ϕ vτ,1 + τ 1/2 |eτ ϕ|∂M tr(v)|τ,1,0  eτ ϕ P vL2 (M) + τ 1/2 |eτ ϕ Bv |∂M |τ,1−k ,

for τ ≥ τ∗ for some τ∗ > 0. We write |eτ ϕ Bv |∂M |τ,1−k  |eτ ϕ χBu|∂M |τ,1−k + |eτ ϕ [B, χ]u|∂M |  |eτ ϕ χ|∂M h|τ,1−k + |eτ ϕ χ|∂M ∂ |τ,1−k + |eτ ϕ [B, χ]u|∂M | Note that

τ,1−k

.

τ,1−k

214

9. QUANTIFIED UNIQUE CONTINUATION ON A RIEMANNIAN MANIFOLD

· [B, χ] vanishes if k = 0. · [B, χ]|∂M is a smooth function supported in A1 ∩ ∂M for k = 1. · [B, χ] = B k−1 +B k−2 ∂ν with B j ∈ D j (∂M) supported in supp(χ) ⊂ A1 ∩ ∂M, j = k − 1, k − 2, for k ≥ 2. Let χ ˜ ∈ C ∞ (∂M) be such that supp(χ) ˜ ⊂ A1 ∩∂M and χ ˜ ≡ 1 on supp(dχ)∩ ˜ and B k−2 = B k−2 χ. ˜ We thus find ∂M. One has B k−1 = B k−1 χ τ 1/2 |eτ ϕ [B, χ]u|∂M |

τ,1−k

 kτ 1/2 |eτ ϕ u|A1 ∩∂M |L2 (∂M) + k(k − 1)τ 1/2 |eτ ϕ B k−1 χu ˜ |∂M |τ,1−k + k(k − 1)τ 1/2 |eτ ϕ B k−2 χ∂ ˜ ν u|∂M |τ,1−k  τ 1/2 |eτ ϕ u|A1 ∩∂M |L2 (∂M) + |eτ ϕ χ∂ ˜ ν u|∂M |τ,−1/2 .

In the cases k = 0, 1, by the assumption made on ∂ , one has τ 1/2 |eτ ϕ χ|∂M ∂ |τ,1−k  kτ 1/2 |eτ ϕ v|∂M |L2 (∂M) . In the case k ≥ 2, since 3/2 − k ≤ 0, we write τ 1/2 |eτ ϕ χ|∂M ∂ |τ,1−k  |eτ ϕ χ|∂M ∂ |τ,3/2−k  |eτ ϕ χ|∂M ∂ |L2 (∂M)  |eτ ϕ v|∂M |L2 (∂M) + |eτ ϕ | D v|g |∂M |

L2 (∂M)

+ |e [D, χ]u|∂M |L2 (∂M) τϕ

 |eτ ϕ v|∂M |L2 (∂M) + |eτ ϕ | D v|g |∂M |

L2 (∂M)

+ |eτ ϕ u|A1 ∩∂M |L2 (∂M) . We thus obtain τ 1/2 |eτ ϕ Bv |∂M |τ,1−k  τ 1/2 |eτ ϕ χ|∂M h|τ,1−k + τ 1/2 |eτ ϕ u|A1 ∩∂M |L2 (∂M) + |eτ ϕ χ∂ ˜ ν u|∂M |τ,−1/2 + τ 1/2 |eτ ϕ v|∂M |L2 (∂M) + |eτ ϕ | D v|g |∂M |

L2 (∂M)

.

With (9.2.3) and (9.2.7), for τ > 0 chosen sufficiently large, we thus obtain τ 1/2 eτ ϕ vτ,1 + τ 1/2 |eτ ϕ|∂M tr(v)|τ,1,0  eτ ϕ χf L2 (V) + τ 1/2 |eτ ϕ χ|∂M h|τ,1−k + eτ ϕ uL2 (A0 ∪A1 ) + eτ ϕ D uL2 Λ1 (A0 ∪A1 ) + τ 1/2 |eτ ϕ u|A1 ∩∂M |L2 (∂M) + |eτ ϕ χ∂ ˜ ν u|∂M |τ,−1/2 .

9.2. UNIQUE CONTINUATION ESTIMATES UP TO BOUNDARIES

215

With the constant C3 defined in (9.2.4)–(9.2.5), we write, by Proposition 7.3, τ 1/2 |eτ ϕ χ|∂M h|τ,1−k  τ 1/2 e

τ supsupp(χ

|∂M )

ϕ

|χ|∂M h|τ,1−k

 eτ C3 |χ|∂M h|H 1−k (∂M) . With the constant C1 also defined in (9.2.4)–(9.2.5) and Proposition 7.3, we obtain (9.2.8) τ 1/2 eτ ϕ vτ,1 + τ 1/2 |eτ ϕ|∂M tr(v)|τ,1,0    eτ C3 f L2 (V) + |χU h|H 1−k (∂M) + uH 1 (Uˆε )   + eτ C1 uH 1 (V) + |u|A1 ∩∂M |L2 (U ∩∂M) + |χ∂ ˜ ν u|∂M |τ,−1/2 , where 0 < ε < r0 , implying A0 ⊂ Uˆε , recalling the definition of the set Uˆε in (9.2.1). Let W ⊂ V be a smooth bounded open set1 such that U ∩ ∂M  ∂W ∩ ∂M. Since u|W ∈ H 2 (W), by Lemma 18.42, we have |χ∂ ˜ ν u|∂M |H −1/2 (∂M)  |∂ν u|∂W |H −1/2 (∂W)  uH 1 (W) + P uL2 (W)  uH 1 (V) + f L2 (V) , using that P u = f + and the assumed pointwise estimation of . From (9.2.8), we thus obtain, as C1 < C3 , τ 1/2 eτ ϕ vτ,1 + τ 1/2 |eτ ϕ|∂M tr(v)|τ,1,0    eτ C3 f L2 (V) + |χU h|H 1−k (∂M) + uH 1 (Uˆε )   + eτ C1 uH 1 (V) + |u|A1 ∩∂M |L2 (U ∩∂M) .     eτ C3 f L2 (V) + |χU h|H 1−k (∂M) + uH 1 (Uˆε ) + eτ C1 uH 1 (V) , using the trace inequality 18.24 of Proposition 18.24. If we restrict the first term of the l.h.s. of (9.2.8) to W defined in (9.2.6), as on this set we have ϕ ≥ C2 , we obtain τ 1/2 eτ ϕ vτ,1  τ 3/2 eτ ϕ uL2 (W ) + τ 1/2 eτ ϕ D uL2 Λ1 (W )  eτ C2 uH 1 (W ) . We have thus obtained

  eτ C2 uH 1 (W )  eτ C3 f L2 (V) + |χU h|H 1−k (∂M) + uH 1 (Uˆε ) 

+ eτ C1 uH 1 (V) , which we write

  uH 1 (W )  eτ (C3 −C2 ) f L2 (V) + |χU h|H 1−k (∂M) + uH 1 (Uˆε ) 

+ e−τ (C2 −C1 ) uH 1 (V) . 1The open set W is introduced since the two open sets U and V are not assumed smooth.

216

9. QUANTIFIED UNIQUE CONTINUATION ON A RIEMANNIAN MANIFOLD

Recalling that C1 < C2 < C3 , by Lemma 5.4, we obtain the local interpolation inequalities (9.2.2) at the boundary.  As in Chapter 5 of Volume 1, with the local results of Lemma 9.2 at the boundary and the result of Theorem 9.1, we deduce the following quantifications of the propagation of “smallness” from a location away from the boundary up to the boundary. Theorem 9.4 (Propagation of “smallness” Up to Boundaries). Let ω be ˇ an open subset of U . The Lopatinski˘ı–Sapiro condition of Definition 2.2 is assumed to hold for (P, B) in V ∩ ∂M. Let C0 > 0 and χU ∈ C ∞ (V ∩ ∂M) such that χU ≡ 1 in a neighborhood of U ∩ ∂M. There exist C > 0 and δ ∈ (0, 1) such that (9.2.9)

δ  + |χ h| + u uH 1 (U ) ≤ Cu1−δ f  k 2 2 U H 1−k ( ∂M) L (V) L (ω) , H 1 (V) 0≤k≤β

for u ∈ H 2 (V) satisfying P u = f +  with f ∈ L2 (V) and   |(m)| ≤ C0 |u(m)| + | D u(m)|g a.e. in V, and Bu|V∩∂M = h|V∩∂M + ∂ with h|k∂M ∈ H 3/2−k (k∂M) and, for all k = 0, . . . , β, ⎧ k ⎪ if k = 0, ⎨∂ ≡ 0 a.e. in V ∩ ∂M k if k = 1, |∂ (m)| ≤ C0 |u(m)| a.e. in V ∩ ∂M   ⎪ ⎩ k |∂ (m)| ≤ C0 |u(m)| + | D u(m)|g a.e. in V ∩ ∂M if k ≥ 2. The geometry, here different from that of Theorem 9.1, is illustrated in Fig. 9.3.

Figure 9.3. Propagation of smallness on a Riemannian manifold up to a part of the boundary. Here, V only meets one connected component of ∂M

9.2. UNIQUE CONTINUATION ESTIMATES UP TO BOUNDARIES

217

Proof. We may assume that  f L2 (V) + (9.2.10) |χU h|H 1−k (k∂M) ≤ uH 1 (V) , 0≤k≤β

since otherwise inequality (9.2.9) is obvious. Let Uˆ be an open set of M such that U  Uˆ  V. Recall the definition of Uˆε given in (9.2.1). With a compactness argument and by Lemma 9.2, we can find a finite number of open sets Wj of V ∩ Uˆ, j ∈ J, such that ∂M ∩ U ⊂ ∪j∈J Wj ,

Wj ∩ ∂M ⊂ k∂M,

for a unique k = k(j) ∈ {0, . . . , β}, and such that, for some values δ = δj ∈ (0, 1) and εj > 0, we have (9.2.11)

δj 1−δj f L2 (V) + |χU h|H 1−k(j) (k(j)∂M) + uH 1 (Uˆε ) uH 1 (Wj )  uH 1 (V) j

δj  1−δj |χU h|H 1−k (k∂M) + uH 1 (Uˆε ) .  uH 1 (V) f L2 (V) + j

0≤k≤β

For ε ∈ (0, 1), we set εU

= {m ∈ U , distg (m, ∂M) < ε}.

There exists ε ∈ (0, 1) such that ε U ⊂ (∪j∈J Wj ). Applying (9.2.11) for each j ∈ J, using now   δ  = min δj ∈ (0, 1), and ε < min min εj , ε ∈ (0, 1), j∈J

j∈J

in place of δj and εj (note that the set Uˆε increases as ε decreases), we obtain (9.2.12)

δ    + |χ h| + u . uH 1 (ε U )  u1−δ f  k 2 1−k 1 1 U ˆ L (V) H ( ∂M) H (V) H (U  ) ε

0≤k≤β

By Theorem 9.1, with U = Uˆε therein, there exists δ0 ∈ (0, 1) such that

δ f  + u , δ ∈ (0, δ0 ]. (9.2.13) uH 1 (Uˆ  )  u1−δ 2 2 L (V) L (ω) H 1 (V) ε

By (9.2.10) and (9.2.13), we have  |χU h|H 1−k (k∂M) + uH 1 (Uˆ  ) (9.2.14) f L2 (V) + ε

0≤k≤β

δ0  0  u1−δ + |χ h| + u . f  k 2 2 1−k 1 U L (V) L (ω) H ( ∂M) H (V) 0≤k≤β

Then, estimates (9.2.12) and (9.2.14) give (9.2.15)

δ  f  + |χ h| + u uH 1 (ε U )  u1−δ k 2 2 1−k 1 U L (V) L (ω) , H ( ∂M) H (V) 0≤k≤β

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9. QUANTIFIED UNIQUE CONTINUATION ON A RIEMANNIAN MANIFOLD

for δ = δ  δ0 . Since ε ∈ (0, ε), we have U ⊂ ε U ∪ Uˆε . Thus, estimate (9.2.13) together with (9.2.15) yields the result.  9.3. Boundary Initiated Unique Continuation: Improved Estimates Here, we extend the result of Section 5.4 of Volume 1. As above, we consider U and V two open sets of M such that U is connected, and U  V  M. One part of the boundary ∂V we consider a first-order differential oper˜ = ∂ν + B ˜  . We assume that (P, B) ˜ fulfills the ˜  and set B ator operator B ˇ Lopatinski˘ı–Sapiro condition of Definition 2.2. Using the improved Carleman estimate of Theorem 8.30 we obtain the following local result. Compare with Lemma 5.12 of Volume 1. Lemma 9.5 (Local Interpolation Inequality with Boundary Observations). Let m0 ∈ ∂V and V 0 be an open neighborhood of m0 where the ˜ fulfills the Lopatinski˘ı–Sapiro ˇ boundary ∂V is smooth and where (P, B) con0 dition of Definition 2.2. Let C0 > 0. There exist W a neighborhood of m0 in V, C > 0, and δ ∈ (0, 1) such that (9.3.1) uH 1 (W 0 ) ≤ Cu1−δ H 1 (V) δ

˜ |V 0 ∩∂V | 2 0 , × f L2 (V) + |u|V 0 ∩∂V |L2 (V 0 ∩∂V) + |Bu L (V ∩∂V) for u ∈ H 2 (V) satisfying P u = f +  with f ∈ L2 (V) and   |(m)| ≤ C0 |u(m)| + |Du(m)| a.e. in V.

The proof can be written mutatis mutandis from that of Lemma 5.12, with an adaptation to the manifold case, by replacing the Carleman estimate of Theorem 3.28 of Volume 1 by that of Theorem 8.30. We then deduce the following result. Theorem 9.6 (Propagation of “smallness” Initiated from the Boundary). Let χU ∈ C ∞ (V ∩∂M) such that χU ≡ 1 in a neighborhood of U ∩∂M. Let also m0 ∈ ∂V and V 0 be an open neighborhood of m0 where the boundary ˇ ∂V is smooth. The Lopatinski˘ı–Sapiro condition of Definition 2.2 is assumed ˜ in V 0 ∩ ∂V. Let C0 > 0. There to hold for (P, B) in V ∩ ∂M and for (P, B) exist C > 0 and δ ∈ (0, 1) such that

 |χU h|H 1−k (k∂M) (9.3.2) uH 1 (U ) ≤ Cu1−δ H 1 (V) f L2 (V) + 0≤k≤β

˜ |V 0 ∩∂V | 2 0 + |u|V 0 ∩∂V |L2 (V 0 ∩∂V) + |Bu L (V ∩∂V)

δ ,

9.4. NOTES

219

for u ∈ H 2 (V) satisfying P u = f +  with f ∈ L2 (V) and   |(m)| ≤ C0 |u(m)| + | D u(m)|g a.e. in V, and Bu|V∩∂M = h|V∩∂M + ∂ with h|k∂M ∈ H 3/2−k (k∂M) and, for all k = 0, . . . , β, ⎧ k ⎪ if k = 0, ⎨∂ ≡ 0 a.e. in V ∩ ∂M k if k = 1, |∂ (m)| ≤ C0 |u(m)| a.e. in V ∩ ∂M   ⎪ ⎩ |∂ (m)| ≤ C0 |u(m)| + | D u(m)|g a.e. in V ∩ k∂M if k ≥ 2. The proof simply follows that of Theorem 5.13 of Volume 1 using Theorem 9.4 and Lemma 9.5. Remark 9.7. (1) In the above results, m0 ∈ ∂V. This allows one to choose m0 in V ∩ ∂M or, if needed, in ∂V \ ∂M. Various choices will be done in applications in Chaps. 10, 11, and 12. ˜ is naturally B ˜ = ∂ν ; see (2) A possible and often natural choice for B Remarks 8.28 and 8.31. The following non-quantified unique continuation result can be simply proven with Lemma 9.5 and an adaptation of Theorem 5.2 of Volume 1 to the manifold case. 2 (M) be such that P u =  with Theorem 9.8. Let u ∈ Hloc   |(m)| ≤ C0 |u(m)| + | D u(m)|g a.e. in V.

If Γ is a bounded open set of ∂M and both ∂ν u|∂M and u|∂M vanish on Γ, then u = 0. Note that for this last result, no particular boundary condition is assumed away from Γ. 9.4. Notes This chapter generalizes results obtained in Chapter 5 of Volume 1 in the case of Dirichlet boundary conditions. A result of quantification of the unique continuation property in the case of Neumann boundary conditions can be found in [219]. There it was derived for the purpose of understanding the stabilization of a boundary-damped wave equation. This topic is covered in the next two chapters. The generalization we provide here encompasses all ˇ boundary operators that fit the framework given by the Lopatinski˘ı–Sapiro conditions recalled in Chap. 2. We do not believe this can be found in the existing literature with the same level of generality. The proof of the results presented in this chapter is based on the Carleman estimates derived in Chap. 8. Once a Carleman estimate is obtained at a boundary, one can apply the method used in the proof of Lemma 9.2 to obtain a quantification of the unique continuation property up to the

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9. QUANTIFIED UNIQUE CONTINUATION ON A RIEMANNIAN MANIFOLD

boundary under such conditions. The mixed Zaremba type conditions are not covered by the analysis we do here. However, with the Carleman estimate proven in [108], a quantification of the unique continuation property as in Lemma 9.2 is derived therein. Here, we consider a boundary operator than can be of arbitrary order in the tangential direction. Ventcel boundary conditions are of order 2 tangentially, and the result we present can be found in [92] for these conditions.

CHAPTER 10

Stabilization of Waves Under Neumann Boundary Damping Contents 10.1. 10.2.

Setting Preliminaries on the Boundary-Damped Wave Equation 10.2.1. Strong Solutions of the Boundary-Damped Wave Equation 10.2.2. Weak Solution of the Boundary-Damped Wave Equation 10.3. Reduced Functional Space and Generator 10.4. Resolvent Estimate and Stabilization Result 10.5. Proof of the Resolvent Estimate 10.6. Notes Appendix 10.A. The Generator of the Boundary-Damped Wave Semigroup

221 222 222 225 229 231 232 235 236 236

10.1. Setting For (M, g) a smooth d-dimensional connected compact Riemannian manifold with boundary we set P0 = −Δg . With a smooth function α ≥ 0 on ∂M, we consider the following wave equation ⎧ 2 ⎪ in (0, +∞) × M, ⎨∂t y + P0 y = 0 (10.1.1) on (0, +∞) × ∂M, ∂ν y + α∂t y = 0 ⎪ ⎩ 0 1 y|t=0 = y , ∂t y|t=0 = y in M, where we denote by ∂ν the outward pointing normal derivative in the sense of the metric g (see (5.3.1) in Sect. 5.3 and (17.4.12) in Sect. 17.4).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 10

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10. NEUMANN BOUNDARY DAMPING

In the case α ≡ 0 the solution exists for t ∈ R with an energy that remains constant with respect to t. Here, we shall assume that α > 0 on a nonempty open subset Γ ⊂ ∂M and we shall prove that in such case the energy decays to zero as t → +∞. The boundary term α∂t y thus acts as a damping and we refer to (10.1.1) as to the boundary-damped wave equation. Similarly to the result of Chapter 6 of Volume 1, without any assumption made on the open set Γ, we prove that the decay rate is at least logarithmic. The outline of this chapter in naturally very similar to that of Chapter 6. 10.2. Preliminaries on the Boundary-Damped Wave Equation To properly define solutions to (10.1.1) we need to provide the proper functional framework, in particular to account for the boundary condition ∂ν y + α∂t y = 0. We address first strong solutions and second weak solution. 10.2.1. Strong Solutions of the Boundary-Damped Wave Equation. We define the boundary operator B∂ by (10.2.1)

B∂ : H 2 (M) × H 1 (M) → H 1/2 (∂M) (v 0 , v 1 ) → ∂ν v 0 |∂M + αv 1 |∂M .

This operator is well defined by the trace formula of Proposition 18.24. We have the following existence and uniqueness result. Theorem 10.1 (Strong Solutions). For (y 0 , y 1 ) ∈ H 2 (M) × H 1 (M) such that B∂ (y 0 , y 1 ) = 0, there exists a unique    y ∈ C 2 [0, +∞); L2 (M) ∩ C 1 [0, +∞);    H 1 (M) ∩ C 0 [0, +∞); H 2 (M) such that (10.2.2) ∂t2 y + P0 y = 0

in L∞ ([0, +∞); L2 (M)),

y|t=0 = y 0 , ∂t y|t=0 = y 1 ,

and B∂ (y(t), ∂t y(t)) = 0 in L∞ ([0, +∞); H 1/2 (∂M)). Moreover, there exists C > 0 such that (10.2.3)   t ≥ 0. y(t)H 2 (M) + ∂t y(t)H 1 (M) ≤ C y 0 H 2 (M) + y 1 H 1 (M) , Solutions given in Theorem 10.1 are called strong solutions of the boundary-damped wave equation. To prove the results of Theorem 10.1 it is convenient to cast the boundarydamped wave equation into a semigroup formalism. We refer to Chapter 12 of Volume 1 for some elements of semigroup theory. We set  0   y y(t) 0 −1 0 (10.2.4) , , Y (t) = , Y = A= ∂t y(t) P0 0 y1

10.2. PRELIMINARIES ON THE BOUNDARY-DAMPED WAVE EQUATION

223

and we consider the Hilbert sum H = H 1 (M) ⊕ L2 (M), naturally endowed with the inner product (U, U  )H = (u0 , u0 )H 1 (M) +(u1 , u1 )L2 (M) ,

U = (u0 , u1 ), U  = (u0 , u1 ),

and norm 2

2

U 2H = u0 H 1 (M) + u1 L2 (M) ,

U = (u0 , u1 ).

If y is a strong solution to (10.2.2) we have     (10.2.5) Y (t) ∈ C 0 [0, +∞); H 2 (M) ⊕ H 1 (M) ∩ C 1 [0, +∞); H , and (10.2.6)

d Y (t) + AY (t) = 0 for t ≥ 0 and Y|t=0 = Y 0 , dt

with moreover B∂ (Y (t)) = 0 for t ≥ 0. Accordingly, we define the unbounded operator A given by (10.2.4) on H with domain   D(A) = V = t (v 0 , v 1 ) ∈ H 2 (M) × H 1 (M); B∂ (V ) = 0 . By Proposition 10.14 and Corollary 10.19 in Appendix 10.A the operator (A, D(A)) is closed on H and moreover D(A) is dense in H . We endow D(A) with the graph norm V 2D(A) = V 2H + AV 2H for V ∈ D(A). In fact, this norm is equivalent to the norm on D(A) inherited from that of H 2 (M) ⊕ H 1 (M) by Corollary 10.26. Theorem 10.2. The unbounded operator (A, D(A)) generates a bounded C0 -semigroup S(t) on H . This result is proven in Appendix 10.A: see Theorem 10.23. Proposition 10.3. Let Y 0 = t (y 0 , y 1 ) ∈ D(A) and set Y (t) = S(t)Y 0 . d Y (t) + AY (t) = 0, for t ≥ 0. If Y (t) = t (y(t), z(t)), then Then, dt    y ∈ C 2 [0, +∞); L2 (M) ∩ C 1 [0, +∞);    H 1 (M) ∩ C 0 [0, +∞); H 2 (M) , z(t) = ∂t y(t), and y is the unique solution of the boundary damped wave equation in the sense of Theorem 10.1. d Y (t) + AY (t) = 0, for t ≥ 0, by applying ProposiNote that we obtain dt tion 12.2 of Volume 1. The proof of Proposition 10.3 is actually contained in that of Theorem 10.1. Using the uniqueness part of Theorem 10.1, Proposition 10.3 shows that the solution of the boundary-damped wave equation is simply given by the first component of Y (t) = S(t)Y 0 .

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Proof of Theorem 10.1. If y is a solution as described in the statement of the theorem, then Y (t) = t (y(t), ∂t y(t))satisfies  (10.2.5) and (10.2.6). Moreover Y (t) ∈ D(A) for t ≥ 0 since B∂ Y (t) = 0. As the graph norm on D(A) is equivalent to the norm on D(A) inherited from that of H 2 (M) ⊕ H 1 (M) by Corollary 10.26, we obtain     Y (t) ∈ C 0 [0, +∞); D(A) ∩ C 1 [0, +∞); H . Then, the second part of Proposition 12.2 of Volume 1 yields Y (t)=S(t)Y (0). This implies the uniqueness of the strong solution. Conversely, let (y 0 , y 1 ) ∈ H 2 (M)×H 1 (M) such that B∂ (y 0 , y 1 ) = 0. We have Y 0 = t (y 0 , y 1 ) ∈ D(A). If we set Y (t) = S(t)Y 0 , by Proposition 12.2 we have Y (t) ∈ C 0 ([0, +∞); D(A)) ∩ C 1 ([0, +∞); H )

and Y (0) = Y 0 .

If we write Y (t) = t (y(t), y 1 (t)) we obtain y(t) ∈ C 0 ([0, +∞); H 2 (M)) ∩ C 1 ([0, +∞); H 1 (M)), and y 1 (t) ∈ C 0 ([0, +∞); H 1 (M)) ∩ C 1 ([0, +∞); L2 (M)),   and in addition B∂ y(t), y 1 (t) = 0 for t ≥ 0. Moreover, the first part of d Y (t) + AY (t) = 0 for t ≥ 0. From the form of A Proposition 12.2 yields dt 1 we find ∂t y = y , which gives    y ∈ C 2 [0, +∞); L2 (M) ∩ C 1 [0, +∞);    H 1 (M) ∩ C 0 [0, +∞); H 2 (M) . d Y (t) + AY (t) = 0 for t ≥ 0 and Y (0) = Y 0 precisely imply the Finally, dt content of (10.2.2). We have thus proven the existence of a strong solution.

Finally, we prove estimate (10.2.3). Let Y 0 ∈ D(A) and Y (t) = S(t)Y 0 ∈ D(A). From the boundedness of the semigroup S(t) on H we have Y (t)H  Y 0 H .

(10.2.7)

In addition, with Proposition 12.2 we deduce, (10.2.8)

AY (t)H = S(t)AY 0 H  AY 0 H .

If Y 0 = t (y 0 , y 1 ) we have Y (t) = t (y(t), ∂t y(t)) and we thus obtain, from Corollary 10.26, y(t)H 2 (M) + ∂t y(t)H 1 (M)  Y (t)H 2 (M)⊕H 1 (M)  Y (t)D(A)  AY (t)H + Y (t)H . With (10.2.7) and (10.2.8) we find y(t)H 2 (M) + ∂t y(t)H 1 (M)  AY 0 H + Y 0 H  Y 0 D(A)  y 0 H 2 (M) + y 1 H 1 (M) , which is precisely (10.2.3).



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225

For a solution to the boundary-damped wave equation we introduce the natural energy:   1  (10.2.9) E(y)(t) = E y(t), ∂t y(t) = ∂t y(t)2L2 (M) + ∇g y(t)2L2 V (M) . 2 If we have    y ∈ C 2 [0, +∞); L2 (M) ∩ C 1 [0, +∞);    H 1 (M) ∩ C 0 [0, +∞); H 2 (M) , we compute   d E(y)(t) = Re(∂t2 y(t), ∂t y(t))L2 (M) + Re ∇g y(t), ∇g ∂t y(t) L2 V (M) . dt With (18.5.5) consequence of the divergence formula of Proposition 18.28 we obtain     ∇g y(t), ∇g ∂t y(t) L2 V (M) = P0 y(t), ∂t y(t) L2 (M) + (∂ν y(t)|∂M , ∂t y(t)|∂M )L2 (∂M)   = P0 y(t), ∂t y(t) L2 (M) − (α∂t y(t)|∂M , ∂t y(t)|∂M )L2 (∂M) , where each term is well defined by the regularity property of y(t) using in particular the trace formula of Theorem 18.25. If moreover we have ∂t2 y + P0 y = 0 in L∞ ([0, +∞); L2 (M)) we obtain, as α ≥ 0, d 2 E(y)(t) = −|α1/2 ∂t y(t)|∂M |L2 (∂M) , dt or rather t

2

E(y)(t ) = E(y)(t) − ∫ |α1/2 ∂t y(σ)|∂M |L2 (∂M) dσ, t

this implies the decay of the energy E(y)(t), viz., (10.2.10)

0 ≤ E(y)(t ) ≤ E(y)(t),

for 0 ≤ t ≤ t .

Observe, however, that the energy E(y 0 , y 1 ) does not provide a norm on H and, thus, it cannot be used to obtain the uniqueness of strong solutions. Compare with Section 6.2 of Volume 1. 10.2.2. Weak Solution of the Boundary-Damped Wave Equation. Weak solutions of the boundary-damped wave equation are solutions that lie in     C 0 [0, +∞); H 1 (M) ∩ C 1 [0, +∞); L2 (M) , that is, Y (t) = t (y(t), ∂t y(t)) ∈ C 0 ([0, +∞); H ). Here, as opposed to the setting in Chapter 6 of Volume 1, the functional setting does not express explicitly the boundary condition. No boundary information is provided by the space H . For strong solutions described in Sect. 10.2.1, the boundary condition that carries the damping term is explicitly given by the domain

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10. NEUMANN BOUNDARY DAMPING

of the semigroup generator A. Weak solutions cannot just be defined by requiring the wave equation to hold in the sense of distribution, as opposed to what is done in Chapter 6. We start by considering y a strong solution of the boundary-damped wave equation as given by Theorem 10.1, with y|t=0 = y 0 and ∂t y|t=0 = y 1 with the requirement Y 0 = t (y 0 , y 1 ) ∈ D(A). For T > 0, we pick       z ∈ C 2 [0, T ]; L2 (M) ∩ C 1 [0, T ]; H 1 (M) ∩ C 0 [0, T ]; H 2 (M) , by integration by parts, we compute (∂t2 y, z)L2 ((0,T )×M) = (y, ∂t2 z)L2 ((0,T )×M) + (∂t y(T ), z(T ))L2 (M) − (y(T ), ∂t z(T ))L2 (M)   − (y 1 , z(0))L2 (M) − (y 0 , ∂t z(0))L2 (M) , and, with the Green formula of Proposition 18.30, we write, for t ≥ 0, (P0 y(t), z(t))L2 (M) = (y(t), P0 z(t))L2 (M) − (∂ν y(t)|∂M , z(t)|∂M )L2 (∂M) + (y(t)|∂M , ∂ν z(t)|∂M )L2 (∂M) . We thus find 0 = (y, (∂t2 + P0 )z)L2 ((0,T )×M) + (∂t y(T ), z(T ))L2 (M) − (y(T ), ∂t z(T ))L2 (M)   − (y 1 , z(0))L2 (M) − (y 0 , ∂t z(0))L2 (M) −(∂ν y(t)|∂M , z(t)|∂M )L2 ((0,T )×∂M) +(y(t)|∂M , ∂ν z(t)|∂M )L2 ((0,T )×∂M) . Since B∂ (y(t), ∂t (y(t)) = 0 we have ∂ν y(t)|∂M = −α∂t y(t)|∂M , yielding, by integration by parts, (∂ν y(t)|∂M , z(t)|∂M )L2 ((0,T )×∂M) = (y(t)|∂M , α∂t z(t)|∂M )L2 ((0,T )×∂M) − (αy(T )|∂M , z(T )|∂M )L2 (∂M) 0 + (αy|∂M , z(0)|∂M )L2 (∂M) .

The above computations thus give the identity (10.2.11) 0 = (y, (∂t2 + P0 )z)L2 ((0,T )×M) + (y(t)|∂M , ∂ν z(t)|∂M − α∂t z(t)|∂M )L2 ((0,T )×∂M) + (∂t y(T ), z(T ))L2 (M) − (y(T ), ∂t z(T ))L2 (M) + (αy(T )|∂M , z(T )|∂M )L2 (∂M)   0 − (y 1 , z(0))L2 (M) − (y 0 , ∂t z(0))L2 (M) + (αy|∂M , z(0)|∂M )L2 (∂M) . Observe now that this last formula isin fact well defined for functions y    0 1 1 2 that lie in C [0, +∞); H (M) ∩ C [0, +∞); L (M) . We are thus led to the following definition of weak solutions of the boundary-damped wave equation.

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227

Definition 10.4 (Weak Solutions). Let (y 0 , y 1 ) ∈ H 1 (M) × L2 (M). A function y ∈ C 0 ([0, +∞); H 1 (M)) ∩ C 1 ([0, +∞); L2 (M)) such that y(0) = y 0 and ∂t y(0) = y 1 is said to be a weak solution to the boundary-damped equation (10.1.1) if identity (10.2.11) holds for all T > 0 and all       z ∈ C 2 [0, T ]; L2 (M) ∩ C 1 [0, T ]; H 1 (M) ∩ C 0 [0, T ]; H 2 (M) . Remark 10.5. The computations above show that a strong solution as given by Theorem 10.1 is also a weak solution. In such a case (y 0 , y 1 ) ∈ H 2 (M) × H 1 (M) and B∂ (y 0 , y 1 ) = 0. The following result further shows that there is existence and uniqueness of a weak solution in the case we wish to consider, that is, (y 0 , y 1 ) ∈ H 1 (M) × L2 (M). Theorem 10.6 (Weak Solutions). Let (y 0 , y 1 ) ∈ H 1 (M) × L2 (M). There exists a unique weak solution     y ∈ C 0 [0, +∞); H 1 (M) ∩ C 1 [0, +∞); L2 (M) to the boundary-damped wave equation in the sense of Definition 10.4. Moreover, there exists C > 0 such that (10.2.12)   y(t)H 1 (M) + ∂t y(t)L2 (M) ≤ C y 0 H 1 (M) + y 1 L2 (M) ,

t ≥ 0.

If we set Y (t) = S(t)Y 0 with Y 0 = t (y 0 , y 1 ), then Y (t) = t (y(t), ∂t y(t)). Finally, for such a weak solution we observe decay of the energy function defined in (10.2.9), that is, (10.2.13)

0 ≤ E(y)(t ) ≤ E(y)(t),

for 0 ≤ t ≤ t .

Remark 10.7. Because of the uniqueness of weak solution and by Remark 10.5, we see that if (y 0 , y 1 ) ∈ H 2 (M) × H 1 (M), then the weak solution given by Theorem 10.6 is in fact a strong solution in the sense of Theorem 10.1. Proof. First, we address uniqueness. We thus consider the case y 0 = 0 and y 1 = 0. For ψ 1 ∈ Cc∞ (Ω), by Theorem 10.1 we consider    ψ ∈ C 2 [0, +∞); L2 (M) ∩ C 1 [0, +∞);    H 1 (M) ∩ C 0 [0, +∞); H 2 (M) such that ∂t2 ψ + P0 ψ = 0 in L∞ ([0, +∞); L2 (M)),

ψ|t=0 = 0, ∂t ψ|t=0 = ψ 1 ,

and B∂ (ψ(t), ∂t ψ(t)) = 0 in L∞ ([0, +∞); H 1/2 (∂M)).

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For T ≥ 0 we set z(t) = ψ(T − t) and we have ∂t2 z + P0 z = 0 in L∞ ([0, T ]; L2 (M)),

z|t=T = 0, ∂t z|t=T = −ψ 1 ,

and ∂ν z(t) − α∂t z(t) = 0 in L∞ ([0, T ]; H 1/2 (∂M)). Identity (10.2.11) then reads (y(T ), ψ 1 )L2 (M) = 0. As ψ 1 ∈ Cc∞ (Ω) and T ≥ 0 are both arbitrary, we conclude that y ≡ 0. Second, we address existence. Setting Y 0 = t (y 0 , y 1 ) we have Y 0 ∈ H and we set Y (t) = S(t)Y 0 . We also pick a sequence Y 0,n ⊂ D(A) such that Y 0,n → Y 0 as n → +∞, using the density of the domain of the semigroup generator, and set Y n (t) = S(t)Y 0,n . By Proposition 10.3, we have Y n (t) = t (y n (t), ∂ y n (t)) and y n (t) is a strong solution of the boundary-damped wave t equation in the sense of Theorem 10.1. The boundedness of the semigroup stated in Theorem 10.2 implies Y (t) − Y n (t)H  Y 0 − Y n,0 H ,

t ≥ 0,

that is, uniform convergence in C 0 ([0, +∞); H ). If Y (t) = t (y(t), z(t)) this gives z = ∂t y. We thus write (10.2.14) y(t) − y n (t)H 1 (M) + ∂t y(t) − ∂t y n (t)L2 (M)  Y 0 − Y n,0 H ,

t ≥ 0.

By remark 10.5, we see that (10.2.11) holds for the strong solution y n (t). Yet, all terms in (10.2.11) converge according to (10.2.14). We thus obtain that y(t) is a weak solution. For strong solution we have obtained the decay of the energy function E(y)(t) given in (10.2.10). Yet, the uniform convergence written in (10.2.14) implies that the energy associated with y n (t) converges to that of y(t). We thus obtain (10.2.13).  The weak formulation hides the damping phenomenon that is expressed by B∂ (y(t), ∂t y(t)) = 0 in the case a strong solution. For such a solution this equality holds in L∞ ([0, +∞); H 1/2 (∂M)). As L∞ ([0, +∞); H 1/2 (∂M)) ⊂ H −1/2 ((0, T ) × ∂M) we define the maps T1 : D(A) → H −1/2 ((0, T ) × ∂M) → ∂ν y|(0,T )×∂M , Y0

T1 : D(A) → H −1/2 ((0, T ) × ∂M) → (∂t y)|(0,T )×∂M , Y0

where t (y(t), ∂t y(t)) = S(t)Y 0 . In the case of a weak solution we have the following result. Proposition 10.8. The maps T1 , T1 can be uniquely extended to H and for a weak solution of the boundary-damped wave equation t (y(t), ∂t y(t)) = S(t)Y 0 , with Y 0 ∈ H , we have (10.2.15)

T1 (Y 0 ) + αT1 (Y 0 ) = 0.

10.3. REDUCED FUNCTIONAL SPACE AND GENERATOR

229

Proof. Let Y 0 ∈ H . As y ∈ C 1 ([0, +∞), L2 (M)) ∩ C 0 ([0, +∞), we have H 1 ((0, T ) × M) and the trace formula of Theorem 18.25 gives y|(0,T )×∂M ∈ H 1/2 ((0, T ) × ∂M). A weak derivation then gives ∂t     y|(0,T )×∂M ∈ H −1/2 ((0, T ) × ∂M). The map T1 : Y 0 → ∂t y|(0,T )×∂M is thus bounded from H into H −1/2 ((0, T ) × ∂M). Let now (Y 0,n )n∈N ⊂ D(A) be such that Y 0,n → Y 0 in H . For t (y n (t), ∂t y n (t)) = S(t)Y 0,n the damping condition (10.2.15) holds. Observe that we have  n   0,n , T1 Y 0,n = (∂t y n )|(0,T )×∂M = ∂t y|(0,T )×∂M = T1 Y H 1 (M))

since y n ∈ C 1 ([0, +∞), H 1 (M)). The boundedness of T1 on H shows that T1 extends uniquely to H and that this extension is precisely T1 . Now since T1 Y 0,n = −αT1 Y 0,n and since α is smooth we find that T1 also uniquely extends to H and that (10.2.15) holds for the two extended maps.  10.3. Reduced Functional Space and Generator Here, we introduce a linear space H˙ of H of codimension one, on which the norm inherited by H is equivalent to the square root of the energy function  1 1 2 2 (10.3.1) U = t (u0 , u1 ). u L2 (M) + ∇g u0 L2 V (M) , E(U ) = 2 The generator of the boundary-damped wave semigroup (A, D(A)) yields an ˙ D(A)) ˙ that generates a C0 -semigroup of contraction unbounded operator (A, ˙ D(A)) ˙ is derived on H˙ . In the next section, a resolvent estimate for (A, leading to the stabilization property of the boundary-damped wave equation. ˙ D(A)) ˙ and provide their properties and Here, we only define H˙ and (A, the connection with the semigroup generated by (A, D(A)). Details are provided in Appendix 10.A. The reader will also note that the results of the previous section rely in fact on the analysis carried on in this appendix. For clarity, in the main text of this chapter, we presented first (A, D(A)) and S(t) as they are related directly to the boundary-damped wave equation. However, their analysis relies essentially on the understanding of the un˙ D(A)) ˙ and the semigroup S(t) ˙ bounded operator (A, it generates as one can see in Appendix 10.A. Below, to ease reading, we provide precise references to the different results of Appendix 10.A. We introduce the continuous linear form (10.3.2)

Fα : H → C  0   v → |α|−1 ∫ αv 0 |∂M μg∂ + ∫ v 1 μg , 1 (∂M) 1 L v M ∂M

230

10. NEUMANN BOUNDARY DAMPING

(recall that we assume that α ≥ 0 is positive on some open subset of the boundary) and the space H˙ = ker(Fα ) = {V ∈ H ; Fα (V ) = 0}. We set ΠN : H → H and ΠH˙ : H → H as the bounded linear maps (10.3.3)

ΠN V = Fα (V )Θ,

ΠH˙ = IdH −ΠN ,

with Θ = t (1, 0). They are projectors onto N = span{Θ} and H˙ , respectively, according to the direct sum (10.3.4) H = H˙ ⊕ N . In particular note that H˙ and N are not orthogonal in H . Note that the continuity of Fα implies that of ΠH˙ and ΠN . By Lemma 10.16, we defined the following norm on H˙ 2

2

V 2H˙ := ∇g v 0 L2 V (M) + v 1 L2 (M) ,

V = t (v 0 , v 1 ),

and we have the following norm equivalence V ∈ H˙ .

V H˙  V H ,

Moreover, for the energy function recalled in (10.3.1), we have, for U ∈ H , 1 (10.3.5) E(U ) = E(U − ΠN U ) = E(ΠH˙ U ) = ΠH˙ U 2H˙ . 2 The operator A has the following properties: Ran(A) ⊂ H˙ and ker(A) = N , by Lemma 10.15 and (10.A.4). In particular, if Y (t) ∈ C 0 ([0, +∞); D(A))∩ d C 1 ([0, +∞); H ) is a strong solution of the semigroup equation dt Y (t) + AY (t) = 0, then we find that t → Fα (Y (t)) is constant for t ∈ [0, +∞); see (10.A.7). One is thus led to introduce the unbounded operator A˙ on H˙ given by the following domain   ˙ = V ∈ H 2 (M) × H 1 (M); B∂ (V ) = 0, Fα (V ) = 0 D(A)   = ΠH˙ D(A) ⊂ D(A), ˙ = AV for V ∈ D(A). ˙ We then have A = A˙ ◦ Π ˙ ; see and such that AV H ˙ D(A)) ˙ as to the the diagram above (10.A.9). Below, we shall refer to (A, reduced generator of the boundary-damped wave equation. ˙ D(A)) ˙ generates a C0 -semigroup of conThe unbounded operator (A, ˙ ˙ traction on H by Theorem 10.22. We denote this semigroup S(t) and we have ˙ ◦ Π ˙ + ΠN . S(t) = S(t) H

where S(t) is the bounded C0 -semigroup1 generated by (A, D(A)). 1Note in fact, that this is the actual definition we give of S(t) in Appendix 10.A; see (10.A.24).

10.4. RESOLVENT ESTIMATE AND STABILIZATION RESULT

If Y 0 ∈ D(A), the solution of the semigroup equation reads

d dt Y

231

(t)+AY (t) = 0

˙ ◦ Π ˙ Y 0 + ΠN Y 0 . Y (t) = S(t)Y 0 = S(t) H ˙ ◦ Π ˙ Y 0 . We have We set Y˙ (t) = ΠH˙ Y (t) = S(t) H  Fα (Y 0 ) . ΠN Y 0 = Fα (Y 0 )Θ = 0 Hence, by (10.3.5) we find that the energy of Y (t) is precisely that of Y˙ (t):       1 2 E Y (t) = E ΠH˙ Y (t) = E Y˙ (t) = Y˙ (t)H˙ . (10.3.6) 2 Understanding the stabilization properties of the semigroup S(t) can thus ˙ be done through that of the semigroup S(t). This is the subject of the next section. As in Chapter 6 of Volume 1 the analysis relies on the derivation of a resolvent estimate for the semigroup generator. 10.4. Resolvent Estimate and Stabilization Result ˙ D(A)) ˙ of the boundary-damped wave For the reduced generator (A, equation we have the following resolvent estimate. Proposition 10.9. Let Γ be a nonempty open subset of ∂M and α ∈ C ∞ (∂M) be such that α > 0 on Γ. Then, the unbounded operator iσ IdH˙ −A˙ is invertible on H˙ for all σ ∈ R and there exist K > 0 and σ0 > 0 such that ˙ −1  ≤ KeK|σ| , (iσ IdH˙ −A) L (H˙ ,H˙ )

σ ∈ R, |σ| ≥ σ0 .

The proof of Proposition 10.9 is given below in Sect. 10.5. The main consequences of this estimate, with the result of Theorem 6.6 of Volume 1, are the following result. Theorem 10.10. Let Γ be a nonempty open subset of ∂M and α ∈ be such that α > 0 on Γ. Let k ∈ N. Then, there exists C > 0 such that, if the initial condition Y 0 is in the domain of the operator A˙ k , then C 0 k 0 ˙ H˙ ≤  (10.4.1) S(t)Y k A˙ Y H˙ . log(2 + t) C ∞ (∂M)

Observing that D(A˙ k ) = ΠH˙ D(Ak ) as A = A˙ ◦ ΠH˙ and using (10.3.6) we obtain the following stabilization result for the boundary-damped wave equation (10.1.1). Corollary 10.11. Let Γ be a nonempty open subset of ∂M and α ∈ C ∞ (∂M) be such that α > 0 on Γ. Let k ∈ N. Then, there exists C > 0 such

232

10. NEUMANN BOUNDARY DAMPING

that, if the initial condition Y 0 = (y 0 , y 1 ) is in the domain of the operator Ak , the energy of the strong solution y(t) to (10.2.2) satisfies C k 0 2 (10.4.2) E(y)(t) ≤  2k A Y H . log(2 + t) By Proposition 6.12 of volume 1, with Theorem 10.10 we obtain Corollary 10.12. The energy of any weak solution to the boundarydamped wave equation (10.1.1) goes to 0 as t → +∞. 10.5. Proof of the Resolvent Estimate Here, we give the proof of the resolvent estimate of Proposition 10.9. ˙ −1 is well defined First, for σ ∈ R, the resolvent operator (iσ IdH˙ −A) ˙ D(A)) ˙ is contained in {z ∈ and continuous on H˙ as the spectrum of (A, C; Re z > 0} by Proposition 10.27. We shall now estimate the operator norm of this resolvent. ˙ and F ∈ H˙ be such that Let thus U ∈ D(A) ˙ (10.5.1) = F, U = t (u0 , u1 ), F = t (f 0 , f 1 ). (iσ Id ˙ −A)U H

Our goal is to find an estimate of the form U H˙ ≤ KeK|σ| F H˙ . The resolvent equation (10.5.1) reads iσu1 − P0 u0 = f 1 ,

iσu0 + u1 = f 0 ,

which we write, with f = iσf 0 − f 1 , (10.5.2) iσu0 + u1 = f 0 , (−σ 2 + P0 )u0 = f,

(∂ν u0 + αu1 )|∂M = 0,

∂ν u0 |∂M = (iσαu0 − αf 0 )|∂M .

Multiplying the second equation by u0 and an integration over M give ((−σ 2 + P0 )u0 , u0 )L2 (M) = (f, u0 )L2 (M) . An integration by parts and the boundary equation in (10.5.2) yield (10.5.3) 2

2

2

2

(f, u0 )L2 (M) = −σu0 L2 (M) + ∇g u0 L2 V (M) − (∂ν u0 |∂M , u0 |∂M )L2 (∂M) = −σu0 L2 (M) + ∇g u0 L2 V (M) 2

− iσ|α1/2 u0 |∂M |L2 (∂M) + (αf 0 |∂M , u0 |∂M )L2 (∂M) , using that α ≥ 0. Computing the imaginary part of (10.5.3) we obtain 2

σ|α1/2 u0 |∂M |L2 (∂M) = Im(αf 0 |∂M , u0 |∂M )L2 (∂M) − Im(f, u0 )L2 (M) . We then write, for |σ| ≥ σ0 > 0, (10.5.4)

2

|α1/2 u0 |∂M |L2 (∂M)  |σ|−1 (f 0 H 1 (M) + f L2 (M) )u0 H 1 (M)  F H˙ u0 H 1 (M) ,

10.5. PROOF OF THE RESOLVENT ESTIMATE

233

using the trace inequality of Proposition 18.24. From the properties of the damping function α on ∂M, there exist Γ0 an open subset of Γ and δ > 0 such that α ≥ δ > 0 in Γ0 . This yields (10.5.5)

2

δ |u0 |∂M |L2 (Γ

0)

 F H˙ u0 H 1 (M) ,

|σ| ≥ σ0 .

The key estimate is given by the following observation lemma. Lemma 10.13. For Γ0 as introduced above, there exists C > 0 such that   u0 H 1 (M) ≤ CeC|σ| F H˙ + |u0 |∂M |L2 (Γ ) . 0

The proof is given below. Then estimate (10.5.5) yields 1 1   2 2 u0 H 1 (M)  eC|σ| F H˙ + u0 H 1 (M) F  ˙ , H

and with the Young inequality we obtain u0 H 1 (M)  eC|σ| F H˙ . Finally, as u1 = f 0 − iσu0 we obtain u0 H 1 (M) + u1 L2 (M)  eC|σ| F H˙ , yielding the resolvent estimate of Proposition 10.9.



Proof of Lemma 10.13. The proof we provide is based on the results on the quantification of the unique continuation property that were obtained in Chap. 9. ˜ = R × M and we denote by n = (s, m) a point of M ˜ with We set M ˜ = R × ∂M. On M ˜ we consider the metric s ∈ R and m ∈ M. We have ∂ M ˜ we consider the = ds ⊗ ds + g, where g is the metric on M. On M gM ˜ 2 operator Q = Ds + P0 , that is, the Laplace–Beltrami operator for gM ˜ . The function u(s, m) = eσs u0 (m) is then a solution to ˜ Qu = h in M,

˜ Bu = h∂ on ∂ M,

˜ that is, Bu = ∂ν u ˜ where B is the Neumann boundary operator on ∂ M, |∂ M and ˜ h(n) = eσs f (m), n = (s, m) ∈ M, h∂ (n) = α(m)eσs (iσu0 − f 0 )|∂M (m),

˜ n = (s, m) ∈ ∂ M.

ˇ In particular, by Example 2.5, the Lopatinski˘ı–Sapiro condition of Defi˜ nition 2.2 holds for (Q, B) at ∂ M. We set U = (−1, 1) × M and V = (−2, 2) × M to fit the setting of Chap. 9. The geometry is illustrated in Fig. 10.1. We consider m0 ∈ Γ0 ⊂ ∂M, with Γ0 as introduced above the statement of Lemma 10.13, n0 = (0, m0 ) and V 0 a neighborhood of n0 in ˜ ⊂ (−1, 1) × Γ0 . We then apply Theorem 9.6 to the U such that V 0 ∩ ∂ M

234

10. NEUMANN BOUNDARY DAMPING

s ˜ M 2

V

1 U 0

∂M

M

∂M

−1

˜ ∂M

−2

˜ ∂M

Figure 10.1. Geometry for the application of the quantified unique continuation property of Theorem 9.6 ˜ in place of M and B = B ˜ = ∂ν therein, yielding, for function u with M some δ ∈ (0, 1), (10.5.6)

uH 1 (U )  u1−δ ˜ + |u|V 0 ∩∂ M ˜| 2 0 H 1 (V) hL2 (V) + |h∂ |L2 (V∩∂ M) ˜ L (V ∩∂ M) δ + |∂ν u|V 0 ∩∂ M ˜| 2 0 ˜ L (V ∩∂ M)

δ  u1−δ + |h | + |u | , h 2 2 0 1 ˜ ˜ ∂ L (V) L (V∩∂ M) |V ∩∂ M 2 0 H (V) ˜ L (V ∩∂ M)

0 using that Bu|∂ M ˜ = ∂ν u|∂ M ˜ = h∂ and that V ⊂ U . We have

uH 1 (V)  e2|σ| u0 H 1 (M) , hL2 (V) ≤ e2|σ| f L2 (M)  e2|σ| F H˙ , |u|M ˜|

˜ L2 (V 0 ∩∂ M)

 e|σ| |u0 |∂M |L2 (Γ ) , 0

and uH 1 (U ) ≥ uL2 (−1,1;H 1 (M))  e−|σ| u0 H 1 (M) . We also write

 0  3|σ| |αu |∂M |L2 (∂M) + |f 0 |∂M |L2 (∂M) |h∂ |L2 (V∩∂ M) ˜ e    e3|σ| |α1/2 u0 |∂M |L2 (∂M) + f 0 H 1 (M) ,

10.6. NOTES

235

using that α is bounded and the trace inequality of Proposition 18.24. With (10.5.4) we obtain   1/2 1/2 3|σ| F  ˙ u0 H 1 (M) + F H˙ . |h∂ |L2 (V∩∂ M) ˜ e H

Collecting the above estimations, with (10.5.6) we find   1/2 1/2 u0 H 1 (M)  eC|σ| F H˙ + |u0 |∂M |L2 (Γ ) + F  ˙ u0 H 1 (M) . 0

The Young inequality yields the sought result.

H



10.6. Notes The results of the present chapter are generalized in Chap. 11 in the ˇ case of a damping associated with Lopatinski˘ı–Sapiro boundary conditions. Thus, the present chapter can very well be omitted. Yet, the generalization provided in Chap. 11 requires the reading of the material of Part 1. Since Neumann boundary conditions are natural in many models, it appeared to us that writing the proof of the logarithmic stabilization in this particular case was sensible. Moreover, the presentation of Chap. 11 follows very much that of the present chapter, which eases the reading of that latter chapter. We refer to Section 6.7 of Volume 1 for references on stabilization. The result proven here is that of [219]. The method is based on [217] (where logarithmic stabilization is proven in the case of an inner damping and Dirichlet boundary conditions). In Sect. 10.5 where the key resolvent estimate of Proposition 10.9 is proven we use a quantified unique continuation property of Chap. 9. One could also proceed as in Section 6.5.2 and obtain the resolvent estimate from the derivation of a global Carleman estimate directly without going through an interpolation type estimate. This is left to the reader and requires the construction of a global Carleman weight function. As explained in Section 6.6 of Volume 1, another approach lies also in the derivation of a Carleman estimate for the operator Pσ = P0 −σ 2 with the boundary operator ∂ν −iσα. Observe that we could also treat the case of an inner damping with homogeneous Neumann boundary conditions: ⎧ 2 ⎪ in (0, +∞) × M, ⎨∂t y + P0 y + α∂t y = 0 on (0, +∞) × ∂M, ∂ν y = 0 ⎪ ⎩ 0 1 y|t=0 = y , ∂t y|t=0 = y in M, with α ≥ 0. Constant functions are solutions to this damped wave equation. The C0 -semigroup associated with this damped wave equation acts on H = H 1 (M) ⊕ L2 (M) and is given by  0 −1 A= P0 α

236

10. NEUMANN BOUNDARY DAMPING

with domain D(A) = {t (v 0 , v 1 ) ∈ H 2 (M) × H 1 (M); ∂ν v 0 |∂M = 0}. Setting Fα : H → C  0 v 0 1 → |α|−1 L1 (M) ∫ (αv + v )μg , v1 M and H˙ = ker(Fα ) we have H = H˙ ⊕ N with N = span{Θ}, Θ = t (1, 0). The space H˙ is invariant by A and is the counterpart of the reduced functional space introduced in Sect. 10.3. This setting allows one to adapt the analysis of the present chapter and the analysis carried out in Chapter 6 of Volume 1. Appendix 10.A. The Generator of the Boundary-Damped Wave Semigroup The boundary-damped wave equation associated with the operator P0 = −Δg reads in a system form (see Sect. 10.2) (10.A.1) with

∂t V + AV = 0,

V (0) = V 0 ∈ H = H 1 (M) ⊕ L2 (M),

0 −1 , A= P0 0   D(A) = V = t (v 0 , v 1 ) ∈ H 2 (M) × H 1 (M); B∂ (V ) = 0 , 

with the operator B∂ defined in (10.2.1). The different results given in this section rely a lot on the analysis of Neumann problem for the Laplace–Beltrami operator P0 on the Riemannian manifold (M, g) whose treatment is given in Sect. 18.7. In particular, some of the notations of that section are used here. Proposition 10.14. The unbounded operators (A, D(A)) is a closed operator on H . Proof. Let (V n )n ⊂ D(A), V n = t (v 0,n , v 1,n ), be such that both V n → V = t (v 0 , v 1 ) and AV n → W = t (w0 , w1 ) in H as n → +∞. We thus have (v 0,n )n ⊂ H 2 (M), (v 1,n )n ⊂ H 1 (M) and v 0,n → v 0 in H 1 (M),

v 1,n → v 1 in L2 (M).

As AV n = t (−v 1,n , P0 v 0,n ) we have −v 1,n → w0 in H 1 (M),

P0 v 0,n → w1 in L2 (M).

We thus obtain v 1 = −w0 ∈ H 1 (M) and in fact v 1,n → v 1 in H 1 (M). It thus remains to prove that v 0 ∈ H 2 (M), P0 v 0 = w1 and B∂ V = 0. Set w1,n = P0 v 0,n ∈ L2 (M). We have w1,n → w1 in L2 (M). With the divergence formula of Proposition 18.28 we have ∫ w1,n μg = − ∫ ∂ν v 0,n |∂M μg∂ = − ∫ hn μg∂ ,

M

∂M

∂M

hn = −αv 1,n |∂M ,

10.A. THE GENERATOR OF THE BOUNDARY-DAMPED WAVE SEMIGROUP

237

as B∂ (V n ) = 0. We recall that for a function w ∈ H k (M), [w] denotes its class in k H (M)/C and w = Πw is the unique representative of [w] with a vanishing average. See the discussion below (18.7.7). Here, we have P0 v 0,n = w1,n and ∂ν v 0,n = hn . As v 1,n → v 1 in H 1 (M) we find that hn → h = −αv 1 |∂M in H 1/2 (∂M) by Proposition 18.24. In particular,   ∫ w1 μg + ∫ hμg∂ = lim ∫ w1,n μg + ∫ hn μg∂ = 0. M

∂M

n→+∞

M

∂M

Then, by Proposition 18.48 and Corollary 18.49 there exists a unique [u] ∈ H 2 (M)/C solution to the Neumann problem, in the sense that P0 u = w1 and ∂ν u|∂M = h, for any u ∈ [u], in particular u. Moreover we have v 0,n − uH 2 (M)  w1,n − w1 L2 (M) + |hn − h|H 1/2 (∂M)  w1,n − w1 L2 (M) + v 1,n − v 1 H 1 (M) , by the trace inequality of Proposition 18.24. Hence, v 0,n → u in H 2 (M), implying, [v 0,n ] → [u] in H 2 (M)/C, as the map Φ : H 2 (M) → H 2 (M)/C is an isometry by Lemma 18.47. Since v 0,n → v 0 in H 1 (M) we have v 0 ∈ [u] by the continuity of the map Φ : H 2 (M) → H 2 (M)/C introduced below (18.7.7). This gives v 0 ∈ H 2 (M), P0 v 0 = w1 , and ∂ν v 0 |∂M = h = −αv 1 |∂M meaning precisely that B∂ (V ) = 0.  Constant functions (in both time and space) are solutions to the boundary-damped wave equation (10.1.1). For such solution the energy (10.2.9) is identically zero. As a result, the energy does not stand as a proper norm to carry out the analysis of the solutions. In the semigroup framework, a constant solution y(t) ≡ Cst is identified to V (t) ≡ CstΘ with Θ = t (1, 0). One observes that indeed Θ ∈ D(A) and AΘ = 0 and thus V (t) = Θ is a trivial solution to (10.A.1). We set N = span{Θ} = C × {0}, with C the space of constant functions on M as introduced above (18.7.7). Observe that by Lemma 18.46, we have in H = H 1 (M) ⊕ L2 (M),

N ⊥ = H 1 (M) ⊕ L2 (M),

and in H 2 (M) ⊕ H 1 (M),

N ⊥ = H 2 (M) ⊕ H 1 (M),

with the spaces H 1 (M) and H 2 (M) as given in (18.7.8). However, even though A vanishes on N this does not imply that it maps H 2 (M) ⊕ H 1 (M) into H 1 (M) ⊕ L2 (M), i.e., A does not preserve the orthogonality with respect to N , the reason being that A is simply not symmetric (and thus not selfadjoint). The orthogonal spaces of N in H and H 2 (M) ⊕ H 1 (M) are thus not the proper spaces to consider if one wishes to ignore constant functions in the analysis.

238

10. NEUMANN BOUNDARY DAMPING

An important observation is the following result. Lemma 10.15. We have ker(A) = N and if V ∈ D(A) and AV = the following relation holds

t (u0 , u1 )

∫ αu0 |∂M μg∂ + ∫ u1 μg = 0. M

∂M

Proof. Above we saw that N ⊂ ker(A). If now V = t (v 0 , v 1 ) ∈ ker(A), then V ∈ D(A) and AV = 0. This reads v 1 = 0 and v 0 ∈ H 2 (M), ∂ν v 0 |∂M = 0,

P0 v 0 = 0.

From Proposition 18.48 and Corollary 18.49 we have v 0 ≡ Cst, which gives V ∈ N. Let now V = t (v 0 , v 1 ) ∈ D(A). We have U = t (u0 , u1 ) = AV = t (−v 1 , P v 0 ) ∈ H . With the divergence formula of Proposition 18.28, we 0 compute ∫ αu0 |∂M μg∂ + ∫ u1 μg = − ∫ αv 1 |∂M μg∂ + ∫ P0 v 0 μg M M ∂M  1  = − ∫ αv |∂M + ∂ν v 0 |∂M μg∂ = 0,

∂M

∂M



since B∂ (V ) = 0.

With the above lemma we are naturally led to introduce the following linear form (10.A.2)

Fα : H → C  0   v → |α|−1 ∫ αv 0 |∂M μg∂ + ∫ v 1 μg , 1 (∂M) 1 L v M ∂M

and the space (10.A.3)

H˙ = ker(Fα ) = {V ∈ H ; Fα (V ) = 0}.

The second part of Lemma 10.15 thus reads (10.A.4) Ran(A) ⊂ H˙ . The form Fα is bounded by the trace inequality of Proposition 18.24 and we note that Fα (Θ) = 1 with Θ = t (1, 0). We set ΠN : H → H and ΠH˙ : H → H as the bounded linear maps (10.A.5)

ΠN V = Fα (V )Θ,

ΠH˙ = IdH −ΠN .

We see that Ran(ΠN ) = ker(ΠH˙ ) = N and Ran(ΠH˙ ) = ker(ΠN ) = H˙ , that both maps ΠN and ΠH˙ are projectors and we have (10.A.6)

H = H˙ ⊕ N ,

N = span{Θ}.

Note that this direct sum is to be understood in the sense of vector spaces only and not in the sense of Hilbert spaces. The space N is not orthogonal to H˙ .

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From the point of view of the semigroup formulation of the boundarydamped wave equation, the introduction of the form Fα and the space H˙ also appear natural with the following argument. Let us assume for now that we know that A generates a C0 -semigroup on H (a proof is given below). For a strong solution U (t) = t (u0 (t), u1 (t)), that is U (t) ∈ C 0 (R+ ; D(A)) ∩ d U (t) + AU (t) = 0, we observe that t → Fα (U (t)) is C 1 (R+ ; H ) such that dt constant. In fact, we write 

d   d Fα (U (t)) = Fα U (t) = −Fα AU (t) = 0, (10.A.7) dt dt by Lemma 10.15. Then, the strong solution associated with the initial condition ΠH˙ U (0) ∈ D(A) ∩ H˙ remains in H˙ for all t > 0. From the above analysis, we have A ◦ ΠH˙ = A. If we define the un˙ D(A)) ˙ with bounded map (A,     ˙ = Π ˙ D(A) = D(A) ∩ H˙ = V ∈ D(A); Fα (V ) = 0 (10.A.8) D(A) H   = V ∈ H 2 (M) × H 1 (M); B∂ (V ) = 0, Fα (V ) = 0 , (we recall that the operator B∂ is defined in (10.2.1) and the operator Fα in ˙ = AU for U ∈ D(A), ˙ we then have A = A˙ ◦ Π ˙ . This is (10.A.2)) by AU H summarized in the following diagram

Note also that since N ⊂ D(A) we have (10.A.9)

˙ ⊕ N. D(A) = D(A)

In particular, with (10.A.6), this gives (10.A.10)

˙ ⇔ V ∈ D(A), ΠH˙ V ∈ D(A)

for V ∈ H .

The space H˙ inherits the norm of H = H 1 (M) ⊕ L2 (M). However, the following equivalent norm will be more adapted to our purpose here. Lemma 10.16. The norm .H˙ defined by 2

2

V 2H˙ := ∇g v 0 L2 V (M) + v 1 L2 (M) 2

2

=  D v 0 L2 Λ1 (M) + v 1 L2 (M) ,

V = t (v 0 , v 1 ),

is equivalent to .H on H˙ . Proof. It suffices to prove the following adapted Poincar´e inequality v 0 L2 (M)  V H˙ ,

V ∈ H˙ ,

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10. NEUMANN BOUNDARY DAMPING

to reach the conclusion. The proof is similar to that of the Poincar´e inequality of Proposition 18.9. Assume that the inequality does not hold. Then, there exists a sequence V n = t (v 0,n , v 1,n )n∈N ⊂ H˙ such that v 0,n L2 (M) = 1,

∇g v 0,n L2 V (M) + v 1,n L2 (M) → 0.

In particular, v 0,n is bounded in H 1 (M) and, thus, there exists v 0 ∈ H 1 (M) such that v 0,n  v 0 in H 1 (M) up to a subsequence. On the one hand, we have ∇g v 0,n  ∇g v 0 in L2 V (M), implying that 0 ∇g v = 0. Because of the form of ∇g given in (17.2.2) this implies that v 0 ≡ Cst in every chart and it thus constant on the whole M since M is connected. On the other hand, from the continuity of the trace map (see Proposi0,n 0  v|∂M in H 1/2 (∂M) (apply, for instance, Propotion 18.24) we have v|∂M sition 35.8 in [320] or Theorem 3.10 in [90]). Consequently 0,n 0 ∫ αv|∂M μg∂ → ∫ αv|∂M μg ∂ .

∂M

∂M

0 μg∂ = 0. As ∫M v 1,n μg → 0 and Fα (V n ) = 0 we obtain in the limit ∫∂M αv|∂M 0 Above we obtained v ≡ Cst; since ∫∂M α μg∂ > 0, this gives v 0 ≡ 0, which yields a contradiction as v 0 L2 (M) = lim v 0,n L2 (M) = 1 since v 0,n → v 0 in L2 (M) from the compact injection ι : H 1 (M) → L2 (M) by the Rellich–Kondrachov theorem (Theorem 18.7). 

Naturally, associated with the norm .H˙ is the following inner product, for U, V ∈ H˙ , (10.A.11)

(V, U )H˙ := (∇g v 0 , ∇g u0 )L2 V (M) + (v 1 , u1 )L2 (M) = (D v 0 , D u0 )L2 Λ1 (M) + (v 1 , u1 )L2 (M) .

˙ D(A)) ˙ on H˙ is closed. Proposition 10.17. The unbounded operator (A, ˙ ˙ ˙ into H˙ is comMoreover, D(A) is dense in H and the injection of D(A) pact. In the proof of Proposition 10.17 we shall need the following lemma. Lemma 10.18. The map ˙ → H 1 (M) D(A) (u0 , u1 ) → u1 is surjective. Moreover, there exists a bounded linear map Ψ : H 1 (M) → ˙ for all w ∈ H 1 (M). H 2 (M) such that (Ψw, w) ∈ D(A) Proof of Lemma 10.18. Let w ∈ H 1 (M) and set h = −αw|∂M ∈ H 1/2 (∂M) using the trace property of Proposition 18.24. Then, with the

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trace theorem (Theorem 18.25), there exists v ∈ H 2 (M) obtained linearly and continuously with respect to h such that v|∂M = 0 and ∂ν v|∂M = h and vH 2 (M)  |h|H 1/2 (∂M)  wH 1 (M) . Since ∫∂M α μg∂ = 0, we can pick c0 ∈ C such that c0 ∫∂M αμg∂ +∫M wμg = 0. We then set Ψw = v + c0 . Note that the map w → Ψw is linear and that ˙ Moreover, we see that we have V = t (Ψw, w) ∈ D(A). c0 L2 (M)  wL1 (M)  wH 1 (M) . Consequently ΨwH 2 (M)  wH 1 (M) . The proof is complete.



˙ be such that both Proof of Proposition 10.17. Let (un )n ⊂ D(A) n n ˙ ˙ (u )n and (Au )n converge in H . Denote by u and v the respective limits. ˙ ⊂ D(A) and Au ˙ n = Aun . As (A, D(A)) is closed, and as Recall that D(A) the topology of H˙ is that of a subspace of H , we find that u ∈ D(A)∩ H˙ = ˙ and v = Au = Au. ˙ The operator (A, ˙ D(A)) ˙ is thus closed on H˙ . D(A) ˙ ˙ Let us Let now U ∈ H be such that (U, V )H˙ = 0 for all V ∈ D(A). ˙ is dense in H˙ . prove that U = 0, which will assert that D(A) 2 Let v ∈ H (M) be such that ∂ν v|∂M = 0. We then set v 0 = v + c0 with 0 c chosen so that ∫∂M αv 0 |∂M μg∂ = 0, which is possible as ∫∂M α μg∂ = 0. ˙ and, with (18.5.5) consequence of Observe then that V = t (v 0 , 0) ∈ D(A) the divergence formula of Proposition 18.28, we obtain 0 = (U, V )H˙ = (∇g u0 , ∇g v 0 )L2 V (M) = (u0 , P0 v 0 )L2 (M) = (u0 , P0 v)L2 (M) . By Lemma 18.51, this implies that u0 ≡ Cst. We thus find that (10.A.12)

0 = (U, V )H˙ = (u1 , v 1 )L2 (M) ,

˙ V = t (v 0 , v 1 ) ∈ D(A).

Let w ∈ H 1 (M). By Lemma 10.18, there exists v ∈ H 2 (M) such that ˙ Then, (10.A.12) yields (u1 , w)L2 (M) = 0. As w ∈ V = t (v, w) ∈ D(A). H 1 (M) is arbitrary, this gives u1 = 0 as H 1 (M) is dense in L2 (M). Thus, U = t (u0 , 0) ∈ H˙ ∩ N since u0 ≡ Cst. By (10.A.6) we finally obtain U = 0. This concludes the proof of the density property. ˙ into H˙ follows from the The compact property of the injection of D(A) ˙ and H˙ (see (10.A.3) and (10.A.8)) and the Rellich– definitions of D(A) Kondrachov theorem (Theorem 18.7).  From (10.A.6) and (10.A.9), and the previous proposition we obtain the following result. Corollary 10.19. The domain D(A) is dense in H . We now make explicit the operator A˙ ∗ adjoint of A˙ with respect to the inner product on H˙ introduced in (10.A.11). We recall that the domain of the adjoint is defined by   ˙ |(V, AU ˙ ) ˙ | ≤ CU  ˙ . D(A˙ ∗ ) = V ∈ H˙ ; ∃C > 0, ∀U ∈ D(A), H H

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10. NEUMANN BOUNDARY DAMPING

Proposition 10.20. We have (10.A.13)   D(A˙ ∗ ) = V = t (v 0 , v 1 ) ∈ H˙ ; v 0 ∈ H 2 (M), v 1 ∈ H 1 (M); B∂∗ (V ) = 0 with B∂∗ (V ) = −∂ν v 0 + αv 1 . For V = t (v 0 , v 1 ) ∈ D(A˙ ∗ ) we have  1  v 0 1 ∗ ˙ (10.A.14) . A˙ ∗ V = , that is, A = −P0 0 −P0 v 0   Proof. Let V ∈ H 2 (M) × H 1 (M) ∩ H˙ be such that B∂∗ (V ) = 0. If ˙ with (18.5.5) consequence of the divergence formula U = t (u0 , u1 ) ∈ D(A), of Proposition 18.28 we compute ˙ V ) ˙ = −(∇g u1 , ∇g v 0 )L2 V (M) + (P0 u0 , v 1 )L2 (M) (AU, H

= −(u1 , P0 v 0 )L2 (M) + (∇g u0 , ∇g v 1 )L2 V (M) − (u1 |∂M , ∂ν v 0 |∂M )L2 (∂M) − (∂ν u0 |∂M , v 1 |∂M )L2 (∂M) . Using that B∂ (U ) = 0, that is, (∂ν u0 + αu1 )|∂M = 0, for the two boundary terms, we obtain (u1 |∂M , ∂ν v 0 |∂M )L2 (∂M) + (∂ν u0 |∂M , v 1 |∂M )L2 (∂M) = (u1 |∂M , ∂ν v 0 |∂M )L2 (∂M) − (αu1 |∂M , v 1 |∂M )L2 (∂M) = (u1 |∂M , (∂ν v 0 − αv 1 )|∂M )L2 (∂M) = 0, since B∂∗ (V ) = (∂ν v 0 − αv 1 )|∂M = 0. We thus obtain (10.A.15)

˙ V ) ˙ = −(u1 , P0 v 0 )L2 (M) + (∇g u0 , ∇g v 1 )L2 V (M) , (AU, H

which yields, as v 0 ∈ H 2 (M) and v 1 ∈ H 1 (M), 0 ˙ V ) ˙ |  u1  2 |(AU, L (M) + ∇g u L2 V (M)  U H˙ . H

Thus V ∈ D(A˙ ∗ ) and (10.A.15) precisely yields the form of A˙ ∗ V given in (10.A.14). ˙ we have Conversely, let V = t (v 0 , v 1 ) ∈ D(A˙ ∗ ). If U = t (u0 , u1 ) ∈ D(A) (10.A.16)

˙ V ) ˙ = −(∇g u1 , ∇g v 0 )L2 V (M) + (P0 u0 , v 1 )L2 (M) , (AU, H

and (10.A.17)

1 ˙ V ) ˙ |  U  ˙  ∇g u0  2 |(AU, L V (M) + u L2 (M) . H H

By the Riesz representation theorem, there exists W = t (w0 , w1 ) ∈ H˙ such that ˙ V ) ˙ = (U, W ) ˙ = (∇g u0 , ∇g w0 )L2 V (M) + (u1 , w1 )L2 (M) . (10.A.18) (AU, H H Let u ∈ H 2 (M) be such that ∂ν u|∂M = 0. We then pick u0 = u + c0 with c0 chosen so that ∫∂M αu0 |∂M μg∂ = 0, which is possible as ∫∂M α μg∂ = 0.

10.A. THE GENERATOR OF THE BOUNDARY-DAMPED WAVE SEMIGROUP

243

˙ With such a choice, we have on Observe that we have U = t (u0 , 0) ∈ D(A). the one hand ˙ V ) ˙ = (P0 u0 , v 1 )L2 (M) , (AU, H

and on the other hand ˙ V ) ˙ = (∇g u0 , ∇g w0 )L2 V (M) = (P0 u0 , w0 )L2 (M) , (AU, H by (18.5.5) consequence of the divergence formula of Proposition 18.28. Consequently, (P0 u, v 1 − w0 )L2 (M) = (P0 u0 , v 1 − w0 )L2 (M) = 0, for all u ∈ H 2 (M) such that ∂ν u|∂M = 0. By Lemma 18.51, this implies v 1 = w0 + Cst. In particular, v 1 ∈ H 1 (M). ˙ we may now write, by From (10.A.16), with U = t (u0 , u1 ) ∈ D(A), (18.5.5), (10.A.19)

˙ V ) ˙ = −(∇g u1 , ∇g v 0 )L2 V (M) − (∂ν u0 |∂M , v 1 |∂M )L2 (∂M) (AU, H + (∇g u0 , ∇g v 1 )L2 V (M) = −(∇g u1 , ∇g v 0 )L2 V (M) + (αu1 |∂M , v 1 |∂M )L2 (∂M) + (∇g u0 , ∇g w0 )L2 V (M) ,

using that B∂ (U ) = 0 and v 1 = w0 + Cst. Note that the boundary L2 -inner product is well defined since both ∂ν u0 |∂M and v 1 |∂M are in H 1/2 (∂M) the trace formula of Theorem 18.25. With (10.A.18) we conclude that (10.A.20) (∇g u1 , ∇g v 0 )L2 V (M) − (u1 |∂M , αv 1 |∂M )L2 (∂M) = −(u1 , w1 )L2 (M) . Let w ∈ H 1 (M). From Lemma 10.18 there exists z ∈ H 2 (M) such that ˙ With (10.A.20) we find that (z, w) ∈ D(A). (10.A.21) (∇g w, ∇g v 0 )L2 V (M) = −(w, w1 )L2 (M) + (w|∂M , αv 1 |∂M )L2 (∂M) , for all w ∈ H 1 (M). Note that choosing w = 1, yields (10.A.22)

− ∫ w1 μg + ∫ αv 1 |∂M μg∂ = 0. M

∂M

Then, Proposition 18.48 applies to the (variational) Eq. (10.A.21) under condition (10.A.22), yielding a unique solution [v] ∈ H 2 (M)/C. As a result v 0 ∈ [v] implying that v 0 ∈ H 2 (M) and ∂ν v 0 = αv 1 on ∂M. As V = t (v 0 , v 1 ) ∈ H˙ we have Fα (V ) = 0. We have therefore obtained that D(A˙ ∗ ) is as described in (10.A.13). The proof is complete.  We now proceed toward the proof that A˙ generates a C0 -semigroup on ˙ H . To that purpose we need following lemma that provides a first resolvent estimate.

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10. NEUMANN BOUNDARY DAMPING

Lemma 10.21. Let B = A˙ or A˙ ∗ and z ∈ C be such that Re z < 0. We have (z IdH˙ −B)U H˙ ≥ | Re z| U H˙ , U ∈ D(B). ˙ We write Proof. Let U = t (u0 , u1 ) ∈ D(A).  0  u zu0 + u1 ˙ , 1 ((z IdH˙ −A)U, U )H˙ = zu1 − P0 u0 u H˙ = zU 2H˙ + (∇g u1 , ∇g u0 )L2 V (M) − (P0 u0 , u1 )L2 (M) = zU 2H˙ + 2i Im(∇g u1 , ∇g u0 )L2 V (M) + (∂ν u0 |∂M , u1 |∂M )L2 (∂M) . We find, computing the real part and using that B∂ (U ) = 0, ˙ (10.A.23) − Re((z IdH˙ −A)U, U )H˙ = − Re(z)U 2H˙ + ∫ α|u1 |2|∂M μg∂ . ∂M

As α ≥ 0 and Re z < 0, we find ˙ U )H˙ | ≥ | Re(z)| U 2H˙ , | Re((z IdH˙ −A)U, ˙ In the case B = A˙ ∗ we similarly which yields the conclusion for B = A. have, for U ∈ D(A˙ ∗ ), ((z IdH˙ −A˙ ∗ )U, U )H˙ = zU 2H˙ − (∇g u1 , ∇g u0 )L2 V (M) + (P0 u0 , u1 )L2 (M) yielding ((z IdH˙ −A˙ ∗ )U, U )H˙ = zU 2H˙ − (∇g u1 , ∇g u0 )L2 V (M) + (∇g u0 , ∇g u1 )L2 V (M) − (∂ν u0 |∂M , u1 |∂M )L2 (∂M) . As B∂∗ (U ) = −∂ν u0 + αu1 = 0 we find − Re((z IdH˙ −A˙ ∗ )U, U )H˙ = − Re(z)U 2H˙ + ∫ α|u1 |2|∂M μg∂ , ∂M

which also yields the conclusion in this case.



We may now state and prove that A˙ generates a C0 -semigroup. ˙ D(A)) ˙ generates a C0 Theorem 10.22. The unbounded operator (A, ˙ −t A ˙ ˙ on H . semigroup of contraction S(t) =e Proof. By Lemma 10.21 , for λ < 0, the operator λ IdH˙ −A˙ is invert˙ −1  ≤ ible with a bounded inverse and moreover we have (λ IdH˙ −A) L (H˙ ) −1 |λ| . We then obtain the result by the Hille-Yosida theorem (see Theo˙ D(A)) ˙ is closed and D(A) ˙ is dense in H˙ by rem 12.6 of Volume 1) as (A, Proposition 10.17. 

10.A. THE GENERATOR OF THE BOUNDARY-DAMPED WAVE SEMIGROUP

245

˙ D(A)) ˙ above Recalling the diagram illustrating the definition of (A, (10.A.9), and that A vanishes on N = span{θ}, we define the following operator on H (10.A.24)

˙ ◦ Π ˙ + ΠN , S(t) := S(t) H

with t ≥ 0 as a parameter, and where ΠH˙ and ΠN are the projectors defined in (10.A.5). Observe that we have the following commutative diagram

˙ The operator S(t) coincides with S(t) on H˙ and coincides with the identity map on its complement N . Theorem 10.23. The map S(t) is a bounded C0 -semigroup on H ; its generator is the unbounded operator (A, D(A)). Proof. First, we see that S(0) = IdH˙ ◦ΠH˙ +ΠN = IdH . Second, from ˙ the strong continuity of S(t) at t = 0+ we deduce that of S(t). Third, we write, using the above commutative diagram, (10.A.5) and the semigroup ˙ property of S(t), for t, t ≥ 0, and U ∈ H . ˙  ) ◦ Π ˙ ◦ S(t)U + ΠN S(t)U S(t ) ◦ S(t)U = S(t H  ˙ ˙ = S(t ) ◦ S(t) ◦ ΠH˙ U + ΠN U ˙  + t) ◦ Π ˙ U + ΠN U = S(t 

H

= S(t + t)U. These three properties show that S(t) is a C0 -semigroup on H . For the boundedness of S(t) we write ˙ ◦ Π ˙ U  + ΠN U  . S(t)U H ≤ S(t) H H H ˙ As ΠN is continuous we have ΠN U H  U H . Next, since S(t)◦Π H˙ U ∈ H˙ we have ˙ ◦ Π ˙ U  ˙  Π ˙ U  ˙  Π ˙ U   U  , ˙ ◦ Π ˙ U   S(t) S(t) H H H H H H H H H ˙ by Lemma 10.16 and the boundedness of S(t) on H˙ and that of ΠH˙ on H . Gathered together, the last three inequalities yield S(t)U H  U H . Finally, to determine the generator (B, D(B)) of S(t), for t > 0 and U ∈ H , we compute  1  1 ˙ U − S(t)U = IdH˙ −S(t) ΠH˙ U, t t

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10. NEUMANN BOUNDARY DAMPING

˙ which allows one to conclude that U ∈ D(B) if and only if ΠH˙ U ∈ D(A), that is, U ∈ D(A) by (10.A.10). In such a case, we have   1 1 ˙ ˙ ˙ U = AU, U − S(t)U = lim IdH˙ −S(t) ΠH˙ U = AΠ BU = lim H t→0+ t t→0+ t as A = A˙ ◦ ΠH˙ ; see the commutative diagram above (10.A.10). As a conclusion the generator of S(t) is precisely (A, D(A)).  Remark 10.24. Note that Theorem 10.23 provides also a proof of Corollary 10.19 by Corollary 1.2.5 in [270]. ˙ the norm inherited from that of H 2 (M)⊕ Proposition 10.25. On D(A) ˙  ˙  AU ˙  . is equivalent to the norm U → AU H H

H 1 (M)

˙ ⊂ H 2 (M) ⊕ H 1 (M). We have Proof. Let U = t (u0 , u1 ) ∈ D(A) ˙ = t (−u1 , P0 u0 ) ∈ H˙ by Lemma 10.15 and as A = A˙ ◦ Π ˙ ; see the AU H commutative diagram above (10.A.10). Since the norm .H˙ is equivalent to the norm .H on H˙ by Lemma 10.16 we have ˙   u1  1 ˙  ˙  AU + P0 u0  2 . AU H

H

H (M)

L (M)

As we have P0 u0 L2 (M)  u0 H 2 (M) by Proposition 18.12, to conclude, it remains to prove that we have u0 H 2 (M)  u1 H 1 (M) + P0 u0 L2 (M) .

(10.A.25)

Since B∂ (U ) = 0 we have ∂ν u0 |∂M = −αu1 |∂M ∈ H 1/2 (∂M),

with αu1 |∂M H 1/2 (∂M)  u1 H 1 (M) ,

by the trace inequality of Proposition 18.24. With the divergence formula of Proposition 18.28, we have ∫ P0 u0 μg + ∫ ∂ν u0 |∂M μg∂ = 0.

M

∂M

Then, by the elliptic regularity result of Proposition 18.48 and Corollary 18.49 for the Neumann problem we obtain (10.A.26) where u0 = u0 −

ffl

Mu

0μ

g.

We thus have

u0 H 2 (M)  u0 H 2 (M) + |

(10.A.27) We set U =

u0 H 2 (M)  P0 u0 L2 (M) + u1 H 1 (M) ,

ffl

ffl M

u0 μg | Volg (M)1/2 .

˙ =U −( M g )Θ ∈ H . Since U ∈ D(A) one has ffl 0 ffl 0 ΠN U = −( u μg )ΠN Θ = −( u μg )Θ.

t (u0 , u1 )

M

u0 μ

M

From the continuity of ΠN we obtain ffl | u0 μg |  U H  u0 H 1 (M) + u1 L2 (M) . M

10.A. THE GENERATOR OF THE BOUNDARY-DAMPED WAVE SEMIGROUP

247

With (10.A.26) and (10.A.27), we thus obtain (10.A.25), which concludes the proof.  Corollary 10.26. On D(A) the norm inherited from that of H 2 (M) ⊕ H 1 (M) is equivalent to the graph norm .D(A) given by V 2D(A) = V 2H + AV 2H .

Proof. Let U = t (u0 , u1 ) ∈ D(A). Using the projectors ΠH˙ and ΠN defined above (10.A.6), by (10.A.9) we have U = ΠH˙ U + ΠN U, ˙ and ΠN U ∈ N = ker(A), using Lemma 10.15. By with ΠH˙ U ∈ D(A) Proposition 10.25 we obtain U H 2 (M)⊕H 1 (M)  ΠH˙ U H 2 (M)⊕H 1 (M) + ΠN U H 2 (M)⊕H 1 (M)  A˙ ◦ ΠH˙ U H + ΠN U H 2 (M)⊕H 1 (M) . As AU = A ◦ ΠH˙ U = A˙ ◦ ΠH˙ U we thus have U H 2 (M)⊕H 1 (M)  AU H + ΠN U H 2 (M)⊕H 1 (M) . Since dim N = 1 one has ΠN U H 2 (M)⊕H 1 (M)  ΠN U H  U H , by the continuity of ΠN on H . We have thus obtained U H 2 (M)⊕H 1 (M)  AU H + U H . As the opposite inequality is clear, the proof is complete.



The study of the stabilization properties of the damped wave equation requires to prove that the imaginary axis {Re z = 0} is in the resolvent set ˙ This is the subject of the following proposition. of the operator A. ˙ D(A)) ˙ is contained in {z ∈ Proposition 10.27. The spectrum of (A, C; Re(z) > 0}. Proof. Let z ∈ C. We consider the two cases. Case 1: Re z < 0. By Lemma 10.21 z IdH˙ −A˙ is injective. Moreover, as its adjoint z IdH˙ −A˙ ∗ is injective and satisfies (z IdH˙ −A˙ ∗ )U H˙  ˙ by Lemma 10.21, using B = A˙ ∗ therein, the map U H˙ for U ∈ D(A) z IdH˙ −A˙ is surjective (see e.g. [90, Theorem 2.20]). The estimation of ˙ then gives the continuity of the operator (z Id ˙ Lemma 10.21, for B = A, H −1 ˙ −A) on H˙ .

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10. NEUMANN BOUNDARY DAMPING

˙ Case 2: Re z = 0. We start by proving the injectivity of z IdH˙ −A. t 0 1 ˙ ˙ Let thus U = (u , u ) ∈ D(A) be such that zU − AU = 0. This gives (10.A.28)

zu0 + u1 = 0,

−P0 u0 + zu1 = 0.

First, if z = 0 one has u1 = 0, P0 u0 = 0, and ∂ν u0 |∂M = 0 as B∂ (U ) = 0. Thus u0 ≡ Cst by Corollary 18.49. As moreover Fα (U ) = 0, we find ∫∂M αu0 μg∂ = 0 meaning that u0 = 0 as ∫∂M αμg∂ > 0. Second, if now z = 0, using (10.A.23) we obtain ˙ U ) ˙ = − ∫ α|u1 |2 μg . 0 = Re((z Id ˙ −A)U, H

H

∂M

|∂M

∂

As α|u1 |2|∂M ≥ 0, this implies that α|u1 |2|∂M vanishes a.e. on ∂M. Then, on the one hand u0 |∂M vanishes a.e. on supp(α) by computing the trace of the first equality in (10.A.28) and, on the other hand, ∂ν u0 |∂M = 0 as B∂ (U ) = 0. We observe that we have P0 u0 = zu1 = −z 2 u0 . With the unique continuation property, initiated from the boundary, as stated in Theorem 9.6, we obtain that u0 vanishes in M and u1 as well. If we now prove that z IdH˙ −A˙ is surjective, the result then follows from the closed graph theorem as A˙ is a closed operator. We write z IdH˙ −A˙ = ˙ By the first part of the proof, T is T + IdH˙ with T = (z − 1) IdH˙ −A. invertible with a bounded inverse. The operator T is unbounded on H˙ . We ˙ equipped with the graph norm denote by T˜ the restriction of T to D(A) ˙ The operator T˜ is bounded by Proposition 10.25. It is associated with A. also invertible. It is thus a bounded Fredholm operator of index ind T˜ = 0 (see Definition 11.6 of Volume 1). Similarly, we denote by ι the injection ˙ into H˙ and A˜ the restriction of A˙ on D(A) ˙ viewed as a bounded of D(A) ˜ ˜ operator. We have zι − A = T + ι. Since ι is a compact operator by Proposition 10.17, we obtain that zι− A˜ is also a bounded Fredholm operator of index 0 by Theorem 11.15 of Volume 1. Hence, zι − A˜ is surjective since z IdH −A˙ is injective as proven above. Consequently, z IdH −A˙ is surjective. 

CHAPTER 11

Stabilization of Waves Under General Boundary Damping Contents 11.1. Setting 11.2. Strong Solutions and Energy 11.3. Reduced Functional Space and Generator 11.4. Resolvent Estimate and Stabilization Result 11.5. Proof of the Resolvent Estimate 11.6. Notes Appendix 11.A. The Generator of the Boundary-Damped Wave Semigroup

249 251 253 256 256 260 260 260

11.1. Setting For (M, g) a smooth d-dimensional connected compact Riemannian manifold with boundary, with d ≥ 2, we consider a second-order differential operator P on M given by P = −Δg + R1 , where R1 is a first-order differential operator. We also consider a differential boundary operator B defined in a neighborhood of ∂M of order 0 or 1 on each connected component of ∂M. On the part of ∂M where the order is 0, denoted by 0∂M, we consider the Dirichlet boundary operator, that is, Bu|0∂M = u|0∂M . We assume that the part of ∂M where the order is 1, denoted by 1∂M, is nonempty, meaning that there exists a connected component of ∂M where B is a genuine first-order operator.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 11

249

250

11. GENERAL BOUNDARY DAMPING

We introduce the unbounded operator P on L2 (M) with domain D(P) = {u ∈ L2 (M); P u ∈ L2 (M) and Bu|∂M = 0}. To ensure Fredholm properties, we require that (P, B) satisfies the Lopatinˇ ski˘ı–Sapiro conditions; see Theorem 3.1. Consequently, as observed in the beginning of Chap. 4, one has D(P) = {u ∈ H 2 (M); Bu|∂M = 0}. We also require the operator to be selfadjoint; by Theorem 4.3, this implies that R1 and B take the following forms: (11.1.1) R1 = iV + f in M

and

B = ∂ν + B  with B  = iX  + h in 1∂M,

where we denote by ∂ν the outward pointing normal derivative in the sense of the metric g (see (5.3.1) in Sect. 5.3 and (17.4.12) in Sect. 17.4), V a smooth real vector field on M, f a complex valued function on M, X  a smooth real vector field on 1∂M, and h a complex valued function on ∂M, with the additional properties (11.1.2) Im f = divg V /2 in M

and

2 Im h − divg X  + g(V, ν)|1∂M = 0 in 1∂M.

Moreover, in the case d = 2, one has |X  |g∂ = 1, and in the case d ≥ 3, we moreover have |X  |g∂ < 1. In the present chapter, we shall furthermore require |X  |g∂ < 1 independently of the dimension. Consequently, the spectrum of P is composed of a sequence of eigenvalues of finite multiplicities that grows to +∞; see Theorems 4.9 and 4.12. If one does not assume |X  |g∂ < 1, in dimension d = 2, the case of eigenvalues that go to −∞ can occur (see Proposition 4.13). This would imply that the associated wave equation is not well posed, having its “generator” with unbounded positive eigenvalues. Finally, if we denote by μ0 ≤ μ1 ≤ · · · ≤ μn ≤ · · · the real eigenvalues of the selfadjoint operator (P, D(P)), we moreover assume that μ0 ≥ 0. With a smooth function α ≥ 0 given on wave equation: ⎧ 2 ∂t y + P y = 0 ⎪ ⎪ ⎪ ⎨y = 0 (11.1.3) ⎪ By + α∂t y = 0 ⎪ ⎪ ⎩ y|t=0 = y 0 , ∂t y|t=0 = y 1

1

∂M, we consider the following in (0, +∞) × M, on (0, +∞) × 0∂M, on (0, +∞) × 1∂M, in M.

We assume that α > 0 on a nonempty open subset Γ ⊂ 1∂M. Below we define a proper energy for solutions of this equation, and we prove that the energy decays to zero as t → +∞. The present chapter thus generalizes the result of Chap. 10 obtained in the case of a Neumann boundary damping.

11.2. STRONG SOLUTIONS AND ENERGY

251

The content of the present chapter is thus very close and adapted from that of Chap. 10. 11.2. Strong Solutions and Energy We define the boundary operator B∂ by (11.2.1)

B∂ : H 2 (M) × H 1 (M) → H 1/2 (1∂M) (v 0 , v 1 ) → Bv 0 |1∂M + αv 1 |1∂M .

This operator is well defined by the trace formula of Proposition 18.24. We recall the notation HD1 (M) = {u ∈ H 1 (M); u|0∂M = 0}, and we introduce the space HD2 (M) = H 2 (M) ∩ HD1 (M). On HD1 (M), we use the usual H 1 -inner product and H 1 -norm. On HD2 (M), we use the usual H 2 -inner product and H 2 -norm. We have the following existence and uniqueness result. Theorem 11.1 (Strong Solutions). For (y 0 , y 1 ) ∈ HD2 (M) × HD1 (M) such that B∂ (y 0 , y 1 ) = 0, there exists a unique    y ∈ C 2 [0, +∞); L2 (M) ∩ C 1 [0, +∞);    HD1 (M) ∩ C 0 [0, +∞); HD2 (M) such that (11.2.2) ∂t2 y + P y = 0 and (11.2.3)

in L∞ ([0, +∞); L2 (M)),

y|t=0 = y 0 , ∂t y|t=0 = y 1 ,

  B∂ (y(t), ∂t y(t)) = 0 in L∞ [0, +∞); H 1/2 (1∂M) .

Moreover, there exists C > 0 such that (11.2.4)   y(t)H 2 (M) + ∂t y(t)H 1 (M) ≤ C y 0 H 2 (M) + y 1 H 1 (M) ,

t ≥ 0.

Solutions given in Theorem 11.1 are called strong solutions of the boundary-damped wave equation (11.1.3). To prove the results of Theorem 11.1 it is convenient to cast the boundarydamped wave equation into a semigroup formalism. Theorem 11.1 then follows from Theorem 11.2 and Proposition 11.3 below similarly to what is done in Chapter 6 of Volume 1 and Chap. 10 (see, for instance, the proof of Theorem 10.1). We refer to Chapter 12 for some elements of semigroup theory. We set  0   y 0 −1 y(t) 0 (11.2.5) , , Y = A= , Y (t) = P 0 ∂t y(t) y1

252

11. GENERAL BOUNDARY DAMPING

and we consider the Hilbert sum H = HD1 (M) ⊕ L2 (M), naturally endowed with the inner product (U, U  )H = (u0 , u0 )H 1 (M) + (u1 , u1 )L2 (M) ,

U = (u0 , u1 ), U  = (u0 , u1 ),

and norm 2

2

U 2H = u0 H 1 (M) + u1 L2 (M) ,

U = (u0 , u1 ).

If y is a strong solution to (11.2.2) and (11.2.3), we have     Y (t) ∈ C 0 [0, +∞); HD2 (M) ⊕ HD1 (M) ∩ C 1 [0, +∞); H , (11.2.6) and d Y (t) + AY (t) = 0 for t ≥ 0 and Y|t=0 = Y 0 , dt with moreover B∂ (Y (t)) = 0 for t ≥ 0. We define the unbounded operator A given by (11.2.5) on H with domain   D(A) = V = t (v 0 , v 1 ) ∈ HD2 (M) × HD1 (M); B∂ (V ) = 0 . (11.2.7)

By Proposition 11.10 and Corollary 11.14 in Appendix 11.A, the operator (A, D(A)) is closed on H , and moreover D(A) is dense in H . We endow D(A) with the graph norm V 2D(A) = V 2H + AV 2H for V ∈ D(A). In fact, this norm is equivalent to the norm on D(A) inherited from that of HD2 (M) ⊕ HD1 (M) by Corollary 11.20. Theorem 11.2. The unbounded operator (A, D(A)) generates a bounded C0 -semigroup S(t) on H . This result is proven in Appendix 11.A; see Theorem 11.18. Proposition 11.3. Let Y 0 = t (y 0 , y 1 ) ∈ D(A) and set Y (t) = S(t)Y 0 . d Then, dt Y (t) + AY (t) = 0, for t ≥ 0. If Y (t) = t (y(t), z(t)), then       y ∈ C 2 [0, +∞); L2 (M) ∩ C 1 [0, +∞); HD1 (M) ∩ C 0 [0, +∞); HD2 (M) , z(t) = ∂t y(t), and y is the unique strong solution (11.2.2) and (11.2.3). d Y (t) + AY (t) = 0, for t ≥ 0, by applying ProposiNote that we obtain dt tion 12.2 of Volume 1. The proof of Proposition 11.3 is actually contained in that of Theorem 11.1. Using the uniqueness part of Theorem 11.1, Proposition 11.3 shows that the solution of (11.2.2) and (11.2.3) is simply given by the first component of Y (t) = S(t)Y 0 .

To analyze stabilization, we need to introduce a proper energy. We set   1  ∂t y(t)2L2 (M) + N (y(t)) , (11.2.8) E(y)(t) = E y(t), ∂t y(t) = 2

11.3. REDUCED FUNCTIONAL SPACE AND GENERATOR

253

with N (.) given by (11.2.9) N (u) = ∇g u2L2 V (M) + (R1 u, u)L2 (M) + B  γ D (u), γ D (¯ u)H −1/2 (1∂M),H 1/2 (1∂M) , as in (4.6.6). This is motivated by the density of D(P) in HD1 (M) and the fact that N (u) = (P u, u)L2 (M) ≥ 0 if u ∈ D(P); hence N (u) ≥ 0 on HD1 (M). Note that this definition of the energy coincides with that given in a Neumann boundary damping in Chap. 10; see (10.2.9). Considering y(t) a strong solution and observing that t → E(y)(t) is differentiable, we find   d ˜ ∂t y(t), y(t) , E(y)(t) = Re(∂t2 y(t), ∂t y(t))L2 (M) + Re N dt ˜ with N (., .) defined in (4.6.5). We then compute   ˜ ∂t y(t), y(t) = (∇g ∂t y(t), ∇g y(t))L2 V (M) Re N + (R1 ∂t y(t), y(t))L2 (M) + B  ∂t y(t)|1∂M , y¯(t)|1∂M H −1/2 (1∂M),H 1/2 (1∂M) = (∂t y(t), −Δg y(t))L2 (M) + (∂t y(t)|1∂M , ∂ν y(t)|1∂M )L2 (1∂M) + (∂t y(t), R1 y(t))L2 (M)   + ∂t y(t)|1∂M , B  y(t)|1∂M L2 (1∂M) , with Lemma 4.7. We then obtain d 2 , E(y)(t) = −|α1/2 ∂t y(t)|1∂M | 2 1 L ( ∂M) dt using that ∂ν y(t) + B  y(t) + α∂t y(t) = 0 on 1∂M. This implies the decay of the energy E(y)(t), viz., (11.2.10)

0 ≤ E(y)(t ) ≤ E(y)(t),

for 0 ≤ t ≤ t .

11.3. Reduced Functional Space and Generator Set N = ker(A), and observe that U ∈ N if and only if U = t (ϕ, 0) with ϕ ∈ ker(P), that is, P ϕ = 0, u|0∂M = 0, and Bϕ|1∂M = 0. If ϕ ∈ ker(P) and if (αϕ|1∂M , ϕ|1∂M )L2 (1∂M) = 0, then, as α > 0 on some open set Γ of 1∂M, one finds ϕ|Γ = 0 and ∂ν ϕ|Γ = 0, using the form of B on 1∂M. Then, the unique continuation result of Theorem 9.8 applies yielding ϕ = 0 in M. Hence, we have the following result. Lemma 11.4. The bilinear map (ϕ, ψ) → (αϕ|1∂M , ψ|1∂M )L2 (1∂M) is an inner product on the finite dimensional space ker(P).

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11. GENERAL BOUNDARY DAMPING

We now consider an orthonormal basis {ϕ0 , . . . , ϕn−1 } of ker(P) associated with this inner product. For ϕ ∈ ker(P), ϕ = 0, we introduce the linear form Fα,ϕ : H → C

(11.3.1)

 0 v → (αϕ|1∂M , ϕ|1∂M )−1 L2 (1∂M) v1   (αv|01∂M , ϕ|1∂M )L2 (1∂M) + (v 1 , ϕ)L2 (M) , and we set Θϕ = t (ϕ, 0) ∈ ker(A). Note that Fα,ϕ is a bounded map. We then set H˙ =

∩

ϕ∈ker(P)

n−1

ker(Fα,ϕ ) = ∩ ker(Fα,ϕj ). j=0

Note that for j = 0, . . . , n − 1, one simply has Fα,ϕj (V ) = (αv|01∂M , ϕ|1∂M )L2 (1∂M) +(v 1 , ϕ)L2 (M) . Setting Θj = t (ϕj , 0), j = 0, . . . , n− 1, one has Fα,ϕj (Θj ) = 1. We define ΠN V =

n−1 

Fα,ϕj (V )Θj ,

V ∈H,

j=0

and ΠH˙ = IdH −ΠN , and we obtain that ΠN and ΠH˙ are continuous projectors associated with the direct sum (11.3.2) H = H˙ ⊕ N and H˙ = ker(ΠN ). Note that H˙ and N are not orthogonal in H . Lemma 11.5. We have Ran(A) ⊂ H˙ . Proof. Let U = t (u0 , u1 ) = AV with V = t (v 0 , v 1 ) ∈ D(A). One has u0 = −v 1 ∈ HD1 (M) and u1 = P v 0 ∈ L2 (M). If ϕ ∈ ker(P), one computes (αϕ|1∂M , ϕ|1∂M )L2 (1∂M) Fα,ϕ (U ) = −(αv|11∂M , ϕ|1∂M )L2 (1∂M) + (P v 0 , ϕ)L2 (M) = −(αv|11∂M , ϕ|1∂M )L2 (1∂M) − (Bv|11∂M , ϕ|1∂M )L2 (1∂M) = 0, applying the Green formula (4.5.3), using that B∂ V = 0. The space H˙ inherits the norm of H = HD1 (M) ⊕ L2 (M). Lemma 11.11, the norm .H˙ defined by 2

V 2H˙ := N (v 0 ) + v 1 L2 (M) ,

 By

V = t (v 0 , v 1 ),

is equivalent to .H on H˙ . This latter norm is more adapted to our purpose here.

11.3. REDUCED FUNCTIONAL SPACE AND GENERATOR

255

 The energy of a strong solution is given in Sect. 11.2 by E(y)(t) = E Y (t) with  1 2 (11.3.3) E(U ) = U = t (u0 , u1 ), N (u0 ) + u1 L2 (M) , 2 and Y = t (y, ∂t y). We note that 1 E(U ) = E(ΠH˙ U ) = ΠH˙ U 2H˙ . 2 In fact, we have ΠH˙ U = U + Θϕ for some ϕ ∈ ker(P). We then have (11.3.4)

2

2E(U + Θϕ ) = N (u0 + ϕ) + u1 L2 (M) , and ˜ (ϕ, u0 ). N (u0 + ϕ) = N (u0 ) + N (ϕ) + 2 Re N ˜ (ϕ, u0 ) = (P ϕ, u0 ) = 0, By (4.6.7), we have N (ϕ) = (P ϕ, ϕ) = 0 and N yielding (11.3.4). Below we shall consider the semigroup generated by the restriction of A on the reduced space H˙ . A proper norm is then ΠH˙ U H˙ and (11.3.4) shows that the chosen energy of the associated solution of the damped wave equation is precisely the norm in the space where evolution occurs. Following Chap. 10, we introduce the unbounded operator A˙ on H˙ given by the following domain: ˙ = D(A) ∩ H˙ D(A)  = V ∈ HD2 (M) × HD1 (M);

 B∂ (V ) = 0, Fα,ϕj (V ) = 0, j = 0, . . . , n − 1 ,

˙ = AV for V ∈ D(A). ˙ We then have A = A˙ ◦ Π ˙ . We and such that AV H ˙ ˙ say that (A, D(A)) is the reduced generator of the damped wave equation we consider here.   ˙ = Π ˙ D(A) since N = ker(A) ⊂ D(A). Thus, one Observe that D(A) H has ˙ ⊕ N. (11.3.5) D(A) = D(A) With (11.3.2) and (11.3.5), one has (11.3.6)

˙ ⇔ V ∈ D(A), ΠH˙ V ∈ D(A)

for V ∈ H .

˙ D(A)) ˙ generates a C0 -semigroup of conThe unbounded operator (A, ˙ ˙ traction on H by Theorem 11.17. We denote this semigroup S(t), and we have ˙ ◦ Π ˙ + ΠN , S(t) = S(t) H

where S(t) is the bounded C0 -semigroup generated by (A, D(A)).

256

11. GENERAL BOUNDARY DAMPING

d If Y 0 ∈ D(A), the solution of the semigroup equation dt Y (t)+AY (t) = 0 reads ˙ ◦ Π ˙ Y 0 + ΠN Y 0 . Y (t) = S(t)Y 0 = S(t) H

˙ By (11.3.4), we find that the energy We set Y˙ (t) = ΠH˙ Y (t) = S(t)◦Π H˙ Y ˙ of Y (t) is precisely that of Y (t):    1    2 (11.3.7) E Y (t) = E ΠH˙ Y (t) = E Y˙ (t) = Y˙ (t)H˙ . 2 The stabilization properties of the semigroup S(t) are thus analyzed by that ˙ of the semigroup S(t) in the next section. 0.

11.4. Resolvent Estimate and Stabilization Result ˙ D(A)) ˙ of the boundary-damped wave For the reduced generator (A, equation, we have the following resolvent estimate. Proposition 11.6. Let Γ be a nonempty open subset of 1∂M, and let α ∈ C ∞ (1∂M) be such that α > 0 on Γ. Then, the unbounded operator iσ IdH˙ −A˙ is invertible on H˙ for all σ ∈ R, and there exist K > 0 and σ0 > 0 such that K|σ| ˙ −1  , σ ∈ R, |σ| ≥ σ0 . (iσ Id ˙ −A) ˙ ˙ ≤ Ke H

L (H ,H )

The proof of Proposition 11.6 is given below in Sect. 11.5. Proposition 11.6 is the counterpart of Proposition 10.9. By Theorem 6.6 of Volume 1, a consequence is the logarithmic asymp˙ totic stability of the semigroup S(t). The statement of this result is the same as in Theorem 10.10, from which one deduces the stabilization result for the damped wave equation (11.1.3). Theorem 11.7. Let Γ be a nonempty open subset of 1∂M, and let α ∈ C ∞ (1∂M) be such that α > 0 on Γ. Let k ∈ N. Then, there exists C > 0 such that, if the initial condition Y 0 = (y 0 , y 1 ) is in the domain of the operator Ak , the energy of the strong solution y(t) to (11.2.2) and (11.2.3) satisfies C k 0 2 (11.4.1) E(y)(t) ≤  2k A Y H . log(2 + t) If one were to introduce weak solutions, one would then obtain the counterpart result of Corollary 10.12. This is left to the reader. 11.5. Proof of the Resolvent Estimate Here, we give the proof of the resolvent estimate of Proposition 11.6. ˙ −1 is well defined First, for σ ∈ R, the resolvent operator (iσ IdH˙ −A) ˙ D(A)) ˙ is contained in {z ∈ and continuous on H˙ as the spectrum of (A, C; Re z > 0} by Proposition 11.21.

11.5. PROOF OF THE RESOLVENT ESTIMATE

257

˙ and F ∈ H˙ be such that Let U ∈ D(A) (11.5.1)

˙ = F, (iσ IdH˙ −A)U

U = t (u0 , u1 ), F = t (f 0 , f 1 ).

The resolvent equation (11.5.1) reads iσu1 − P u0 = f 1 ,

iσu0 + u1 = f 0 ,

(Bu0 + αu1 )|1∂M = 0,

which we write, with f = iσf 0 − f 1 , (11.5.2) iσu0 + u1 = f 0 ,

(−σ 2 + P )u0 = f,

Bu0 |1∂M = α(iσu0 − f 0 )|1∂M .

Writing ((−σ 2 + P )u0 , u0 )L2 (M) = (f, u0 )L2 (M) , with the integration by parts formula (4.6.4), we find (11.5.3) 2

(f, u0 )L2 (M) = ∇g u0 L2 V (M) + (R1 u0 , u0 )L2 (M) 2

− (∂ν u0 |1∂M , u0 |1∂M )L2 (1∂M) − σu0 L2 (M) 2

= ∇g u0 L2 V (M) + (R1 u0 , u0 )L2 (M) + (B  u0|1∂M , u0|1∂M )L2 (1∂M) − iσ|α1/2 u0 |1∂M |

2 L2 (1∂M)

+ (αf 0 |1∂M , u0 |1∂M )L2 (1∂M)

− σu0 2L2 (M) = N (u0 ) − iσ|α1/2 u0 |1∂M |

2 L2 (1∂M)

+ (αf 0 |1∂M , u0 |1∂M )L2 (1∂M) − σu0 2L2 (M) , using that α ≥ 0, where N (.) as given in (11.2.9). Recall that N (.) is real (and nonnegative) on HD1 (M). Computing the imaginary part of (11.5.3), we obtain σ|α1/2 u0 |1∂M |

2 L2 (1∂M)

= Im(αf 0 |1∂M , u0 |1∂M )L2 (1∂M) − Im(f, u0 )L2 (M) .

We then write, for |σ| ≥ σ0 > 0, (11.5.4)

|α1/2 u0 |1∂M |

2 L2 (1∂M)

 |σ|−1 (f 0 H 1 (M) + f L2 (M) )u0 H 1 (M)  F H˙ u0 H 1 (M) ,

using the trace inequality of Proposition 18.24. From the properties of the damping function α on 1∂M, there exist Γ0 an open subset of Γ and δ > 0 such that α ≥ δ > 0 in Γ0 . This yields (11.5.5)

δ |u0 |1∂M |

2 L2 (Γ0 )

 F H˙ u0 H 1 (M) ,

|σ| ≥ σ0 .

The key estimate is given by the following observation lemma.

258

11. GENERAL BOUNDARY DAMPING

Lemma 11.8. For Γ0 as introduced above, there exists C > 0 such that   u0 H 1 (M) ≤ CeC|σ| F H˙ + |u0 |1∂M | 2 . L (Γ0 )

The proof is given below. Then estimate (11.5.5) yields 1 1   2 u0 H 1 (M)  eC|σ| F H˙ + F  2 ˙ u0 H 1 (M) , H

and with the Young inequality, we obtain u0 H 1 (M)  eC|σ| F H˙ . Finally, as u1 = f 0 − iσu0 , we obtain u0 H 1 (M) + u1 L2 (M)  eC|σ| F H˙ , 

yielding the resolvent estimate of Proposition 11.6.

˜ = R × M, and we denote by n = Proof of Lemma 11.8. We set M ˜ ˜ = R × ∂M. We (s, m) a point of M with s ∈ R and m ∈ M. We have ∂ M 0 1 0 1 ˜ = R × ∂M and ∂ M ˜ = R × ∂M. set ∂ M ˜ On M, we consider the metric g˜ = ds ⊗ ds + g, where g is the metric on ˜ we consider the operator Q = Ds2 + P . The function u(s, m) = M. On M, σs 0 e u (m) is then a solution to ˜ Qu = h in M,

0 1 ˜ , and Bu 1 ˜ u = 0 on ∂ M | ∂M = h∂ on ∂ M ,

where h(n) = eσs f (m),

˜ n = (s, m) ∈ M,

h∂ (n) = α(m)eσs (iσu0 − f 0 )|1∂M (m),

1 ˜. n = (s, m) ∈ ∂ M

The principal symbol of Q is q(s, m, σ, ω) = σ 2 + |ω|2gm = |(σ, ω)|2g˜m . Let s ∈ R, σ ∈ R, and (m, ω  ) ∈ T ∗ ∂M. Following Sect. 2.2, we set qˇ(s, m, σ, ω  , z) = |(σ, ω  ) − znm |g˜m . 2

As we have (nm , (σ, ω  ))g˜m = 0, we find qˇ(m, s, ω  , σ, z) = z 2 + |(σ, ω  )|g˜m = (z − i|(σ, ω  )|g˜m )(z + i|(σ, ω  )|g˜m ). 2

ˇ The Lopatinski˘ı–Sapiro condition holds on 0∂M by Example 2.5–(1). With ˇb(m, ω  , z) = b(m, ω  − znm ), on the one hand, having the Lopatinˇ ski˘ı–Sapiro condition for (P, B) on 1∂M reads ˇb(m, ω  , i|ω  |g∂ ) = 0 for all (m, ω  ) ∈ T ∗1∂M with ω  = 0 by Proposition 2.3. This condition is part of the assumption formulated here. On the other hand, the Lopatinski˘ı– ˇ Sapiro condition for (Q, B) on R× 1∂M reads ˇb(m, ω  , i|(σ, ω  )|g˜m ) = 0 for all 1 ˜ with (σ, ω  ) = (0, 0) also by applying Proposition 2.3. (s, m, σ, ω  ) ∈ T ∗ ∂ M We wish to prove that this latter condition holds too.

11.5. PROOF OF THE RESOLVENT ESTIMATE

259

On 1∂M, we have Bu|1∂M = ∂ν u|1∂M + (iX  + h)u|1∂M , where X  is a  , ω  . In real valued vector field. This gives b(m, ω  − znm ) = −iz − Xm 1  the case d ≥ 3, we have |X |g∂ < 1 on ∂M by Proposition 4.6. In the case d = 2, this property is assumed here. Because of the latter property, if ω  = 0, observe that we have ˇb(m, ω  , i|(σ, ω  )| ) = |(σ, ω  )| − X  , ω   ≥ |ω  | − X  , ω   > 0. m m g˜m g˜m g∂ If ω  = 0 and σ = 0, then ˇb(m, ω  , i|(σ, ω  )|g˜m ) = |σ| = 0. Consequently, the ˇ Lopatinski˘ı–Sapiro condition holds for (Q, B) on R × 1∂M. We set U = (−1, 1)×M and V = (−2, 2)×M to fit the setting of Chap. 9. The geometry is the same as in Fig. 10.1. We consider m0 ∈ Γ0 ⊂ 1∂M, with Γ0 as introduced above the statement of Lemma 11.8, n0 = (0, m0 ) and ˜ ⊂ (−1, 1) × Γ0 ⊂ U ∩ ∂ M. ˜ V 0 a neighborhood of n0 in U such that V 0 ∩ ∂ M ˜ in place of M and We then apply Theorem 9.6 to the function u with M ˜ B = B therein, yielding, for some δ ∈ (0, 1),

uH 1 (U )  u1−δ ˜ + |u|V 0 ∩∂ M ˜| 2 0 H 1 (V) hL2 (V) + |h∂ |L2 (V∩∂ M) ˜ L (V ∩∂ M) δ + |Bu|V 0 ∩∂ M ˜| 2 0 ˜ L (V ∩∂ M)

δ  u1−δ + |h | + |u | . h 2 2 0 1 ˜ ˜ 2 0 ∂ L (V∩∂ M) L (V) |V ∩∂ M H (V) ˜ L (V ∩∂ M)

With this estimate, we conclude following the end of the proof of Lemma 10.13 starting from estimate (10.5.6).  Remark 11.9. Here, in the case d = 2, we have assumed that |X  |g∂ < 1. A reason associated with the nature of the spectrum of the generator of the wave semigroup is put forward in the introductory Sect. 11.1. In the proof of Lemma 11.8, the condition |X  |g∂ < 1 also appears in a ˇ crucial way for the Lopatinski˘ı–Sapiro condition to hold for the pair (Q, B) if it holds for the pair (P, B). We further discuss this point here. In fact, assume now that |X  |g∂ > 1. If one assumes that P , along with the boundary condition B, yields a selfadjoint operator, this is the alternative case as shown in Theorem 4.3 in the case d = 2. ˇ In the proof of Lemma 11.8, the Lopatinski˘ı–Sapiro condition holds for (Q, B) at m if and only if ˇb(m, ω  , i|(σ, ω  )| ) = |(σ, ω  )| − X  , ω   = 0, m g˜m g˜m  for (σ, ω  ) = (0, 0). This follows from Proposition 2.3. Here, ω  and Xm both lie in a one-dimensional space. Thus, one can choose ω  = 0 such that  , ω   = −|X  | |ω  | . Then, the condition ˇ b(m, ω  , i|(σ, ω  )|g˜m ) = 0 Xm g∂ m g∂ reads  2 |g∂ |ω  |2g∂ . σ 2 + |ω  |2g∂ = |Xm

260

11. GENERAL BOUNDARY DAMPING

   |2 − 1 > 0, if one chooses σ = ± |X  |2 − 1 1/2 |ω  | , one obtains Since |Xm g∂ g∂ m g∂ ˇb(m, ω  , i|(σ, ω  )| ) = 0 meaning that the Lopatinski˘ı–Sapiro ˇ condition does g˜m not hold for (Q, B) at m. The proof based on the unique continuation results ˇ of Chap. 9 cannot be carried out, since the Lopatinski˘ı–Sapiro condition is  | assumed at the boundary therein. One thus sees that the condition |Xm g∂ < 1 is also an important technical assumption in the proof scheme we carry out. 11.6. Notes The present chapter generalizes the result of Chap. 10 obtained in the case of a Neumann boundary damping. Chapter 10 could have thus been omitted. Yet the Neumann boundary condition appears as quite classical and their independent treatment seemed important for that respect. The generalization we provide here encompasses all boundary operators that ˇ fit the framework given by the Lopatinski˘ı–Sapiro conditions recalled in Chap. 2 with the additional requirement that the Laplace–Beltrami operator be selfadjoint. Up to our knowledge, this cannot be found in the existing literature with the same level of generality. Here damping occurs at the boundary. Observe that we could also treat the case of an inner damping ⎧ 2 ⎪ in (0, +∞) × M, ⎨∂t y + P0 y + α∂t y = 0 By = 0 on (0, +∞) × ∂M, ⎪ ⎩ 0 1 y|t=0 = y , ∂t y|t=0 = y in M, with some α ≥ 0. Such a result relies on the analysis used in the Chapter 6 of Volume 1 and the present chapter. Appendix 11.A. The Generator of the Boundary-Damped Wave Semigroup Proposition 11.10. The unbounded operators (A, D(A)) are a closed operator on H . Proof. Let (V n )n ⊂ D(A), V n = t (v 0,n , v 1,n ), be such that both V n → V = t (v 0 , v 1 ) and AV n → W = t (w0 , w1 ) in H as n → +∞. We thus have (v 0,n )n ⊂ HD2 (M), (v 1,n )n ⊂ HD1 (M) and v 0,n → v 0 in HD1 (M),

v 1,n → v 1 in L2 (M).

As AV n = t (−v 1,n , P v 0,n ), we have −v 1,n → w0 in HD1 (M),

P v 0,n → w1 in L2 (M).

We thus obtain v 1 = −w0 ∈ HD1 (M) and in fact v 1,n → v 1 in HD1 (M). It thus remains to prove that v 0 ∈ HD2 (M), P v 0 = w1 , and B∂ V = 0.

11.A. THE GENERATOR OF THE BOUNDARY-DAMPED WAVE SEMIGROUP

261

Set w1,n = P v 0,n ∈ L2 (M). We have w1,n → w1 in L2 (M). With the Green formula (4.5.3), for ϕ ∈ ker(P), one writes , ϕ|1∂M )L2 (1∂M) = (αv|1,n , ϕ|1∂M )L2 (1∂M) , (w1,n , ϕ)L2 (M) = −(Bv|0,n 1 1 ∂M ∂M = 0 and B∂ (V n ) = 0. We set hn = −αv|1,n ∈ H 1/2 (1∂M). using that v|0,n 0 1 ∂M ∂M

As v 1,n → v 1 in HD1 (M), we find that hn → h = −αv 1 |1∂M in H 1/2 (1∂M) by the trace inequality of Proposition 18.24. In particular, (w1 , ϕ)L2 (M) + (h, ϕ|1∂M )L2 (1∂M)   = lim (w1,n , ϕ)L2 (M) + (hn , ϕ|1∂M )L2 (1∂M) = 0, n→+∞

if ϕ ∈ ker(P). Then, by Theorem 4.18, there exists a unique u ∈ H 2 (M) ∩ HD1 (M) solution to P u = w1 and Bu|1∂M = h. Moreover, one has v 0,n − uH 2 (M)  w1,n − w1 L2 (M) + |hn − h|H 1/2 (1∂M)  w1,n − w1 L2 (M) + v 1,n − v 1 H 1 (M) , by Proposition 18.24; see (4.7.3) for the definition of H 2 (M). Since v 0,n → v 0 in HD1 (M), we have v 0,n → v 0 in HD1 (M) implying that v 0 = u, that is, v 0 − u ∈ ker(P). This gives v 0 ∈ HD2 (M), P v 0 = P u = w1 , and Bv 0 |1∂M = Bu|1∂M = h = −αv 1 |1∂M meaning precisely that B∂ (V ) = 0.  Lemma 11.11. The norm .H˙ defined by 2

V 2H˙ := N (v 0 ) + v 1 L2 (M) ,

V = t (v 0 , v 1 ),

is equivalent to the norm .H on H˙ . Proof. The proof follows that of Lemma 10.16 by proving that  V  ˙ , V = t (v 0 , v 1 ) ∈ H˙ . v 0  2 L (M)

H

Assume that the inequality does not hold. Then, there exists a sequence V n = t (v 0,n , v 1,n )n∈N ⊂ H˙ such that v 0,n L2 (M) = 1,

N (v 0,n )1/2 + v 1,n L2 (M) → 0,

as n → ∞.

For λ ∈ R well chosen, Nλ (.)1/2 is a norm on HD1 (M ) by Proposition 4.14 with Nλ (.) = N (.) + λ.2L2 (M) . Thus the sequence (v 0,n )n∈N is bounded in HD1 (M), and, hence, there exists v 0 ∈ HD1 (M) such that v 0,n  v 0 in ˜ (v, u) is a HD1 (M) up to a subsequence. If u ∈ HD1 (M), the map v → N ˜ (v 0 , u) as n → +∞. ˜ (v 0,n , u) → N continuous form. We thus find N Yet by Cauchy–Schwarz like inequality (4.6.8), one finds ˜ (v 0,n , u) → 0, as n → +∞, N since N (v 0,n ) → 0. One thus obtains ˜ (v 0 , u) = 0, for all u ∈ H 1 (M). N D Then, by Theorems 4.18 and 4.19, one has v 0 ∈ ker(P).

262

11. GENERAL BOUNDARY DAMPING

Since V n ⊂ H˙ , one has Fα,v0 (V n ) = 0, which reads (αv|0,n , v0 ) 2 1 + (v 1,n , v 0 )L2 (M) = 0. 1 ∂M |1∂M L ( ∂M)  v|01∂M in H 1/2 (1∂M) and v 1,n → 0 in L2 (M), one obtains Since v|0,n 1 ∂M (αv|01∂M , v|01∂M )L2 (1∂M) = 0. By Lemma 11.4, we conclude that v 0 = 0 since v 0 ∈ ker(P). This is in  contradiction with having v 0,n L2 (M) = 1 since v 0,n → v 0 in L2 (M). Associated with the norm .H˙ is the following inner product, for U, V ∈ H˙ , ˜ (v 0 , u0 ) + (v 1 , u1 )L2 (M) . (11.A.1) (V, U ) ˙ := N H

˙ D(A)) ˙ on H˙ is closed. Proposition 11.12. The unbounded operator (A, ˙ is dense in H˙ and the injection of D(A) ˙ into H˙ is comMoreover, D(A) pact. In the proof of Proposition 11.12, we shall need the following lemma. Lemma 11.13. The map ˙ → H 1 (M) D(A) D (u0 , u1 ) → u1 is surjective. Moreover, there exists a bounded linear map Ψ : HD1 (M) → ˙ for all w ∈ H 1 (M). HD2 (M) such that (Ψw, w) ∈ D(A) D Proof of Lemma 11.13. Let w ∈ HD1 (M) and set h = −αw|1∂M ∈ H 1/2 (1∂M) using the trace property of Proposition 18.24. Pick then f = n−1 j=0 fj φj ∈ ker(P) such that (f, φj )L2 (M) + (h, φj |1∂M )L2 (1∂M) = 0, j = 0, . . . , n − 1. The map w → f is linear, and we have f L2 (M)  |h|L2 (1∂M)  wH 1 (M) . By Theorem 4.18, there exists v ∈ H 2 (M) ∩ HD1 (M) such that P v = f and Bv|1∂M = h. Moreover, we have vH 2 (M)  f L2 (M) + |h|H 1/2 (1∂M)  wH 1 (M) . With a Hilbert basis {ϕ0 , . . . , ϕn−1 } of ker(P) associated with the inner product of Lemma 11.4, we set aj = (αv|1∂M , ϕj |1∂M )L2 (1∂M) + (w|1∂M , ϕj |1∂M ) n−1 L2 (1∂M) and Ψw = v− j=0 aj ϕj . We then have (αΨw|1∂M , ϕj |1∂M )L2 (1∂M) + (w|1∂M , ϕj |1∂M )L2 (1∂M) = 0. The map w → Ψw is linear and V = t (Ψw, w) ∈ ˙ Moreover, we see that D(A).   n−1   a j ϕj   v|1∂M  2 1 + w|1∂M  2 1  j=0

L2 (M)

L ( ∂M)

L ( ∂M)

 vH 2 (M) + wH 1 (M)  wH 1 (M) .

11.A. THE GENERATOR OF THE BOUNDARY-DAMPED WAVE SEMIGROUP

263

Consequently, ΨwH 2 (M)  wH 1 (M) . The proof is complete.



˙ D(A)) ˙ is closed, one Proof of Proposition 11.12. To prove that (A, can write the proof given for Proposition 10.17 without any change. The ˙ into H˙ is also the argument for the compactness of the injection of D(A) same. ˙ in H˙ . We consider We now address the question of the density of D(A) ˙ and we prove that U = 0, U ∈ H˙ such that (U, V )H˙ = 0 for all V ∈ D(A), ˙ ˙ which asserts that D(A) is dense in H . Let f ∈ L2 (M) = ker(P)⊥ . By Proposition 4.17, one has f ∈ Ran(P). Thus there exists v ∈ D(P) such that Pv = f . Note that v ∈ HD2 (M) and Bv|1∂M = 0. With a Hilbert basis {ϕ0 , . . . , ϕn−1 } of ker(P) associated with the inner product of Lemma 11.4, we set aj = (αv|1∂M , ϕj |1∂M )L2 (1∂M) and  0 v 0 = v − n−1 j=0 aj ϕj ∈ D(P). Observe that (αv|1∂M , ϕj |1∂M )L2 (1∂M) = 0 for j = 0, . . . , n − 1. ˙ Then, with (4.6.7), Set V = t (v 0 , 0) and observe that V ∈ D(A). ˜ (u0 , v 0 ) = (u0 , Pv 0 )L2 (M) = (u0 , f )L2 (M) . 0 = (U, V ) ˙ = N H

Since f ∈ find that (11.A.2)

ker(P)⊥

is arbitrary, one concludes that u0 ∈ ker(P). We thus

˜ (u0 , v 0 ) + (u1 , v 1 )L2 (M) 0 = (U, V )H˙ = N = (Pu0 , v 0 )L2 (M) + (u1 , v 1 )L2 (M) = (u1 , v 1 )L2 (M) ,

˙ Let w ∈ HD1 (M). By Lemma 11.13, there for V = t (v 0 , v 1 ) ∈ D(A). 2 ˙ Then, (11.A.2) yields exists v ∈ HD (M) such that V = t (v, w) ∈ D(A). 1 1 (u , w)L2 (M) = 0. As w ∈ HD (M) is arbitrary, this gives u1 = 0 as HD1 (M) is dense in L2 (M). Thus, U = t (u0 , 0) ∈ H˙ ∩ N since u0 ∈ ker(P). By (11.3.2), we finally obtain U = 0. This concludes the proof of the density property.  From (11.3.2), (11.3.5), and Proposition 11.12, we obtain the following result. Corollary 11.14. The domain D(A) is dense in H . We now make explicit the operator A˙ ∗ adjoint of A˙ with respect to the inner product on H˙ introduced in (11.A.1). We recall that the domain of the adjoint is defined by   ˙ |(V, AU ˙ ) ˙ | ≤ CU  ˙ . D(A˙ ∗ ) = V ∈ H˙ ; ∃C > 0, ∀U ∈ D(A), H H Proposition 11.15. We have (11.A.3)   D(A˙ ∗ ) = V = t (v 0 , v 1 ) ∈ H˙ ; v 0 ∈ HD2 (M), v 1 ∈ HD1 (M); B∂∗ (V ) = 0

264

11. GENERAL BOUNDARY DAMPING

with B∂∗ (V ) = −Bv 0 + αv 1 . For V = t (v 0 , v 1 ) ∈ D(A˙ ∗ ), we have  1  v 0 1 ∗ ∗ ˙ ˙ , that is, A = A V = (11.A.4) . −P 0 −P v 0   Proof. Let V ∈ HD2 (M) × HD1 (M) ∩ H˙ be such that B∂∗ (V ) = 0. If ˙ we have U = t (u0 , u1 ) ∈ D(A), ˙ V ) ˙ = −N ˜ (u1 , v 0 ) + (P u0 , v 1 )L2 (M) . (AU, H ˜ in (4.6.5), Lemma 4.7, and formula (18.5.5), and With the definition of N the form of B, we compute ˜ (u1 , v 0 ) = (∇g u1 , ∇g v 0 )L2 V (M) + (u1 , R1 v 0 )L2 (M) N + (u1 |1∂M , B  v 0 |1∂M )L2 (1∂M) = (u1 , −Δg v 0 )L2 (M) + (u1 , R1 v 0 )L2 (M) + (u1 |1∂M , ∂ν v 0 |1∂M )L2 (1∂M) + (u1 |1∂M , B  v 0 |1∂M )L2 (1∂M) = (u1 , P v 0 )L2 (M) + (u1 |1∂M , Bv 0 |1∂M )L2 (1∂M) . Similarly, ˜ (u0 , v 1 ) − (Bu0 1 , v 1 1 ) 2 1 (P u0 , v 1 )L2 (M) = N | ∂M | ∂M L ( ∂M) , yielding ˜ (u0 , v 1 ) ˙ V ) ˙ = −(u1 , P v 0 )L2 (M) + N (AU, H − (u1 |1∂M , Bv 0 |1∂M )L2 (1∂M) . − (Bu0 |1∂M , v 1 |1∂M )L2 (1∂M) . Using that B∂ (U ) = 0, that is, (Bu0 + αu1 )|1∂M = 0, for the two boundary terms, we obtain (u1 |1∂M , Bv 0 |1∂M )L2 (1∂M) + (Bu0 |1∂M , v 1 |1∂M )L2 (1∂M) = (u1 |1∂M , Bv 0 |1∂M )L2 (1∂M) − (αu1 |1∂M , v 1 |1∂M )L2 (1∂M) = (u1 |1∂M , (Bv 0 − αv 1 )|1∂M )L2 (1∂M) = 0, since B∂∗ (V ) = (Bv 0 − αv 1 )|1∂M = 0. We thus have (11.A.5)

˜ (u0 , v 1 ), ˙ V ) ˙ = −(u1 , P v 0 )L2 (M) + N (AU, H

which yields, as v 0 ∈ HD2 (M) and v 1 ∈ HD1 (M), 0 1/2 ˙ V ) ˙ |  u1  2 |(AU,  U H˙ , L (M) + N (u ) H

by (4.6.8) and the form of the norm .H˙ defined in Lemma (11.11). Thus V ∈ D(A˙ ∗ ) and (11.A.5) precisely yields the form of A˙ ∗ V given in (11.A.4).

11.A. THE GENERATOR OF THE BOUNDARY-DAMPED WAVE SEMIGROUP

265

˙ we have Conversely, let V = t (v 0 , v 1 ) ∈ D(A˙ ∗ ). If U = t (u0 , u1 ) ∈ D(A), (11.A.6)

˜ (u1 , v 0 ) + (P u0 , v 1 )L2 (M) , ˙ V ) ˙ = −N (AU, H

and (11.A.7)

˙ V ) ˙ |  U  ˙  N (u0 )1/2 + u1  2 |(AU, L (M) . H H

By the Riesz representation theorem, there exists W = t (w0 , w1 ) ∈ H˙ such that (11.A.8)

˜ (u0 , w0 ) + (u1 , w1 )L2 (M) . ˙ V ) ˙ = (U, W ) ˙ = N (AU, H H

Let f ∈ Ran(P), and let u ∈ HD2 (M) be such that Bu|1∂M = 0 and P u = f by Theorem 4.18. With the orthonormal basis {ϕ0 , . . . , ϕn−1 } of ker(P)  introduced in Sect. 11.3, we set u0 = u + n−1 j=0 νj ϕj , with νj chosen such that (u0|1∂M , ϕj |1∂M )L2 (1∂M) = 0,

j = 0, . . . , n − 1.

Observe that we have u0 ∈ D(P) with P u0 = f and moreover U = t (u0 , 0) ∈ ˙ With such a choice, we have on the one hand D(A). ˙ V ) ˙ = (P u0 , v 1 )L2 (M) , (AU, H and on the other hand, ˜ (u0 , w0 ) = (P u0 , w0 )L2 (M) , ˙ V) ˙ =N (AU, H by (4.6.7). Consequently, (f, v 1 − w0 )L2 (M) = (P u0 , v 1 − w0 )L2 (M) = 0, for all f ∈ Ran(P) implying that v 1 − w0 ∈ Ran(P)⊥ = ker(P) by Proposition 4.17. In particular, v 1 ∈ HD1 (M). ˙ we may now write, by (4.6.4), From (11.A.6), with U = t (u0 , u1 ) ∈ D(A), ˜ (u1 , v 0 ) + (P u0 , v 1 )L2 (M) ˙ V ) ˙ = −N (AU, H ˜ (u1 , v 0 ) − (∂ν u0 1 , v 1 1 = −N

| ∂M )L2 (1∂M) + (∇g u , ∇g v )L2 V (M) + (R1 u0 , v 1 )L2 (M) ˜ (u1 , v 0 ) + (αu1 1 , v 1 1 ) 2 1 = −N | ∂M | ∂M L ( ∂M) + | ∂M

0

1

˜ (u0 , v 1 ), N

˜ (., .) in (4.6.5). We then find using that B∂ (U ) = 0 and the definition of N (11.A.9) ˜ (u1 , v 0 ) + (αu1 1 , v 1 1 ) 2 1 ˜ 0 0 ˙ V ) ˙ = −N (AU, | ∂M | ∂M L ( ∂M) + N (u , w ), H ˜ (z, z  ) = 0 for all z ∈ H 1 (M) if z  ∈ ker(P); see (4.6.7). since N D With (11.A.8), we conclude that (11.A.10)

1 1 ˜ (u1 , v 0 ) + (αu1 1 , v 1 1 ) 2 1 −N | ∂M | ∂M L ( ∂M) = (u , w )L2 (M) .

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11. GENERAL BOUNDARY DAMPING

Let w ∈ HD1 (M). From Lemma 11.13, there exists z ∈ HD2 (M) such that t (z, w) ∈ D(A). Then, from (11.A.10), one deduces (11.A.11) 1 ˜ (w, v 0 ) + (w 1 , αv 1 1 ) 2 1 −N | ∂M | ∂M L ( ∂M) = (w, w )L2 (M) ,

w ∈ HD1 (M).

Note that choosing w ∈ ker(P), one finds (11.A.12)

−(w, w1 )L2 (M) + (w|1∂M , αv 1 |1∂M )L2 (1∂M) = 0.

Then, Theorem 4.19 applies to the variational equation (11.A.11) under condition (11.A.12), yielding a unique solution v ∈ H 2 (M) ∩ HD1 (M) (use also Theorem 4.18). As a result, v 0 − v ∈ ker(P) implying that v 0 ∈ HD2 (M) and Bv 0 = αv 1 on 1∂M. As V = t (v 0 , v 1 ) ∈ H˙ , we have Fα (V ) = 0. We have therefore obtained that D(A˙ ∗ ) is as described in (11.A.3). The proof is complete.  The resolvent estimate of the following lemma is key in the proof that A˙ generates a C0 -semigroup on H˙ . Lemma 11.16. Let B = A˙ or A˙ ∗ and z ∈ C be such that Re z < 0. We have (z IdH˙ −B)U H˙ ≥ | Re z| U H˙ , U ∈ D(B). ˙ We write Proof. Let U = t (u0 , u1 ) ∈ D(A).   0 zu0 + u1 u ˙ ((z IdH˙ −A)U, U )H˙ = , 1 1 0 zu − P0 u u H˙ 2 1 0 ˜ (u , u ) − (P u0 , u1 )L2 (M) = zU  ˙ + N

H ˜ (u1 , u0 ) − (R1 u0 , u1 )L2 (M) = zU 2H˙ + N − (∇g u0 , ∇g u1 )L2 V (M) + (∂ν u0 |∂M , u1 |∂M )L2 (∂M) ,

using formula (18.5.5). Since we have B∂ (U ) = (∂ν +B  )u0 |∂M +αu1 |∂M = 0, we obtain ˙ ˜ (u1 , u0 ) − (αu0 |∂M , u1 |∂M )L2 (∂M) , U ) ˙ = zU 2 ˙ + 2i Im N ((z Id ˙ −A)U, H

H

H

˜ (., .) in (4.6.5). Computing the real part of the recalling the definition of N above equality, one obtains ˙ (11.A.13) − Re((z IdH˙ −A)U, U )H˙ = − Re(z)U 2H˙ + ∫ α|u1 |2|∂M μg∂ . ∂M

As α ≥ 0 and Re z < 0, we find ˙ U )H˙ | ≥ | Re(z)| U 2H˙ , | Re((z IdH˙ −A)U, ˙ which yields the conclusion for B = A. The conclusion follows similarly in the case B = A˙ ∗ . We may now state and prove that A˙ generates a C0 -semigroup.



11.A. THE GENERATOR OF THE BOUNDARY-DAMPED WAVE SEMIGROUP

267

˙ D(A)) ˙ generates a C0 Theorem 11.17. The unbounded operator (A, ˙ −t A ˙ ˙ semigroup of contraction S(t) = e on H . The argument, based on the Hille–Yosida theorem, is that of the proof of Theorem 10.22 without modification. We define the following bounded operator on H : (11.A.14)

˙ ◦ Π ˙ + ΠN , S(t) := S(t) H

with t ≥ 0 as a parameter, and where ΠH˙ and ΠN are the projectors associated with the direct sum in (11.3.2). Observe that we have the following commutative diagram:

˙ The operator S(t) coincides with S(t) on H˙ and coincides with the identity map on its complement N . Theorem 11.18. The map S(t) is a bounded C0 -semigroup on H ; its generator is the unbounded operator (A, D(A)). The proof is mutatis mutandis that of Theorem 10.23. ˙ the norm inherited from that of H 2 (M)⊕ Proposition 11.19. On D(A) ˙  ˙  AU ˙  . H 1 (M) is equivalent to the norm U → AU H H ˙ ⊂ H 2 (M) ⊕ H 1 (M). We have Proof. Let U = t (u0 , u1 ) ∈ D(A) ˙ = t (−u1 , P u0 ) ∈ H˙ by Lemma 11.5 and as A = A˙ ◦ Π ˙ . Following the AU H proof of the counterpart Proposition 10.25, it suffices to prove that we have (11.A.15)

u0 H 2 (M)  u1 H 1 (M) + P u0 L2 (M) .

Since B∂ (U ) = 0 we have Bu0 |1∂M = −αu1 |1∂M ∈ H 1/2 (1∂M), with αu1 |1∂M 

H 1/2 (1∂M)

 u1 H 1 (M) ,

by the trace inequality of Proposition 18.24. If ϕ ∈ ker(P) with the identity (4.5.3), one has (P u0 , ϕ)L2 (M) + (Bu0|1∂M , ϕ|1∂M )L2 (1∂M) = 0. Then, by the elliptic regularity result of Theorem 4.18, one has (11.A.16)

u0 H 2 (M)  P u0 L2 (M) + u1 H 1 (M) ,

268

11. GENERAL BOUNDARY DAMPING

where u0 = u0 − Πker(P) u0 , where Πker(P) is the orthogonal projection onto ker(P) in L2 (M). We thus have (11.A.17)

u0 H 2 (M)  u0 H 2 (M) + Πker(P) u0 H 2 (M) .

˙ one has We set U = t (u0 , u1 ) = U − t (Πker(P) u0 , 0) ∈ H . Since U ∈ D(A), ΠN U = −t (Πker(P) u0 , 0), since t (Πker(P) u0 , 0) ∈ N . Since dim ker(P) < ∞ one has Πker(P) u0 H 2 (M)  Πker(P) u0 H 1 (M)  ΠN U H  U H  u0 H 1 (M) + u1 L2 (M) , from the continuity of ΠN on H . With (11.A.16) and (11.A.17), we thus obtain (11.A.15), which concludes the proof.  From Proposition 11.19 arguing as in the proof of Corollary 10.26, one obtains the following result. Corollary 11.20. On D(A), the norm inherited from that of H 2 (M)⊕ H 1 (M) is equivalent to the graph norm .D(A) given by V 2D(A) = V 2H + AV 2H .

The study of the stabilization properties of the damped wave equation requires to prove that the imaginary axis {Re z = 0} is in the resolvent set ˙ This is the subject of the following proposition. of the operator A. ˙ D(A)) ˙ is contained in Proposition 11.21. The spectrum of (A, {z ∈ C; Re(z) > 0}. Proof. Let z ∈ C. We consider the two cases. Case 1: Re z < 0. Replacing Lemma 10.21 by Lemma 11.16, one concludes that z is in the resolvent set of A˙ as in the counterpart case in the proof of Proposition 10.27. ˙ Case 2: Re z = 0. We start by proving the injectivity of z IdH˙ −A. ˙ be such that zU − AU ˙ = 0. This gives Let thus U = t (u0 , u1 ) ∈ D(A) zu0 + u1 = 0,

−P u0 + zu1 = 0.

First, if z = 0, one has u1 = 0, P u0 = 0, and Bu0 |∂M = 0 as B∂ (U ) = 0. Thus u0 ∈ ker(P), meaning that U = t (u0 , 0) ∈ N . Yet ΠN U = 0 since U ∈ H˙ = ker(ΠN ). Thus U = 0.

11.A. THE GENERATOR OF THE BOUNDARY-DAMPED WAVE SEMIGROUP

269

Second, if z = 0, using (11.A.13), we obtain ˙ 0 = Re((z IdH˙ −A)U, U )H˙ = − ∫ α|u1 |2|∂M μg∂ . ∂M

With the same argument as that used in the counterpart case in the proof of Proposition 10.27, one obtains P0 u0 = −z 2 u0 , and u0|1∂M = ∂ν u0|1∂M = 0 on Γ, where Γ is an open set of 1∂M where α > 0. With the unique continuation property, initiated from the boundary, as stated in Theorem 9.8, we obtain that u0 vanishes in M and u1 as well. The proof of the surjectivity of z IdH˙ −A˙ is the same as in the end of the proof of Proposition 10.27; it is based on the Fredholm index arguments. The result then follows from the closed graph theorem using that A˙ is a closed operator. 

CHAPTER 12

Spectral Inequality for General Boundary Conditions and Application Contents 12.1. 12.2. 12.3. 12.3.1. 12.3.2. 12.4. 12.5.

Setting Spectral Inequality The Parabolic Semigroup Spectral Representation and Sobolev Scale Weak Solutions of the Nonhomogeneous Parabolic Cauchy Problem Null-Controllability of the Associated Parabolic Equation Notes

271 273 275 276 278 278 280

12.1. Setting For (M, g) a smooth d-dimensional connected compact Riemannian manifold with boundary, with d ≥ 2, we consider a second-order differential operator P on M given by P = −Δg + R1 where R1 is a first-order differential operator. We also consider a differential boundary operator B defined in a neighborhood of ∂M of order 0 or 1 on each connected component of ∂M. On the part of ∂M where the order is 0, denoted by 0∂M, we consider the Dirichlet boundary operator, that is, Bu|0∂M = u|0∂M .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 12

271

272

12. SPECTRAL INEQUALITY FOR GENERAL BOUNDARY CONDITIONS

We introduce the unbounded operator P on L2 (M) with domain D(P) = {u ∈ L2 (M); P u ∈ L2 (M) and Bu|∂M = 0}. As in the previous chapter we consider the case where • the map u → (P u, Bu) has Fredholm properties; • the unbounded operator (P, D(P)) is selfadjoint; • the spectrum of P is composed of a sequence of eigenvalues μ0 ≤ μ1 ≤ · · · ≤ μn ≤ · · · , each of finite multiplicity, that grows to +∞. Those properties are fulfilled under assumptions made on R1 and B. Fredholm properties are equivalent to having (P, B) satisfying the ˇ Lopatinski˘ı–Sapiro conditions; see Theorem 3.1. In such case, as observed in the beginning of Chap. 4 one has D(P) = {u ∈ H 2 (M); Bu|∂M = 0}. By Theorem 4.3 the selfadjointness of (P, D(P)) is equivalent to having R1 = iV + f in M

and

B = ∂ν + B  with B  = iX  + h in 1∂M,

where we denote by ∂ν the outward pointing normal derivative in the sense of the metric g (see (5.3.1) in Sect. 5.3 and (17.4.12) in Sect. 17.4), V a smooth real vector field on M, f a complex-valued function on M, X  a smooth real vector field on 1∂M, and h a complex-valued function on ∂M, with the additional properties Im f = divg V /2 in M

and

2 Im h−divg X  +g(V, ν)|1∂M = 0 in 1∂M.

Moreover, in the case d = 2, one has |X  |g∂ = 1 and in the case d ≥ 3, we moreover have |X  |g∂ < 1. We furthermore require |X  |g∂ < 1 independently of the dimension. Consequently, the spectrum of P is composed of a sequence of eigenvalues of finite multiplicities that grows to +∞; see Theorems 4.9 and 4.12. If one does not assume |X  |g∂ < 1, in dimension d = 2 the case of eigenvalues that go to −∞ can occur (see Proposition 4.13). The operator P could not be the generator of a C0 -semigroup in such case. This case is not of relevance for us as we aim to study the null-controllability of the associated parabolic equation. In Sect. 12.2 we prove a spectral inequality associated with the spectral decomposition of P of the form of the spectral inequality proven in Chapter 7 of Volume 1. In Sect. 12.3 we define the parabolic semigroup associated with P and weak solution of the nonhomogeneous Cauchy problem that are needed for the definition of the controlled parabolic equation and its solutions

12.2. SPECTRAL INEQUALITY

(12.1.1)

⎧ ⎪ ⎨ ∂ t y + P y = 1ω v By = 0 ⎪ ⎩ y(0) = y 0

273

in (0, T ) × M, on (0, T ) × ∂M, in M,

where v is the control function that only acts in an open region ω of M. Finally, in Sect. 12.4 we prove the null-controllability of (12.1.1). The proof schemes are similar to those given in Chapter 7 of Volume 1. Arguments are adapted to take into account the more general boundary conditions and operator involved here. 12.2. Spectral Inequality One of the main results of this chapter is the following spectral inequality, counterpart of Theorem 7.10 of Volume 1. Theorem 12.1 (Spectral Inequality). Let P and B be such that the ˇ Lopatinski˘ı–Sapiro condition holds on ∂M and (P, D(P)) is selfadjoint. In the case d = 2 assume furthermore that X  is such that |X  |g∂ < 1 on 1∂M. Let (φj )j∈N be a Hilbert basis of L2 (M) made of eigenfunctions of (P, D(P)) associated with the sequence μ0 ≤ μ1 ≤ · · · ≤ μn ≤ · · · of eigenvalues. Let ω be an open set in M. There exists K > 0 such that for all μ ≥ max(μ0 , 0) one has (12.2.1)

wL2 (M) ≤ KeK

√

μ

wL2 (ω) ,

w ∈ span{φj ; μj ≤ μ}.

Proof. The proof is along the lines of that of Theorem 7.10, yet it uses the unique continuation property of Chap. 9 consequence of the Carleman ˇ estimate of Chap. 8 derived under the Lopatinski˘ı–Sapiro condition. If μ0 ≤ 0, we replace P by Pλ = P + λ Id with λ > −μ0 . Then, the eigenvalues of Pλ are simply 0 < μ0 + λ ≤ μ1 + λ ≤ · · · . Observe that if one derives a spectral inequality as (12.2.1) for Pλ , then one obtains also such an inequality for P, since (μ + λ)1/2  μ1/2 for μ ≥ 0. We may therefore assume that μ0 > 0. Let S0 > 0 and α ∈ (0, S0 /2) and set V = (0, S0 ) × M and U = (α, S0 − α) × M. Set also z = (s, x) with s ∈ (0, S0 ) and x ∈ M. We define the following augmented elliptic operator Q := Ds2 + P in V. We equip ˜ = R × M with the metric g˜ = ds ⊗ ds + g. M With the same argument as in the proof of Lemma 11.8 in Sect. 11.5 we ˇ find that the Lopatinski˘ı–Sapiro condition holds for (Q, B) on R × ∂M = ˜. ∂M

274

12. SPECTRAL INEQUALITY FOR GENERAL BOUNDARY CONDITIONS

s ˜ M

˜ ∂M

˜ ∂M

S0 V S0 − α U

α 0

∂M

M

ω

∂M

Figure 12.1. Geometry for the application of the quantified unique continuation property of Theorem 9.6 Since ω is an open set in M, from Theorem 9.6 there exist C > 0 and δ ∈ (0, 1) such that for u ∈ H 2 (V) that satisfies Bu|∂ M ˜ = 0 for s ∈ (0, S0 ) one has (12.2.2)

δ . uH 1 (U ) ≤ Cu1−δ H 1 (V) QuL2 (V) + |u|s=0 |H 1 (ω) + |∂s u|s=0 |L2 (ω) The geometry associated with the application of Theorem 9.6 is illustrated in Fig. 12.1.  Let μ ≥ μ0 and w ∈ span(φj ; μj ≤ μ), meaning that w = μj ≤μ αj φj . We set  1/2   αj sinh sμj φj . u(s, .) = 1/2 μj ≤μ μj It is in H 2 (V) and Qu = 0,

u|s=0 = 0.

Inequality (12.2.2) applied to u reads

As ∂s u|s=0 = (12.2.3)



δ uH 1 (U ) ≤ Cu1−δ H 1 (V) |∂s u|s=0 |L2 (ω) . μj ≤μ αj φj

= w, we have

|∂s u|s=0 |L2 (ω) = wL2 (ω) .

12.3. THE PARABOLIC SEMIGROUP

Note that (12.2.4)

275

√  1/2   1/2  1/2 μj | sinh sμj | + | cosh sμj |  eC μ .

This gives u2L2 (V) + ∂s u2L2 (V)  eC

√

μ

 μj ≤μ

|αj |2 .

We also write with Corollary 3.3 ∇g u(s, .)L2 V (M)  u(s, .)H 2 (M)  P u(s, .)L2 (M) + u(s, .)L2 (M) , With (12.2.4) one has S0



0

μj ≤μ

∫ P u(s, .)2L2 (M) ds =

S0

1/2

|αj |2 ∫ μj | sinh(sμj )|2 ds  eC 0

√

μ

 μj ≤μ

|αj |2 ,

yielding S0

∫ ∇g u(s, .)2L2 V (M) ds  eC

√

0

We thus conclude that u2H 1 (V)  eC

(12.2.5)

√

μ

 μj ≤μ

μ

 μj ≤μ

|αj |2 .

|αj |2 . 1/2

1/2

Observe now that for s ≥ α we have sinh(sμj )/μj ≥ α. Then, for some κ > 0 we have S0 −α   1/2 2 (12.2.6) |αj |2 ≤ |αj |2 ∫ μ−1 κ j | sinh(sμj )| ds μj ≤μ

0 0. Under the above assumption, we have the following results. Proofs are the same as in the classical case of homogeneous Dirichlet boundary conditions presented in Section 10.2 of Volume 1, using the analysis of the elliptic and spectral properties of (P, D(P)) in Sects. 4.6 and 4.7.

276

12. SPECTRAL INEQUALITY FOR GENERAL BOUNDARY CONDITIONS

Theorem 12.2. Assume that μ0 > 0. The operator (P, D(P)) is maximal monotone. Let T ∈ R+ ∪ {+∞}. The operator P generates C0 -semigroup of contraction S(t) = e−tP on L2 (M). If y 0 ∈ D(P), then y(t) = S(t)y 0 is the unique solution in C 0 ([0, T ]; D(P)) ∩ C 1 ([0, T ]; L2 (M)), such that y(0) = y 0 and d y(t) + Py(t) = 0 dt holds in L2 (M) for all 0 ≤ t ≤ T . The definition of monotone operators is recalled in Definition 12.22 of Volume 1 in the case of Hilbert spaces. That of maximal monotone operators is given in Definition 12.10. Here, [0, T ] means [0, +∞) if T = +∞. 12.3.1. Spectral Representation and Sobolev Scale. The spectral representation of the semigroup S(t) is the same as in Section 10.2.1 of Volume 1 and is based on the spectral characterization of L2 (M) by means of a Hilbert basis (φj )j∈N of eigenfunction of (P, D(P)), and that of HD1 (M) and D(P) given in Proposition 4.16. If y 0 ∈ L2 (M) and we set yj (t) = e−tμj (y 0 , φj )L2 (M) , for t ≥ 0 and j ∈ N, then (yj (t))j ∈ C 0 ([0, T ], 2 (C)) and  S(t)y 0 = yj (t)φj , t ≥ 0, j∈N

L2 (M).

with convergence in As in Section 10.1.3 of Volume 1 we can define the following spaces associated with the spectral family: for s ≥ 0, we set  s/2 uj φj ; (μj uj )j ∈ 2 (C)}. K s (M) = {u = j∈N

K 2 (M)

K 1 (M)

= D(P), = HD1 (M); see Proposition 4.16. MoreWe have   over, one has K s = D(Ps/2 ). For s ≥ 0 we also set K −s (M) = K s (M) and we find that if u ∈ K −s (M), then   −s/2 μj u, φj K −s (M),K s (M) j ∈ 2 (C). Using L2 (M) as a pivot space, we thus identify  −s/2 uj φj ; (μj uj )j ∈ 2 (C)}. K −s (M) = {u = j∈N

Similarly to Theorem 10.22 of Volume 1 we have the following result. Theorem 12.3. Let T ∈ R+ ∪ {+∞}. The semigroup S(t) is analytic and for y 0 ∈ L2 (M), the function y(t) = S(t)y 0 is in C 0 ([0, T ]; L2 (M)) ∩ C ∞ ((0, T ]; K s (M)),

s ∈ R,

12.3. THE PARABOLIC SEMIGROUP

277

and is such that (12.3.1) y(0) = y 0 and

d y(t) + Py(t) = 0 holds in L2 (M) for 0 < t ≤ T. dt

Moreover, y(t) = S(t)y 0 is the unique solution of (12.3.1) in C 0 ([0, T ]; L2 (M)) ∩ C 1 ((0, T ]; L2 (M)) ∩ C 0 ((0, T ]; D(P)). For r ∈ R and s ≥ 0, we also define the unbounded operator Psr : → K r (M) with domain K r+2s (M) given by  s  Psr u = μj u j φ j , u = uj φj ∈ K r+2s (M).

K r (M)

j∈N

j∈N

For s = 1 we write Pr instead of Psr . We have P0 = P. If r ≥ 0, the operator Psr is a restriction of Ps0 to K r (M) ⊂ K 0 (M) = L2 (M). If r < 0, the operator Psr is an extension of Ps0 to K r (M) ⊃ L2 (M). For any r ∈ R the operator Pr generates a C0 -semigroup on K r (M). For these operators we have counterpart results to Lemma 10.28 and Theorem 10.29 of Volume 1. For the nonhomogeneous parabolic Cauchy problem we can prove a result counterpart to Theorem 10.44 (also of Volume 1). Theorem 12.4. Let T ∈ R+ ∪{+∞} and r ∈ R. Let f ∈ L2 (0, T ; K r (M)) and y 0 ∈ K r+1 (M). There exists a unique function y ∈ C 0 ([0, T ]; K r+1 (M)) ∩L2 (0, T ; K r+2 (M)) ∩ H 1 (0, T ; K r (M)) that is solution of the parabolic equation d y + Pr y = f dt in L2 (0, T ; K r (M)) and satisfies moreover y(0) = y 0 . The solution is given by t

y(t) = S(t)y 0 + ∫ S(t − σ)f (σ)dσ. 0

Moreover, there exists C > 0 such that d  yL∞ (0,T ;K r+1 (M)) + yL2 (0,T ;K r+2 (M)) +  y  dt L2 (0,T ;K r (M)) 

≤ C y 0 K r+1 (M) + f L2 (0,T ;K r (M)) . For r = 0 we refer to strong solution in the case y 0 ∈ K 1 (M) and f ∈ L2 ((0, T ) × M) or in the case y 0 ∈ D(P) and f ∈ L2 (0, T ; D(P0 )). Statements of Corollaries 10.46 and 10.47 of Volume 1 can be adapted.

278

12. SPECTRAL INEQUALITY FOR GENERAL BOUNDARY CONDITIONS

12.3.2. Weak Solutions of the Nonhomogeneous Parabolic ˇ Cauchy Problem. Weak solutions in the case of Lopatinski˘ı–Sapiro type boundary condition can be introduced similarly to the more classical case of homogeneous Dirichlet conditions, as presented in Section 10.3.4 of Volume 1. Definition 12.5. Let T ∈ R+ ∪ {+∞}. Let y 0 ∈ L2 (M) and f ∈ One says that y ∈ C 0 ([0, T ]; L2 (M)) ∩ L2 (0, T ; K 1 (M)) is a weak solution to the parabolic equation d y + Py = f, y(0) = y 0 , dt if we have L2 (0, T ; K −1 (M)).

(y(t), ψ)L2 (M) + (y, Pψ)L2 ((0,t)×M) = (y 0 , ψ)L2 (M) t

+ ∫ f (σ), ψK −1 (M),K 1 (M) dσ, 0

for all ψ ∈ D(P) and for all t ∈ [0, T ]. An inspection of the proof of Theorem 10.49 of Volume 1 shows the following result can be proven mutatis mutandis. Theorem 12.6. Let T ∈ R+ ∪ {+∞}. Let y 0 ∈ L2 (M) and f ∈ L2 (0, T ; K −1 (M)). There exists a unique weak solution y to the parabolic equation d y + Py = f, y(0) = y 0 , dt in the sense of Definition 12.5. It coincides with the solution of the semigroup equation d y + P−1 y = f, y(0) = y 0 , dt given by Theorem 12.4 in the case r = −1. In particular we have y ∈ C 0 ([0, T ]; L2 (M)) ∩ L2 (0, T ; K 1 (M)) ∩ H 1 (0, T ; K −1 (M)). Naturally, uniqueness implies that a strong solution is also a weak solution. 12.4. Null-Controllability of the Associated Parabolic Equation For some T > 0 we consider the null-controllability problem of the parabolic problem associated with (P, B) with an inner control function v that acts on an open subset ω of M, ⎧ ⎪ in (0, T ) × M, ⎨ ∂ t y + P y = 1ω v (12.4.1) By = 0 on (0, T ) × ∂M, ⎪ ⎩ 0 in M. y(0) = y

12.4. NULL-CONTROLLABILITY OF THE ASSOCIATED PARABOLIC EQUATION279

The technique we use is that of Chapter 7 of Volume 1, based on the spectral inequality derived in Sect. 12.2. For y 0 ∈ L2 (M) and v ∈ L2 ((0, T )×M) solutions to the inner-controlled parabolic equation (12.4.1) are of weak type and given by Definition 12.5 and Theorem 12.6. In particular, such a solution y is unique in C 0 ([0, T ]; L2 (M)) ∩ L2 (0, T ; K 1 (M)) ∩ H 1 (0, T ; K −1 (M)) and is given by the Duhamel formula t

(12.4.2)

y(t) = S(t)y 0 + ∫ S(t − σ)1ω v(σ)dσ, 0

with the semigroup defined in Sect. 12.3. Definition 12.7. We say that the system in (12.4.1) is null-controllable  2 (M) at time T > 0 if for all y 0 ∈ L2 (M) one can choose v ∈ L2 (0, T )× in L  M such that the weak solution given by (12.4.2) satisfies y(T ) = 0. Above we saw that the C0 -semigroup Sλ (t) generated by Pλ = P + (λ − μ0 ) Id is given by Sλ (t) = e−(λ−μ0 )t S(t). Hence, with (12.4.2) one sees that the parabolic equation associated with P is null-controllable if and only if the parabolic equation associated with Pλ is null-controllable. We may thus assume as above that μ0 > 0. Null-controllability is equivalent to an observability inequality according to Proposition 7.7 of Volume 1. Here, it reads (12.4.3)

S(T )zL2 (M) ≤ Cobs 1ω S(t)zL2 ((0,T )×M) ,

z ∈ L2 (M).

We prove this observability inequality with the dual technique of Section 7.8 of Volume 1. Lemma 12.8. Let T > 0. Let f : [0, T ] → R be given by    t f (t) = exp − K(K + K 2 + 2t ln 6)/t , 4 where K is the constant in the spectral inequality of Theorem 12.1. We then have t

f (t)S(t)z2L2 (M) − f (t/2)z2L2 (M) ≤ ∫ 1ω S(σ)z2L2 (M) dσ. 0

The proof is exactly that of Lemma 7.20 of Volume 1 replacing the spectral inequality of Theorem 7.10 by that of Theorem 12.1; see Remark 7.21 (recall that (P, D(P)) is such that μ0 > 0 here). With Lemma 7.19 of Volume 1 we conclude that the observability inequality holds. Theorem 12.9 (Observability Under General Boundary Conditions). Let T > 0. There exists CT such that, S(T )z 0 L2 (M) ≤ CT 1ω S(t)z 0 L2 ((0,T )×M) ,

z 0 ∈ L2 (M).

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12. SPECTRAL INEQUALITY FOR GENERAL BOUNDARY CONDITIONS

Equivalently, we have the following null-controllability result. Theorem 12.9 (Null-Controllability Under General Boundary Conditions). Let T > 0. There exists CT > 0 such that for all initial conditions y 0 ∈ L2 (M), there exists v ∈ L2 ((0, T ) × M), with vL2 ((0,T )×M) ≤ CT y 0 L2 (M) , such that the weak solution y(t) ∈ C 0 ([0, T ]; L2 (M))∩L2 (0, T ; K 1 (M)) ∩ H 1 (0, T ; K −1 (M)) to d y + Py = 1ω v, y(0) = y 0 , dt satisfies y(T ) = 0. 12.5. Notes In the case d = 2, let P = −Δg + iV + f be as in the main text and B = ∂ν + iX  + h. Assume that the conditions of Theorem 4.3 hold meaning that (P, D(P)) is selfadjoint. Assume that |X  |g∂ > 1 on some connected component N of 1∂M, which is permitted by Lemma 4.5. Such an example is given in Proposition 4.13. Such an operator does not fit the framework of Theorem 12.1. If a spectral inequality were to hold for (P, D(P)), in the case of some discrete spectrum going to −∞ the inequality should naturally take the following form (12.5.1)

wL2 (M) ≤ KeK

√

μ

wL2 (ω) ,

w ∈ span{φj ; |μj | ≤ μ}.

ˇ While the Lopatinski˘ı–Sapiro condition holds for (P, B) on N one can check with Proposition 2.8 that the same condition fails to hold for (Q, B) with the operator Q given by Q = Ds2 + P . Note that the latter pair of operators is to ˇ be considered in dimension three as far as the Lopatinski˘ı–Sapiro condition is concerned. The proof of Theorem 12.1 does not apply to this case. To our knowledge having a spectral inequality for the operator (P, D(P)) in the form of (12.5.1) is an open question. Here, we have only considered first-order boundary conditions. There exist higher-order boundary conditions such that together with the Laplace– ˇ Beltrami operator, the Lopatinski˘ı–Sapiro condition holds and one obtains a selfadjoint operator. An example is the so-called Ventcel boundary condition B = ∂ν − hΔg∂M for h > 0, see Example 2.5-(4). With the results of R. Buffe in [92] obtained for a proof of logarithmic stabilization of the wave equation, it is very likely that a spectral inequality as in Theorem 12.1 can be derived and the null-controllability of the associated parabolic equation can be achieved.

Part 4

Further Aspects of Carleman Estimates

CHAPTER 13

Carleman Estimates with Source Terms of Weaker Regularity Contents 13.1. 13.2. 13.2.1. 13.2.2. 13.3. 13.3.1. 13.3.2.

Setting and Main Result Local Setting at the Boundary Choice of Coordinates Conjugation A Microlocal Estimate Case 1: One Root in the Upper Complex Half-Plane Case 2: Roots Are Different and Both in the Open Lower Complex Half-Plane 13.3.3. Case 3: Roots Coincide in the Open Lower Complex Half-Plane 13.4. Patching Estimates Together 13.4.1. Patching Microlocal Estimates 13.4.2. Patching Local Estimates 13.5. Shifted Estimate 13.6. Estimates Without Prescribed Boundary Conditions 13.6.1. Estimate for a Weight Function with Nonvanishing Neumann Trace 13.6.2. Estimate for a Weight Function with a Vanishing Neumann Trace 13.7. Global Estimates 13.7.1. Global Estimate with an Inner Observation 13.7.2. Global Estimate with a Boundary Observation 13.8. Notes Appendix 13.A. Some Technical Proofs 13.A.1. A Trace Result

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 13

284 288 288 289 290 291 295 297 301 302 303 305 305 305 315 324 324 324 325 326 326 326

283

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13. SOURCE TERMS OF WEAKER REGULARITY

ˇ 13.A.2. Trace Estimate Under the Lopatinski˘ı–Sapiro Condition 13.A.3. Sub-ellipticity of a First-Order Factor 13.A.4. Proof of the Shifted Estimate

326 328 328

13.1. Setting and Main Result As in Chap. 8, on a d-dimensional compact Riemannian manifold (M, g), we consider the following elliptic problem P u = f,

Bu|∂M = f∂ ,

where P = −Δg +R1 with R1 a first-order differential operator on M and B is differential operator of order at most k ∈ N in a neighborhood of a point of ˇ ∂M, where we assume that P and B satisfy the Lopatinski˘ı–Sapiro condition (see Definition 2.2). Here, the novelty lies in the regularity assumed for both f and f∂ . We shall in fact assume that f takes the form (13.1.1)

f = F0 + divg F,

F0 ∈ L2 (M), F ∈ L2 V (M),

that is, f ∈ H −1 (M). We recall that L2 V (M) is the space of L2 -vector fields on M (see Sect. 18.1). Then, one cannot expect the solution to have a regularity higher than H 1 (M) in general. In fact, away from the boundary we have the following local result. Proposition 13.1. Let (M, g) be a compact Riemannian manifold with boundary and let P = −Δg + R1 with R1 a first-order differential operator with bounded coefficients on M. Let m0 ∈ M \ ∂M, V 0 be an open neighborhood of m0 in M such that V 0 ∩ ∂M = ∅, and let ϕ ∈ C ∞ (M) be such that the pair (P, ϕ) has the sub-ellipticity property of Definition 5.1 in V 0 . Then there exist an open neighborhood V 1 of m0 in M and two constants C and τ∗ > 0 such that   (13.1.2) τ 1/2 eτ ϕ uτ,1 ≤ C eτ ϕ F0 L2 (M) + τ eτ ϕ F L2 V (M) , for τ ≥ τ∗ and u ∈ H 1 (M), F0 ∈ L2 (M), F ∈ L2 V (M) such that P u = F0 + divg F, and supp(u) ∪ supp(F0 ) ∪ supp(F ) ⊂ V 1 . A proof is given below. If u ∈ H 1 (M), if a Carleman estimate is derived at the boundary, the norms associated with the boundary terms need to be adapted. Naturally, for u ∈ H 1 (M) the Dirichlet trace u|∂M ∈ H 1/2 (∂M). The Neumann trace ∂ν u|∂M is not defined in general. Here, ν is the outward pointing unit vector field at ∂M. However, one has the following result. Proposition 13.2. Let u ∈ H 1 (M) be such that P u = F0 + divg F with F0 ∈ L2 (M) and F ∈ L2 V (M). Then, ∂ν u + g(F, ν) admits a trace at ∂M that lies in H −1/2 (∂M). Moreover, there exists C > 0 such that        ∂ν u + g(F, ν) ≤ C F0 L2 (M) + F L2 V (M) + uH 1 (M) . |∂M −1/2 H

(∂M)

13.1. SETTING AND MAIN RESULT

285

With the expression “admits a trace” one means that the trace map well defined for u and F smooth extends uniquely for the regularity we have here. A proof is given in Appendix 13.A.1. As in Chaps. 2 and 8, we consider a boundary operator B of order at most one in the direction normal to the boundary, that is, if B is of order k ∈ N we have Bu|∂M = B k−1 ∂ν u|∂M + B k u|∂M , where B k−1 and B k are smooth differential operators on ∂M of order k − 1 and k, respectively. If k = 0, that is, Bu|∂M is the Dirichlet trace, we have B k−1 = 0 and B k = 1. Here, we also define the boundary operator (13.1.3)

B(u, h) = B k−1 (∂ν u + h)|∂M + B k u|∂M ,

for a function h such that the trace (∂ν u + h)|∂M makes sense. Proposition 13.2 indicates that one can use h = g(F, ν). In the neighborhood of a point m0 of the boundary ∂M, we prove the following estimate. Proposition 13.3. Let (M, g) be a compact Riemannian manifold with boundary and let P = −Δg + R1 with R1 a first-order differential operator with bounded coefficients on M. Let m0 ∈ ∂M, V 0 be an open set of M such that m0 ∈ V 0 , and let ϕ ∈ C ∞ (M) be such that the pair (P, ϕ) has the sub-ellipticity property of Definition 5.1 in V 0 . Consider B a differential operator of order k ∈ N in V 0 . Moreover, assume that (P, B, ϕ) satisfies ˇ the Lopatinski˘ı–Sapiro condition at m0 (Definition 8.1). Then there exist an open neighborhood V 1 of m0 in M and two constants C and τ∗ > 0 such that (13.1.4) τ 1/2 eτ ϕ uτ,1 + τ |eτ ϕ u|∂M |τ,1/2 + τ |eτ ϕ (∂ν u + g(F, ν))|∂M |τ,−1/2   ≤ C eτ ϕ F0 L2 (M) + τ eτ ϕ F L2 V (M) + τ |eτ ϕ|∂M B(u, g(F, ν))|τ,1/2−k , for τ ≥ τ∗ and u ∈ H 1 (M), F0 ∈ L2 (M), F ∈ L2 V (M) such that P u = F0 + divg F, and supp(u) ∪ supp(F0 ) ∪ supp(F ) ⊂ V 1 . The norms used in the theorem are the Sobolev norms with a large parameter on M and ∂M introduced in Chap. 7. Following Chap. 8, the proof of this proposition is carried out by first the introduction of local coordinates in Sect. 13.2, second the derivation of microlocal estimates in Sect. 13.3 and, third the patching of these estimates in Sect. 13.4.1. The main result of this chapter follows from Propositions 13.1 and 13.3 by a patching argument in Sect. 13.4.2 already used in Section 3.5.

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13. SOURCE TERMS OF WEAKER REGULARITY

Theorem 13.4. Let (M, g) be a compact Riemannian manifold with boundary and let P = −Δg + R1 with R1 a first-order differential operator on M. Let V be an open set of M and set V∂ = V ∩ ∂M. Let ϕ ∈ C ∞ (M) be such that the pair (P, ϕ) has the sub-ellipticity property of Definition 5.1 in V . If V∂ = ∅, consider B a differential operator of order k ∈ N in V and ˇ assume moreover that (P, B, ϕ) satisfies the Lopatinski˘ı–Sapiro condition at all points m ∈ V∂ (Definition 8.1). Then, there exist C and τ∗ > 0 such that (13.1.5) τ 1/2 eτ ϕ uτ,1 + τ |eτ ϕ u|∂M |τ,1/2 + τ |eτ ϕ (∂ν u + g(F, ν))|∂M |τ,−1/2   ≤ C eτ ϕ F0 L2 (M) + τ eτ ϕ F L2 V (M) + τ |eτ ϕ|∂M B(u, g(F, ν))|τ,1/2−k , for τ ≥ τ∗ and u ∈ H 1 (M), F0 ∈ L2 (M), F ∈ L2 V (M) such that P u = F0 + divg F, and supp(u) ∪ supp(F0 ) ∪ supp(F ) ⊂ V . Remark 13.5. With commutator arguments, as in Chapter 3 of Volume 1, we have eτ ϕ uτ,1  τ eτ ϕ uL2 (M) + eτ ϕ D uL2 Λ1 (M) , where D u stands for the covariant derivative on M of u defined by means of the Levi-Civita connection. These notions are recalled in Sect. 17.4. Remark 13.6. The estimate of Theorem 13.4 is not a global estimate in the sense of Sections 3.6 and 5.4.1: the open set V cannot be equal to M since the sub-ellipticity condition on the whole M and the Lopatinˇ ski˘ı–Sapiro condition on the whole boundary ∂M are incompatible. The argument is similar to that given at the beginning of Section 3.6. If one has a global estimate, then Theorem 4.5 of Volume 1 used in all local charts implies that |dϕ| ≥ C > 0 in M. Hence ϕ reaches its maximum on ∂M, say at m0 ∈ ∂M. Then, ∂ν ϕ(m0 ) ≥ 0, which is compatible with having the ˇ Lopatinski˘ı–Sapiro condition holding at m0 by the first item of Theorem 8.7. A global estimate requires the introduction of an observation term as in Sections 3.6 and 5.4.1. Result of this type are given in Sect. 13.7. Remark 13.7. Compared with Theorem 8.14 the norm τ −1/2 eτ ϕ uτ,2  τ 3/2 eτ ϕ uL2 (M) + τ 1/2 eτ ϕ D uL2 Λ1 (M) + τ −1/2 eτ ϕ H uL2 Λ2 (M) is replaced by τ 1/2 eτ ϕ uτ,1  τ 3/2 eτ ϕ uL2 (M) + τ 1/2 eτ ϕ D uL2 Λ1 (M) . This is consistent with the smoothness of the function u that is limited to H 1 because of the nature of the source term F0 + divg F ∈ H −1 (M).

13.1. SETTING AND MAIN RESULT

287

Similarly, for the estimation of the trace u|∂M the norm |eτ ϕ tr(u)|τ,3/2 is replaced by τ |eτ ϕ u|∂M |τ,1/2 since here u|∂M ∈ H 1/2 (∂M). We finish this introductory section by providing the proof of the Carleman estimate away from the boundary. The proof is rather simple; in fact, the main difficulties arise at the boundary and do not appear here. They are treated in the sections below where Proposition 13.3 is proven. Proof of Proposition 13.1. Consider C = (O, κ) a local chart such that m0 ∈ O. It is sufficient to prove an estimate in an open neighborhood U of x0 = κ(m0 ) in Rd with U ⊂ κ(O). In fact, we take U an open set of Rd such that U  κ(O ∩ V ). Here, we simply write ϕ and P for the local representatives of the weight function and the operator. The conjugated operator is Pϕ = eτ ϕ P e−τ ϕ . Let u ∈ H01 (U ) such that P u = F0 +divg F where F0 ∈ L2 (Rd ) and F ∈ L2 V (R2 ) are such that supp(F0 ) ∪ supp(F ) ⊂ U . We set v = eτ ϕ u and G = eτ ϕ F . We then have Pϕ v = G0 + divg G with G0 = eτ ϕ F0 − τ eτ ϕ g(∇g ϕ, F ), using (17.2.6). Since U is bounded we have G0 ∈ L2 (U ) and G ∈ L2 V (U ). Let ψ ∈ Cc∞ (κ(O ∩ V )) be such that ψ ≡ 1 in a neighborhood of U . Setting w = ψΛ−1 τ v, we write −1 −1 Pϕ w = ψΛ−1 τ Pϕ v + [Pϕ , ψΛτ ]v = ψΛτ (G0 + divg G) 2 + [Pϕ , ψΛ−1 τ ]v ∈ L (U ),

and we find (13.1.6)

Pϕ wL2 (U )  τ −1 G0 L2 (U ) + GL2 V (U ) + vL2 (U ) .

−1 Because of the respective supports of ψ and v, we write w = Λ−1 τ v+[ψ, Λτ ]v. This yields

(13.1.7)

vτ,1 = Λ−1 τ vτ,2  wτ,2 + vL2 (U ) .

Since ϕ and P have the sub-ellipticity property in U . As a result, Theorem 3.11 of Volume 1 applies. We rather use the equivalent form (3.3.3), that is, τ −1/2 wτ,2  Pϕ wL2 (U ) that, together with (13.1.6) and (13.1.7), implies τ −1/2 vτ,1  τ −1 G0 L2 (U ) + GL2 V (U ) + vL2 (U ) . Taking τ > 0 sufficiently large and multiplying by τ we obtain τ 1/2 vτ,1  G0 L2 (U ) + τ GL2 V (U )  eτ ϕ F0 L2 (U ) + τ eτ ϕ F L2 V (U ) , which concludes the proof with V 1 = κ−1 (U ).



288

13. SOURCE TERMS OF WEAKER REGULARITY

13.2. Local Setting at the Boundary 13.2.1. Choice of Coordinates. At a point m0 ∈ ∂M, we use the local chart C = (O, κ) associated with normal geodesic coordinates introduced in Sect. 8.3.2; see also Sections 9.4 and 17.6. We have κ(∂M ∩ O) = {xd = 0} ∩ κ(O),

κ(M ∩ O) = {xd ≥ 0} ∩ κ(O).

Set x0 = κ(m0 ). In this chart, the principal symbol of the Laplace–Beltrami operator takes the form  pC (x, ξ) = ξd2 + r(x, ξ  ), r(x, ξ  ) = g C,ij (x)ξi ξj  |ξ  |2 . 1≤i,j≤d−1

As ∂ν = −iDd , the local representative of the principal symbol of B is given by bC (x, ξ) = bk (x, ξ  ) − ibk−1 (x, ξ  )ξd , where bj (x, ξ  ), j = k, k − 1 are polynomial and homogeneous of degree j in ξ  . We set P0 (x, D) = Op(pC ) = Dd2 + R0 (x, D ),

with R0 = OpT (r),

and B0 = Op(bC ) = OpT (bk ) − iOpT (bk−1 )Dd . In agreement with (13.1.3), we then define B0 (u, h) = OpT (bk )u|xd =0+ − iOpT (bk−1 )(Dd u + ih)|xd =0+ .

(13.2.1)

Consider u ∈ H 1 (Rd+ ) that satisfies P0 u = F0 + divg F, with F0 ∈ L2 (Rd+ ), F ∈ L2 V (Rd+ ).

(13.2.2)

From Proposition 13.2 one has (Dd u − iF d )|xd =0+ = i(∂ν u + g(F, ν)) ∈ H −1/2 (Rd−1 ),

(13.2.3)

implying B0 (u, −F d ) ∈ H 1/2−k (Rd−1 ). The following result is equivalent to Proposition 13.3. Proposition 13.8. Let P0 = Op(pC ), B0 = Op(bC ). Let m0 ∈ ∂M, V 0 be an open neighborhood of m0 in M that meets one connected component of ∂M. Assume that (P0 , ϕC ) has the sub-ellipticity property of Definition 3.2 of Volume 1 in U 0 with U 0 = κ(V 0 ∩O). Moreover, assume that (P0 , B0 , ϕC ) ˇ satisfies the Lopatinski˘ı–Sapiro condition at x0 (Definition 8.1). Then, there exists a bounded neighborhood U+ of x0 in Rd+ such that U+ ⊂ U 0 and two constants C and τ∗ > 0 such that (13.2.4) C

C

C

τ 1/2 eτ ϕ uτ,1 + τ |eτ ϕ u|xd =0+ |τ,1/2 + τ |eτ ϕ (Dd u − iF d )|xd =0+ |τ,−1/2 

C C C ≤ C eτ ϕ F0 + + τ eτ ϕ F + + τ |eτ ϕ B0 (u, −F d )|τ,1/2−k ,

13.2. LOCAL SETTING AT THE BOUNDARY

289

for τ ≥ τ∗ and u ∈ H 1 (Rd+ ), F0 ∈ L2 (Rd+ ), F ∈ L2 V (Rd+ ) such that P0 u = F0 + divg F, and supp(u) ∪ supp(F0 ) ∪ supp(F ) ⊂ U+ . 13.2.2. Conjugation. In the local chart C, we define the conjugated operators of P0 and B0 : C

C

P0,ϕ = eτ ϕ P0 e−τ ϕ ,

C

C

B0,ϕ = eτ ϕ B0 e−τ ϕ .

In what follows, to ease notation we shall write ϕ and p in place of ϕC and pC , respectively. We have B0,ϕ = eτ ϕ OpT (bk )e−τ ϕ − ieτ ϕ OpT (bk−1 )e−τ ϕ (Dd + iτ ∂d ϕ) ˜ k (x, D , τ ) − iB k−1 (x, D , τ )Dd , =B ϕ

ϕ

with Bϕk−1 (x, D , τ ) = eτ ϕ OpT (bk−1 )e−τ ϕ , ˜ϕk (x, D , τ ) = eτ ϕ OpT (bk )e−τ ϕ + τ Bϕk−1 (x, D , τ )∂d ϕ, B with respective principal symbols  k−1 bk−1 (x, ξ  + iτ dx ϕ(x)), ϕ ( ) = b ˜bk ( ) = bk (x, ξ  + iτ dx ϕ(x)) + τ bk−1 ( )∂d ϕ(x), ϕ



ϕ

(x, ξ  , τ ).

˜bk ( ) ϕ

 bk−1 ϕ ( )

= Note that and are polynomial and howith  mogeneous of degree k and k − 1 in (ξ , τ ), respectively. In agreement with (13.2.1) we define (13.2.5) ˜ϕk (x, D , τ )u|x =0+ − iBϕk−1 (x, D , τ )(Dd u + ih)|x =0+ . B0,ϕ (u, h) = B d d With u satisfying (13.2.2), setting v = eτ ϕ u and   ˜ 0 = eτ ϕ F0 − τ g(∇g ϕ, F ) , G = eτ ϕ F, G ˜ 0 + divg G. From the form of the divergence with (17.2.6) we have P0,ϕ v = G  in (17.2.5) we find P0,ϕ v = G0 + 1≤i≤d ∂j Gj , with ˜0 + G0 = G

 1 (∂j det gx )Gj . 2 det gx 1≤i≤d

With u, F0 , F as in Proposition 13.8 we have v ∈ H 1 (Rd+ ), G0 ∈ L2 (Rd+ ), G ∈ L2 V (Rd+ ) with supports in U+ . From (13.2.3) we have   (Dd v − iGd )|xd =0+ = eτ ϕ Dd u − iF d − iτ ∂d ϕu |x =0+ ∈ H −1/2 (Rd−1 ), d

yielding C

|eτ ϕ u|xd =0+ |τ,1/2 + |e

τ ϕC

|xd =0+

(Dd u − iF d )|xd =0+ |τ,−1/2

 |v|xd =0+ |τ,1/2 + |(Dd v − iGd )|xd =0+ |τ,−1/2 .

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13. SOURCE TERMS OF WEAKER REGULARITY

We shall prove the following proposition that is equivalent to Proposition 13.8. Proposition 13.9. With the same setting and assumption as in Proposition 13.8, there exist there exists a bounded neighborhood U+ of x0 in Rd+ such that U+ ⊂ U 0 , C > 0, and τ∗ > 0 such that (13.2.6) τ 1/2 vτ,1 + τ |v|xd =0+ |τ,1/2 + τ |(Dd v − iGd )|xd =0+ |τ,−1/2

 ≤ C G0 + + τ G+ + τ |B0,ϕ (v, −Gd )|τ,1/2−k , for τ ≥ τ∗ and v ∈ H 1 (Rd+ ), G0 ∈ L2 (Rd+ ), G ∈ L2 V (Rd+ ) such that  P0,ϕ v = G0 + ∂ j Gj , supp(v) ∪ supp(G0 ) ∪ supp(G) ⊂ U+ . 1≤i≤d

In the next section we prove microlocal versions of this result. The proof of Proposition 13.9 is given in Sect. 13.4.1 with a patching argument. 13.3. A Microlocal Estimate ˇ Here, we shall only assume that the Lopatinski˘ı–Sapiro condition holds at one point in the cotangent bundle T ∗ ∂M × [0, +∞). Introducing a cutoff function χ ∈ ST0,τ in phase space near this point we prove an estimate for the microlocalized function OpT (χ)v. Proposition 13.10. Let U 0 = κ(V 0 ∩ O). Assume that (P0 , ϕ) has the sub-ellipticity property of Definition 3.2 of Volume 1 in U 0 . Let ω 0 ∈ ∗ ∂M, with local representative (ω 0 )C = ξ 0 ∈ Rd−1 , and τ 0 ≥ 0 such Tm 0 ˇ that (ξ 0 , τ 0 ) = 0, and assume that the Lopatinski˘ı–Sapiro condition of Def0 0 0 inition 8.1 holds at (m , ω , τ ). Then, there exists U a conic open neighborhood of 0 = (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ such that for χ ∈ ST0,τ , homogeneous of degree zero, with supp(χ) ⊂ U , there exist C > 0 and τ∗ > 0 such that (13.3.1) τ 1/2 OpT (χ)vτ,1 + τ |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |τ,−1/2

+ τ |OpT (χ)v|xd =0+ |τ,1/2 ≤ C G0 + + τ G+ + τ |B0,ϕ (v, −Gd )|τ,1/2−k  + vτ,1 + |(Dd v − iGd )|xd =0+ |τ,−3/2 , for τ ≥ τ∗ and v ∈ H 1 (Rd+ ), G0 ∈ L2 (Rd+ ), G ∈ L2 V (Rd+ ) such that  ∂ j Gj . P0,ϕ v = G0 + 1≤i≤d

With the normal geodesic coordinates we introduced locally, we may use some of the notation of Sect. 8.3.2 that we recall now. We have (13.3.2)

P0,ϕ = (Dd + iτ ∂d ϕ(x))2 + R0,ϕ ,

13.3. A MICROLOCAL ESTIMATE

291

where the principal symbol of R0,ϕ = eτ ϕ R0 e−τ ϕ ∈ Ψ2T,τ is precisely rϕ ( ) = r(x, ξ  + iτ dx ϕ(x))

 = (x, ξ  , τ ),

We introduced α( ) such that Re α( ) ≥ 0 and α2 ( ) = rϕ ( ) and we set γj ( ) = −iτ ∂d ϕ(x) + i(−1)j α( ),

 = (x, ξ  , τ ).

This yields

   pϕ () = pˇϕ (m, ω  , τ, z) = z − γ1 ( ) z − γ2 ( ) ,  = (x, ξ  , ξd = z, τ ).

ˇ By the first result of Theorem 8.7, if the Lopatinski˘ı–Sapiro condition 0 0 holds for (P, B, ϕ) at m , then ∂d ϕ(x ) > 0. This implies that we have Im γ1 (0 ) < 0. For the proof of Proposition 13.10 we shall treat differently the situations γ2 (0 ) = γ1 (0 ) and γ2 (0 ) = γ1 (0 ). Following (8.3.10), we prove the estimate of Proposition 13.10 in the following three (exhaustive) cases: (1) either Im γ2 (0 ) ≥ 0; (2) or Im γ2 (0 ) < 0 and γ2 (0 ) = γ1 (0 ); (3) or Im γ2 (0 ) < 0 and γ2 (0 ) = γ1 (0 ). 0 0 0 0 0 0 In case (1), we have pˇ+ ϕ (m , ω , z, τ ) = z − γ2 (x , ξ , τ ), while in cases (2) 0 0 0 and (3), we have pˇ+ ϕ (m , ω , z, τ ) = 1. In Sect. 8.3.4.2, Cases (2) and (3) were treated together. However, because of the low regularity of the source terms, the proof in this case (2) require to factorize the principal symbol of Pϕ in a smooth way and this can only be done in the case where roots do not cross. This explains the different treatments of Cases (2) and (3). 13.3.1. Case 1: One Root in the Upper Complex Half-Plane. Here, we have 0 0 0 0 0 0 pˇ+ ϕ (m , ω , z, τ ) = z − γ2 (x , ξ , τ ),

that is, Im γ1 (x0 , ξ 0 , τ 0 ) < 0 and Im γ2 (x0 , ξ 0 , τ 0 ) ≥ 0. With the Lopatinˇ ski˘ı–Sapiro condition holding for (P, B, ϕ) at the considered point, by (8.3.11), we have moreover bϕ (x0 , ξ 0 , ξd = γ2 , τ 0 ) = b(x0 , ξ 0 + iτ 0 dx ϕ(x0 ), γ2 + iτ 0 ∂d ϕ(x0 )) = 0. As the roots γ1 and γ2 are locally smooth with respect to  = (x, ξ  , τ ) and homogeneous of degree one in (ξ  , τ ) by Proposition 6.28, there exists U a conic open neighborhood of (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ and C, C  > 0 such that (13.3.3) γ1 ( ) = γ2 ( ), Im γ2 ( ) ≥ −CλT,τ , and Im γ1 ( ) ≤ −C  λT,τ , and (13.3.4)

bϕ (x, ξ  , ξd = γ2 (x, ξ  , τ ), τ ) = 0,

if  = (x, ξ  , τ ) ∈ U . Without any loss of generality we may moreover assume that SU is compact.

292

13. SOURCE TERMS OF WEAKER REGULARITY

We let χ ∈ ST0,τ be as in the statement and χ ˜ ∈ ST0,τ be homogeneous of degree zero and be such that supp(χ) ˜ ⊂ U and χ ˜ ≡ 1 on supp(χ). From the smoothness and the homogeneity of the roots, we have χγ ˜ j ∈ ST1,τ , j = 1, 2. We set ˜ 2 ) and P − = Dd − OpT (χγ ˜ 1 ). P + = Dd − OpT (χγ Since OpT (χ)P0,ϕ = OpT (χ)P − P + + R with R ∈ Ψτ1,0 , we write OpT (χ)P − P + v = OpT (χ)P0,ϕ v − Rv = OpT (χ)G0  + OpT (χ) ∂j Gj − Rv, 1≤j≤d

and we obtain OpT (χ)P − (P + v − iGd ) = OpT (χ)G0 + OpT (χ)



∂j Gj + iOpT (χ)OpT (χγ ˜ 1 )Gd − Rv,

1≤j≤d−1

which yields (13.3.5) OpT (χ)P − (P + v − iGd )τ,0,−1  G0 τ,0,−1 + G+ + vτ,1,−1  τ −1 G0 + + G+ + vτ,1,−1 , using Remark 6.8. From Lemma 6.20 (with s = −1), we have for τ chosen sufficiently large, (13.3.6) OpT (χ)z+ + |OpT (χ)z|xd =0+ |τ,−1/2  P − OpT (χ)zτ,0,−1 + zτ,0,−N , for z ∈ S (Rd+ ). With a density argument, estimate (13.3.6) applies to z = P + v − iGd , yielding OpT (χ)(P + v − iGd )+ + |OpT (χ)(P + v − iGd )|xd =0+ |τ,−1/2  P − OpT (χ)(P + v − iGd )τ,0,−1 + P + v − iGd τ,0,−N  OpT (χ)P − (P + v − iGd )τ,0,−1 + vτ,1,−1 + Gd + , using that P − OpT (χ) = OpT (χ)P − mod Ψ0T,τ . With (13.3.5), we then obtain OpT (χ)(P + v − iGd )+ + |OpT (χ)(P + v − iGd )|xd =0+ |τ,−1/2  τ −1 G0 + + G+ + vτ,1,−1 .

13.3. A MICROLOCAL ESTIMATE

293

To lighten notation we set w = OpT (χ)v and we write P + w+ + |(P + w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  OpT (χ)P + v+ + |OpT (χ)(P + v − iGd )|xd =0+ |τ,−1/2 + v+ + |v|xd =0+ |τ,−1/2  OpT (χ)(P + v − iGd )+ + |OpT (χ)(P + v − iGd )|xd =0+ |τ,−1/2 + Gd + + vτ,1,−1 , using the trace inequality of Corollary 6.10 (with m = 0 and s = −1/2). We thus obtain (13.3.7)

P + w+ + |(P + w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  τ −1 G0 + + G+ + vτ,1,−1 .

We now use the following lemma. Lemma 13.11. There exist C > 0 and τ∗ > 0 such that for any N ∈ N, there exists CN > 0 such that |OpT (χ)|τ,1/2 + |OpT (χ) |τ,−1/2

˜ k OpT (χ) − iB k−1 OpT (χ) | ≤ C |B ϕ ϕ τ,1/2−k     + |OpT (χ) − OpT (χγ ˜ 2 )OpT (χ)|τ,−1/2 + CN ||τ,−N + | |τ,−N , for τ ≥ τ∗ ,  ∈ H 1/2 (Rd−1 ), and  ∈ H −1/2 (Rd−1 ). A proof is given in Appendix 13.A.2. This is precisely where the Lopatinˇ ski˘ı–Sapiro condition is used. We set  = v|xd =0+ ∈ H 1/2 (Rd−1 ) and  = (Dd v − iGd )|xd =0+ ∈ −1/2 (Rd−1 ). As [Dd , OpT (χ)] ∈ Ψ0T,τ , we note that H |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |τ,−1/2  |OpT (χ)|xd =0+ |

τ,−1/2

+ |v|xd =0+ |τ,−1/2 , and ˜ϕk OpT (χ) − iBϕk−1 OpT (χ) | |B τ,1/2−k ˜ϕk OpT (χ)v|x =0+ − iBϕk−1 (Dd OpT (χ)v − iOpT (χ)Gd )|x =0+ |  |B d d τ,1/2−k + |v|xd =0+ |τ,−1/2  |B0,ϕ (w, −OpT (χ)Gd )|τ,1/2−k + |v|xd =0+ |τ,−1/2 ,

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13. SOURCE TERMS OF WEAKER REGULARITY

recalling the form of B0,ϕ given in (13.2.5). We also have |OpT (χ) − OpT (χγ ˜ 2 )OpT (χ)|τ,−1/2  |(Dd w − OpT (χγ ˜ 2 )w)|xd =0+ − iOpT (χ)Gd|xd =0+ |  |P + w|xd =0+ − iOpT (χ)Gd|xd =0+ |

τ,−1/2

τ,−1/2

+ |v|xd =0+ |τ,−1/2

+ |v|xd =0+ |τ,−1/2 .

With these observations, the result of Lemma 13.11 reads (13.3.8) |w|xd =0+ |τ,1/2 + |(Dd w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  |B0,ϕ (w, −OpT (χ)Gd )|τ,1/2−k + |(P + w − iOpT (χ)Gd )|xd =0+ |τ,−1/2 + vτ,1,−1 + |(Dd v − iGd )|xd =0+ |τ,−N , for τ ≥ 1 chosen sufficiently large, using the trace inequality of Corollary 6.10 (with m = 0 and s = −1/2). With a commutator argument we find |B0,ϕ (w, −OpT (χ)Gd )|τ,1/2−k  |B0,ϕ (v, −Gd )|τ,1/2−k + |v|xd =0+ |τ,−1/2 + |(Dd v − iGd )|xd =0+ |τ,−3/2 , and we write |v|xd =0+ |τ,−1/2  τ −1 |v|xd =0+ |τ,1/2  τ −1 vτ,1 , with the trace inequality of Proposition 6.9. Estimate (13.3.8) thus becomes (13.3.9) |w|xd =0+ |τ,1/2 + |(Dd w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  |B0,ϕ (w, −OpT (χ)Gd )|τ,1/2−k + |(P + w − iOpT (χ)Gd )|xd =0+ |τ,−1/2 + τ −1 vτ,1 + |(Dd v − iGd )|xd =0+ |τ,−3/2 . With (13.3.7) we thus obtain (13.3.10) P + w+ + |w|xd =0+ |τ,1/2 + |(Dd w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  τ −1 G0 + + G+ + |B0,ϕ (w, −OpT (χ)Gd )|τ,1/2−k + τ −1 vτ,1 + |(Dd v − iGd )|xd =0+ |τ,−3/2 .

P+

We recall that γ˜2 = χγ ˜ 2 ∈ ST1,τ and P + = Dd − OpT (˜ γ2 ). We write + + = P2 + iP1 with   1 + 1 γ2 ) + OpT (˜ γ 2 )∗ , P + (P + )∗ = Dd − OpT (˜ 2 2   1 1 + + ∗ P − (P ) = − OpT (˜ = γ2 ) − OpT (˜ γ 2 )∗ . 2i 2i

P2+ = P1+

13.3. A MICROLOCAL ESTIMATE

295

Observe that P2+ ∈ Ψτ1,0 and P1+ ∈ Ψ1T,τ are both formally selfadjoint. Their respective principal symbols are p+ ˜2 ( ) ∈ Sτ1,0 , 2 () = ξd − Re γ

 p+ ˜2 ( ) ∈ ST1,τ . 1 ( ) = − Im γ

+  The principal symbol of P + is p+ () = p+ 2 () + ip1 ( ).

Rd+

Lemma 13.12. There exists a conic open neighborhood W of supp(χ) in × Rd−1 × R+ such that χ ˜ ≡ 1 in W and

p+ () = 0 ⇒

1 + +  +    {p , p }( ) = {p+ 2 , p1 }( ) > 0,  ∈ W ,  = ( , ξd ). 2i

A proof is given in Appendix 13.A.3. With Lemma 13.12, applying Lemma 6.22 with a density argument, for τ ≥ 1 sufficiently large, we obtain τ −1/2 wτ,1  P + w+ + |w|xd =0+ |τ,1/2 + vτ,0,−N , yielding, with (13.3.10), τ −1/2 wτ,1 + |w|xd =0+ |τ,1/2 + |(Dd w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  τ −1 G0 + + G+ + |B0,ϕ (w, −OpT (χ)Gd )|τ,1/2−k + τ −1 vτ,1 + |(Dd v − iGd )|xd =0+ |τ,−3/2 . Upon multiplication by τ we obtain (13.3.1). This concludes the proof of Proposition 13.10 in Case 1.  13.3.2. Case 2: Roots Are Different and Both in the Open Lower Complex Half-Plane. Here, we have Im γ1 (0 ) < 0 and Im γ2 (0 ) < 0, and γ1 (0 ) = γ2 (0 ), where 0 = (x0 , ξ 0 , τ 0 ). As the roots γ1 and γ2 are locally smooth with respect to  = (x, ξ  , τ ) and homogeneous of degree one in (ξ  , τ ) by Proposition 6.28, there exists U a conic open neighborhood of (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ and C > 0 such that γ1 ( ) = γ2 ( ), Im γ2 ( ) ≤ −CλT,τ , and Im γ1 ( ) ≤ −CλT,τ , if  ∈ U . Without any loss of generality we may moreover assume that SU is compact. ˜ ∈ ST0,τ be homogeneous of We let χ ∈ ST0,τ be as in the statement and χ degree zero and be such that supp(χ) ˜ ⊂ U and χ ˜ ≡ 1 on supp(χ). From the smoothness and the homogeneity of the roots, we have χγ ˜ j ∈ ST1,τ , j = 1, 2. We set ˜ 2 ) and P − = Dd − OpT (χγ ˜ 1 ). P + = Dd − OpT (χγ

296

13. SOURCE TERMS OF WEAKER REGULARITY

Since OpT (χ)P0,ϕ = OpT (χ)P − P + + R with R ∈ Ψτ1,0 , we write OpT (χ)P − P + v = OpT (χ)P0,ϕ v − Rv = OpT (χ)G0  + OpT (χ) ∂j Gj − Rv, 1≤j≤d

and we obtain OpT (χ)P − (P + v − iGd ) = OpT (χ)G0 + OpT (χ)



∂j Gj + iOpT (χ)OpT (χγ ˜ 1 )Gd − Rv,

1≤j≤d−1

which yields (13.3.11) OpT (χ)P − (P + v − iGd )τ,0,−1  G0 τ,0,−1 + G+ + vτ,1,−1  τ −1 G0 + + G+ + vτ,1,−1 , using Remark 6.8. From Lemma 6.20 (with s = −1), we have for τ chosen sufficiently large, (13.3.12) OpT (χ)z+ + |OpT (χ)z|xd =0+ |τ,−1/2  P − OpT (χ)zτ,0,−1 + zτ,0,−N , for z ∈ S (Rd+ ). With a density argument we apply estimate (13.3.12) for z = P + v −iGd , yielding OpT (χ)(P + v − iGd )+ + |OpT (χ)(P + v − iGd )|xd =0+ |τ,−1/2  P − OpT (χ)(P + v − iGd )τ,0,−1 + P + v − iGd τ,0,−N  OpT (χ)P − (P + v − iGd )τ,0,−1 + vτ,1,−1 + Gd + , using that P − OpT (χ) = OpT (χ)P − mod Ψ0T,τ . With (13.3.11), we then obtain OpT (χ)(P + v − iGd )+ + |OpT (χ)(P + v − iGd )|xd =0+ |τ,−1/2  τ −1 G0 + + G+ + vτ,1,−1 . We set w = OpT (χ)v and we write P + w+ + |(P + w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  OpT (χ)P + v+ + |OpT (χ)(P + v − iGd )|xd =0+ |τ,−1/2 + vτ,0 + |v|xd =0+ |τ,−1/2  OpT (χ)(P + v − iGd )+ + |OpT (χ)(P + v − iGd )|xd =0+ |τ,−1/2 + Gd + + vτ,1,−1 ,

13.3. A MICROLOCAL ESTIMATE

297

using the trace inequality of Corollary 6.10 (with m = 0 and s = −1/2). We thus obtain P + w+ + |(P + w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  τ −1 G0 + + G+ + vτ,1,−1 . Next, from Lemma 6.20 (with s = 0), with a density argument, we have for τ chosen sufficiently large, wτ,1 + |w|xd =0+ |τ,1/2  P + w+ + vτ,0,−N . This yields wτ,1 + |w|xd =0+ |τ,1/2 + |(P + w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  τ −1 G0 + + G+ + vτ,1,−1 . Observing that |(Dd w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  |w|xd =0+ |τ,1/2 + |(P + w − iOpT (χ)Gd )|xd =0+ |τ,−1/2 , we finally obtain 

τ wτ,1 + |w|xd =0+ |τ,1/2 + |(Dd w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  G0 + + τ G+ + τ vτ,1,−1 , which concludes the proof of Proposition 13.10 in Case 2.



Remark 13.13. We observe that this microlocal estimate is in fact stronger than the estimate in the statement of Proposition 13.10. One has τ wτ,1 instead of τ 1/2 wτ,1 on the left-hand side and the term τ |B0,ϕ (v, −Gd )|τ,1/2−k is not present in the right-hand side. This is consistent with the result obtained in Sect. 8.3.4.2 where the roots are also in the lower complex half-plane; see Remark 8.21. This is natural as this implies a perfectly elliptic estimate, meaning that all relevant traces can be microlocally estimated. Here, a particular treatment is needed because of the low regularity of the source terms; the smooth factorization pϕ = p+ p− in the neighborhood of supp(χ) is crucial here; it was needed in Sect. 8.3.4.2. The need for this smooth factorization forces us to consider separately the case of roots that coincide. This is the subject of the next section. 13.3.3. Case 3: Roots Coincide in the Open Lower Complex Half-Plane. Here, we consider the case Im γ2 (0 ) < 0 and γ2 (0 ) = γ1 (0 ), which means α(0 ) = 0. We recall that α is defined through its square, viz., α( )2 = r(x, ξ  + iτ dx ϕ(x)),

 = (x, ξ  , τ ).

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13. SOURCE TERMS OF WEAKER REGULARITY

At the set {α = 0} roots may not be smooth (whereas they are smooth away from this set by Proposition 6.28). Here, we shall concentrate our analysis close that set. Note that Re α( )2 = r(x, ξ  ) − τ 2 r(x, dx ϕ(x)), implying λ2T,τ  τ 2 + r(x, ξ  )  τ 2 (1 + r(x, dx ϕ(x)) + Re α( )2  τ 2 + |α( )|2 . Thus, for ε0 > 0 chosen sufficiently small, if |α2 ( )| ≤ ε0 λ2T,τ , then λT,τ  τ.

(13.3.13)

Hence, there exists U 0 a conic open neighborhood of 0 in U 0 × Rd−1 × R+ where ∂d ϕ ≥ C > 0 and |α2 ( )| ≤ ε0 λ2T,τ , implying that (13.3.13) holds there. Without any loss of generality we may moreover assume that SU 0 is compact. From the form of the operator P0,ϕ , since we consider a microlocal region where |α| is small, we can foresee that the estimation we shall obtain near that region is almost given by an estimate for the conjugated operator Qϕ = (Dd + iτ ∂d ϕ(x))2 . Lemma 13.14. Let χ ˜ ∈ ST0,τ , homogeneous of degree zero, be such that supp(χ) ˜ ⊂ U 0 . There exist C > 0 and τ∗ > 0 such that  ˜ τ,1 + |(Dd OpT (χ)v ˜ − iOpT (χ)H ˜ d )|xd =0+ |τ,−1/2 (13.3.14) τ OpT (χ)v

  + |OpT (χ)v ˜ |xd =0+ |τ,1/2 ≤ C H0 + + τ H+ + τ vτ,1,−1 , for τ ≥ τ∗ and v ∈ H 1 (Rd+ ), H0 ∈ L2 (Rd+ ), H ∈ L2 V (Rd+ ) such that  ∂j H j . Qϕ v = H 0 + 1≤i≤d

A proof is given below. For ε > 0 to be fixed below, we let Uε be the conic open set of U 0 × × R+ given by

Rd−1

Uε = { ∈ U 0 ; |α2 ( )| < ελ2T,τ }. We take 0 < ε ≤ ε0 /2 allowing us to choose χε ∈ ST0,τ homogeneous of degree zero such that supp(χε ) ⊂ Uε . In Uε ⊂ U 0 one has λT,τ  τ . With Proposition 6.11, there exists C1 > 0 such that OpT (χε )vτ,0,2 ≤ C1 τ 2 OpT (χε )v+ + Cε,N vτ,0,−N , for any N ∈ N, with some Cε,N > 0.

13.3. A MICROLOCAL ESTIMATE

299

As |α2 ( )| ≤ ελ2T,τ in Uε , one has by Corollary 2.51, for C0 > 1 and C1 = C0 C1 , (13.3.15)

 vτ,0,−N R0,ϕ OpT (χε )v+ ≤ C0 εOpT (χε )vτ,0,2 + Cε,N  ≤ 2εC1 τ 2 OpT (χε )v+ + Cε,N vτ,0,−N .

Recalling (13.3.2), we write Qϕ OpT (χε )v = P0,ϕ OpT (χε )v − R0,ϕ OpT (χε )v = OpT (χε )P0,ϕ v + [P0,ϕ , OpT (χε )]v − R0,ϕ OpT (χε )v  = H0 + ∂j H j , 1≤j≤d

with H = OpT (χε )G and H0 = OpT (χε )G0 +



[OpT (χε ), ∂j ]Gj + [P0,ϕ , OpT (χε )]v

1≤j≤d

− R0,ϕ OpT (χε )v. We have H+  G+ and, with (13.3.15),   H0 + ≤ Cε G0 + + G+ + vτ,1 + Cετ 2 OpT (χε )v+ . ˜ ⊂U0 We choose χ ˜ ∈ ST0,τ homogeneous of degree zero such that supp(χ) and χ ˜ ≡ 1 in a neighborhood of Uε0 /2 . Applying Lemma 13.14 we thus ˜ ◦ χε , obtain, with χ ˜ε = χ

˜ε )vτ,1 + |(Dd OpT (χ ˜ε )v − iOpT (χ ˜ε )Gd )|xd =0+ |τ,−1/2 τ OpT (χ 

 + |OpT (χ ˜ε )v|xd =0+ |τ,1/2 ≤ Cε G0 + + τ G+ + vτ,1 + Cετ 2 OpT (χε )v+ . Since χ ˜ ≡ 1 in a neighborhood of supp(χε ) we have χ ˜ε = χε mod ST−N ,τ for any N ∈ N by pseudo-differential calculus. We then find |(Dd OpT (χε )v − iOpT (χε )Gd )|xd =0+ |τ,−1/2 ≤ |OpT (χε )(Dd v − iGd )|xd =0+ |τ,−1/2 + |[Dd , OpT (χε )]v|xd =0+ |τ,−1/2 ≤ |OpT (χ ˜ε )(Dd v − iGd )|xd =0+ |τ,−1/2   + Cε |(Dd v − iGd )|xd =0+ |τ,−N + vτ,1,−1 ˜ε )v − iOpT (χ ˜ε )Gd )|xd =0+ |τ,−1/2 ≤ |(Dd OpT (χ + |[Dd , OpT (χ ˜ε )]v|xd =0+ |τ,−1/2   + Cε |(Dd v − iGd )|xd =0+ |τ,−N + vτ,1,−1 ≤ |(Dd OpT (χ ˜ε )v − iOpT (χ ˜ε )Gd )|xd =0+ |τ,−1/2   + Cε |(Dd v − iGd )|xd =0+ |τ,−N + vτ,1,−1 ,

300

13. SOURCE TERMS OF WEAKER REGULARITY

using the trace inequality of Corollary 6.10 (with m = 0 and s = −1/2). ˜ε )v|xd =0+ |τ,1/2 , and writing Arguing similarly the term |OpT (χ ˜ε )vτ,1 + Cε vτ,1,−N , OpT (χε )vτ,1 ≤ OpT (χ we then obtain

τ OpT (χε )vτ,1 + |(Dd OpT (χε )v − iOpT (χε )Gd )|xd =0+ |τ,−1/2  + |OpT (χε )v|xd =0+ |τ,1/2 ≤ Cετ 2 OpT (χε )v+

 + Cε G0 + + τ G+ + vτ,1 + |(Dd v − iGd )|xd =0+ |τ,−N . Choosing ε > 0 sufficiently small, to be kept fixed and setting χ = χε we finally obtain

(13.3.16) τ OpT (χ)vτ,1 + |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |τ,−1/2  + |OpT (χ)v|xd =0+ |τ,1/2  G0 + + τ G+ + vτ,1 + |(Dd v − iGd )|xd =0+ |τ,−N , which concludes the proof of Proposition 13.10 in Case 3.



Proof of Lemma 13.14. We have d + iτ ∂d ϕ(x).  We set L = D ˜ 2 v = OpT (χ)H ˜ 0 + 1≤j≤d OpT (χ)∂ ˜ j H j , which we write OpT (χ)L  ˜ − iH d ) = OpT (χ)H ˜ 0+ OpT (χ)∂ ˜ jHj OpT (χ)L(Lv 1≤j≤d−1

+ τ OpT (χ)∂ ˜ d ϕH d . This yields (13.3.17) ˜ − iH d )τ,0,−1  H0 τ,0,−1 + H+  τ −1 H0 + + H+ , OpT (χ)L(Lv using Remark 6.8. Since τ ∂d ϕ  λT,τ in U 0 , from Lemma 6.20 (with s = −1), we have for τ chosen sufficiently large, (13.3.18) OpT (χ)z ˜ + + |OpT (χ)z ˜ |xd =0+ |τ,−1/2  LOpT (χ)z ˜ τ,0,−1 + zτ,0,−N , for z ∈ S (Rd+ ). We apply estimate (13.3.18) for z = Lv −H d , with a density argument, yielding ˜ − iH d )+ + |OpT (χ)(Lv ˜ − iH d )|xd =0+ |τ,−1/2 OpT (χ)(Lv  LOpT (χ)(Lv ˜ − iH d )τ,0,−1 + Lv − iH d τ,0,−N  OpT (χ)L(Lv ˜ − iH d )τ,0,−1 + vτ,1,−1 + H d + ,

13.4. PATCHING ESTIMATES TOGETHER

301

using that [L, OpT (χ)] ˜ ∈ Ψ0T,τ . With (13.3.17), we then obtain ˜ − iH d )+ + |OpT (χ)(Lv ˜ − iH d )|xd =0+ |τ,−1/2 OpT (χ)(Lv  τ −1 H0 + + H+ + vτ,1,−1 . We then write ˜ + + |(LOpT (χ)v ˜ − iOpT (χ)H ˜ d )|xd =0+ |τ,−1/2 LOpT (χ)v  OpT (χ)Lv ˜ ˜ − iH d )|xd =0+ |τ,−1/2 + + |OpT (χ)(Lv + v+ + |v|τ,−1/2  OpT (χ)(Lv ˜ − iH d )+ + |OpT (χ)(Lv ˜ − iH d )|xd =0+ |τ,−1/2 + H d + + vτ,1,−1 , using the trace inequality of Corollary 6.10 (with m = 0 and s = −1/2). We thus obtain ˜ + + |(LOpT (χ)v ˜ − iOpT (χ)H ˜ d )|xd =0+ |τ,−1/2 LOpT (χ)v  τ −1 H0 + + H+ + vτ,1,−1 . Next, invoking Lemma 6.20 a second time (with s = 0 this time), we have for τ chosen sufficiently large, ˜ τ,1 + |OpT (χ)v ˜ |xd =0+ |τ,1/2  LOpT (χ)v ˜ + + vτ,0,−N . OpT (χ)v This yields ˜ τ,1 + |OpT (χ)v ˜ |xd =0+ |τ,1/2 OpT (χ)v + |(LOpT (χ)v ˜ − OpT (χ)H ˜ d )|xd =0+ |τ,−1/2  τ −1 H0 + + H+ + vτ,1,−1 . Observing that ˜ − OpT (χ)H ˜ d )|xd =0+ |τ,−1/2 |(Dd OpT (χ)v  |OpT (χ)v ˜ |xd =0+ |τ,1/2 + |(LOpT (χ)v ˜ − OpT (χ)H ˜ d )|xd =0+ |τ,−1/2 , which concludes the proof.



13.4. Patching Estimates Together First, we show how microlocal estimates given by Proposition 13.10 can be patched together to form the local estimate of Proposition 13.9. Second, we show how such local estimate can be patched to obtain the result of Theorem 13.4.

302

13. SOURCE TERMS OF WEAKER REGULARITY

13.4.1. Patching Microlocal Estimates. Here, we prove Proposition 13.9 by means of the microlocal result obtained in Sect. 13.3. Equivalently, this gives the result of Propositions 13.3 and 13.8. We use the argument and the notation of the proof of Proposition 8.22. We recall the setting introduced therein and we only detail the argument for the terms that only occur in the present case. With x0 as in the statement of Propositions 13.8 and 13.9 the Lopatinˇ ski˘ı–Sapiro condition for (P0 , B0 , ϕ) holds at (x0 , ξ 0 , τ 0 ) for all ξ 0 ∈ Tx∗0 Rd−1 d−1 ∼ and τ 0 ≥ 0 such that (ξ 0 , τ 0 ) = 0. It is in fact sufficient to con=R sider (ξ 0 , τ 0 ) in the half-unit sphere Sd−1 + , using the notation introduced in (1.7.3). there exists a conic open By Proposition 13.10 for all (ξ 0 , τ 0 ) ∈ Sd−1 + neighborhood Uy0 of 0 = (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ such that estimate (13.3.1) holds. In fact, by reducing U0 we can choose U0 = O0 × Γ0 where O0 is an open set in U 0 and Γ0 is a conic open set we can thus find finitely in Rd−1 × R+ . With the compactness of Sd−1 + ⊂ ∪j∈J Γj . We many such open sets Uj = Oj × Γj , j ∈ J, such that Sd−1 + then set O = ∩j∈J Oj that is an open neighborhood of x0 in U 0 and we set Vj = O × Γj ⊂ Uj . We also choose an open neighborhood U of x0 in Rd such that U+ = U ∩ U 0  O. We then choose a smooth partition of unity, χj , j ∈ J, of the closed set F = U+ × Sd−1 in the manifold N = Rd × Sd−1 subordinated to the + + covering by the open sets Vj ∩ N according to Theorem 15.14. We then extend each χj smoothly to Rd × Rd−1 × R+ by homogeneity of order 0 for |(ξ  , τ )| ≥ 1 > 0. We have supp(χj ) ⊂ Vj and  χj ( ) = 1, j∈J

conic neighborhood of U+ ×Rd−1 ×R+ and |(ξ  , τ )| ≥ 1. for  = (x, ξ  , τ ) in a  We also set χ = 1 − j∈J χj Let v ∈ H 1 (Rd+ ), G0 ∈ L2 (Rd+ ), and G ∈ L2 V (Rd+ ) be such that supp(v) ∪ supp(G0 ) ∪ supp(G) ⊂ U+ and  ∂ j Gj . P0,ϕ v = G0 + 1≤i≤d

As supp(χj ) ⊂ Uj , we can apply the microlocal estimate of Proposition 13.10 τ 1/2 OpT (χj )vτ,1 + τ |(Dd OpT (χj )v − iOpT (χj )Gd )|xd =0+ |τ,−1/2 + τ |OpT (χj )v|xd =0+ |τ,1/2  G0 + + τ G+ + τ |B0,ϕ (v, −Gd )|τ,1/2−k + vτ,1 + |(Dd v − iGd )|xd =0+ |τ,−N , for τ > 0 chosen sufficiently large.

13.4. PATCHING ESTIMATES TOGETHER

303

Arguing as in the proof of Proposition 8.22 we have  OpT (χj )vτ,1 + vτ,1,−N , vτ,1  j∈J

and |v|xd =0+ |τ,1/2 

 j∈J

|OpT (χj )v|xd =0+ |τ,1/2 + vτ,1,−N .

We now consider the term that is specific to the estimates proven in this chapter:  |(Dd OpT (χj )v − iOpT (χj )Gd )|xd =0+ |τ,−1/2 |(Dd v − iGd )|xd =0+ |τ,−1/2  j∈J

+ |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |τ,−1/2 . We then write |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |τ,−1/2  |OpT (χ)(Dd v − iGd )|xd =0+ |τ,−1/2 + |[Dd , OpT (χ)]v|xd =0+ |τ,−1/2  |(Dd v − iGd )|xd =0+ |τ,−N + |v|xd =0+ |τ,−N  |(Dd v − iGd )|xd =0+ |τ,−N + vτ,1,−N , since χ is supported away from U+ and using the trace inequality of Corollary 6.10. We thus obtain τ 1/2 vτ,1 + τ |v|xd =0+ |τ,1/2 + τ |(Dd v − iGd )|xd =0+ |τ,−1/2  G0 + + τ G+ + τ |B0,ϕ (v, −Gd )|τ,1/2−k + |(Dd v − iGd )|xd =0+ |τ,−N + vτ,1 . We conclude the proof of Proposition 13.9 by choosing τ > 0 sufficiently large.  13.4.2. Patching Local Estimates. Here we show how the patching method first described in Section 3.5 allows one to obtain our main result, Theorem 13.4, from Proposition 13.3. For all m ∈ V , by Lemma 3.9 of Volume 1 used in a local chart (upon a smooth extension of the coefficients and the weight function locally in Rd− if needed), there exists an open neighborhood Vm0 of m where ϕ and P have the sub-ellipticity property and where either the result of Proposition 13.1 or Proposition 13.3 applies: there exists an open neighborhood Vm1 ⊂ Vm0 , with m ∈ Vm1 , where either the local estimate (13.1.2) or (13.1.4) holds. From the covering of the compact set V by the open sets Vm1 , m ∈ V , we can extract a finite covering (Vi )i∈I , such that for all i ∈ I the Carleman

304

13. SOURCE TERMS OF WEAKER REGULARITY

estimate in Vi holds for τ ≥ τi > 0, C = Ci > 0: (13.4.1) τ 1/2 eτ ϕ wτ,1 + τ |eτ ϕ w|∂M |τ,1/2 + τ |eτ ϕ|∂M (∂ν w + g(H, ν))|∂M |τ,−1/2  ≤ C eτ ϕ H0 L2 (M) + τ eτ ϕ HL2 V (M)  + τ |eτ ϕ|∂M B(w, g(H, ν))|τ,1/2−k , for w ∈ H 1 (M), H0 ∈ L2 (M), H ∈ L2 V (M) such that P w = H0 + divg H, and supp(w) ∪ supp(H0 ) ∪ supp(H) ⊂ Vi . Note that the boundary terms on the l.h.s. and the r.h.s. vanish if Vi ∩∂M = ∅. Let ( χ(i) )i∈I be a partition of unity of V subordinated to the open covering Vi , i ∈ I, as given by Theorem 15.14, that is, χ(i) ∈ C ∞ (M), with supp( χ(i) ) ⊂ Vi ,

0 ≤ χ(i) ≤ 1, i ∈ I,

 and χ = i∈I χ(i) ≡ 1 in a neighborhood of V . Let now u, F0 , and F be as in the statement. For all i ∈ I, we set (i) u(i) = χ(i) u ∈ H 1 (M) with supp(u(i) ) ⊂ Vi . We have P u(i) = H0 +divg F (i) with (i)

H0 = χ(i) F0 − [P, χ(i) ]u − g(∇g χ(i) , F ), F (i) = χ(i) F. We have (i)

eτ ϕ H0 L2 (M) + τ eτ ϕ F (i) L2 V (M)  eτ ϕ F0 L2 (M) + τ eτ ϕ F L2 V (M) + eτ ϕ uL2 (M) + eτ ϕ ∇g uL2 V (M)  eτ ϕ F0 L2 (M) + τ eτ ϕ F L2 V (M) + eτ ϕ uτ,1 , and |eτ ϕ|∂M B(u(i) , g(F (i) , ν))|τ,1/2−k  |eτ ϕ|∂M B(u, g(F, ν))|τ,1/2−k + |eτ ϕ u|∂M |τ,−1/2  |eτ ϕ|∂M B(u, g(F, ν))|τ,1/2−k + τ −1 eτ ϕ uτ,1 . As u = χu =

 i∈I

u(i) , F = χF =

 i∈I

F (i) we find

τ 1/2 eτ ϕ uτ,1 + τ |eτ ϕ u|∂M |τ,1/2 + τ |eτ ϕ|∂M (∂ν u + g(F, ν))|∂M |τ,−1/2   1/2 τ ϕ (i) (i) τ e u τ,1 + τ |eτ ϕ u|∂M |  τ,1/2

i∈I

+ τ |e

τ ϕ|∂M

(∂ν u

(i)

+ g(F

(i)

 , ν))|∂M |τ,−1/2 .

13.6. ESTIMATES WITHOUT PRESCRIBED BOUNDARY CONDITIONS

305

Since, for each u(i) we have a local Carleman estimate of the form given by (13.4.1), we obtain τ 1/2 eτ ϕ uτ,1 + τ |eτ ϕ u|∂M |τ,1/2 + τ |eτ ϕ|∂M (∂ν u + g(F, ν))|∂M |τ,−1/2  ≤ C eτ ϕ F0 L2 (M) + τ eτ ϕ F L2 V (M)

 + τ |eτ ϕ|∂M B(u, g(F, ν))|τ,1/2−k + eτ ϕ uτ,1 .

We conclude the proof of Theorem 13.4 by choosing τ > 0 sufficiently large.  13.5. Shifted Estimate Similarly to Theorem 8.24 we have the following shifted estimate. Theorem 13.15. Let (M, g) be a compact Riemannian manifold with boundary and let P = −Δg + R1 with R1 a first-order differential operator on M. Let V be an open set of M and set V∂ = V ∩ ∂M. Let ϕ ∈ C ∞ (M) be such that the pair (P, ϕ) has the sub-ellipticity property of Definition 5.1 in V . If V∂ = ∅, consider B a differential operator of order k ∈ N in V and ˇ assume moreover that (P, B, ϕ) satisfies the Lopatinski˘ı–Sapiro condition at all points m ∈ V∂ (Definition 8.1). Then, there exist C and τ∗ > 0 such that (13.5.1) τ eτ ϕ uτ,0,1/2 + τ 3/2 |eτ ϕ u|∂M |L2 (∂M) + τ 3/2 |eτ ϕ|∂M (∂ν u + g(F, ν))|∂M |τ,−1   ≤ C eτ ϕ F0 L2 (M) + τ eτ ϕ F L2 V (M) + τ 3/2 |eτ ϕ|∂M B(u, g(F, ν))|τ,−k , for τ ≥ τ∗ and u ∈ H 1 (M), F0 ∈ L2 (M), F ∈ L2 V (M) such that P u = F0 + divg F, and supp(u) ∪ supp(F0 ) ∪ supp(F ) ⊂ V . A proof based on Proposition 13.9 is given in Appendix 13.A.4. 13.6. Estimates Without Prescribed Boundary Conditions Let B = ∂ν + B  be a first-order boundary operator such that (P, B) ˇ satisfies the Lopatinski˘ı–Sapiro condition. As in Sect. 8.4.2 we prove estimates with terms involving both the Dirichlet trace u|∂M and the trace Bu|∂M . Yet, one does not impose any condition on the weight function at the boundary, in particular in connection with B. Firsts, we consider the case a weight function ϕ such that ∂ν ϕ = 0. We discuss the case where ∂ν ϕ can vanish in Sect. 13.6.2 below. 13.6.1. Estimate for a Weight Function with Nonvanishing Neumann Trace. Here we prove an estimate in an open set V that meets the boundary and where ∂ν ϕ does not vanish.

306

13. SOURCE TERMS OF WEAKER REGULARITY

Theorem 13.16. Let (M, g) be a compact Riemannian manifold with boundary and let P = −Δg + R1 with R1 a first-order differential operator with bounded coefficients on M. Let V be an open set of M such that V∂ = V ∩ ∂M = ∅. Let B  be a differential operator of order one on ∂M ˇ such that (P, B) fulfills the Lopatinski˘ı–Sapiro condition of Definition 2.2 on  ∞ V∂ for B = ∂ν + B . Let ϕ ∈ C (M) be such that the pair (P, ϕ) has the sub-ellipticity property of Definition 5.1 in V and such that ∂ν ϕ = 0 in V∂ . Then, there exist C and τ∗ > 0 such that (13.6.1) τ 1/2 eτ ϕ uτ,1 + τ |eτ ϕ u|∂M |τ,1/2 + τ |eτ ϕ|∂M (∂ν u + g(F, ν))|∂M |τ,−1/2

≤ C eτ ϕ F0 L2 (M) + τ eτ ϕ F L2 V (M)  + τ 3/2 |eτ ϕ u|∂M |L2 (∂M) + τ |eτ ϕ|∂M B(u, g(F, ν))|τ,−1/2 , for τ ≥ τ∗ and u ∈ H 1 (M), F0 ∈ L2 (M), F ∈ L2 V (M) such that P u = F0 + divg F, and supp(u) ∪ supp(F0 ) ∪ supp(F ) ⊂ V . Note that one does not assume that (P, B, ϕ) fulfills the Lopatinski˘ı– ˇ Sapiro condition of Definition 8.1. The same is done in Sect. 8.4.2. We shall prove a microlocal version of this estimate. With the notation of Sect. 13.2, in a local chart C = (O, κ) near a point of ∂M we set P0 = Op(pC ), where pC is the local representative of the principal symbol of the Laplace–Beltrami operator. The associated conjugated operator is P0,ϕ = C C eτ ϕ P0 e−τ ϕ . We also set B0 = OpT (bC ) where bC is the local representative of the principal symbol of B  . Here, B0,ϕ (v, h) takes the form (13.6.2)

˜ϕ u|x =0+ − i(Dd u + ih)|x =0+ , B0,ϕ (v, h) = B d d C

C

˜ϕ = B  + τ ∂d ϕC with B  = eτ ϕ B  e−τ ϕ ∈ DT1,τ . This is consiswhere B 0,ϕ 0,ϕ 0 tent with the definition of B0,ϕ (v, h) given in (13.2.5). Proposition 13.17. Let m0 ∈ ∂M and V 0 be an open neighborhood of m0 in M. Let U 0 = κ(V 0 ∩ O) and x0 = κ(m0 ). Assume that (P0 , ϕ) has the sub-ellipticity property of Definition 3.2 of Volume 1 in U 0 and ∂d ϕC = 0 in U 0 ∩ {xd = 0}. Let B  be a differential operator of order one on ∗ ∂M, with local representative ∂M ∩ U 0 and set B = ∂ν + B  . Let ω 0 ∈ Tm 0 0 C 0 d−1 0 (ω ) = ξ ∈ R , and τ ≥ 0 such that (ξ 0 , τ 0 ) = 0. If τ 0 = 0 assume ˇ moreover that the Lopatinski˘ı–Sapiro condition of Definition 2.2 holds at 0 0 C (m , ω ) for (P0 , B0 , ϕ ). Then, there exists U a conic open neighborhood of 0 = (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ such that for χ ∈ ST0,τ , homogeneous

13.6. ESTIMATES WITHOUT PRESCRIBED BOUNDARY CONDITIONS

307

of degree zero, with supp(χ) ⊂ U , there exist C > 0 and τ∗ > 0 such that (13.6.3) τ 1/2 OpT (χ)vτ,1 + τ |OpT (χ)v|xd =0+ |τ,1/2 + τ |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |τ,−1/2

≤ C G0 + + τ G+ + τ 3/2 |v|xd =0+ |∂ + τ |B0,ϕ (v, −Gd )|τ,−1/2

 + vτ,1 + |(Dd v − iGd )|xd =0+ |τ,−3/2 ,

for τ ≥ τ∗ and v ∈ H 1 (Rd+ ), G0 ∈ L2 (Rd+ ), G ∈ L2 V (Rd+ ) such that  ∂ j Gj . P0,ϕ v = G0 + 1≤i≤d

Estimates of this sort can be patched together as in Sect. 13.4.1 yielding a local version of Theorem 13.16. Local such versions can then be patched together as in Sect. 13.4.2 yielding the result of Theorem 13.16. With the normal geodesic coordinates we introduced in Sect. 13.2, we may use some of the notation of Sect. 8.3.2 that we recall now. In particular, to ease notation we shall write ϕ in place of ϕC , for example, in what follows, with a similar simplification for other symbols. We have (13.6.4)

P0,ϕ = (Dd + iτ ∂d ϕ(x))2 + R0,ϕ ,

where the principal symbol of R0,ϕ = eτ ϕ R0 e−τ ϕ ∈ Ψ2T,τ is precisely (13.6.5)

rϕ ( ) = r(x, ξ  + iτ dx ϕ(x))

 = (x, ξ  , τ ).

We introduced α( ) such that Re α( ) ≥ 0 and α2 ( ) = rϕ ( ) and we set (13.6.6)

γj ( ) = −iτ ∂d ϕ(x) + i(−1)j α( ),

 = (x, ξ  , τ ).

This yields, for the principal symbol of P0,ϕ ,    pϕ () = pˇϕ (m, ω  , τ, z) = z − γ1 ( ) z − γ2 ( ) ,  = (x, ξ  , ξd = z, τ ). Note that if τ = 0 and ξ  = 0, then rϕ ( ) = r(x, ξ  ) > 0. With γj ( ) = i(−1)j r(x, ξ  )1/2 in this case one finds that having γ1 ( ) = γ2 ( ) and (τ, ξ  ) = 0 excludes τ = 0. Below, we prove the estimate of Proposition 13.17 in the following three (exhaustive) cases: (1) τ 0 = 0; (2) τ 0 = 0 and γ2 (0 ) = γ1 (0 ); (3) γ2 (0 ) = γ1 (0 ) (and necessarily τ 0 = 0). 13.6.1.1. Case 1: Vanishing Carleman Parameter. Here, τ 0 = 0. The ˇ Lopatinski˘ı–Sapiro condition of Definition 2.2 holds at (m0 , ω 0 ) for (P0 , ϕ), ˇ meaning that the Lopatinski˘ı–Sapiro condition of Definition 8.1 holds at 0 0 0 (m , ω , τ ). Hence, Proposition 13.10 applies and yields an estimation stronger than that of Proposition 13.17.

308

13. SOURCE TERMS OF WEAKER REGULARITY

13.6.1.2. Case 2: τ 0 = 0 and γ2 (0 ) = γ1 (0 ). As the roots γ1 and γ2 are locally smooth with respect to  = (x, ξ  , τ ) and homogeneous of degree one in (ξ  , τ ) by Proposition 6.28, there exists U a conic open neighborhood of (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ and C > 0 such that (13.6.7)

γ1 ( ) = γ2 ( ) and τ ≥ CλT,τ ,

 = (x, ξ  , τ ) ∈ U .

Without any loss of generality we may moreover assume that SU is compact. ˜ ∈ ST0,τ be homogeneous of We let χ ∈ ST0,τ be as in the statement and χ degree zero and be such that supp(χ) ˜ ⊂ U and χ ˜ ≡ 1 on supp(χ). From the smoothness and the homogeneity of the roots, we have χγ ˜ j ∈ ST1,τ , j = 1, 2. We set ˜ 2 ) and Q1 = Dd − OpT (χγ ˜ 1 ). Q2 = Dd − OpT (χγ We set z2 = Q1 v and z1 = Q2 v. Since OpT (χ)P0,ϕ = OpT (χ)Q1 Q2 + R1 = OpT (χ)Q2 Q1 + R2 with R1 , R2 ∈ Ψτ1,0 , we write OpT (χ)Q1 z1 = OpT (χ)P0,ϕ v − R1 v = OpT (χ)G0  + OpT (χ) ∂j Gj − R1 v, 1≤j≤d

and OpT (χ)Q2 z2 = OpT (χ)P0,ϕ v − R2 v = OpT (χ)G0  + OpT (χ) ∂j Gj − R2 v. 1≤j≤d

˜ j )Gd , one obtains Writing Dd Gd = Qj Gd + OpT (χγ OpT (χ)Q1 (z1 − iGd )+ + OpT (χ)Q2 (z2 − iGd )τ,+   G0 + + OpT (χ)OpT (χγ ˜ j )Gd + +



j=1,2

1≤j≤d−1

 G0 + + +



OpT (χ)∂j Gj + + vτ,1 

j=1,2

1≤j≤d−1

OpT (χγ ˜ j )OpT (χ)Gd +

∂j (OpT (χ)Gj )+ + G+ + vτ,1 ,

˜ j ), OpT (χ)] ∈ Ψ0T,τ and [∂j , OpT (χ)] ∈ Ψ0T,τ . Since τ  λT,τ in U , as [OpT (χγ with Proposition 6.12 we obtain OpT (χ)Q1 (z1 − iGd )+ + OpT (χ)Q2 (z2 − iGd )τ,+  G0 + + τ G+ + vτ,1

13.6. ESTIMATES WITHOUT PRESCRIBED BOUNDARY CONDITIONS

309

Since [Qj , OpT (χ)] ∈ Ψ0T,τ we obtain (13.6.8) Q1 OpT (χ)(z1 − iGd )+ + Q2 OpT (χ)(z2 − iGd )+  G0 + + τ G+ + vτ,1 + (z1 − iGd )+ + (z2 − iGd )+  G0 + + τ G+ + vτ,1 . We choose a conic open neighborhood W of supp(χ) such that W ⊂ {χ ˜ ≡ 1}. We then have W ⊂ U . We denote by pϕ , q2 , q1 the respective ˜ j ( ), j = 1, 2. principal symbols of P0,ϕ , Q2 , Q1 . We have qj () = ξd − χγ   ˜ ) ≡ 1, we have pϕ () = q2 q1 (), for  = ( , ξd ). For  ∈ W , as χ( By (13.6.7) one has γ2 ( ) = γ1 ( ) in W and thus Lemma 13.25 applies: 1 {qj , qj }( ) > 0,  ∈ V ,  = ( , ξd ), j = 1, 2, 2i that is, both factors Q2 and Q1 are sub-elliptic in W . Thus, Lemma 6.22 applies, yielding qj () = 0 ⇒

τ −1/2 OpT (χ)(zj − iGd )τ,1  Qj OpT (χ)(zj − iGd )+ + |OpT (χ)(zj − iGd )|xd =0+ |τ,1/2 + (zj − iGd )τ,0,−N , for j = 1, 2. With (13.6.8) we obtain (13.6.9)  τ −1/2 OpT (χ)(zj − iGd )τ,1 j=1,2

 G0 + + τ G+ +

 j=1,2

|OpT (χ)(zj − iGd )|xd =0+ |τ,1/2 + vτ,1 .

Next, we observe that

  OpT (χ)(z1 − z2 ) = OpT (χ)(Q2 − Q1 )v = OpT (χ)OpT χ(γ ˜ 1 − γ2 ) v   ˜ 1 − γ2 ) OpT (χ)v + R12 v, = OpT χ(γ  ˜ 1 − γ2 )] ∈ Ψ0T,τ . We also write where R12 = [OpT (χ), OpT χ(γ   Dd OpT (χ)(z1 − z2 ) = Dd OpT χ(γ ˜ 1 − γ2 ) OpT (χ)v + Dd R12 v    = OpT χ(γ v, ˜ 1 − γ2 ) OpT (χ)Dd v + R12  ∈ Ψ1,0 . with R12 τ Since χ(γ ˜ 1 − γ2 ) ∈ ST1,τ is elliptic in a neighborhood of supp(χ), with Proposition 2.37 of volume 1 (and Remark 2.38 both adapted to tangential operators) we then obtain   ˜ 1 − γ2 ) OpT (χ)wτ,0,s + wτ,0,−N , OpT (χ)wτ,0,s+1  OpT χ(γ

310

13. SOURCE TERMS OF WEAKER REGULARITY

and   |OpT (χ)w|xd =0+ |τ,s+1  |OpT χ(γ ˜ 1 − γ2 ) OpT (χ)w|xd =0+ |τ,s + |w|xd =0+ |τ,−N , for s ∈ R, N ∈ N. It yields OpT (χ)vτ,0,2 + OpT (χ)Dd vτ,0,1    OpT χ(γ ˜ 1 − γ2 ) OpT (χ)vτ,0,1   + OpT χ(γ ˜ 1 − γ2 ) OpT (χ)Dd v+ + vτ,1,−N  OpT (χ)(z1 − z2 )τ,1 + vτ,1   OpT (χ)(zj − iGd )τ,1 + vτ,1 , j=1,2

and similarly   ˜ 1 − γ2 ) OpT (χ)v|xd =0+ |τ,1/2 |OpT (χ)v|xd =0+ |τ,3/2  |OpT χ(γ + |v|xd =0+ |τ,−N  |OpT (χ)(z1 − z2 )|xd =0+ |τ,1/2 + |v|xd =0+ |τ,1/2   |OpT (χ)(zj − iGd )|xd =0+ |τ,1/2 + |v|xd =0+ |τ,1/2 . j=1,2

We then write τ OpT (χ)vτ,1  OpT (χ)vτ,1,1  OpT (χ)vτ,0,2 + Dd OpT (χ)vτ,0,1  OpT (χ)vτ,0,2 + OpT (χ)Dd vτ,0,1 + vτ,0,1   OpT (χ)(zj − iGd )τ,1 + vτ,1 . j=1,2

With (13.6.9) we obtain τ 1/2 OpT (χ)vτ,1 + |OpT (χ)v|xd =0+ |τ,3/2   τ −1/2 OpT (χ)(zj − iGd )τ,1 +



j=1,2

j=1,2

|OpT (χ)(zj − iGd )|xd =0+ |τ,1/2

+ τ −1/2 vτ,1 + |v|xd =0+ |τ,1/2   G0 + + τ G+ + |OpT (χ)(zj − iGd )|xd =0+ |τ,1/2 + vτ,1 , j=1,2

13.6. ESTIMATES WITHOUT PRESCRIBED BOUNDARY CONDITIONS

311

using the trace inequality of Corollary 6.10 (with m = 0 and s = 1/2). Because of the form of zj we have, with commutator arguments as above, τ 1/2 OpT (χ)vτ,1 + |OpT (χ)v|xd =0+ |τ,3/2 + |OpT (χ)(Dd v − iGd )|xd =0+ |τ,1/2   G0 + + τ G+ + |OpT (χ)(zj − iGd )|xd =0+ |τ,1/2 + vτ,1 , j=1,2

yielding, again with commutator arguments, (13.6.10) τ 1/2 OpT (χ)vτ,1 + |OpT (χ)v|xd =0+ |τ,3/2 + |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |τ,1/2   G0 + + τ G+ + |OpT (χ)(zj − iGd )|xd =0+ |τ,1/2 + vτ,1 . j=1,2

With Proposition 6.12 as τ  λT,τ in U and with the form of zj and that of B0,ϕ (v, h) given in (13.6.2) we have  |OpT (χ)(zj − iGd )|xd =0+ |τ,1/2 j=1,2

 τ 3/2

 j=1,2

+



j=1,2

|OpT (χ)(zj − iGd )|xd =0+ |τ,−1

|(zj − iGd )|xd =0+ |τ,−N

 ˜ϕ v)|x =0+ |  τ 3/2 |(Dd v − iGd + iB d τ,−1        ˜ + ˜ j ) v|xd =0+ τ,−1 iBϕ + OpT (χγ j=1,2

+ |(Dd v − iGd )|xd =0+ |τ,−N + |v|xd =0+ |τ,−N  τ |B0,ϕ (v, −Gd )|τ,−1/2 + τ 3/2 |v|xd =0+ |∂ + |(Dd v − iGd )|xd =0+ |τ,−N . With (13.6.10) we thus obtain an estimate that is stronger than the sought result. 13.6.1.3. Case 3: γ2 (0 ) = γ1 (0 ). If γ2 (0 ) = γ1 (0 ), then by (13.6.5)–(13.6.6) one has rϕ (0 ) = 0 and γ2 (0 ) = γ1 (0 ) = −iτ ∂d ϕ(x0 ). As ∂d ϕ(x0 ) = 0 this double root is not real here. Hence, near 0 the operator P0,ϕ is elliptic. Let ε0 > 0. Since τ 0 = 0, there exists U 0 a conic open neighborhood of 0  in U 0 × Rd−1 × R+ where |rϕ ( )| ≤ ε0 λ2T,τ , and |∂d ϕ(x)| ≥ C > 0, and λT,τ  τ.

312

13. SOURCE TERMS OF WEAKER REGULARITY

Without any loss of generality we may moreover assume that SU 0 is compact. From the form of the operator P0,ϕ , since we consider a microlocal region where |α| is small, we can foresee that the estimation we shall obtain near that region is almost given by an estimate for the conjugated operator Qϕ = (Dd + iτ ∂d ϕ(x))2 . In the case ∂d ϕ(x) ≥ C > 0 an perfect elliptic estimate for the operator Qϕ is given by Lemma 13.14. In the case ∂d ϕ(x) ≤ −C < 0, the counterpart result is given by the following lemma. Lemma 13.18. Let χ ˜ ∈ ST0,τ , homogeneous of degree zero, be such that supp(χ) ˜ ⊂ U 0 . There exist C > 0 and τ∗ > 0 such that

(13.6.11) τ OpT (χ)v ˜ τ,1 ≤ C H0 + + τ H+ ˜ − iOpT (χ)H ˜ d )|xd =0+ |τ,−1/2 + τ |(Dd OpT (χ)v  + τ |OpT (χ)v ˜ |xd =0+ |τ,1/2 + τ vτ,1,−1 , for τ ≥ τ∗ and v ∈ H 1 (Rd+ ), H0 ∈ L2 (Rd+ ), H ∈ L2 V (Rd+ ) such that  ∂j H j . Qϕ v = H 0 + 1≤i≤d

A proof is given below. Note that the estimate given by Lemma 13.14 in the case ∂d ϕ ≥ C > 0 implies (13.6.11). Here, in any case only an estimate as in (13.6.11) is needed. We shall thus apply this estimate either in both the cases ∂d ϕ ≥ C > 0 and ∂d ϕ ≤ −C < 0. For ε > 0 to be fixed below, we let Uε be the conic open set of U 0 × × R+ given by

Rd−1

Uε = { ∈ U 0 ; |α2 ( )| < ελ2T,τ }. We take 0 < ε ≤ ε0 /2 allowing us to choose χε ∈ ST0,τ homogeneous of degree zero such that supp(χε ) ⊂ Uε . In Uε ⊂ U 0 one has λT,τ  τ . With Proposition 6.11, there exists C1 > 0 such that OpT (χε )vτ,0,2 ≤ C1 τ 2 OpT (χε )v+ + Cε,N vτ,0,−N , for any N ∈ N, with some Cε,N > 0. C1

As |α2 ( )| ≤ ελ2T,τ in Uε , one has by Corollary 2.51, for C0 > 1 and = C0 C1 ,

(13.6.12)

 R0,ϕ OpT (χε )v+ ≤ C0 εOpT (χε )vτ,0,2 + Cε,N vτ,0,−N  vτ,0,−N . ≤ 2εC1 τ 2 OpT (χε )v+ + Cε,N

13.6. ESTIMATES WITHOUT PRESCRIBED BOUNDARY CONDITIONS

313

Recalling (13.6.4), we write Qϕ OpT (χε )v = P0,ϕ OpT (χε )v − R0,ϕ OpT (χε )v = OpT (χε )P0,ϕ v + [P0,ϕ , OpT (χε )]v − R0,ϕ OpT (χε )v  ∂j H j , = H0 + 1≤j≤d

with H = OpT (χε )G and H0 = OpT (χε )G0 +



[OpT (χε ), ∂j ]Gj + [P0,ϕ , OpT (χε )]v

1≤j≤d

− R0,ϕ OpT (χε )v. We have H+  G+ and, with (13.6.12),   H0 + ≤ Cε G0 + + G+ + vτ,1 + Cετ 2 OpT (χε )v+ . We choose χ ˜ ∈ ST0,τ homogeneous of degree zero such that supp(χ) ˜ ⊂U0 and χ ˜ ≡ 1 in a neighborhood of Uε0 /2 . Applying Lemma 13.18 we thus ˜ ◦ χε , obtain, with χ ˜ε = χ

τ OpT (χ ˜ε )vτ,1 ≤ Cε G0 + + τ G+ + vτ,1 + τ |OpT (χ ˜ε )v|xd =0+ |τ,1/2  + τ |(Dd OpT (χ ˜ε )v − iOpT (χ ˜ε )Gd )|xd =0+ |τ,−1/2 + Cετ 2 OpT (χε )v+ . We write ˜ε )v − iOpT (χ ˜ε )Gd )|xd =0+ |τ,−1/2 |(Dd OpT (χ ≤ |OpT (χ ˜ε )(Dd v − iGd )|xd =0+ |τ,−1/2 + |[Dd , OpT (χ ˜ε )]v|xd =0+ |τ,−1/2 ≤ C|OpT (χε )(Dd v − iGd )|xd =0+ |τ,−1/2 + Cε vτ,1,−1  ≤ C |(Dd OpT (χε )v − iOpT (χε )Gd )|xd =0+ |τ,−1/2  + |[Dd , OpT (χε )]v|xd =0+ |τ,−1/2 + Cε vτ,1,−1 ≤ C|(Dd OpT (χε )v − iOpT (χε )Gd )|xd =0+ |τ,−1/2 + Cε vτ,1,−1 , using the trace inequality of Corollary 6.10 (with m = 0 and s = −1/2). We also have ˜ε )v|xd =0+ |τ,1/2  |OpT (χε )v|xd =0+ |τ,1/2 . |OpT (χ ˜ε = χε mod ST−N Since χ ˜ ≡ 1 in a neighborhood of supp(χε ) we have χ ,τ for any N ∈ N by pseudo-differential calculus. We then find ˜ε )vτ,1 + Cε vτ,1,−N . OpT (χε )vτ,1 ≤ OpT (χ

314

13. SOURCE TERMS OF WEAKER REGULARITY

We have thus obtained

τ OpT (χε )vτ,1 ≤ Cε G0 + + τ G+ + |(Dd OpT (χε )v − iOpT (χε )Gd )|xd =0+ |τ,−1/2  + |OpT (χε )v|xd =0+ |τ,1/2 + vτ,1 + Cετ 2 OpT (χε )v+ .

Choosing ε > 0 sufficiently small, to be kept fixed and setting χ = χε we finally obtain τ OpT (χ)vτ,1  G0 + + τ G+ + |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |τ,−1/2 + |OpT (χ)v|xd =0+ |τ,1/2 + vτ,1 , which concludes the proof of Proposition 13.10 in Case 3.



Proof of set L = Dd +iτ ∂d ϕ(x). We have OpT (χ)L ˜ 2v = Lemma 13.18. We j ˜ 0 + 1≤j≤d OpT (χ)∂ ˜ j H , which we write OpT (χ)H  OpT (χ)L(Lv ˜ − iH d ) = OpT (χ)H ˜ 0+ OpT (χ)∂ ˜ jHj 1≤j≤d−1

+ τ OpT (χ)∂ ˜ d ϕH d . This yields (13.6.13) ˜ − iH d )τ,0,−1  H0 τ,0,−1 + H+  τ −1 H0 + + H+ , OpT (χ)L(Lv using Remark 6.8. Since τ ∂d ϕ  −λT,τ in U 0 , from Lemma 6.21 (with s = −1), we have for τ chosen sufficiently large, (13.6.14) ˜ +  LOpT (χ)z ˜ τ,0,−1 + |OpT (χ)z ˜ |xd =0+ |τ,−1/2 + zτ,0,−N , OpT (χ)z for z ∈ S (Rd+ ). We apply estimate (13.6.14) for z = Lv −H d , with a density argument, yielding ˜ − iH d )+ OpT (χ)(Lv  LOpT (χ)(Lv ˜ − iH d )τ,0,−1 + |OpT (χ)(Lv ˜ − iH d )|xd =0+ |τ,−1/2 + Lv − iH d τ,0,−N  OpT (χ)L(Lv ˜ − iH d )τ,0,−1 + |OpT (χ)(Lv ˜ − iH d )|xd =0+ |τ,−1/2 + vτ,1,−1 + H d + ,

13.6. ESTIMATES WITHOUT PRESCRIBED BOUNDARY CONDITIONS

315

using that [L, OpT (χ)] ˜ ∈ Ψ0T,τ . With (13.6.13), we then obtain ˜ − iH d )+ OpT (χ)(Lv  τ −1 H0 + + |OpT (χ)(Lv ˜ − iH d )|xd =0+ |τ,−1/2 + H+ + vτ,1,−1 . We then write ˜ +  OpT (χ)Lv ˜ LOpT (χ)v + + v+  OpT (χ)(Lv ˜ − iH d )+ + H d + + v+ , and ˜ − iH d )|xd =0+ |τ,−1/2 |OpT (χ)(Lv  |(LOpT (χ)v ˜ − iOpT (χ)H ˜ d )|xd =0+ |τ,−1/2 + |v|xd =0+ |τ,−1/2  |(LOpT (χ)v ˜ − iOpT (χ)H ˜ d )|xd =0+ |τ,−1/2 + vτ,1,−1 using the trace inequality of Corollary 6.10 (with m = 0 and s = −1/2). We thus obtain ˜ +  τ −1 H0 + + H+ LOpT (χ)v + |(LOpT (χ)v ˜ − iOpT (χ)H ˜ d )|xd =0+ |τ,−1/2 + vτ,1,−1 . Next, invoking Lemma 6.21 a second time (with s = 0 this time), we have for τ chosen sufficiently large, ˜ τ,1  LOpT (χ)v ˜ + + |OpT (χ)v ˜ |xd =0+ |τ,1/2 + vτ,0,−N . OpT (χ)v This yields ˜ τ,1  τ −1 H0 + + H+ + |OpT (χ)v ˜ |xd =0+ |τ,1/2 OpT (χ)v + |(LOpT (χ)v ˜ − OpT (χ)H ˜ d )|xd =0+ |τ,−1/2 + vτ,1,−1 . Observing that ˜ − OpT (χ)H ˜ d )|xd =0+ |τ,−1/2 |(LOpT (χ)v  |OpT (χ)v ˜ |xd =0+ |τ,1/2 + |(Dd OpT (χ)v ˜ − OpT (χ)H ˜ d )|xd =0+ |τ,−1/2 , which concludes the proof.



13.6.2. Estimate for a Weight Function with a Vanishing Neumann Trace. Here, we place ourselves in the neighborhood of a point of ∂M where ∂ν ϕ may vanish. Theorem 13.19. Let (M, g) be a compact Riemannian manifold with boundary and let P = −Δg + R1 with R1 a first-order differential operator with bounded coefficients on M. Let V be an open set of M such that V∂ = V ∩ ∂M = ∅. Let B  be a differential operator of order one on ∂M ˇ such that (P, B) fulfills the Lopatinski˘ı–Sapiro condition of Definition 2.2 on  ∞ V∂ for B = ∂ν + B . Let ϕ ∈ C (M) be such that the pair (P, ϕ) has the sub-ellipticity property of Definition 5.1 in V .

316

13. SOURCE TERMS OF WEAKER REGULARITY

Assume that ∂ν ϕ|∂M vanishes in some point of V∂ and let V  ⊂ V be a neighborhood in M of such points. Then, there exist C and τ∗ > 0 such that (13.6.15) τ 1/2 eτ ϕ uτ,1 + τ |eτ ϕ u|∂M |τ,1/2 + τ |eτ ϕ|∂M (∂ν u + g(F, ν))|∂M |τ,−1/2

≤ C eτ ϕ F0 L2 (M) + τ eτ ϕ F L2 V (M) + τ 3/2 eτ ϕ g(ν, F )L2 V (M)  + τ 3/2 |eτ ϕ u|∂M |L2 (∂M) + τ |eτ ϕ|∂M B(u, g(F, ν))|τ,−1/2 , for τ ≥ τ∗ and u ∈ H 1 (M), F0 ∈ L2 (M), F ∈ L2 V (M) such that P u = F0 + divg F, and supp(u) ∪ supp(F0 ) ∪ supp(F ) ⊂ V  . The difference with Theorem 13.16 lies in the term eτ ϕ g(ν, F )L2 V (M) on the right-hand side on the estimate. Here, it appears with the power τ 3/2 of the large parameter. In Theorem 13.16 one only has τ implying that the present estimate is weaker. For the proof of Theorem 13.19 one can proceed as in Sect. 13.6.1. Near each point ∂M ∩ V  one proves a local estimate. For that, one uses a local chart C = (κ, O) as above. The local estimate is itself based on microlocal estimates. For a point x0 where ∂d ϕ(x0 ) = 0 one uses the microlocal estimate of Proposition 13.17. Here, it suffices to consider points x0 such that ∂d ϕ(x0 ) = 0. Since the sub-ellipticity property holds, in this case one has dx ϕ(x0 ) = 0. This property holds locally. We thus consider 0 = (x0 , ξ 0 , τ 0 ) with (ξ 0 , τ 0 ) ∈ Rd−1 × [0, +∞). If τ 0 = 0, then Case 1 in the proof of Proposition 13.17 applies.  2 We may now assume that τ 0 > 0. One writes pϕ () = ξd +iτ ∂d ϕ(x) + rϕ ( ). One has pϕ (0 , ξd ) = ξd2 + rϕ (0 ). If rϕ (0 ) =0, then in a conic neighborhood of 0 one can write pϕ () = ξd − γ1 ( ) ξd − γ2 ( ) with γ1 ( ) = γ2 ( ). Then, one observes that Case 2 in the proof of Proposition 13.17 applies. Then, the only remaining case to consider here is rϕ (0 ) = 0. We recall that rϕ ( ) = r(x, ξ  + iτ dx ϕ(x)), with r(x, ξ  )  |ξ  |2 . If r˜(x, ., .) is the associated bilinear form, rϕ ( ) vanishes if and only if (13.6.16)

r(x, ξ  ) = τ 2 r(x, dx ϕ(x)) and r˜(x, ξ  , dx ϕ(x)) = 0.

If d = 2, then ξ  ∈ R and rϕ ( ) = 0 cannot occur if (ξ  , τ ) = (0, 0) as dx ϕ(x) = 0 locally. However, if d ≥ 3, with ξ  ∈ Rd−1 there always exists (ξ  , τ ) = (0, 0) such that (13.6.16) holds. In the following proposition we give a microlocal estimate in a conic neighborhood of such a point. Note that in such case ξ  = 0 if τ = 0.

13.6. ESTIMATES WITHOUT PRESCRIBED BOUNDARY CONDITIONS

317

The following proposition yields the additional microlocal estimate near 0 to complete the proof of Theorem 13.19. Proposition 13.20. Let m0 ∈ ∂M and V 0 be an open neighborhood of m0 in M. Let U 0 = κ(V 0 ∩O) and x0 = κ(m0 ). Assume that (P0 , ϕ) has the sub-ellipticity property of Definition 3.2 of Volume 1 in U 0 and ∂d ϕ(x0 ) = 0. Let B  be a differential operator of order one on ∂M∩U 0 and set B = ∂ν +B  . ∗ ∂M, with local representative (ω 0 )C = ξ 0 ∈ Rd−1 , and τ 0 > 0 Let ω 0 ∈ Tm 0 such that (ξ 0 , τ 0 ) = 0 and rϕ (x0 , ξ 0 , τ 0 ) = 0. Then, there exists U a conic open neighborhood of 0 = (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ such that for χ ∈ ST0,τ , homogeneous of degree zero, with supp(χ) ⊂ U , there exist C > 0 and τ∗ > 0 such that (13.6.17) τ 1/2 OpT (χ)vτ,1 + τ |OpT (χ)v|xd =0+ |τ,1/2 + τ |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |τ,−1/2

≤ C G0 + + τ G + + τ 3/2 Gd + + τ 3/2 |v|xd =0+ |∂ + τ |B0,ϕ (v, −Gd )|τ,−1/2  + vτ,1 + |(Dd v − iGd )|xd =0+ |τ,−3/2 , for τ ≥ τ∗ and v ∈ H 1 (Rd+ ), G0 ∈ L2 (Rd+ ), G ∈ L2 V (Rd+ ), and G = (G1 , . . . , Gd−1 ), such that  ∂ j Gj . P0,ϕ v = G0 + 1≤i≤d

Remark 13.21. Note that this estimate only differs from that of Proposition 13.17 by the term τ 3/2 Gd + that replaces the term τ Gd + . Proof. We first observe that rϕ ( ) satisfies a strong sub-elliptic property for  = (x, ξ  , τ  ) in a conic neighborhood of 0 = (x0 , ξ 0 , τ 0 ). Indeed, one has rϕ (0 ) = 0. With 0 = (0 , ξd = 0) one has pϕ (0 ) = 0 and thus, since (P0 , ϕ) has the sub-ellipticity property one has {pϕ , pϕ }(0 )/i > 0. Observe that   {pϕ , pϕ }() = 2 ξd − iτ ∂d ϕ(x) {ξd − iτ ∂d ϕ, pϕ }()   + 2 ξd + iτ ∂d ϕ(x) {rϕ , ξd + iτ ∂d ϕ} + {rϕ , rϕ }, which yields 1 1 0 < {pϕ , pϕ }(0 ) = {rϕ , rϕ }(0 ). i i By continuity and homogeneity there exists a conic neighborhood V of 0 such that 1 (13.6.18) {rϕ , rϕ }( ) ≥ C0 λ3T,τ and τ ≥ C0 λT,τ , i for some C0 > 0. Without any loss of generality we may moreover assume that SV is compact. Let 0 < ε < 1; there exists a conic neighborhood

318

13. SOURCE TERMS OF WEAKER REGULARITY

Uε ⊂ V of 0 such that (13.6.19)

|rϕ ( )| ≤ ελ2T,τ , and |∂d ϕ(x)| ≤ ε,

 = (x, ξ  , τ ) ∈ Uε .

We consider χ ∈ Sτ0 supported in Uε . The value of ε be set below. We define 1 ∗ ) and R2 = (R0,ϕ + R0,ϕ 2 1 ∗ ), R1 = (R0,ϕ − R0,ϕ 2i with respective principal symbols r2 ( ) = r(x, ξ  ) − τ 2 r(x, dx ϕ) ∈ Sτ2 , and r1 ( ) = 2τ r˜(x, ξ  , dx ϕ) ∈ τ Sτ1 ⊂ Sτ2 . Note that (13.6.18) reads (13.6.20)

{r2 , r1 }( )  λ3T,τ  τ 3 ,

 ∈ V ,

. using that τ  λT,τ in V From P0,ϕ v = G0 + 1≤j≤d ∂j Gj , we write  ∂j Gj + [P0,ϕ , OpT (χ)]v, P0,ϕ OpT (χ)v = OpT (χ)G0 + OpT (χ) 1≤j≤d

yielding w2 + iw1 = H0 +



∂j H j ,

1≤j≤d−1

with w2 = Dd (Dd OpT (χ)v − iOpT (χ)Gd ) − (τ ∂d ϕ)2 OpT (χ)v + R2 OpT (χ)v,   w1 = 2τ ∂d ϕ Dd OpT (χ)v − iOpT (χ)Gd + R1 OpT (χ)v and H0 = OpT (χ)G0 +



[OpT (χ), ∂j ]Gj + [P0,ϕ , OpT (χ)]v

1≤j≤d

+ 2τ (∂d ϕ)OpT (χ)Gd − τ ∂d2 ϕOpT (χ)v, and H j = OpT (χ)Gj . One has   H0 + ≤ Cε G0 + + G + + τ Gd + + vτ,1 , and ∂j H j + ≤ Cε τ Gj + , using Proposition 6.12 since τ  λT,τ in V . Exploiting the standard Carleman approach we then write (13.6.21)   w2 2+ + w1 2+ + 2 Re(w2 , iw1 )+ ≤ Cε G0 2+ + τ 2 G2+ + v2τ,1 .

13.6. ESTIMATES WITHOUT PRESCRIBED BOUNDARY CONDITIONS

319

We focus our attention on the computation of the term Re(w2 , iw1 )+ in (13.6.21) that we write as a sum of six terms Iij , 1 ≤ i ≤ 3, 1 ≤ j ≤ 2, where Iij is the inner product of the ith term in the expression of w2 and the jth term in the expression of iw1 . Term I11 . With an integration by parts, we have   (τ )−1 I11 = 2 Re Dd (Dd OpT (χ)v − iOpT (χ)Gd ), i∂d ϕ Dd OpT (χ)v  − iOpT (χ)Gd + 2  = − ∫ ∂d ϕ∂d (Dd OpT (χ)v − iOpT (χ)Gd ) dx Rd+

 2 = ∫ ∂d ϕ(Dd OpT (χ)v − iOpT (χ)Gd )|x Rd−1

+ d =0

dx

 2 + ∫ ∂d2 ϕ(Dd OpT (χ)v − iOpT (χ)Gd ) dx, Rd+

yielding 2

|I11 |  τ |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |∂

(13.6.22)

2

+ τ Dd OpT (χ)v2+ + τ Gd + . Term I12 . With an integration by parts, we have I12 = Re(Dd (Dd OpT (χ)v − iOpT (χ)Gd ), iR1 OpT (χ)v)+ = Re((Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ , R1 OpT (χ)v|xd =0+ )∂ + Re((Dd OpT (χ)v − iOpT (χ)Gd ), iDd R1 OpT (χ)v)+ With R1 ∈ τ Ψ1T,τ we find    Re((Dd OpT (χ)v − iOpT (χ)Gd )|x =0+ , R1 OpT (χ)v|x =0+ )∂  d d 2

 τ |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |∂ + τ |OpT (χ)v|xd =0+ |2τ,1 . We also write Re(Dd OpT (χ)v − iOpT (χ)Gd , iDd R1 OpT (χ)v)+ = − Re(iOpT (χ)Gd , iR1 Dd OpT (χ)v)+ + Re(Dd OpT (χ)v − iOpT (χ)Gd , i[Dd , R1 ]OpT (χ)v)+ , using that iR1 is formally anti-adjoint. Since R1 ∈ τ Ψ1τ , [Dd , R1 ] ∈ τ Ψ0T,τ one finds    Re((Dd OpT (χ)v − iOpT (χ)Gd ), iDd R1 OpT (χ)v)+  2

 τ Dd OpT (χ)v2+ + τ OpT (χ)Gd τ,1 + τ OpT (χ)v2+ .

320

13. SOURCE TERMS OF WEAKER REGULARITY

Consequently, one has 2

I12  τ Dd OpT (χ)v2+ + τ OpT (χ)Gd τ,1 + τ OpT (χ)v2+ 2

+ τ |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |∂ + τ |OpT (χ)v|xd =0+ |2τ,1 . Term I21 . We write    τ −3 I21 = −2 Re (∂d ϕ)3 OpT (χ)v, i Dd OpT (χ)v − iOpT (χ)Gd + = − ∫ (∂d ϕ)3 ∂d |OpT (χ)v|2 dx Rd+

  − 2 Re (∂d ϕ)3 OpT (χ)v, OpT (χ)Gd + . With an integration by parts one has − ∫ (∂d ϕ)3 ∂d |OpT (χ)v|2 dx = 3 ∫ (∂d ϕ)2 (∂d2 ϕ)|OpT (χ)v|2 dx Rd+

Rd+

+ ∫ (∂d ϕ)3 |OpT (χ)v|2|xd =0+ dx . Rd−1

We thus find  2 |I21 |  ετ 3 OpT (χ)v2+ + |OpT (χ)v|xd =0+ |2∂ + Gd + , since |∂d ϕ| ≤ ε < 1 in supp(χ). Term I22 . We write, using that R1 is formally selfadjoint,   I22 = −τ 2 Re (∂d ϕ)2 OpT (χ)v, iR1 OpT (χ)v +   = −τ 2 i[(∂d ϕ)2 , R1 ]OpT (χ)v, OpT (χ)v + . The principal symbol of i[(∂d ϕ)2 , R1 ] is given by s1 = ∂d ϕ{∂d ϕ, r1 } ∈ τ ∂d ϕST0,τ yielding |s1 |  ετ in Uε . Thus, Corollary 2.51 yields |I22 |  ετ 3 OpT (χ)v2+ + v20,−N , for any N ∈ N. Term I31 . We write   I31 = 2τ Re R2 OpT (χ)v, i(∂d ϕ)(Dd OpT (χ)v − iOpT (χ)Gd ) + First, one has     2τ Re R2 OpT (χ)v, (∂d ϕ)OpT (χ)Gd   ετ OpT (χ)v2 τ,0,1 + 2

+ τ OpT (χ)Gd τ,0,1 ,

13.6. ESTIMATES WITHOUT PRESCRIBED BOUNDARY CONDITIONS

321

since |∂d ϕ| ≤ ε in supp(χ). Second, one computes   Re R2 OpT (χ)v, i(∂d ϕ)Dd OpT (χ)v +   = (∂d ϕ)R2 OpT (χ)v, iDd OpT (χ)v +   + i(∂d ϕ)Dd OpT (χ)v, R2 OpT (χ)v +   = − (∂d ϕ)R2 OpT (χ)v|xd =0+ , OpT (χ)v|xd =0+ ∂   + iAOpT (χ)v, OpT (χ)v + , with A = R2 (∂d ϕ)Dd − Dd (∂d ϕ)R2 with principal symbol given by a2 + a1 ξd with a2 ∈ ST2,τ and a1 ∈ ST1,τ with the additional property a2  ελ2T,τ in Uε where |∂d ϕ| + |r2 |λ−2 T,τ  ε. Consequently, with the Young inequality, one obtains      Op (χ)v, i(∂ ϕ)D Op (χ)v 2τ Re R  2 d d T T +    τ |OpT (χ)v|xd =0+ |2τ,1 + ε + ε1/2 τ OpT (χ)v2τ,0,1 + ε−1/2 τ Dd OpT (χ)v2+ , yielding

  |I31 |  τ |OpT (χ)v|xd =0+ |2τ,1 + ε + ε1/2 τ OpT (χ)v2τ,0,1 2

+ ε−1/2 τ Dd OpT (χ)v2+ + τ OpT (χ)Gd τ,0,1 . Term I32 . Since R1 and R2 are both tangential and formally selfadjoint we have I32 = Re(R2 OpT (χ)v, iR1 OpT (χ)v)+ = (i[R2 , R1 ]OpT (χ)v, OpT (χ)v)+ , where i[R2 , R1 ] ∈ Ψ3T,τ has {r2 , r1 } for principal symbol. With (13.6.20) and the microlocal G˚ arding inequality of Theorem 2.50 of Volume 1 one finds I32 ≥ COpT (χ)v2τ,0,3/2 − CN v2τ,0,−N , for any N ∈ N. With (13.6.21) and collecting the estimations obtained for the six terms we find OpT (χ)vτ,0,3/2 + w2 +   ≤ Cε G0 + + τ G+ + vτ,1

+ Cτ 1/2 |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |∂ + |OpT (χ)v|xd =0+ |τ,1 + (1 + ε−1/4 )Dd OpT (χ)v+   + (1 + ε1/2 τ )OpT (χ)v+ + ε1/2 + ε1/4 OpT (χ)vτ,0,1  + OpT (χ)Gd τ,1 .

322

13. SOURCE TERMS OF WEAKER REGULARITY

For ε > 0 chosen sufficiently small and for τ ≥ 1 chosen sufficiently large we find OpT (χ)vτ,0,3/2 + w2 +   ≤ Cε G0 + + τ G+ + vτ,1

+ Cτ 1/2 |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |∂ + |OpT (χ)v|xd =0+ |τ,1 + (1 + ε−1/4 )Dd OpT (χ)v+  + OpT (χ)Gd τ,1 . The following lemma explains how the L2 -norm of Dd OpT (χ)v can be recovered from that of OpT (χ)v and w2 . Lemma 13.22. One has τ 1/2 Dd OpT (χ)v+  ε1/2 τ 3/2 OpT (χ)v+ + (ετ )−1/2 w2 + + τ 1/2 Gd + + τ 1/2 |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |∂ + τ 1/2 |OpT (χ)v|xd =0+ |∂ + Cε,N vτ,0,−N . A proof is given below. With the lemma, for τ ≥ ε−2 , we obtain ε−1/2 τ 1/2 Dd OpT (χ)v+ + OpT (χ)vτ,0,3/2 + w2 +

≤ Cε G0 + + τ G+ + vτ,1 + τ 1/2 |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |∂  + τ 1/2 |OpT (χ)v|xd =0+ |τ,1

 + Cτ 1/2 (1 + ε−1/4 )Dd OpT (χ)v+ + OpT (χ)Gd τ,1 . For ε > 0 chosen sufficiently small one obtains τ 1/2 Dd OpT (χ)v+ + OpT (χ)vτ,0,3/2 + w2 +  G0 + + τ G+ + τ 1/2 OpT (χ)Gd τ,1 + vτ,1 + τ 1/2 |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |∂ + τ 1/2 |OpT (χ)v|xd =0+ |τ,1 . With the form of B0,ϕ given in (13.6.2) we then obtain (13.6.23) τ 1/2 Dd OpT (χ)v+ + OpT (χ)vτ,0,3/2 + w2 +  G0 + + τ G+ + τ 1/2 OpT (χ)Gd τ,1 + vτ,1 + τ 1/2 |B0,ϕ (OpT (χ)v, OpT (χ)Gd )|∂ + τ 1/2 |OpT (χ)v|xd =0+ |τ,1 .

13.6. ESTIMATES WITHOUT PRESCRIBED BOUNDARY CONDITIONS

323

We write B0,ϕ (OpT (χ)v, −OpT (χ)Gd ) ˜ϕ OpT (χ)v|x =0+ − i(Dd OpT (χ)v − iOpT (χ)Gd )|x =0+ =B d d   d ˜ = OpT (χ)B0,ϕ (v, −G ) + [Bϕ , OpT (χ)] − i[Dd , OpT (χ)] v. With Proposition 6.11, since τ  λT,τ in V , one finds τ 1/2 |B0,ϕ (OpT (χ)v, −OpT (χ)Gd )|∂  τ |B0,ϕ (v, −Gd )|τ,−1/2 + τ 1/2 |v|xd =0+ |∂  τ |B0,ϕ (v, −Gd )|τ,−1/2 + v1 , using the trace inequality of Corollary 6.10. With the same arguments applied to other terms in the right-hand side of (13.6.23) yielding τ 1/2 OpT (χ)Gd τ,1  τ 3/2 Gd + , τ 1/2 |OpT (χ)v|xd =0+ |τ,1  τ 3/2 |v|xd =0+ |∂ , 

we then obtain the sought result. Proof of Lemma 13.22. We write Dd (Dd OpT (χ)v − iOpT (χ)Gd ) = w2 + (τ ∂d ϕ)2 OpT (χ)v − R2 OpT (χ)v. We estimate    τ  w2 + (τ ∂d ϕ)2 OpT (χ)v − R2 OpT (χ)v, OpT (χ)v +   (ετ )−1 w2 2+ + ετ 3 OpT (χ)v2+ + Cε,N v2τ,0,−N ,

using Corollary 2.51 as |r2 |  ελ2T,τ and |∂d ϕ| ≤ ε in supp(χ). We compute τ (Dd (Dd OpT (χ)v − iOpT (χ)Gd ), OpT (χ)v)+ = iτ ((Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ , OpT (χ)v|xd =0+ )∂ + τ Dd OpT (χ)v2+ − iτ (OpT (χ)Gd , Dd OpT (χ)v)+ . This yields τ Dd OpT (χ)v2+  2 2  τ (Dd (Dd OpT (χ)v − iOpT (χ)Gd ), OpT (χ)v)+  + τ Gd + 2

+ τ |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |∂ + τ |OpT (χ)v|xd =0+ |2∂ , which gives the result.



324

13. SOURCE TERMS OF WEAKER REGULARITY

13.7. Global Estimates 13.7.1. Global Estimate with an Inner Observation. With the same arguments as in Chapters 3 and 5 we have the following global estimate with an inner observation. Theorem 13.23. Let M be a smooth compact connected Riemannian manifold. Let P = −Δg + R1 where R1 is a first-order differential operator with bounded coefficients. Let ω, ω0 be two nonempty open subsets of M such that ω0  ω. Let ϕ be a global weight function adapted to Γ0 = ∂M and ω0 in the sense of Definition 5.7. Consider also B a differential operator of order k ∈ N ˇ in ∂M and assume moreover that (P, B, ϕ) satisfies the Lopatinski˘ı–Sapiro condition on ∂M (Definition 8.1). Then, there exist τ∗ > 0 and C ≥ 0 such that τ 1/2 eτ ϕ uτ,1 + τ |eτ ϕ u|∂M |τ,1/2 + τ |eτ ϕ (∂ν u + g(F, ν))|∂M |τ,−1/2  ≤ C eτ ϕ F0 L2 (M) + τ eτ ϕ F L2 V (M)

 + τ |eτ ϕ|∂M B(u, g(F, ν))|τ,1/2−k + τ 3/2 eτ ϕ uL2 (ω) ,

for τ ≥ τ∗ and u ∈ H 1 (M), F0 ∈ L2 (M), F ∈ L2 V (M) such that P u = F0 + divg F. 13.7.2. Global Estimate with a Boundary Observation. Consider a neighborhood O of ∂M where one has global normal geodesic coordinates as given by Theorem 17.22 using that M is compact here: for some z0 > 0 one has a diffeomorphism Φ, such that (13.7.1)

Φ : ∂M × [0, z0 ) → O (m , z) → Φ(m , z),

allowing one to use (m , z) to parameterize O, and the pullback of g takes  (z) ⊗ 1 + d ⊗ d . In O we denote by ν the image of −∂ by the the form gm  z z z z tangent map T Φ(m , z) (see Sect. 15.3.2). At the boundary ν is precisely the outward pointing unit vector field. With ν thus introduced we can give a precise meaning of g(F, ν) for a vector field F corresponding to the normal part of F near the boundary. With the results of Theorems 13.4 and 13.16 and Proposition 13.20 we deduce the following global estimates. Theorem 13.24. Let P = −Δg + R1 where R1 is a first-order differential operator with bounded coefficients. Let Γ0 , Γobs be two nonempty open subsets of ∂M such that Γ0 ∪ Γobs = ∂M and Γobs \ Γ0 = ∅. Let ϕ be a global weight function adapted to Γ0 in the sense of Definition 5.10. Consider also B a differential operator of order k ∈ N in Γ0 and ˇ assume moreover that (P, B, ϕ) satisfies the Lopatinski˘ı–Sapiro condition on Γ0 (Definition 8.1).

13.8. NOTES

325

˜ = Consider also B  a first-order differential operator on Γobs and B ˜ fulfills the Lopatinski˘ı–Sapiro ˇ condition of ∂ν + B  and assume that (P, B) Definition 2.2 on Γobs . Let also χ0 , χobs ∈ C ∞ (M) be such that χ0 |∂M and χobs |∂M form a partition of unity of ∂M associated with the covering by Γ0 and Γobs , that is, supp(χ0 |∂M ) ⊂ Γ0 , supp(χobs |∂M ) ⊂ Γobs , and χ0 |∂M + χobs |∂M ≡ 1 in ∂M. Moreover, we assume that χobs is supported in O (neighborhood of ∂M introduced above). Then, there exist τ∗ > 0 and C ≥ 0 such that τ 1/2 eτ ϕ uτ,1 + τ |eτ ϕ u|∂M |τ,1/2 + τ |eτ ϕ (∂ν u + g(F, ν))|∂M |τ,−1/2  ≤ C eτ ϕ F0 L2 (M) + τ eτ ϕ F L2 V (M) + τ 3/2 χobs eτ ϕ g(ν, F )L2 (M) + τ |χ0 eτ ϕ|∂M B(u, g(F, ν))|τ,1/2−k + τ 3/2 |χobs eτ ϕ u|∂M |L2 (∂M)

 ˜ g(F, ν))| + τ |χobs eτ ϕ|∂M B(u, τ,−1/2 ,

for τ ≥ τ∗ and u ∈ H 1 (M), F0 ∈ L2 (M), F ∈ L2 V (M) such that P u = F0 + divg F. ˜ .) has the same definition as B(., .) given In the theorem statement, B(., ˜ in (13.1.3) with B replaced by B. Note that results in the form given in Theorems 5.8 and 5.11 could also be written. Imposing limitation on the supports of the functions then allows for nonsmooth parts of the boundary. This is, for instance, used in Sections 6.5.2 and 7.5.2 for the derivation of a resolvent estimate and a spectral inequality. If one compares the result of Theorems 13.23 and 13.24 one sees that the term χobs eτ ϕ g(ν, F )L2 (M) appears in the latter with a factor τ 3/2 instead of τ . The reason lies in the observation that the chosen weight function ϕ is such that ∂ν ϕ = 0 at places in Γobs . Indeed, if it is not the case, then ∂ν ϕ < 0 on ∂M and necessarily ϕ reaches a maximum in M \ ∂M which is excluded for a Carleman estimate to hold by Theorems 4.5 and 4.7. It would be satisfying to be able to obtain the factor τ , in particular if needed in applications. 13.8. Notes A Carleman estimate for a parabolic operator with a source term in H −1 appears in the work of O. Yu. Imanuvilov and M. Yamamoto [181]. There, homogeneous Dirichlet boundary conditions are considered. The case of an elliptic operator, also with homogeneous Dirichlet boundary conditions, can be found in the work of O. Yu. Imanuvilov and J.P. Puel [179]. In the work of E. Fern´andez-Cara et al. [144], Neumann

326

13. SOURCE TERMS OF WEAKER REGULARITY

boundary conditions are considered for a parabolic operator. Motivations for these works are inverse problems and controllability of Navier-Stokes equations. Here, we prove estimation with a source term in H −1 for any boundary ˇ operator for which the Lopatinski˘ı–Sapiro condition holds. As in Chap. 8 the proofs we give are based on the analysis of the conjugated operator viewed in certain microlocal regions as a product of two first-order operators. This allows us to circumvent the difficulty raised by not having a well defined Neumann trace of the solution; only a modified version of the Neumann trace make sense; see Proposition 13.2. This modified Neumann trace appears in the results proven here; see Theorem 13.4. Appendix 13.A. Some Technical Proofs 13.A.1. A Trace Result. Here, we prove Proposition 13.2. Since P u = F0 + divg F and P = −Δg + R1 one has divg (∇g u + F ) = −F0 + R1 u ∈ L2 (M). This implies that V = ∇g u + F is in the space of L2 -vector fields that have a L2 -divergence.  This space is H(divg , M). By Lemma 18.43, the trace of ∂ν u + g(F, ν) |∂M = g(V, ν)|∂M makes sense and is in H −1/2 (M). Moreover, one has     ∂ν u + g(F, ν)  |∂M −1/2 H

(∂M)

 V H(divg ,M)  V L2 V (M) +  divg V L2 (M) . One has V L2 V (M)  ∇g uL2 V (M) + F L2 V (M)  uH 1 (M) + F L2 V (M) , and  divg V L2 (M)  F0 L2 (M) + R1 uL2 (M)  F0 L2 (M) + uH 1 (M) .    We thus obtain the expected estimation for  ∂ν u + g(F, ν) |∂M 

H −1/2 (∂M)

. 

ˇ 13.A.2. Trace Estimate Under the Lopatinski˘ı–Sapiro Condition. Here, we prove Lemma 13.11 by adapting the proof of Lemma 8.19 in Sect. 8.A.3 and using its notation. In particular, with σ = 1 therein, we introduce 

1/2−k k−1 k   1 ˜ B1ϕ = Λ1/2−k Bϕ (x, D , τ ) −iΛT,τ Bϕ (x, D , τ )ΛT,τ , T,τ 

1/2 P1,+ = −Λ−1/2 ˜ 2 ) ΛT,τ , T,τ OpT (χγ

13.A. SOME TECHNICAL PROOFS

327

with respective principal symbols  

1/2 ˜bk ( ) −iλ3/2−k bk−1 ( ) , p1,+ = −λ−1/2 χγ . b1ϕ ( ) = λ1/2−k ˜ λ 2 ϕ ϕ T,τ T,τ T,τ T,τ We also set M1 = (B1ϕ )∗ B1ϕ + (P1,+ )∗ P1,+ . It is 2 × 2 matrix operator of order one. Let u1 ∈ H 1/2 (Rd−1 ) and u2 ∈ H −1/2 (Rd−1 ). Setting U = t (u , Λ−1 u ) ∈ H 1/2 (Rd−1 )2 , we have 1 T,τ 2 (13.A.1)

M1 U, U H −1/2 (Rd−1 )2 ,H 1/2 (Rd−1 )2 2

2

= |B1ϕ U |∂ + |P1ϕ U |∂ 2

˜ϕk (x, D , τ )u1 − iBϕk−1 (x, D , τ )u2 | = |B τ,1/2−k + |u2 − OpT (χγ ˜ 2 )u1 |τ,−1/2 . Let W ⊂ U be an open conic neighborhood of supp(χ) where χ ˜ ≡ 1. Let χ ˆ ∈ ST0,τ be homogeneous of degree zero and be supported in W and such that χ ˆ ≡ 1 on supp(χ). As W ⊂ U we have bϕ (x, ξ  , ξd = γ2 (x, ξ  , τ ), τ ) = 0,

 = (x, ξ  , τ ) ∈ W .

Lemma 8.36, for σ = 1 and the order of the boundary operator equal to k therein, gives with a density argument ˆ OpT (χ)U ˆ H −1/2 (Rd−1 )2 ,H 1/2 (Rd−1 )2 M1 OpT (χ)U, ≥ C|OpT (χ)U ˆ |2τ,1/2 − CN |U |2τ,−N , for N ∈ N and τ ≥ 1 chosen sufficiently large.  We now set u1 = OpT (χ) and u2 = OpT (χ) , that is U = OpT (χ),   ˆ and χ and from Λ−1 T,τ OpT (χ) . Because of the support conditions of χ pseudo-differential calculus, we find, for any N ∈ N, M1 U, U H −1/2 (Rd−1 )2 ,H 1/2 (Rd−1 )2 ≥ ReM1 OpT (χ)U, ˆ OpT (χ)U ˆ H −1/2 (Rd−1 )2 ,H 1/2 (Rd−1 )2  2  2 − CN ||τ,−N + | |τ,−N , and

  2 |OpT (χ)U ˆ |2τ,1/2 ≥ |U |2τ,1/2 − CN ||2τ,−N + | |τ,−N ,

yielding

  M1 U, U H −1/2 (Rd−1 )2 ,H 1/2 (Rd−1 )2 ≥ C|U |2τ,1/2 − CN ||2τ,−N + |h|2τ,−N   2 = C |OpT (χ)|2τ,1/2 + |OpT (χ) |τ,−1/2   2 − CN ||2τ,−N + | |τ,−N .

By (13.A.1), this concludes the proof of Lemma 13.11.



328

13. SOURCE TERMS OF WEAKER REGULARITY

13.A.3. Sub-ellipticity of a First-Order Factor. Here, we prove the sub-ellipticity property of the first-order operator P + stated in Lemma 13.12. We start with the following lemma Lemma 13.25. Let V be a conic open set of U 0 × Rd−1 × R+ . Assume that pϕ read pϕ = q1 q2 in V with q1 and q2 in the form qj () = ξd − rj ( ), j = 1, 2, where r1 , r2 ( ) ∈ ST1,τ and are such that r1 ( ) = r2 ( ) for  ∈ V . If the sub-ellipticity property holds for (P, ϕ) in U 0 , then one has 1  ∈ V ,  = ( , ξd ), j = 1, 2. {qj , qj }( ) > 0, qj () = 0 ⇒ 2i Proof. We treat the case j = 1. We compute {pϕ , pϕ } = |q1 |2 {q2 , q2 } + |q2 |2 {q1 , q1 } + q1 q2 {q2 , q1 } + q2 q1 {q1 , q2 }. We recall that the sub-ellipticity property reads (see Definition 3.2 of Volume 1) 1 {pϕ , pϕ }() > 0. pϕ () = 0 ⇒ 2i Let  = ( , ξd ) with  ∈ V be such that q1 () = 0. Then pϕ ( ) = 0 and q2 () = 0 since r1 ( ) = r2 ( ). We thus find 1 {q1 , q1 }( ) > 0, 2i  since {pϕ , pϕ }() = |q2 |2 (){q1 , q1 }( ). We now proceed with the proof of Lemma 13.12. We choose the conic open neighborhood W of supp(χ) such that W ⊂ {χ ˜ ≡ 1}. We then have W ⊂ U . We denote by pϕ , p− the respective principal symbols of P0,ϕ , P − . We have p− () = ξd − γ˜1 ( ). We recall that ˜  ) ≡ 1, we have γ˜2 = γ2 and γ˜1 = γ1 p+ () = ξd − γ˜2 ( ). For  ∈ W , as χ( + − and thus pϕ () = p p (), for  = ( , ξd ). By (13.3.3) one has γ˜2 ( ) = γ˜1 ( ) in W and thus Lemma 13.25 applies.  13.A.4. Proof of the Shifted Estimate. Here, we prove Theorem 13.15. In the framework of the normal geodesic coordinates introduced in Sect. 13.2.1 we prove in fact the following proposition. Proposition 13.26. With the same setting and assumption as in Proposition 13.8, there exist U a bounded open neighborhood of x0 in U 0 , C > 0, and τ∗ > 0 such that (13.A.2) τ vτ,1,−1/2 + τ 3/2 |v|xd =0+ |∂ + τ 3/2 |(Dd v − iGd )|xd =0+ |τ,−1 

≤ C G0 + + τ G+ + τ 3/2 |B0,ϕ (v, −Gd )|τ,−k ,

13.A. SOME TECHNICAL PROOFS

329

for τ ≥ τ∗ and v ∈ H 1 (Rd+ ), G0 ∈ L2 (Rd+ ), G ∈ L2 V (Rd+ ) such that  P0,ϕ v = G0 + ∂ j Gj , supp(v) ∪ supp(G0 ) ∪ supp(G) ⊂ U+ . 1≤i≤d

It implies the estimate (13.A.3) τ eτ ϕ uτ,1,−1/2 + τ 3/2 |eτ ϕ u|∂M |∂ + τ 3/2 |eτ ϕ|∂M (∂ν u + g(F, ν))|∂M |τ,−1   ≤ C eτ ϕ F0 L2 (M) + τ eτ ϕ F L2 V (M) + τ 3/2 |eτ ϕ|∂M B(u, g(F, ν))|τ,−k , for τ ≥ τ∗ and u ∈ H 1 (M), F0 ∈ L2 (M), F ∈ L2 V (M) such that P u = F0 + divg F, and supp(u) ∪ supp(F0 ) ∪ supp(F ) ⊂ V 1 . Then, following the patching argument of Sect. 13.4.2 we deduce Theorem 13.15. ˜ be the open set of Rd subset of Proof of Proposition 13.26. Let U U 0 given by Proposition 13.9 and let U be a second open set of Rd such that ˜ ˜ . We choose χ ∈ C ∞ U U c (U+ ) be such that χ ≡ 1 in a neighborhood of U+ . −1/2 We set w = τ 1/2 χΛT,τ v. One has  −1/2 −1/2  Pϕ w = τ 1/2 χΛT,τ Pϕ + [Pϕ , χΛT,τ ] v  −1/2  −1/2 −1/2  χ∂j (ΛT,τ Gj ) + [Pϕ , χΛT,τ ]v = τ 1/2 χΛT,τ G0 + ˜0 +  G ˜j , =G

1≤i≤d

1≤i≤d

with

  ˜ 0 = τ 1/2 χΛ−1/2 G0 +  [χ, ∂j ](Λ−1/2 Gj ) + [Pϕ , χΛ−1/2 ]v , G T,τ T,τ T,τ 1≤i≤d

˜j = and G

−1/2 τ 1/2 χΛT,τ Gj .

We apply Proposition 13.9 to w:

−1/2

−1/2

(13.A.4) τ χΛT,τ vτ,1 + τ 3/2 |χΛT,τ v|xd =0+ |τ,1/2 −1/2

−1/2

+ τ 3/2 |(Dd (χΛT,τ v)|xd =0+ − iχΛT,τ Gd )|xd =0+ |τ,−1/2 

˜ 0  + τ G ˜ + τ 3/2 |B0,ϕ (χΛ−1/2 v, −χΛ−1/2 Gd )| , ≤ C G T,τ T,τ + + τ,1/2−k We write −1/2

vτ,1,−1/2 = χvτ,1,−1/2 = ΛT,τ χvτ,1 −1/2

−1/2

 χΛT,τ vτ,1 + [ΛT,τ , χ]vτ,1 −1/2

 χΛT,τ vτ,1 + vτ,1,−3/2 .

330

13. SOURCE TERMS OF WEAKER REGULARITY

Thus, for τ > 0 chosen sufficiently large we obtain −1/2

vτ,1,−1/2  χΛT,τ vτ,1 .

(13.A.5) Similarly one has

−1/2

|v|xd =0+ |∂  |χΛT,τ v|τ,1/2 .

(13.A.6) Next, one writes

|(Dd v − iGd )|xd =0+ |τ,−1 = |(Dd (χv) − iχGd )|xd =0+ |τ,−1 −1/2

= |ΛT,τ (Dd (χv) − iχGd )|xd =0+ |τ,−1/2 , and −1/2

ΛT,τ (Dd (χv) − iχGd ) −1/2

−1/2

−1/2

−1/2

= Dd (χΛT,τ v) − iχΛT,τ Gd + Dd [ΛT,τ , χ]v − i[ΛT,τ , χ]Gd  −1/2 −1/2 −1/2 −1/2 = Dd (χΛT,τ v) − iχΛT,τ Gd + [ΛT,τ , χ](Dd v − iGd ) + Dd , [ΛT,τ , χ] v.  −1/2 −1/2 −3/2 Since [ΛT,τ , χ] and Dd , [ΛT,τ , χ] are in ΨT,τ , it yields   −1/2 −1/2 |(Dd v − iGd )|xd =0+ |τ,−1   Dd (χΛT,τ v) − iχΛT,τ Gd |x

+ d =0

  τ,−1/2

+ |(Dd v − iG )|xd =0+ |τ,−2 + |v|xd =0+ |τ,−2 , d

and with (13.A.6) and for τ > 0 chosen sufficiently large we obtain (13.A.7) |v|xd =0+ |∂ + |(Dd v − iGd )|xd =0+ |τ,−1   −1/2 −1/2 −1/2  |χΛT,τ v|τ,1/2 +  Dd (χΛT,τ v) − iχΛT,τ Gd |x

d

=0+

  τ,−1/2

.

Combining (13.A.4), (13.A.5) and (13.A.7) we obtain   (13.A.8) τ vτ,1,−1/2 + τ 3/2 |v|xd =0+ |∂ + |(Dd v − iGd )|xd =0+ |τ,−1 

˜ + τ 3/2 |B0,ϕ (χΛ−1/2 v, −χΛ−1/2 Gd )| ˜ 0  + τ G . ≤ C G T,τ T,τ + + τ,1/2−k ˜ 0  . As one has [Pϕ , χΛ−1/2 ] ∈ Ψτ1,−1/2 one obtains Next, we estimate G T,τ + (13.A.9)

˜ 0   G0  + G + +

 1≤i≤d

G+ + τ 1/2 vτ,1,−1/2 .

We also have (13.A.10)

˜ j   Gj   G . G + + +

13.A. SOME TECHNICAL PROOFS

331

With the definition of B0,ϕ in (13.2.5) one has −1/2

−1/2

B0,ϕ (χΛT,τ v, −χΛT,τ Gd )

  ˜ϕk (χΛ−1/2 v)|x =0+ − iBϕk−1 Dd (χΛ−1/2 v) − iχΛ−1/2 Gd =B . T,τ T,τ T,τ d |xd =0+

   −1/2 ˜ k k−1 = ΛT,τ B Dd v − iGd |x =0+ ϕ v|xd =0+ − iBϕ d

˜ϕk χ, Λ−1/2 ]v|x =0+ + [B T,τ d −1/2

−1/2

− i[Bϕk−1 Dd χ, ΛT,τ ]v|xd =0+ − [Bϕk−1 χ, ΛT,τ ]Gd |xd =0+ −1/2 ˜ϕk χ, Λ−1/2 ]v|x =0+ = ΛT,τ B0,ϕ (v, −Gd ) + [B T,τ d 

−1/2 −1/2 k−1 k−1 − i [Bϕ Dd χ, ΛT,τ ]v|xd =0+ − [Bϕ χ, ΛT,τ ]Dd v|xd =0+  −1/2  − i[Bϕk−1 χ, ΛT,τ ] Dd v − iGd |x =0+ d

Observe that −1/2

−1/2

[Bϕk−1 Dd χ, ΛT,τ ]v|xd =0+ − [Bϕk−1 χ, ΛT,τ ]Dd v|xd =0+ −1/2

= Bϕk−1 [Dd , χ]ΛT,τ v|xd =0+ . If fact, in a neighborhood of supp(v), one has χ ≡ 1 implying that χDd v|xd =0+ = Dd χv|xd =0+ = Dd v|xd =0+ yielding −1/2

−1/2

[Bϕk−1 Dd χ, ΛT,τ ]v|xd =0+ − [Bϕk−1 χ, ΛT,τ ]Dd v|xd =0+ −1/2

−1/2

= Bϕk−1 Dd χΛT,τ v|xd =0+ − ΛT,τ Bϕk−1 Dd χv|xd =0+ −1/2

−1/2

− Bϕk−1 χΛT,τ Dd v|xd =0+ + ΛT,τ Bϕk−1 χDd v|xd =0+ −1/2

−1/2

= Bϕk−1 Dd χΛT,τ v|xd =0+ − Bϕk−1 χΛT,τ Dd v|xd =0+ −1/2

= Bϕk−1 [Dd , χ]ΛT,τ v|xd =0+ . −1/2

k−3/2

˜ k χ, Λ Since [B ϕ T,τ ] ∈ ΨT,τ −1/2

k−5/2

ΛT,τ ] ∈ ΨT,τ

−1/2

, Bϕk−1 [Dd , χ]ΛT,τ

k−3/2

∈ ΨT,τ

, and [Bϕk−1 χ,

one obtains

(13.A.11) −1/2 −1/2 |B0,ϕ (χΛT,τ v, −χΛT,τ Gd )|τ,1/2−k

   |B0,ϕ (v, −Gd )|τ,−k + |v|xd =0+ |τ,−1 + | Dd v − iGd |x

+ d =0

|

Combining (13.A.8)–(13.A.11) we obtain   (13.A.12) τ vτ,1,−1/2 + τ 3/2 |v|xd =0+ |∂ + |(Dd v − iGd )|xd =0+ |τ,−1 

≤ C G0 + + τ G+ + τ 3/2 |B0,ϕ (v, −Gd )|τ,−k    + τ 1/2 vτ,1,−1/2 + τ 3/2 |v|xd =0+ |τ,−1 + | Dd v − iGd |x =0+ | d

We then obtain the result by choosing τ > 0 sufficiently large.

τ,−2

τ,−2

.

 . 

CHAPTER 14

Optimal Estimates at the Boundary Contents 14.1. 14.1.1. 14.1.2. 14.1.3. 14.2. 14.2.1. 14.2.2. 14.2.3. 14.2.4. 14.3. 14.4. 14.4.1. 14.4.2. 14.4.3. 14.5. 14.6. 14.6.1. 14.6.2. 14.6.3. 14.7. 14.7.1. 14.7.2.

Statement and Proof Scheme Local Estimates and Patching Microlocal Estimate An Improved Estimate for a First-Order Sub-elliptic Factor Some Elements of H¨ormander Calculus Metrics and Order Functions Symbols and Pseudo-Differential Operators Symbol Calculus Sobolev Bounds and Positivity Inequalities Proof of the First-Order Estimate Proof of the Microlocal Estimate Case 1: One Root in the Open Upper Complex Half-Plane Case 2: One Root on the Real Axis Case 3: Both Roots in the Open Lower Complex Half-Plane Optimality Aspect A Refined Estimate Without Any Prescribed Boundary Operator Case 1: τ 0 = 0 Case 2: τ 0 > 0 and γ1 (0 ) = γ2 (0 ) Case 3: τ 0 > 0 and γ1 (0 ) = γ2 (0 ) Optimal Estimate with Source Terms of Weaker Regularity Microlocal Estimate Shifted Estimate

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 14

334 335 336 336 337 337 339 341 343 344 348 349 351 353 353 358 360 360 362 362 363 367

333

334

14. OPTIMAL ESTIMATES AT THE BOUNDARY

14.8. Notes Appendix 14.A. H¨ormander Calculus Properties 14.A.1. Slowly Varying and Temperate Metrics 14.A.2. Order Functions 14.B. Symbol Properties ˜ k Im f , k˜ Im f 14.B.1. Properties of k, k, 14.B.2. Applicability of the Fefferman–Phong–Bony Inequality 14.B.3. Proof of Theorem 14.17 14.B.4. Proof of Fefferman–Phong–Bony Inequality

368 368 368 368 369 371 371 372 373 376

In this chapter, we provide a Carleman estimate near the boundary ˇ under Lopatinski˘ı–Sapiro conditions. Compared to the results obtained in Chap. 8, the estimate we obtain improves at the level of the boundary terms. We also prove that this improvement is in fact optimal. 14.1. Statement and Proof Scheme Let (M, g) be a smooth compact Riemannian manifold with boundary, and let P = −Δg + R1 with R1 a first-order differential operator with bounded coefficients on M. Theorem 14.1. Let V be an open set of M and set V∂ = V ∩ ∂M. Let ϕ ∈ C ∞ (M) be such that the pair (P, ϕ) has the sub-ellipticity property of Definition 5.1 in V . If V∂ = ∅, consider B a differential operator of order β in V . For 0 ≤ k ≤ β, denote by k∂M the union of the connected components of ∂M where B is of order k. Moreover, assume that (P, B, ϕ) satisfies the ˇ Lopatinski˘ı–Sapiro condition at all points m ∈ V∂ (Definition 8.1). Then, there exist C and τ∗ > 0 such that (14.1.1) τ −1/2 eτ ϕ uτ,2 + τ −1/4 |eτ ϕ|∂M tr(u)|τ,1,1/2

 ≤ C eτ ϕ P uL2 (M) + τ −1/4 |eτ ϕ Bu|k∂M | 1≤k≤β

 τ,3/2−k

,

for all u ∈ C ∞ (M), with supp(u) ⊂ V , and τ ≥ τ∗ . If one compares the estimates of Theorem 14.1 to that of Theorem 8.14, one finds an improvement of a factor τ −1/4 associated with the trace terms on the r.h.s. of the estimates in the present results. Naturally, with such a factor on the r.h.s., the same factor occurs for the trace term on the l.h.s. We prove in Sect. 14.5 that the estimate in Theorem 14.1 is optimal with respect to the boundary terms. Note that optimality with respect to the volume terms is proven in Section 4.1.2 of Volume 1. Arguing as for the proof of Theorem 8.24, we deduce from Theorem 14.1 the following shifted estimate.

14.1. STATEMENT AND PROOF SCHEME

335

Theorem 14.2. Under the assumptions of Theorem 14.1, there exist C and τ∗ > 0 such that τ 1/2 eτ ϕ uτ,1 + τ 1/4 |eτ ϕ|∂M tr(u)|τ,1,0

 ≤ C eτ ϕ P uL2 (M) + τ 1/4 |eτ ϕ Bu|k∂M | 1≤k≤β

 τ,1−k

,

for all u ∈ C ∞ (M), with supp(u) ⊂ V , and τ ≥ τ∗ . 14.1.1. Local Estimates and Patching. As in Chap. 8, we prove a similar estimate in the neighborhood of a point m0 of the boundary ∂M. Proposition 14.3. Let m0 ∈ ∂M and V 0 be an open neighborhood of in M that meets one connected component of ∂M. Let ϕ ∈ C ∞ (M) be such that the pair (P, ϕ) has the sub-ellipticity property of Definition 5.1 in V 0 . Consider B a differential operator of order k in V 0 of the form ˇ of (8.1.2). Moreover, assume that (P, B, ϕ) satisfies the Lopatinski˘ı–Sapiro 0 1 condition at m (Definition 8.1). Then there exist a neighborhood V of m0 in M and two constants C and τ∗ > 0 such that m0

(14.1.2) τ −1/2 eτ ϕ uτ,2 + τ −1/4 |eτ ϕ|∂M tr(u)|τ,1,1/2 

τϕ −1/4 τ ϕ ≤ C e P uL2 (M) + τ |e Bu|∂M |τ,3/2−k , for all u ∈ C ∞ (M), with supp(u) ⊂ V 1 , and τ ≥ τ∗ . Then, arguing as in Sect. 5.3, for Theorems 5.5 and 5.6, patching together estimates as above, we obtain the result of Theorem 14.1. We follow the setting and notation of Sect. 8.3. We consider m0 ∈ ∂M, V an open neighborhood of m0 in M where the sub-ellipticity property for the pair (P, ϕ) holds, and C = (O, κ) a local chart at the boundary associated with coordinates (x , xd ) as described by (8.3.3), with m0 ∈ O. With the local representative of the principal symbols pC , bC of P , and B as given in (8.3.4), it suffices to prove the following estimate. 0

Proposition 14.4. Let P0 = Op(pC ) and B0 = Op(bC ). Let m0 ∈ ∂M and V 0 be an open neighborhood of m0 in M that meets one connected component of ∂M. Set x0 = κ(m0 ). Assume that (P0 , ϕC ) has the subellipticity property of Definition 3.2 of Volume 1 in U 0 with U 0 = κ(V 0 ∩O). ˇ condition Moreover, assume that (P0 , B0 , ϕC ) satisfies the Lopatinski˘ı–Sapiro 0 at x (Definition 8.1). Then, there exist a bounded open neighborhood U+ of x0 in Rd+ such that U+ ⊂ U 0 and two constants C and τ∗ > 0 such that C

τ ϕC

(14.1.3) τ −1/2 eτ ϕ uτ,2 + τ −1/4 |e |xd =0+ tr(u)|τ,1,1/2 

C C ≤ C eτ ϕ P0 u+ + τ −1/4 |eτ ϕ B0 u|xd =0+ |τ,3/2−k ,

336

14. OPTIMAL ESTIMATES AT THE BOUNDARY ∞

for all u ∈ C c (U+ ) and τ ≥ τ∗ . ∞

We recall that for an open subset U+ of Rd+ , the space C c (U+ ) is introduced in (8.3.6). The proof of Proposition 14.4 relies on a microlocal version of the estimate that we state below and it follows the proof scheme of Proposition 8.16 by patching together such microlocal estimates as in Sect. 8.3.5. 14.1.2. Microlocal Estimate. Using the local setting and notation of Sect. 8.3, we prove the following result. In particular, in the considered local chart, we define the conjugated operators of P0 and B0 : C

C

P0,ϕ = eτ ϕ P0 e−τ ϕ ,

C

C

B0,ϕ = eτ ϕ B0 e−τ ϕ ,

with principal symbols p0,ϕ and b0,ϕ , respectively. Proposition 14.5. Let x0 = κ(m0 ) be such that x0d = 0. Assume that (P0 , ϕC ) has the sub-ellipticity property of Definition 3.2 of Volume 1 in ∗ ∂M, with local representative U 0 with U 0 = κ(V 0 ∩ O). Let ω 0 ∈ Tm 0 0 C 0 d−1 0 (ω ) = ξ ∈ R , and τ ≥ 0 such that (ξ 0 , τ 0 ) = 0, and assume that the ˇ Lopatinski˘ı–Sapiro condition of Definition 8.1 holds at (m0 , ω 0 , τ 0 ). Then, there exists U a conic open neighborhood of (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ such that for χ ∈ ST0,τ , homogeneous of degree 0, with supp(χ) ⊂ U , there exist C > 0 and τ∗ > 0 such that (14.1.4) τ −1/2 OpT (χ)vτ,2 + τ −1/4 | tr(OpT (χ)v)|τ,1,1/2 

≤ C P0,ϕ v+ + τ −1/4 |B0,ϕ v|xd =0+ |τ,3/2−k + vτ,2,−1 , for τ ≥ τ∗ , v ∈ S (Rd+ ). The proof of Proposition 14.5 is given in Sect. 14.4. In some microlocal regions, the key aspect of the proof lies in the use of an estimate for a first-order operator that improves upon what can be found in Sect. 6.4. 14.1.3. An Improved Estimate for a First-Order Sub-elliptic Factor. Let f ( ) ∈ ST1,τ be homogeneous of degree one and L = Dd − OpT (f ) with principal symbol () = ξd − f ( ). We introduce Re F = 1 1 ∗ ∗ 2 (OpT (f ) + OpT (f ) ) and Im F = 2i (OpT (f ) − OpT (f ) ) both formally selfadjoint and  1 L2 = L + L∗ = Dd − Re F ∈ Ψτ1,0 , 2  1 L − L∗ = − Im F ∈ Ψ1T,τ L1 = 2i also (formally) selfadjoint. Their respective principal symbols are 2 () = Re () = ξd − Re f ( ) ∈ Sτ1,0 , 1 ( ) = Im () = − Im f ( ) ∈ ST1,τ .

¨ 14.2. SOME ELEMENTS OF HORMANDER CALCULUS

337

Let U be a conic open set of Rd+ × Rd−1 × R+ . We assume that we have, for some C > 0, (14.1.5)

λT,τ ( ) ≤ Cτ,

 ∈ U ,

and that the following sub-ellipticity property holds: (14.1.6) 1 {, }( ) = {2 , 1 }( ) > 0,  ∈ U ,  = ( , ξd ). () = 0 ⇒ 2i As in Sect. 6.4.3, this latter property reads (14.1.7)

1 ( ) = 0 ⇒ {2 , 1 }( ) > 0,

 ∈ U .

We have the following estimate. Proposition 14.6. Let U be as above with the additional assumption that SU is compact. Assume that (14.1.5) and (14.1.7) hold. Let χ ∈ ST0,τ , homogeneous of degree 0, be such that supp(χ) ⊂ U . Let s ∈ R. There exist C > 0 and τ∗ > 0 such that for any N ∈ N, there exists CN > 0 such that (14.1.8) τ −1/2 OpT (χ)uτ,1,s 

≤ C LOpT (χ)uτ,0,s + τ −1/4 |OpT (χ)u|xd =0+ |τ,s+1/2 + CN uτ,0,−N , for τ ≥ τ∗ and u ∈ S (Rd+ ). If one compares with the result of Lemma 6.22, in the case s = 0, one observes the improvement by the factor τ −1/4 for the trace term. This improvement is optimal; we give an indirect explanation of this optimality in Remark 14.28. The proof of Proposition 14.6 is quite involved and is given in Sect. 14.3 below. It relies on general and sharp pseudo-differential methods introduced by L. H¨ormander. 14.2. Some Elements of H¨ ormander Calculus Most of the material presented here is adapted from Sections 18.4 to 18.6 in the book by L. H¨ormander [175]. The calculus of pseudo-differential operators introduced therein is very general. For example, the calculus of operators with a large operators presented in Chapter 2 of Volume 1 and used throughout both Volumes 1 and 2 can be expressed in terms of the H¨ ormander calculus. Note, however, that, because of its complexity, it would have been a bit awkward to introduce this calculus at an early stage of this book. 14.2.1. Metrics and Order Functions. As in the previous chapters, we write  = (x, ξ, τ ) ∈ R2d × [1, +∞) and  = (x, ξ  , τ ) ∈ R2d−1 × [1, +∞). Recalling that λT,τ () = λT,τ ( ) = (|ξ  |2 + τ 2 )1/2 , we introduce some notation. We define the following metrics in phase space:  2  2 and gγ = τ γ −1 |dx|2 + τ γ −1 λ−2 g = |dx|2 + λ−2 T,τ ( )|dξ| T,τ ( )|dξ| ,

338

14. OPTIMAL ESTIMATES AT THE BOUNDARY

for γ ≥ 1. For the first metric, this reads  2 g (z, ζ) = |z|2 + λ−2 T,τ ( )|ζ| ,

(z, ζ) ∈ R2d .

In proofs below, the parameter γ will be chosen large. The parameter τ will also be chosen large as in the other chapters with τ ≥ γ. Note that one has g ≤ gγ . For such a metric g , the associated dual metric is defined through  2 (y, η) · (z, ζ) σ   , g (y, η) = sup g A(z, ζ) (z,ζ)  0 Id the (2d) × (2d) matrix associated with the symplectic with A = −Id 0 2-form σ on R2d (see Sect. 15.7.1), that is,     σ (z, ζ), (y, η) = A(z, ζ) · (y, η) = ζ · y − z · η. The dual metrics gσ and (g γ )σ are given by (14.2.1) gσ = λ2T,τ ( )|dx|2 + |dξ|2 and (g γ )σ = λ2T,τ ( )τ −1 γ|dx|2 + τ −1 γ|dξ|2 , and one defines h() and hγ () by  h()2 = sup (g/g σ ) (y, η) = λ−2 T,τ ( ) ≤ 1, (y,η  )

 2 hγ ()2 = sup (g γ /(g γ )σ ) (y, η) = (τ /γ)2 λ−2 T,τ ( ) ≤ 1/γ ≤ 1. (y,η)

Note that (g γ )σ ≤ gσ . As we shall see below, the metric g (respectively, g γ ) and its dual metric g σ (respectively, (g γ )σ ) satisfy the necessary properties given by L. H¨ormander [175, Sections 18.4-18.5] allowing one to define symbol classes and the associated pseudo-differential operators and calculus. With the metric g, one recovers in fact the tangential symbol classes STm,τ introduced in Section 2.10 of Volume 1 and the associated operators (see below). Let g be a metric acting on R2d as above. Following Definition 18.4.1 in [175], it is said to be slowly varying if there exist ε > 0 and C > 0 such that (14.2.2) g (y, η) < ε ⇒ g˜ ≤ Cg ,

for ˜ = (x + y, ξ + η, τ ),

∀  = (x, ξ, τ ) ∈ R2d × [1, +∞), (y, η), ∈ R2d .

¨ 14.2. SOME ELEMENTS OF HORMANDER CALCULUS

339

Following Definition 18.4.7 in [175], it is said to be σ-temperate if there exist C > 0 and N > 0 such that   ˜ N g˜, ˜, ξ − ξ) (14.2.3) g ≤ C 1 + gσ (x − x ˜ τ ) ∈ R2d × [1, +∞). ∀  = (x, ξ, τ ), ˜ = (˜ x, ξ, Proposition 14.7. The metrics g and g γ are slowly varying and σtemperate. We refer to Sect. 14.A.1 for a proof. Let g be a slowly varying and σ-temperate metric on R2d , here g or g γ . A positive function m() (possibly depending on the parameter γ) is said to be g-continuous if there exist ε > 0 and C > 0 such that (14.2.4) ) ≤ Cm(), g (y, η) < ε ⇒ m()/C ≤ m(˜

with ˜ = (x + y, ξ + η, τ ),

∀∈R

2d

× [1, +∞), (y, η) ∈ R2d .

Remark 14.8. Since g ≤ gγ , a function that is g-continuous is also g γ -continuous. Note also that if m() is independent of γ and is g γ -continuous, then it is g-continuous, as can be seen by taking γ = τ . A positive function m() is said to be σ, g-temperate if there exist C > 0 and N > 0 such that   ˜ N m(˜ ˜, ξ − ξ) ), (14.2.5) m() ≤ C 1 + gσ (x − x ˜ τ ) ∈ R2d × [1, +∞). ∀  = (x, ξ, τ ), ˜ = (˜ x, ξ, Remark 14.9. Since (g γ )σ ≤ gσ , a function that is σ, g γ -temperate is also σ, g-temperate. A function m() that is both g-continuous and σ, g-temperate will be called a g-admissible order function or simply an order function if there is no ambiguity with respect to the used metric. Proposition 14.10. Let g = g or g γ . Let r ∈ R. The functions λτ ()r and λT,τ ( )r are both g-admissible order functions. If m1 () and m2 () are g-admissible order functions, then so is m1 m2 (). We refer to Sect. 14.A.2 for a proof. In what follows, we shall mainly use τ a γ b λT,τ ( )c , with a, b, c ∈ R, as an order function. 14.2.2. Symbols and Pseudo-Differential Operators. Definition 14.11 (Symbols for the Metrics g and g γ ). Consider m() a positive g-admissible order function. Recall that  = (x, ξ, τ ). Let a() ∈

340

14. OPTIMAL ESTIMATES AT THE BOUNDARY

C ∞ (R2d ) with τ ∈ [1, +∞) acting as a parameter. One says that a() ∈ S(m, g) if one has, for α, β ∈ Nd , (14.2.6) |∂xα ∂ξβ a()| ≤ Cα,β m()λT,τ ( )−|β| ,  = (x, ξ, τ ) ∈ R2d × [1, +∞),  = (x, ξ  , τ ). Consider m(, γ) a positive g γ -admissible order function. Let a(, γ) ∈ with τ, γ ∈ [1, +∞), acting as parameters. One says that a(, γ) ∈ S(m, g γ ) if one has, for α, β ∈ Nd , C ∞ (R2d )

(14.2.7) |∂xα ∂ξβ a(, γ)| ≤ Cα,β m(, γ)(τ γ −1 )(|α|+|β|)/2 λT,τ ( )−|β| ,  ∈ R2d × [1, +∞), γ ∈ [1, +∞), τ ≥ γ. Here, we are interested in the counterpart tangential symbols for the metrics g and g γ . Consider mT ( ) a positive g-admissible order function. Recall that  = (x, ξ  , τ ). Let a( ) ∈ S(mT , g). Observe that Definition 14.11 reads in this case (14.2.8)    ∈ R2d−1 × [1, +∞), |∂xα ∂ξβ a( )| ≤ Cα,β  mT ( )λT,τ ( )−|β | , for α ∈ Nd and β  ∈ Nd−1 . Finally, for mT ( , γ) a positive g γ -admissible order function, if a( , γ) ∈ ST (mT , g γ ), then 





(14.2.9) |∂xα ∂ξβ a( , γ)| ≤ Cα,β  mT ( , γ)(τ γ −1 )(|α|+|β |)/2 λT,τ ( )−|β | ,  ∈ R2d−1 × [1, +∞), γ ∈ [1, +∞), τ ≥ γ ≥ 1, for α ∈ Nd and β  ∈ Nd−1 . m Remark 14.12. In the case mT ( ) = λm T,τ , (14.2.8) means that S(λT,τ , g) = with the notation of Definition 2.39 of Volume 1. m m Note that in the case m() = λm τ , one does not have S(λτ , g) = Sτ , with Sτm as in Definition 2.1 (also in Volume 1).

STm,τ ,

Remark 14.13. Observe that if m() (respectively, mT ( )) is both gadmissible and g γ -admissible, then one has S(m, g) ⊂ S(m, g γ ) (respectively, S(mT , g) ⊂ S(mT , g γ )). An example is given by m() = τ a γ b λτ () (respectively, mT ( ) = τ a γ b λT,τ ( )). For a() ∈ S(m, g) (respectively, a(, γ) ∈ S(m, g γ )), one sets the pseudo-differential operator ˆ(ξ) dξ a(x, D, τ )u(x) = Op(a)u(x) := (2π)−d ∫ eix·ξ a() u Rd

(resp. a(x, D, τ, γ)u(x) = Op(a)u(x) := (2π)−d ∫ eix·ξ a(, γ) u ˆ(ξ) dξ). Rd

¨ 14.2. SOME ELEMENTS OF HORMANDER CALCULUS

341

We denote by Ψ(m, g) (respectively, Ψ(m, g γ )) the space of these operators. For mT ( ) a tangential g-admissible order function and a( ) ∈ S(mT , g), one sets the tangential pseudo-differential operator a(x, D , τ )u(x) = OpT (a)u(x) 



:= (2π)1−d ∫ eix ·ξ a( ) u ˆ(ξ  , xd ) dξ  Rd−1

= (2π)



∫∫

1−d

Rd−1 ×Rd−1





ei(x −y )·ξ a( ) u(y  , xd ) dy  dξ  .

We denote by Ψ(mT , g) the space of these operators. For mT ( , γ) a tangential g γ -admissible order function and a(, γ) ∈ S(mT , g γ ), one sets 



ˆ(ξ  , xd ) dξ  a(x, D , τ, γ)u(x) = OpT (a)u(x) := (2π)1−d ∫ eix ·ξ a( , γ) u Rd−1

∫∫

= (2π)1−d



Rd−1 ×Rd−1





ei(x −y )·ξ a( , γ) u(y  , xd ) dy  dξ  .

We denote by Ψ(mT , g γ ) the space of these operators. In what follows, we shall only be interested in tangential symbols and operators. 14.2.3. Symbol Calculus. For the results of this section, we refer to Sections 18.4 and 18.5 in [175]. Proposition 14.14 (Formal Adjoint). Let mT ( , γ) be a g γ -admissible order function and a( , γ) ∈ S(mT , g γ ), with  = (x, ξ  , τ ). Then OpT (a)∗ = OpT (a∗ ) for a certain a∗ ∈ S(mT , g γ ) and a∗ ∼

 α

1 ∂ α ∂ α a ¯, i|α| α! x ξ

meaning, for any N ∈ N, a∗ =

 |α|≤N

1 ∂ α ∂ α a ¯ |α| i α! x ξ

+ RN ,

−1 γ with RN ( , γ) ∈ S(mT (hγ )N +1 , g γ ) = S(mT (τ /γ)N +1 λ−N , g ). T,τ

In particular, a∗ − a ∈ S(mT τ (γλT,τ )−1 , g γ ). Proposition 14.15. Let mT,1 ( , γ) and mT,2 ( , γ) be two g γ -admissible order functions and a1 ( , γ) ∈ S(mT,1 , g γ ) and a2 ( , γ) ∈ S(mT,2 , g γ ). Then, OpT (a1 ) ◦ OpT (a2 ) = OpT (a) with a ∈ S(mT,1 mT,2 , g γ ) and (14.2.10)

a = a1 ◦ a2 ∼



1

α ∈Nd−1

i|α | (α )!





∂ξα a1 ∂xα a2 ,

342

14. OPTIMAL ESTIMATES AT THE BOUNDARY

meaning, for any N ∈ N, a( , γ) =

 |α |≤N

1  ∂ξα a1 | |α  i (α )!



∂xα a2 + RN ,

with the remainder RN ∈ S(mT,1 mT,2 (hγ )N +1 , g γ ) = S(mT,1 mT,2 (τ /γ)N +1 −1 γ λ−N , g ). T,τ We use the notation “◦” as introduced in the statement of Theorem 2.22 of Volume 1. −1 if |ξ  | ≤ τ . In such Remark 14.16. Observe that hγ = γ −1 τ λ−1 T,τ  γ a region, one has RN behaving as a symbol in S(mT,1 mT,2 γ −N −1 , g γ ). Thus, for example, for mT,1 = λaT,τ and mT,1 = λbT,τ , one finds the behavior of a −N −1 , g γ ). This is in contrast with the symbol calculus in symbol in S(λa+b T,τ γ a+b−N −1 , g) = STm,τ = S(λm T,τ , g) for which the counterpart remainder is in S(λT,τ a+b−N −1 . Consequently, the symbol calculus associated with the metric g γ ST,τ yields a weaker gain in the symbol decay in the microlocal region |ξ  | ≤ τ since there γ −1 ≥ τ −1 ≥ λ−1 T,τ .

In sections below, we shall need to compose operators associated with different calculi, say for an operator in Ψ(mT,1 , g) and an operator in Ψ(mT,2 , g γ ). This is provided by the following theorem. Theorem 14.17. Let mT,1 ( ) be a g-admissible order function, and let mT,2 ( , γ) be a g γ -admissible order function. If a1 ( ) ∈ S(mT,1 , g) and a2 ( , γ) ∈ S(mT,2 , g γ ), then OpT (a1 ) ◦ OpT (a2 ) = OpT (a) and OpT (a2 ) ◦ a) with both a = a1 ◦ a2 and a ˜ = a2 ◦ a1 in S(mT,1 mT,2 , g γ ), OpT (a1 ) = OpT (˜ with an asymptotic expansion as in (14.2.10) with, in each case, a remainder −1 γ , g ). term in S(mT,1 mT,2 (τ /γ)(N +1)/2 λ−N T,τ In particular, one has (1) (2) (3) (4) (5)

γ a1 ◦ a2 = a1 a2 mod S(mT,1 mT,2 (τ /γ)1/2 λ−1 T,τ , g ). −1 a2 ◦ a1 = a1 a2 mod S(mT,1 mT,2 (τ /γ)1/2 λT,τ , g γ ). γ a1 ◦ a2 = a1 a2 − i∇ξ a1 · ∇x a2 mod S(mT,1 mT,2 τ γ −1 λ−2 T,τ , g ). γ a2 ◦ a1 = a1 a2 − i∇ξ a2 · ∇x a1 mod S(mT,1 mT,2 τ γ −1 λ−2 T,τ , g ). γ a1 ◦ a2 − a2 ◦ a1 = −i{a1 , a2 } mod S(mT,1 mT,2 τ γ −1 λ−2 T,τ , g ).

The notation {., .} refers to the Poisson bracket (see Sect. 15.7.2). For a proof, we refer to Appendix 14.B.3. Remark 14.18. If mT,1 is both g-admissible and g γ -admissible and a1 ∈ S(mT,1 , g), then one also has a1 ∈ S(mT,1 , g γ ) by Remark 14.13. Then, the composition symbol a1 ◦ a2 can also be analyzed by means of Proposition 14.15. However, the results therein are weaker than those provided by Theorem 14.17 that exploits the differences in the metrics.

¨ 14.2. SOME ELEMENTS OF HORMANDER CALCULUS

343

14.2.4. Sobolev Bounds and Positivity Inequalities. Concerning Sobolev bounds, the basic result is the following one. Theorem 14.19. Let a( , γ) ∈ S(1, g γ ), then OpT (a) is bounded on

L2 (Rd+ ).

For a proof in the case of a general slowly varying σ-temperate metric, we refer to Theorem 18.6.3 in [175]. Corollary 14.20. Let s, r ∈ R and a( , γ) ∈ S(λrT,τ , g γ ). Then, there exists C > 0 such that OpT (a)uτ,0,s ≤ Cuτ,0,s+r ,

u ∈ S (Rd+ ), τ ≥ γ ≥ 1.

The tangential norm .τ,0,s is defined in Sect. 6.2. For the operator classes we have introduced, the G˚ arding inequality takes the following form. Theorem 14.21 (G˚ arding Inequality). Let mT ( , γ) be a g γ -admissible ∈ S(mT , g γ ) be such that order function, and let a( , γ) Re a( , γ) ≥ C0 mT ( , γ), for  = (x, ξ  , τ ), τ ≥ γ ≥ 1, with |ξ  | ≥ R for some C0 > 0 and R > 0. Then, for any 0 < C1 < C0 , there exist τ∗ ≥ 1 and γ∗ ≥ 1 such that 1/2

Re(OpT (a)u, u)+ ≥ C1 OpT (mT )u+ ,

u ∈ S (Rd+ ), γ ≥ γ∗ ,

τ ≥ max(τ∗ , γ). Such an inequality is however not sufficient to achieve the Carleman estimates in the present chapter. Instead, we shall rely on much stronger inequalities, namely, a sharp Fefferman–Phong inequality, as proven by J.M. Bony [77, Theorem 3.2]. Theorem 14.22 (Fefferman–Phong–Bony Inequality). Let a( , γ) ∈ × Rd−1 ) with τ and γ as parameters in [1, +∞). We assume that there exists C > 0 such that a( , γ) satisfies (1) a ≥ 0. (2) |a| ≤ Cλ2T,τ .

C ∞ (Rd

2−|β|

(3) |∂xα ∂ξβ a| ≤ CλT,τ (4)

|∂xα ∂ξβ a|

≤

if |α| + |β| = 2.

(|α|−|β|)/2 Cα,β λT,τ

if |α| + |β| ≥ 4 where Cα,β > 0.

Then, there exists C0 > 0 such that (14.2.11)

Re(OpT (a)u, u)+ + C0 u2+ ≥ 0, u ∈ S (Rd+ ).

As the statement of this inequality we give here is different from that of J.-M. Bony, in particular because the pseudo-differential calculus and operators are not written in the same quantification, we show in Appendix 14.B.4 how this theorem follows from J.-M. Bony’s results.

344

14. OPTIMAL ESTIMATES AT THE BOUNDARY

Remark 14.23. Observe that a as given in the statement of Theorem 14.22 does not lie in the symbol classes introduced above. In fact,  2 ˜ is slowly a( , γ) ∈ S(λ2T,τ , g˜) with g˜ = λT,τ |dx|2 + λ−1 T,τ |dξ | . The metric g varying and σ-temperate and λT,τ is a g˜-admissible order function, thus yielding a proper symbol class and calculus. In particular, OpT (a) is well defined. However, the usual Fefferman–Phong inequality (see Theorem 18.6.8 in [175]) in this symbol/operator class does not yield any useful estimate. 14.3. Proof of the First-Order Estimate Here, we prove Proposition 14.6, relying on the material introduced in Sect. 14.2. The proof we provide is based on a multiplier method and uses both the g and g γ -calculus and the Fefferman–Phong–Bony inequality of Theorem 14.22. Proof of Proposition 14.6. Since SU is compact, we deduce from (14.1.7) that there exist C0 > 0 and C1 > 0 such that (see Lemma 3.8 of Volume 1 and its proof) (14.3.1)

 2  C0 λ−1 T,τ 1 ( ) + {2 , 1 }( ) ≥ C1 λT,τ ,

 ∈ U .

We introduce a C ∞ -function σ0 that is a regularized version of the sign function, viz., 1 σ0 ≥ 0, σ0 is odd, σ0 (t) = 1 if t ≥ 1, σ(t) = 2t, if |t| ≤ , 4 and we set     (14.3.2) k = σ0 (τ /γ)1/2 λ−1 Im f and k˜ = σ  (τ /γ)1/2 λ−1 Im f . 0

T,τ

T,τ

The function σ0 is illustrated in Fig. 14.1. σ0 (t) 1 1/2 −1 1/4

1

t

1

Figure 14.1. Regularized version of the sign function used to form symbol in the H¨ ormander calculus associated with γ the metric g An important point in what follows lies in the following observation.

14.3. PROOF OF THE FIRST-ORDER ESTIMATE

345

Lemma 14.24. One has k, k˜ ∈ S(1, g γ ). Moreover,   k˜ Im f ∈ S (τ /γ)−1/2 λT,τ , g γ . For α ∈ Nd and β ∈ Nd−1 , with |α| = |β| = 1, one also has ∂xα (k Im f ) ∈ S(λT,τ , g γ ) and ∂ξβ (k Im f ) ∈ S(1, g γ ). A proof is given in Appendix 14.B.1. Remark 14.25. Note that the last two properties are equivalent to having (14.3.3)

1−|β|

|∂xα ∂ξβ (k Im f )| ≤ Cα,β (τ /γ)(|α|+|β|−1)/2 λT,τ ,

for α ∈ Nd and β ∈ Nd−1 such that |α| + |β| ≥ 1. With K = (OpT (k) + OpT (k)∗ )/2 (formally) selfadjoint, we use −iK as a multiplier. For v = OpT (χ)u, we write (14.3.4) 2 Re(Lv, −iKv)+ = (i Im F v, iKv)+ + (iKv, i Im F v)+ − 2 Re((Dd − Re F )v, iKv)+   = (K Im F + Im F K − i[Dd − Re F, K])v, v + − (Kv|xd =0+ , v|xd =0+ )∂ . Observe that the two terms on the right-hand side are real. We have, writing formally Dd∗ = Dd , 2[Dd , K] = [Dd , OpT (k)] + [Dd , OpT (k)∗ ] = OpT (Dd k) − [Dd , OpT (k)]∗ = OpT (Dd k) − OpT (Dd k)∗ , yielding

      2 i[Dd , K]v, v + = iOpT (Dd k)v, v + − iOpT (Dd k)∗ v, v +     = iOpT (Dd k)v, v + + v, iOpT (Dd k)v +   = 2 Re iOpT (Dd k)v, v + .

One has iOpT (Dd k) = OpT (∂d k) = OpT ({ξd , k}). By symbol calculus, we have Re F = OpT (Re f ) mod Ψ(1, g), and from Theorem 14.17, we have γ i[Re F, K] = OpT ({Re f, k}) mod Ψ(τ γ −1 λ−1 τ , g ). We thus have (14.3.5) (i[Dd − Re F, K]v, v)+ = Re(OpT ({ξd − Re f, k})v, v)+ + (OpT (r0 )v, v)+ γ with r0 ∈ S(τ γ −1 λ−1 τ , g ). Observing that   Re (OpT (k)∗ Im F + Im F OpT (k)∗ )v, v +   = Re (OpT (k) Im F + Im F OpT (k))v, v + ,

we have (14.3.6)     (K Im F + Im F K)v, v + = Re (OpT (k) Im F + Im F OpT (k))v, v + .

346

14. OPTIMAL ESTIMATES AT THE BOUNDARY

From usual symbol calculus, Lemma 14.24, and Theorem 14.17, we have Im F = OpT (Im f )

mod Ψ(1, g),

k ◦ Im f = k Im f − i∇ξ k · ∇x Im f

γ mod S(τ γ −1 λ−1 T,τ , g ),

Im f ◦ k = k Im f − i∇ξ Im f · ∇x k

γ mod S(τ γ −1 λ−1 T,τ , g ).

Then, we deduce that (14.3.7) OpT (k) Im F + Im F OpT (k) = OpT (2k Im f + i(τ /γ)1/2 r1 + r1 ),  where r1 , r1 ∈ S 1, g γ ) with r1 real valued. From calculus in the symbol classes associated with the metric g γ , we have (14.3.8) 2 Re(iOpT (r1 )v, v)+ = i((OpT (r1 ) − OpT (r1 )∗ )v, v)+ = (OpT (r2 )v, v)+ , where r2 ∈ S(τ /(γλT,τ ), g γ ). From (14.3.5)–(14.3.8), we obtain   (14.3.9) (K Im F + Im F K − i[Dd − Re F, K])v, v +   ≥ Re OpT (2k Im f − {ξd − Re f, k})v, v + − C(τ 1/2 γ −3/2 + 1)v2+ . We compute, with the form of k and k˜ given in (14.3.2),

 −1 ˜ {Re f, Im f } − Im f {Re f, λ } k. {Re f, k} = (τ /γ)1/2 λ−1 T,τ T,τ γ From Lemma 14.24, the term (τ /γ)1/2 k˜ Im f ∇x Re f ·∇ξ (λ−1 T,τ ) is in S(1, g ). −1 ˜ 1/2 We also have {ξd , k} = (τ /γ) λT,τ k{ξd , Im f }. From (14.3.9), we thus have   (14.3.10) (K Im F + Im F K − i[Dd − Re F, K])v, v +   ≥ Re OpT (q0 )v, v + − C(τ 1/2 γ −3/2 + 1)v2+ ,

where (14.3.11)

γ ˜ q0 = 2k Im f − (τ /γ)1/2 λ−1 T,τ {ξd − Re f, Im f }k ∈ S(λT,τ , g ).

We now study the positivity of the symbol q0 in U , and we split the analysis into two cases. Case  ∈ U and | Im f | ≤ (τ /γ)−1/2 λT,τ /4. We then have k˜ = 2 and −1 k = 2(τ /γ)1/2 λ−1 T,τ Im f . We choose τ0 > 0 such that 2 ≥ C0 λT,τ for τ ≥ τ0 with C0 as in (14.3.1). Since 2 = ξd − Re f and 1 = − Im f in (14.3.1), one finds 2 1/2 −1 λT,τ {ξd − Re f, Im f } q0 ≥ 4(τ /γ)1/2 λ−1 T,τ (Im f ) − 2(τ /γ) 

2 + {ξ − Re f, − Im f } = 2(τ /γ)1/2 λ−1 2(Im f ) d T,τ 

−1 2 λ (Im f ) + {ξ − Re f, − Im f } ≥ 2(τ /γ)1/2 λ−1 C 0 d T,τ T,τ

≥ 2C1 (τ /γ)1/2 .

14.3. PROOF OF THE FIRST-ORDER ESTIMATE

347

Case  ∈ U and | Im f | ≥ (τ /γ)−1/2 λT,τ /4. We then have −k Im f ≥ | Im f |/2 ≥ (τ /γ)−1/2 λT,τ /8. ˜ Observe that (τ /γ)−1/2 λT,τ /8 ≥ τ 1/2 γ 1/2 /8 and |(τ /γ)1/2 λ−1 T,τ k{ξd − Re f, 1/2 Im f }|  (τ /γ) . Thus for γ ≥ 1 chosen sufficiently large, we obtain q0 ≥ C2 τ 1/2 γ 1/2 , for C2 > 0. Both cases yield q0 ≥ C3 (τ /γ)1/2 in U for some C3 > 0, if γ ≥ γ0 for γ0 ≥ 1 chosen sufficiently large. ˜ ⊂ U and Let χ ˜ ∈ ST0,τ , homogeneous of degree 0, be such that supp(χ) χ ˜ ≡ 1 on supp(χ). We then set   a = γ 3/2 τ −1/2 q0 − C3 (τ /γ)1/2 and a ˜ = χa. ˜ One has a, a ˜ ∈ S(γ 3/2 τ −1/2 λT,τ , g γ ). Yet, the most important is the following proposition. Lemma 14.26. The function a ˜ fulfills the properties of Theorem 14.22 if τ ≥ γ ≥ γ0 ≥ 1 with γ0 as chosen above. A proof is given in Appendix 14.B.2. With the Fefferman–Phong–Bony inequality of Theorem 14.22, we thus obtain a)v, v)+ ≥ −C0 v2+ . Re(OpT (˜ By Theorem 14.17, for any N ∈ N, one has γ a ˜ ◦ χ − a ◦ χ ∈ S((τ /γ)(N +1)/2 λ−N T,τ , g ) ⊂ S(λT,τ

1−N/2

, g γ ).

Thus, since v = OpT (χ)u, for any N ∈ N, there exists CN > 0 such that Re(OpT (a)v, v)+ ≥ −C0 v2+ − CN u2τ,0,−N . Multiplying by τ 1/2 γ −3/2 , we deduce (14.3.12)  Re(OpT (q0 )v, v)+ ≥ (C3 τ 1/2 γ −1/2 − C0 τ 1/2 γ −3/2 )v2+ − CN u2τ,0,−N  u2τ,0,−N ,  τ 1/2 γ −1/2 v2+ − CN  > 0, if γ ≥ γ is chosen sufficiently large. for some CN 0 From (14.3.10) and (14.3.12) for γ ≥ γ0 chosen sufficiently large and to be kept fixed in what follows, we obtain   (K Im F + Im F K − i[Dd − Re F, K])v, v +

≥ Cτ 1/2 v2+ − C  v2+ − CN u2τ,0,−N . With τ chosen sufficiently large, we conclude that    u2τ,0,−N . (K Im F + Im F K − i[Dd − Re F, K])v, v +  τ 1/2 v2+ − CN

348

14. OPTIMAL ESTIMATES AT THE BOUNDARY

With (14.3.4), we find  Re(Lv, −iKv)+ + |v|xd =0+ |2  τ 1/2 v2+ − CN u2τ,0,−N ,

using that K is bounded on L2 (Rd−1 ). With the Young inequality, for ε > 0, we write |(Lv, −iKv)+ |  ε−1 τ −1/2 Lv2+ + ετ 1/2 v2+ . Thus, for ε > 0 chosen sufficiently small, we obtain

 τ 1/2 v+ ≤ C Lv+ + τ 1/4 |v|xd =0+ |∂ + CN uτ,0,−N . Replacing u by τ s u and using Proposition 6.11 as τ  λT,τ in U by (14.1.5), one obtains, for any N ∈ N, τ −1/2 vτ,0,s+1 − CN uτ,0,−N  τ s Lv+ + τ s+1/4 |v|xd =0+ |∂  Lvτ,0,s + τ −1/4 |v|xd =0+ |τ,s+1/2 . Observing that Dd vτ,0,s  Lvτ,0,s + vτ,0,s+1 , one concludes the proof of Proposition 14.6.



14.4. Proof of the Microlocal Estimate In what follows, to ease the notation, we shall write ϕ, p, b, and bϕ in place of ϕC , pC , bC , and bCϕ = b0,ϕ , respectively. We recall some of the setting of Sect. 8.3 that is used for the microlocal estimate of Proposition 14.5. In the chosen coordinates, the principal symbol of P takes the form  r(x, ξ  ) = g ij (x)ξi ξj  |ξ  |2 . p(x, ξ) = ξd2 + r(x, ξ  ), 1≤i,j≤d−1

The symbol p0,ϕ of the conjugated operator P0,ϕ can be written as    p0,ϕ (x, ξ  , z, τ ) = z − γ1 ( ) z − γ2 ( ) ,  = (x, ξ  , τ ), with γj ( ) = −iτ ∂d ϕ(x) + i(−1)j α(y  ), where α( )2 = r(x, ξ  + iτ dx ϕ(x)) = r(x, ξ  ) − τ 2 r(x, dx ϕ(x)) + 2iτ r˜(x, ξ  , dx ϕ(x)), with r˜(x, ., .) the bilinear form associated with the quadratic form r(x, .). As in Sect. 8.2, α is chosen such that Re α ≥ 0. ˇ By the first result of Theorem 8.7, if the Lopatinski˘ı–Sapiro condition 0 0 holds for (P, B, ϕ) at m , then ∂d ϕ(x ) > 0, and locally we have ∂d ϕ(x) > 0, implying that Im γ1 ( ) < 0.

14.4. PROOF OF THE MICROLOCAL ESTIMATE

With the definition of pˇ+ ϕ in Sect. 8.2, one has  1 if Im γ2 ( ) < 0,  pˇ+ (m, ω , z, τ ) = ϕ z − γ2 ( ) if Im γ2 ( ) ≥ 0,

349

 = (x, ξ  , τ ),

∗ ∂M. One with x and ξ  the local representatives of m ∈ ∂M and ω  ∈ Tm   has Im γ2 (x, ξ , τ ) ≥ 0 equivalent to having Re α(x, ξ , τ ) ≥ τ ∂d ϕ(x).

We prove the estimate of Proposition 14.5 in the following three (exhaustive) cases: (1) Im γ2 (0 ) > 0. (2) or Im γ2 (0 ) = 0. (3) or Im γ2 (0 ) < 0. 14.4.1. Case 1: One Root in the Open Upper Complex HalfPlane. Here, we have Im γ1 (x0 , ξ 0 , τ 0 ) < 0, Im γ2 (x0 , ξ 0 , τ 0 ) > 0, and 0 0 0 0 0 0 pˇ+ ϕ (m , ω , z, τ ) = z − γ2 (x , ξ , τ ).

ˇ With the Lopatinski˘ı–Sapiro condition holding at the considered point, by (8.3.11), we have moreover bϕ (x0 , ξ 0 , ξd = γ2 , τ 0 ) = b(x0 , ξ 0 + iτ 0 dx ϕ(x0 ), γ2 + iτ 0 ∂xd ϕ(x0 )) = 0. As the roots γ1 and γ2 are locally smooth with respect to (x, ξ  , τ ) and homogeneous of degree one in (ξ  , τ ) by Proposition 6.28, there exist U a conic open neighborhood of (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ and C, C  > 0 such that SU is compact and Im γ2 ( ) ≥ CλT,τ , and Im γ1 ( ) ≤ −C  λT,τ , and (14.4.1)

bϕ (x, ξ  , ξd = γ2 ( ), τ ) = 0,

if  = (x, ξ  , τ ) ∈ U . ˜ ∈ ST0,τ be homogeneous of We let χ ∈ ST0,τ be as in the statement and χ degree zero and be such that supp(χ) ˜ ⊂ U and χ ˜ ≡ 1 on supp(χ). From the smoothness and the homogeneity of the roots, we have χγ ˜ j ∈ ST1,τ , j = 1, 2. We set ˜ 2 ) and P − = Dd − OpT (χγ ˜ 1 ). P + = Dd − OpT (χγ From Lemma 6.21, we have, for some τ∗ > 0, (14.4.2) OpT (χ)uτ,1,s 

≤ C P + OpT (χ)uτ,0,s + |OpT (χ)u|xd =0+ |τ,s+1/2 + CN uτ,0,−N ,

350

14. OPTIMAL ESTIMATES AT THE BOUNDARY

for τ ≥ τ∗ , u ∈ S (Rd+ ) and s ∈ R. From Lemma 6.20 (with s = 0), we have, for τ chosen sufficiently large, (14.4.3) OpT (χ)uτ,1 + |OpT (χ)u|xd =0+ |τ,1/2  P − OpT (χ)u+ + uτ,0,−N , for u ∈ S (Rd+ ). Let now v ∈ S (Rd+ ). We apply estimate (14.4.3) for u = P + v, yielding OpT (χ)P + vτ,1 + |OpT (χ)P + v|xd =0+ |τ,1/2  P − OpT (χ)P + v+ + vτ,1,−N P0,ϕ v+ + vτ,1 , 1,0 = OpT (χ)P0,ϕ mod using that P − OpT (χ)P + = OpT (χ)P − P + mod Ψτ,ph 1,0 . We set w = OpT (χ)v. We observe that we have Ψτ,ph

|P + w|xd =0+ |τ,1/2  |OpT (χ)P + v|xd =0+ |τ,1/2 + |v|xd =0+ |τ,1/2  |OpT (χ)P + v|xd =0+ |τ,1/2 + vτ,1 , using the trace inequality of Proposition 6.9. We also have P + wτ,1  OpT (χ)P + vτ,1 + vτ,1 . We thus obtain P + wτ,1 + |P + w|xd =0+ |τ,1/2  P0,ϕ v+ + vτ,1 . Here, since (14.4.1) holds in U , Lemma 8.19 applies and yields | tr(w)|τ,1,1/2  |B0,ϕ w|xd =0+ |τ,3/2−k + |P + w|xd =0+ |τ,1/2 + vτ,1 . We thus obtain P + wτ,1 + | tr(w)|τ,1,1/2  P0,ϕ v+ + |B0,ϕ w|xd =0+ |τ,3/2−k + vτ,1 . With estimate (14.4.2) (for s = 1), we find wτ,1,1 + | tr(w)|τ,1,1/2  P0,ϕ v+ + |B0,ϕ w|xd =0+ |τ,3/2−k + vτ,1 . With the form of P0,ϕ , that is, P0,ϕ = (Dd + iτ ∂d ϕ)2 mod DT2,τ , we can estimate Dd2 u+ from P0,ϕ v+ and u1,1 , and we obtain wτ,2 + | tr(w)|τ,1,1/2  P0,ϕ v+ + |B0,ϕ w|xd =0+ |τ,3/2−k + vτ,1 . As w = OpT (χ)v, with a commutator argument, we have (14.4.4)

|B0,ϕ w|xd =0+ |τ,3/2−k  |B0,ϕ v|xd =0+ |τ,3/2−k + | tr(v)|τ,1,−1/2  |B0,ϕ v|xd =0+ |τ,3/2−k + vτ,2,−1 ,

using the trace inequality of Corollary 6.10. We then obtain wτ,2 + | tr(w)|τ,1,1/2  P0,ϕ v+ + |B0,ϕ v|xd =0+ |τ,3/2−k + vτ,2,−1 ,

14.4. PROOF OF THE MICROLOCAL ESTIMATE

351

which is stronger than (14.1.4). This concludes the proof for Case 1. 14.4.2. Case 2: One Root on the Real Axis. Here, Im γ1 (x0 , ξ 0 , τ 0 ) < 0 and Im γ2 (x0 , ξ 0 , τ 0 ) = 0. In particular, we have 0 0 0 0 0 0 pˇ+ ϕ (m , ω , z, τ ) = z − γ2 (x , ξ , τ ).

Note that this implies τ 0 = 0 (since in the case τ = 0, one has Im γ2 > 0). Thus there exists C0 such that λT,τ (ξ 0 , τ 0 ) ≤ C0 τ 0 . ˇ With the Lopatinski˘ı–Sapiro condition holding at the considered point, by (8.3.11), we have moreover bϕ (x0 , ξ 0 , ξd = γ2 , τ 0 ) = b(x0 , ξ 0 + iτ 0 dx ϕ(x0 ), γ2 + iτ 0 ∂xd ϕ(x0 )) = 0. As the roots γ1 and γ2 are locally smooth with respect to (x, ξ  , τ ) and homogeneous of degree one in (ξ  , τ ) by Proposition 6.28, there exist U a conic open neighborhood of (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ and C, C  > 0 such that SU is compact and γ1 ( ) = γ2 ( ), Im γ2 ( ) ≥ −CλT,τ , and Im γ1 ( ) ≤ −C  λT,τ , and λT,τ ( ) ≤ 2C0 τ, and bϕ (x, ξ  , ξd = γ2 ( ), τ ) = 0,

(14.4.5)

if  = (x, ξ  , τ ) ∈ U . ˜ ∈ ST0,τ be homogeneous of We let χ ∈ ST0,τ be as in the statement and χ degree zero and be such that supp(χ) ˜ ⊂ U and χ ˜ ≡ 1 on supp(χ). From the smoothness and the homogeneity of the roots, we have χγ ˜ j ∈ ST1,τ , j = 1, 2. We set ˜ 2 ) and P − = Dd − OpT (χγ ˜ 1 ). P + = Dd − OpT (χγ From Lemma 6.20 (with s = 0), we have, for τ chosen sufficiently large, (14.4.6) OpT (χ)uτ,1 + |OpT (χ)u|xd =0+ |τ,1/2  P − OpT (χ)u+ + uτ,0,−N , for u ∈ S (Rd+ ). We write P + = P2+ + iP1+ with   1 1 γ2 ) + OpT (˜ γ 2 )∗ , P2+ = P + + (P + )∗ = Dd − OpT (˜ 2 2   1 1 + + + ∗ P − (P ) = − OpT (˜ P1 = γ2 ) − OpT (˜ γ 2 )∗ . 2i 2i Observe that P2+ ∈ Ψτ1,0 and P1+ ∈ Ψ1T,τ are both formally selfadjoint. Their respective principal symbols are ˜2 ( ) ∈ Sτ1,0 , p+ 2 () = ξd − Re γ

 p+ ˜2 ( ) ∈ ST1,τ . 1 ( ) = − Im γ

352

14. OPTIMAL ESTIMATES AT THE BOUNDARY

+  The principal symbol of P + is p+ () = p+ 2 () + ip1 ( ). By Lemma 13.12, + the operator P is sub-elliptic: there exists a conic open neighborhood W ˜ ≡ 1 in W and of supp(χ) in Rd+ × Rd−1 × R+ such that χ

1 + +  +  {p , p }( ) = {p+ 2 , p1 }( ) > 0, 2i  ∈ W ,  = ( , ξd ).

p+ () = 0 ⇒

Since we also have λT,τ  τ in U , from Lemma 14.6, we have, for some τ∗ > 0, (14.4.7) τ −1/2 OpT (χ)uτ,1,s  P + OpT (χ)uτ,0,s + τ −1/4 |OpT (χ)u|xd =0+ |τ,s+1/2 + uτ,0,−N , for τ sufficiently large, u ∈ S (Rd+ ) and s ∈ R. Let now v ∈ S (Rd+ ). We apply estimate (14.4.6) for u = P + v, yielding OpT (χ)P + vτ,1 + |OpT (χ)P + v|xd =0+ |τ,1/2  P − OpT (χ)P + v+ + vτ,1,−N  P0,ϕ v+ + vτ,1 , 1,0 = OpT (χ)P0,ϕ mod using that P − OpT (χ)P + = OpT (χ)P − P + mod Ψτ,ph 1,0 . We set w = OpT (χ)v. We observe that we have Ψτ,ph

|P + w|xd =0+ |τ,1/2  |OpT (χ)P + v|xd =0+ |τ,1/2 + |v|xd =0+ |τ,1/2  |OpT (χ)P + v|xd =0+ |τ,1/2 + vτ,1 , using the trace inequality of Proposition 6.9. We also have P + wτ,1  OpT (χ)P + vτ,1 + vτ,1 . We thus obtain P + wτ,1 + |P + w|xd =0+ |τ,1/2  P0,ϕ v+ + vτ,1 . Here, since (14.4.5) holds in U , Lemma 8.19 applies and yields | tr(w)|τ,1,1/2  |B0,ϕ w|xd =0+ |τ,3/2−k + |P + w|xd =0+ |τ,1/2 + vτ,1 . We thus obtain P + wτ,1 + τ −1/4 | tr(w)|τ,1,1/2  P0,ϕ v+ + τ −1/4 |B0,ϕ w|xd =0+ |τ,3/2−k + vτ,1 . With estimate (14.4.7) (for s = 1), we find τ −1/2 wτ,1,1 + τ −1/4 | tr(w)|τ,1,1/2  P0,ϕ v+ + τ −1/4 |B0,ϕ w|xd =0+ |τ,3/2−k + vτ,1 .

14.5. OPTIMALITY ASPECT

353

With the form of P0,ϕ and (14.4.4), we finally have τ −1/2 wτ,2 + τ −1/4 | tr(w)|τ,1,1/2  P0,ϕ v+ + τ −1/4 |B0,ϕ v|xd =0+ |τ,3/2−k + vτ,2,−1 . This is the sought estimate (14.1.4). This concludes the proof for Case 2. 14.4.3. Case 3: Both Roots in the Open Lower Complex HalfPlane. Here, one applies without any change the argument of Sect. 8.3.4.2 that gives OpT (χ)vτ,2 + | tr(OpT (χ)v)|τ,1,1/2  P0,ϕ v+ + vτ,2,−1 , for τ > 0 chosen sufficiently large as in (8.3.31). This is stronger than (14.1.4). This concludes the proof for Case 3. 14.5. Optimality Aspect As already pointed out above, in Theorem 14.1, the trace term in the r.h.s. has an additional factor τ −1/4 if compared to the result of Theorem 8.14. A natural question is then the optimality of such a result. With the following theorem, we answer this question positively. Theorem 14.27. Let (M, g) be a compact Riemannian manifold with boundary, and let P = −Δg + R1 with R1 a first-order differential operator with smooth coefficients on M. Let V be an open set of M with V∂ = V ∩ ∂M = ∅ and ϕ be a smooth real function defined on V . Let k ∈ N and B = B k−1 ∂ν + B k with B k−1 and B k differential operators on ∂M of order k − 1 and k, respectively. Assume that there exist C > 0, τ∗ > 0, and α > 0 such that 

(14.5.1) τ 3/2 eτ ϕ uL2 (M) ≤ C eτ ϕ P uL2 (M) + τ −α |eτ ϕ Bu|∂M |τ,3/2−k , for all u ∈ C ∞ (M), with supp(u) ⊂ V , and τ ≥ τ∗ . Then, α ≤ 1/4. Remark 14.28. Because of the proof scheme of Theorem 14.1, the optimality of the estimate therein obtained in Theorem 14.27 implies the optimality of estimation for a first-order operator Dd − OpT (f ) given in Proposition 14.6 with respect to the boundary term. Proof. We consider a local chart C = (κ, O) at the boundary with O ⊂ V , that is, O ∩ ∂M = ∅, κ(O) ⊂ {xd ≥ 0} and κ(O ∩ ∂M) ⊂ {xd = 0} (see Sect. 15.1). By abuse of notation, we still denote by P and ϕ the local representatives of the elliptic operator and weight function under consideration. Let U+ be a bounded open set in Rd+ that intersects {xd = 0} and such that U+ ⊂ κ(O). The assumed estimate (14.5.1) reads (14.5.2)

τ 3/2 eτ ϕ u+  eτ ϕ P u+ + τ −α |eτ ϕ Bu|xd =0+ |τ,3/2−k ,

354

14. OPTIMAL ESTIMATES AT THE BOUNDARY ∞

u ∈ C c (U+ ) and τ ≥ τ∗ . We first proceed as in the proof of Theorem 4.4 of Volume 1 in the case d ≥ 2 therein. Let x0 ∈ U + ∩ {xd = 0} and ξ 0 ∈ Rd be such that p(x0 , ξ 0 + idϕ(x0 )) = 0, that is, (14.5.3)

p(x0 , ξ 0 ) = p(x0 , dϕ(x0 )) and p˜(x0 , ξ 0 , dϕ(x0 )) = 0.

Without any loss of generality, we can take x0 = 0 and ϕ(x0 ) = 0. We set ζ 0 = ξ 0 + idϕ(0), w(x) = x · ζ 0 and 1 2 ∂ ϕ(0)xj xk . G(x) = 2 j,k xj xk We pick f ∈ Cc∞ (Rd ), f ≡ 0, and set uτ (x) = eiτ w(x) f (τ 1/2 x). Arguing as in the proof of Theorem 4.4, one finds (see (4.1.21) and (4.1.25)) τ 3/2 eτ ϕ uτ + ∼ τ 3/2−d/4 eG f + and eτ ϕ P uτ + ∼ τ 3/2−d/4 eG M f + , with M = y · px (0, ζ 0 ) + pξ (0, ζ 0 ) · Dx . Moreover, one has M f = 0 by (4.1.24). We now give an asymptotic estimation of the boundary term |eτ ϕ Buτ |xd =0+ |τ,3/2−k . Lemma 14.29. There exists C > 0 such that |eτ ϕ Buτ |xd =0+ |τ,3/2 ≤ Cτ 3/2−(d−1)/4 |eG f|xd =0+ |∂

as τ → +∞.

A proof is given below. From (14.5.2) and the asymptotic estimate obtained for each term, if α > 1/4, letting τ → +∞, one obtains eG f +  eG M f + , for any f ∈ Cc∞ (Rd ), even in the case supp(f ) intersects {xd = 0}. Changing f into e−G f yields f +  Lf + ,   where L = x · px (0, ζ 0 ) + pξ (0, ζ 0 ) · Dx − (Dx G) is a first-order differential operator with a principal part with constant coefficients. We shall now prove that the estimate in (14.5.4) cannot hold. This implies the conclusion, that is, α ≤ 1/4. (14.5.4)

Above we considered any local chart C = (κ, O). Yet as in Sect. 8.3.2, we can choose a local chart associated with normal geodesic coordinates. In such a chart, one has p(x, ξ) = ξd2 + r(x, ξ  ), with r(x, ξ  ) a positive definite quadratic form in ξ  associated with the bilinear form (ξ  , η  ) → r˜(x, ξ  , η  ). One thus finds that the principal part of L is given by 2ζd0 Dd + 2˜ r(0, ζ 0, , Dx ). Recall that ζ 0 = ξ 0 + idϕ(0).

14.5. OPTIMALITY ASPECT

355

Lemma 14.30. One can choose ξ0 in (14.5.3) so that the principal symbol  of L reads (ξ) = z 0 ξd −˜ r(0, η 0 , ξ  ) , with z 0 ∈ C\{0} and η 0 ∈ Cd−1 \Rd−1 . A proof is given below.   The principal symbol of L thus reads (ξ) = z 0 ξd − b · ξ  ) , with b ∈ Cd−1 \ {0}, and since η 0 ∈ / Rd−1 and r˜ has real coefficients, one sees that for a well-chosen ξ 1 ∈ Rd−1 \ {0}, one has Im b · ξ 1 = Im(b) · ξ 1 > 0. One sets ρ = b · ξ 1 . Observe that L takes the form z0 (Dd − b · Dx ) + x · A, where A ∈ Cd is constant. Let h ∈ Cc∞ (Rd ) and χ ∈ Cc∞ (−1, 1) be such that χ = 1 in a neighborhood of 0. For λ ≥ 1, one sets f (x , xd ) = eiλ(ξ One has Lf (x) = eiλ(ξ

1 ·x +ρx

d)

1 ·x +ρx

d)

h(x)χ(xd ).

  g(x) = (Lh)(x)χ(xd ) + z0 h(x)Dd χ(xd ) .

g(x),

Estimate (14.5.4) reads e−λxd Im ρ χh+  e−λxd Im ρ g+ . For a smooth function w with compact support, one has 2

e−λxd Im ρ w+ = ∫ e−2λxd Im ρ |w(., xd )|2L2 (Rd−1 ) dxd . R+

With the Taylor formula for a(xd ) = |w(., xd )|2L2 (Rd−1 ) , one has   ∫ e−2λxd Im ρ a(xd )dxd = ∫ e−2λxd Im ρ a(0) + O(xd ) dxd R+

R+

1 1 a(0) + 2 O(1) 2λ Im ρ λ 1 1 |w(., 0)|2L2 (Rd−1 ) + 2 O(1), = 2λ Im ρ λ

=

as λ → +∞. Applied for w(x) = χ(xd )h(x) and w = g, one obtains |h|xd =0+ |L2 (Rd−1 )  |g|xd =0+ |L2 (Rd−1 ) = |(Lh)|xd =0+ |L2 (Rd−1 ) .   With χ1 , χ2 ∈ Cc∞ (Rd−1 ), we set h(x) = χ1 (x ) + xd χ2 (x ) χ(xd ), and we compute (14.5.5)

(Lh)|xd =0+ = −z0 b · Dx χ1 (x ) + (x · A )χ1 (x ) − iz0 χ2 (x ), where A = (A1 , . . . , Ad−1 ). Observe that if we choose χ2 = ib · Dx χ1 (x ) − i(x · A )χ1 (x )/z0 , we find (Lh)|xd =0+ = 0. However one has h|xd =0+ = χ1 . Thus for χ1 = 0, estimate (14.5.5) cannot hold. We have reached a contradiction. 

356

14. OPTIMAL ESTIMATES AT THE BOUNDARY

Proof of Lemma 14.29. We consider three cases regarding the order of the boundary operator B: (1) k = 0, that is, a Dirichlet-type boundary operator, meaning B k−1 = 0, (2) k = 1, and (3) k ≥ 2. In all the three cases, the estimation is rather similar, yet it is better to separate them for the sake of exposition. Case 1: k = 0. One has a Dirichlet-type boundary condition, that is, Bu|∂M = B 0 u|∂M , with B 0 = b0 a smooth function, and one writes |eτ ϕ B 0 uτ |xd =0+ |τ,3/2  τ −1/2 |eτ ϕ b0 uτ |xd =0+ |τ,2     τ −1/2 τ 2−|β | |Dxβ (eτ ϕ b0 uτ )|xd =0+ |∂ β  ∈Nd−1 |β  |≤2

 τ −1/2  τ −1/2



β  ∈Nd−1 |β  |≤2



β  ∈Nd−1 |β  |≤2

   τ 2−|β | |eτ (ϕ+iw) Dxβ f (τ 1/2 .) |x

+ d =0

|

∂





τ 2−|β |/2 |eτ (ϕ+iw) (Dxβ f )(τ 1/2 .)|xd =0+ |∂ .

Then, arguing as in (4.1.21) in the proof of Theorem 4.4 of Volume 1, yet in dimension d − 1, one obtains the result. Case 2: k = 1. One has B = B 1 − iB 0 Dd , with B 1 and B 0 differential operators of order 1 and 0 on ∂M. One writes |eτ ϕ Buτ |xd =0+ |τ,1/2  τ −1/2 |eτ ϕ Buτ |xd =0+ |τ,1     τ −1/2 τ 1−|β | |Dxβ (eτ ϕ Buτ )|xd =0+ |∂ β  ∈Nd−1 |β  |≤1

 τ −1/2



β  ∈Nd−1 , =0,1 |β  |+≤2

    τ 2−|β |− |eτ (ϕ+iw) Dxβ Dd f (τ 1/2 .) |x

+ d =0

| , ∂

using that the coefficients of B and their derivatives are bounded. This yields |eτ ϕ Buτ |xd =0+ |τ,1/2   τ 3/2

β  ∈Nd−1 , =0,1 |β  |+≤2





τ −(|β |+)/2 |eτ (ϕ+iw) (Dxβ Dd f )(τ 1/2 .)|xd =0+ |∂ .

As above, arguing as in (4.1.21), one obtains the result.

14.5. OPTIMALITY ASPECT

357

Case 3: k ≥ 2. One has B = B k −iB k−1 Dd , with B k and B k−1 differential operators of order k and k − 1 on ∂M. As 3/2 − k < 0, one writes |eτ ϕ Buτ |xd =0+ |τ,3/2−k  τ 3/2−k |eτ ϕ Buτ |xd =0+ |∂       τ 3/2−k τ k−|β |− |eτ (ϕ+iw) Dxβ Dd f (τ 1/2 .) |x β  ∈Nd−1 , =0,1 |β  |+≤k

+ d =0

| , ∂

using that the coefficients of B are bounded. This yields |eτ ϕ Buτ |xd =0+ |τ,3/2−k     τ 3/2 τ −(|β |+)/2 |eτ (ϕ+iw) (Dxβ Dd f )(τ 1/2 .)|xd =0+ |∂ . β  ∈Nd−1 , =0,1 |β  |+≤k

As above, arguing as in (4.1.21), one obtains the result.



Proof of Lemma 14.30. In the chosen normal geodesic coordinates, (14.5.3) reads (14.5.6)

p(0, ξ 0 ) = p(0, dϕ(0)),

ξd0 ∂d ϕ(0) + r˜(0, ξ 0 , dx ϕ(0)) = 0.

If ∂d ϕ(0) = 0, one can choose ξ 0 = 0 and ξd0 = 0 in (14.5.6). Consequently, one can choose ζd0 = ξd0 = 0 and ζ 0 = dx ϕ(0) = 0 since dϕ(0) = 0 by Theorem 4.5 of Volume 1. Thus, with z 0 = 2ξd0 = 0 and / Rd−1 , one obtains the sought form for the principal η 0 = idx ϕ(0)/ξd0 ∈ symbol (ξ). If ∂d ϕ(0) = 0 and d ≥ 3, we choose ξ 0 such that ξd0 = 0 and ξ 0 = 0  with r˜ 0, ξ 0 , dx ϕ(0) = 0 in (14.5.6). Then with z 0 = 2i∂d ϕ(0) = 0 and     η 0 = ξ 0 + idx ϕ(0) / i∂d ϕ(0) ∈ / Rd−1 , one obtains the sought form for the principal symbol (ξ). 2 2 If d = 2, then p(x, ξ) takes locally  0 the form p(x,0ξ) = ξ2 + r(x)ξ10 , where r(x) ≥ C with C > 0. One has p˜ 0, ξ , dx ϕ(0) = ξ2 ∂2 ϕ(0) + r(0)ξ1 ∂1 ϕ(0). The second condition in (14.5.6) reads

ξ20 ∂2 ϕ(0) + r(0)ξ10 ∂1 ϕ(0) = 0. If ∂d ϕ(0) = ∂2 ϕ(0) = 0, one finds ξ20 = −r(0)ξ10 ∂1 ϕ(0)/∂2 ϕ(0), with ξ10 = 0 with z 0 = 2ξ20 + 2i∂2 ϕ(0) = 0 chosen such that p(x, ξ 0 )=p(x, dϕ(0)). Then,  0 0 0 and η = ξ1 + i∂1 ϕ(0) / ξ2 + i∂2 ϕ(0) , one finds that Im η 0 = 0 if and only if ξ20 ∂1 ϕ(0) − ξ10 ∂2 ϕ(0) = 0. With the form of ξ20 given above, one has  ξ10  ξ20 ∂1 ϕ(0) − ξ10 ∂2 ϕ(0) = − r(0)(∂1 ϕ(0))2 + (∂2 ϕ(0))2 = 0. ∂2 ϕ(0) Thus, one obtains the sought form for the principal symbol (ξ).



358

14. OPTIMAL ESTIMATES AT THE BOUNDARY

14.6. A Refined Estimate Without Any Prescribed Boundary Operator Here, we give the counterpart of the results obtained in Sect. 8.4.2. One aims to obtain an estimation without prescribing any particular boundary condition. We only wish to obtain bounds by means of norms on the Dirichlet trace u|∂M and a first-order trace Bu|∂M , that is, where B = ∂ν + B  is a first-order boundary operator. For this to make sense, one assumes that B ˇ fulfills the Lopatinski˘ı–Sapiro condition of Definition 2.2 along with P . Such boundary operators are characterized in Proposition 2.8. Here, the only assumption on the weight function made at the boundary is ∂ν ϕ = 0. Yet, ˇ one does not assume that (P, B, ϕ) fulfills the Lopatinski˘ı–Sapiro condition of Definition 8.1. Theorem 14.31. Let (M, g) be a compact Riemannian manifold with boundary, and let P = −Δg + R1 with R1 a first-order differential operator with bounded coefficients on M. Let V be an open set of M such that V∂ = V ∩ ∂M = ∅. Let B  be a differential operator of order one on ∂M ˇ such that (P, B) fulfills the Lopatinski˘ı–Sapiro condition of Definition 2.2 on  ∞ V∂ for B = ∂ν + B . Let ϕ ∈ C (M) be such that the pair (P, ϕ) has the sub-ellipticity property of Definition 5.1 in V and ∂ν ϕ = 0 in V∂ . Then, there exist C and τ∗ > 0 such that (14.6.1) τ −1/2 eτ ϕ uτ,2 + τ −1/4 |eτ ϕ tr(u)|τ,1,1/2 

≤ C eτ ϕ P uL2 (M) + τ 5/4 |eτ ϕ u|∂M |L2 (∂M) + τ −1/4 |eτ ϕ Bu|∂M |τ,1/2 , for all u ∈ C ∞ (M), with supp(u) ⊂ V , and τ ≥ τ∗ . In the framework of the normal geodesic coordinates introduced through the local chart C = (κ, O) in Sect. 8.3.2 and briefly recalled in Sect. 14.1.1, we prove the following proposition. Then, the proof of Theorem 14.31 is obtained through a patching procedure1 of such local estimations as performed in Section 3.5. Proposition 14.32. Let P0 = Op(pC ) and B0 = Op(bC ). Let m0 ∈ ∂M, V be an open neighborhood of m0 in M. Set x0 = κ(m0 ). Assume that ˇ (P0 , B0 ) satisfies the Lopatinski˘ı–Sapiro condition of Definition 2.2 at x0 . Assume that (P0 , ϕC ) has the sub-ellipticity property of Definition 3.2 of Volume 1 in U 0 with U 0 = κ(V 0 ∩O) and that ∂d ϕC|xd =0 = 0 in U 0 ∩{xd = 0}. 0

There exist a bounded neighborhood U+ of x0 in Rd+ such that U+ ⊂ U 0 and

1This patching procedure is also invoked for the proofs of Theorems 5.5, 5.6, and 8.14 for instance.

14.6. A REFINED ESTIMATE WITHOUT ANY PRESCRIBED . . .

359

two constants C and τ∗ > 0 such that τ −1/2 vτ,2,0 + τ −1/4 | tr(v)|τ,1,1/2

 ≤ C P0,ϕ v+ + τ 5/4 |v|xd =0+ |∂ + τ −1/4 |B0,ϕ v|xd =0+ |τ,1/2 , ∞

for all v ∈ C c (U+ ) and τ ≥ τ∗ . The proof of Proposition 14.32 relies on a microlocal version of the estimate that we state below and it follows the proof scheme of Proposition 8.16 by patching together such microlocal estimates as in Sect. 8.3.5. Proposition 14.33. Let x0 be as in the statement of Proposition 14.32. ˇ condition of DefiniAssume that (P0 , B0 ) satisfies the Lopatinski˘ı–Sapiro 0 C tion 2.2 at x . Assume that (P0 , ϕ ) has the sub-ellipticity property of Definition 3.2 of Volume 1 in U 0 with U 0 = κ(V 0 ∩ O) and that ∂d ϕC|xd =0 = 0 ∗ ∂M, with local representative (ω 0 )C = in U 0 ∩ {xd = 0}. Let ω 0 ∈ Tm 0 0 d−1 0 ξ ∈ R , and τ ≥ 0 such that (ξ 0 , τ 0 ) = 0. Then, there exists U a conic open neighborhood of (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ such that for χ ∈ ST0,τ , homogeneous of degree 0, with supp(χ) ⊂ U , there exist C > 0 and τ∗ > 0 such that

(14.6.2) τ −1/2 OpT (χ)vτ,2 + τ −1/4 | tr(OpT (χ)v)|τ,1,1/2 

≤ C P0,ϕ v+ + τ 5/4 |v|xd =0+ |∂ + τ −1/4 |B0,ϕ v|xd =0+ |τ,1/2 + vτ,2,−1 , for τ ≥ τ∗ , v ∈ S (Rd+ ). We follow the setting and notation of Sect. 14.4. In particular, to ease the notation, we write ϕ, p, b, and bϕ in place of ϕC , pC , bC , and bCϕ = b0,ϕ , respectively. We write     = (x, ξ  , τ ), p0,ϕ (x, ξ  , z, τ ) = z − γ1 ( ) z − γ2 ( ) , with γj ( ) = −iτ ∂d ϕ(x) + i(−1)j α(y  ), where α( )2 = r(x, ξ  + iτ dx ϕ(x)) = r(x, ξ  ) − τ 2 r(x, dx ϕ(x)) + 2iτ r˜(x, ξ  , dx ϕ(x)). For the proof of the microlocal estimate of Proposition 14.33, we shall consider three (exhaustive) cases for 0 = (x0 , ξ 0, , τ 0 ) in the statement: (1) τ 0 = 0. (2) τ 0 > 0 and γ1 (0 ) = γ2 (0 ). (3) τ 0 > 0 and γ1 (0 ) = γ2 (0 ).

360

14. OPTIMAL ESTIMATES AT THE BOUNDARY

Remark 14.34. Here, we assume that ∂d ϕ(x0 ) = 0. As a result, the roots γ1 (0 ) and γ2 (0 ) cannot be both real; otherwise, one has Im γ1 (0 ) = −iτ ∂d ϕ(x0 ) − i Re α(0 ) = 0 Im γ2 (0 ) = −iτ ∂d ϕ(x0 ) + i Re α(0 ) = 0, implying ∂d ϕ(x0 ) = 0 which is excluded. 14.6.1. Case 1: τ 0 = 0. Here we consider 0 = (x0 , ξ 0, , τ 0 ) with ˇ = 0 and ξ 0 ∈ Rd−1 \ {0}. Observe that the Lopatinski˘ı–Sapiro condi0 tion for (P0 , B0 , ϕ) (Definition 8.1) holds at  for any weight function ϕ, ˇ since the Lopatinski˘ı–Sapiro condition of Definition 8.1 coincides with the ˇ Lopatinski˘ı–Sapiro condition of Definition 2.2 at such a point. By Proposition 14.5, there exists U a conic open neighborhood of 0 in U 0 ×Rd−1 ×R+ such that for χ ∈ Sτ0 , homogeneous of degree 0 with supp(χ) ⊂ U , one has τ0

(14.6.3) τ −1/2 OpT (χ)vτ,2 + τ −1/4 | tr(OpT (χ)v)|τ,1,1/2  P0,ϕ v+ + τ −1/4 |B0,ϕ v|xd =0+ |τ,1/2 + vτ,2,−1 , for τ > 0 chosen sufficiently large. This is stronger than (14.6.2). 14.6.2. Case 2: τ 0 > 0 and γ1 (0 ) = γ2 (0 ). By remark 14.34, / R (the other one of the two roots is not real. Assume here that γ2 (0 ) ∈ case is treated by exchanging the roles of the two roots). Since the roots γ1 and γ2 are locally smooth with respect to (x, ξ  , τ ) and homogeneous of degree one in (ξ  , τ ) by Proposition 6.28, there exist U a conic open neighborhood of 0 = (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ and C, C  > 0 such that SU is compact and |γ1 ( ) − γ2 ( )| ≥ CλT,τ , | Im γ2 ( )| ≥ CλT,τ , and λT,τ ( ) ≤ C  τ,  ∈ U . ˜ ∈ ST0,τ be homogeneous of We let χ ∈ ST0,τ be as in the statement and χ degree zero and be such that supp(χ) ˜ ⊂ U and χ ˜ ≡ 1 on supp(χ). From the smoothness and the homogeneity of the roots, one has χγ ˜ j ∈ ST1,τ , j = 1, 2. We set ˜ 2 ) and L1 = Dd − OpT (χγ ˜ 1 ). L2 = Dd − OpT (χγ If Im γ2 ( )  λT,τ , one may apply Lemma 6.21 (with s = 1 therein). If Im γ2 ( )  −λT,τ , one may apply (the stronger) Lemma 6.20 (with s = 1 therein). Either way, one has (14.6.4)

OpT (χ)vτ,1,1  Op(L2 )OpT (χ)vτ,0,1 + |OpT (χ)v|xd =0+ |τ,3/2 + vτ,0,−N ,

for τ ≥ τ0 , for some τ0 > 0.

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By Lemma 13.25, one sees that the operator L1 fulfills the required properties of Proposition 14.6. With s = 0 therein, one finds (14.6.5) τ −1/2 OpT (χ)wτ,1,0  Op(L1 )OpT (χ)w+ + τ −1/4 |OpT (χ)w|xd =0+ |τ,1/2 + wτ,0,−N , for τ ≥ τ1 and w ∈ S (Rd+ ) for some τ1 > 0. In (14.6.5), we take w = Op(L2 )v: τ −1/2 OpT (χ)Op(L2 )vτ,1,0  Op(L1 )OpT (χ)Op(L2 )v+ + τ −1/4 |OpT (χ)Op(L2 )v|xd =0+ |τ,1/2 + Op(L2 )vτ,0,−N . As [OpT (χ), Op(L2 )] ∈ Ψ0T,τ and OpT (χ)P0,ϕ = Op(L1 )OpT (χ)Op(L2 ) mod Ψ1,0 T,τ , one obtains τ −1/2 Op(L2 )OpT (χ)vτ,1,0  OpT (χ)P0,ϕ v+ + τ −1/4 |OpT (χ)Op(L2 )v|xd =0+ |τ,1/2 + vτ,1,0  P0,ϕ v+ + τ −1/4 |Op(L2 )OpT (χ)v|xd =0+ |τ,1/2 + vτ,1,0 , by using the trace inequality of Corollary 6.10. With estimation (14.6.4), one writes τ −1/2 OpT (χ)vτ,1,1 + τ −1/2 Dd Op(L2 )OpT (χ)v+  P0,ϕ v+ + τ −1/4 |Op(L2 )OpT (χ)v|xd =0+ |τ,1/2 + τ −1/2 |OpT (χ)v|xd =0+ |τ,3/2 + vτ,1,0 . From the forms of L2 and B0,ϕ , one then obtains τ −1/2 OpT (χ)vτ,2,0  P0,ϕ v+ + τ −1/4 |B0,ϕ OpT (χ)v|xd =0+ |τ,1/2 + τ −1/4 |OpT (χ)v|xd =0+ |τ,3/2 + vτ,1,0 . Since B0,ϕ − ∂ν ∈ Ψ1T,τ and ∂ν = −∂d = −iDd , one also has (14.6.6)

τ −1/2 OpT (χ)vτ,2,0 + τ −1/4 |OpT (χ)v|xd =0+ |τ,1,1/2  P0,ϕ v+ + τ −1/4 |B0,ϕ OpT (χ)v|xd =0+ |τ,1/2 + τ −1/4 |OpT (χ)v|xd =0+ |τ,3/2 + vτ,1,0 .

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Using that [B0,ϕ , OpT (χ)] ∈ Ψ0T,τ and the trace inequality of Corollary 6.10, one finds τ −1/2 OpT (χ)vτ,2,0 + τ −1/4 |OpT (χ)v|xd =0+ |τ,1,1/2  P0,ϕ v+ + τ −1/4 |B0,ϕ v|xd =0+ |τ,1/2 + τ −1/4 |OpT (χ)v|xd =0+ |τ,3/2 + vτ,1,0 . 3/2

Finally, as ΛT,τ OpT (χ) ∈ τ 3/2 Ψ0T,τ since λT,τ  τ in supp(χ), we achieve the estimate τ −1/2 OpT (χ)vτ,2,0 + τ −1/4 |OpT (χ)v|xd =0+ |τ,1,1/2  P0,ϕ v+ + τ −1/4 |B0,ϕ v|xd =0+ |τ,1/2 + τ 5/4 |v|xd =0+ |∂ + vτ,1,0 , which is only stronger than (14.6.2) with respect to the remainder term. 14.6.3. Case 3: τ 0 > 0 and γ1 (0 ) = γ2 (0 ). By Remark 14.34, / R and there exist U a conic open neighborhood of γ1 (0 ) = γ2 (0 ) ∈ 0 = (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ and C, C  > 0 such that SU is compact and | Im γ1 ( )| ≥ CλT,τ , | Im γ2 ( )| ≥ CλT,τ , and λT,τ ( ) ≤ C  τ,  ∈ U . We let χ ∈ ST0,τ be as in the statement. Then, Lemma 8.25 applies to OpT (χ)v. One has OpT (χ)vτ,2 + | tr(OpT (χ)v)|τ,1,1/2  P0,ϕ v+ + | tr(OpT (χ)v)|τ,1,1/2 + vτ,2,−1 . Since B0,ϕ − ∂ν ∈ Ψ1T,τ and ∂ν = −∂d = −iDd , one also has (14.6.7)

OpT (χ)vτ,2 + | tr(OpT (χ)v)|τ,1,1/2  P0,ϕ v+ + |B0,ϕ OpT (χ)v|xd =0+ |τ,1/2 + |OpT (χ)v|xd =0+ |τ,3/2 + v2τ,2,−1 .

Arguing as we did from (14.6.6) in Case 2, from (14.6.7), we obtain estimate (14.6.2) in Case 3. 14.7. Optimal Estimate with Source Terms of Weaker Regularity Here, we prove an improved version of Theorem 13.4 that allows one to provide an estimate for H 1 -functions solutions to P u = F0 + divg F with F0 ∈ L2 (M), F ∈ L2 V (M). As above, the (optimal) improvement lies in the presence of a factor τ −1/4 for the trace terms. We let B(., .) be as given in (13.1.3), and more generally we use the notation introduced in Chap. 13.

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Theorem 14.35. Let (M, g) be a compact Riemannian manifold with boundary, and let P = −Δg + R1 with R1 a first-order differential operator on M. Let V be an open set of M and set V∂ = V ∩ ∂M. Let ϕ ∈ C ∞ (M) be such that the pair (P, ϕ) has the sub-ellipticity property of Definition 5.1 in V . If V∂ = ∅, consider B a differential operator of order k ∈ N in V , and ˇ assume moreover that (P, B, ϕ) satisfies the Lopatinski˘ı–Sapiro condition at all points m ∈ V∂ (Definition 8.1). Then, there exist C and τ∗ > 0 such that (14.7.1) τ 1/2 eτ ϕ uτ,1 + τ 3/4 |eτ ϕ|∂M u|∂M |τ,1/2

 + τ 3/4 |eτ ϕ|∂M (∂ν u + g(F, ν))|∂M |τ,−1/2 ≤ C eτ ϕ F0 L2 (M)

 + τ eτ ϕ F L2 V (M) + τ 3/4 |eτ ϕ|∂M B(u, g(F, ν))|τ,1/2−k ,

for τ ≥ τ∗ and u ∈ H 1 (M), F0 ∈ L2 (M), F ∈ L2 V (M) such that P u = F0 + divg F, and supp(u) ∪ supp(F0 ) ∪ supp(F ) ⊂ V . Under the setting and notation of Sect. 13.2, we prove microlocal versions of the result in the next section, and the proof of Theorem 14.35 then follows from a patching argument as done in Sect. 13.4. 14.7.1. Microlocal Estimate. With the notation of Sects. 13.2 and 14.1.1, we set P0 = Op(pC ), B0 = Op(bC ). The following proposition is an improved version of Proposition 13.10 with respect to the boundary terms. Proposition 14.36. Let m0 ∈ ∂M and V 0 be an open neighborhood of in M that meets one connected component of ∂M. Set x0 = κ(m0 ). Assume that (P0 , ϕ) has the sub-ellipticity property of Definition 3.2 of Vol∗ ∂M, with local represenume 1 in U 0 with U 0 = κ(V 0 ∩ O). Let ω 0 ∈ Tm 0 0 C 0 d−1 0 tative (ω ) = ξ ∈ R , and τ ≥ 0 such that (ξ 0 , τ 0 ) = 0, and assume ˇ that the Lopatinski˘ı–Sapiro condition of Definition 8.1 holds at (m0 , ω 0 , τ 0 ). Then, there exists U a conic open neighborhood of 0 = (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ such that for χ ∈ ST0,τ , homogeneous of degree zero, with supp(χ) ⊂ U , there exist C > 0 and τ∗ > 0 such that m0

(14.7.2) τ 1/2 OpT (χ)vτ,1 + τ 3/4 |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |τ,−1/2

+ τ 3/4 |OpT (χ)v|xd =0+ |τ,1/2 ≤ C G0 + + τ G+ + τ 3/4 |B0,ϕ (v, −Gd )|xd =0+ |τ,1/2−k

 + vτ,1 + τ 3/4 |(Dd v − iGd )|xd =0+ |τ,−3/2 ,

for τ ≥ τ∗ and v ∈ H 1 (Rd+ ), G0 ∈ L2 (Rd+ ), G ∈ L2 V (Rd+ ) such that  P0,ϕ v = G0 + ∂ j Gj . 1≤i≤d

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We prove the estimate of Proposition 14.36 in the following four (exhaustive) cases: (1) (2) (3) (4)

Im γ2 (0 ) > 0. or Im γ2 (0 ) = 0. or Im γ2 (0 ) < 0 and γ2 (0 ) = γ1 (0 ). or Im γ2 (0 ) < 0 and γ2 (0 ) = γ1 (0 ).

14.7.1.1. Case 1: One Root in the Open Upper Complex Half-Plane. Here, we have Im γ1 (x0 , ξ 0 , τ 0 ) < 0, Im γ2 (x0 , ξ 0 , τ 0 ) > 0, and 0 0 0 0 0 0 pˇ+ ϕ (m , ω , z, τ ) = z − γ2 (x , ξ , τ ).

ˇ With the Lopatinski˘ı–Sapiro condition of Definition 8.1 holding at the considered point, by (8.3.11), we have moreover bϕ (x0 , ξ 0 , ξd = γ2 , τ 0 ) = b(x0 , ξ 0 + iτ 0 dx ϕ(x0 ), γ2 + iτ 0 ∂xd ϕ(x0 )) = 0. As the roots γ1 and γ2 are locally smooth with respect to (x, ξ  , τ ) and homogeneous of degree one in (ξ  , τ ) by Proposition 6.28, there exist U a conic open neighborhood of (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ and C, C  > 0 such that SU is compact and Im γ2 ( ) ≥ CλT,τ , and Im γ1 ( ) ≤ −C  λT,τ , and bϕ (x, ξ  , ξd = γ2 ( ), τ ) = 0, if  = (x, ξ  , τ ) ∈ U . ˜ ∈ ST0,τ be homogeneous of We let χ ∈ ST0,τ be as in the statement and χ degree zero and be such that supp(χ) ˜ ⊂ U and χ ˜ ≡ 1 on supp(χ). From the smoothness and the homogeneity of the roots, we have χγ ˜ j ∈ ST1,τ , j = 1, 2. We set ˜ 2 ) and P − = Dd − OpT (χγ ˜ 1 ). P + = Dd − OpT (χγ From Lemma 6.21, we have, for some τ∗ > 0, (14.7.3) OpT (χ)uτ,1,s 

≤ C P + OpT (χ)uτ,0,s + |OpT (χ)u|xd =0+ |τ,s+1/2 + CN uτ,0,−N , for τ ≥ τ∗ , u ∈ S (Rd+ ) and s ∈ R. Let now v ∈ S (Rd+ ). We set w = OpT (χ)v. Estimate (13.3.7) that reads (14.7.4)

P + w+ + |(P + w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  τ −1 G0 + + G+ + vτ,1,−1

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365

applies here since the present setting is a subcase of Case 1 for the proof of Proposition 13.10. For the same reason, estimate (13.3.9) also holds in the present case: (14.7.5) |w|xd =0+ |τ,1/2 + |(Dd w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  |B0,ϕ (w, −OpT (χ)Gd )|τ,1/2−k + |(P + w − iOpT (χ)Gd )|xd =0+ |τ,−1/2 + τ −1 vτ,1 + |(Dd v − iGd )|xd =0+ |τ,−3/2 . With (14.7.4) and (14.7.5), we obtain (14.7.6) P + w+ + τ −1/4 |w|xd =0+ |τ,1/2 + τ −1/4 |(Dd w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  τ −1/4 |B0,ϕ (w, −OpT (χ)Gd )|τ,1/2−k + τ −1 G0 + + G+ + τ −1 vτ,1 + τ −1/4 |(Dd v − iGd )|xd =0+ |τ,−3/2 . With (14.7.3) in the case s = 0, we obtain τ −1/4 wτ,1 + τ −1/4 |w|xd =0+ |τ,1/2 + τ −1/4 |(Dd w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  τ −1/4 |B0,ϕ (w, −OpT (χ)Gd )|τ,1/2−k + τ −1 G0 + + G+ + τ −1 vτ,1 + τ −1/4 |(Dd v − iGd )|xd =0+ |τ,−3/2 , yielding τ 3/4 wτ,1 + τ 3/4 |w|xd =0+ |τ,1/2 + τ 3/4 |(Dd w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  τ 3/4 |B0,ϕ (w, −OpT (χ)Gd )|τ,1/2−k + G0 + + τ G+ + vτ,1 + τ 3/4 |(Dd v − iGd )|xd =0+ |τ,−3/2 , which is stronger than (14.7.2). This concludes the proof for Case 1. 14.7.1.2. Case 2: One Root on the Real Axis. Here, Im γ1 (x0 , ξ 0 , τ 0 ) < 0 and Im γ2 (x0 , ξ 0 , τ 0 ) = 0. In particular, we have 0 0 0 0 0 0 pˇ+ ϕ (m , ω , z, τ ) = z − γ2 (x , ξ , τ ).

Note that this implies τ 0 = 0 (since in the case τ = 0, one has Im γ2 > 0). Thus, there exists C0 such that λT,τ (ξ 0 , τ 0 ) ≤ C0 τ 0 . ˇ With the Lopatinski˘ı–Sapiro condition of Definition 8.1 holding at the considered point, by (8.3.11), we have moreover bϕ (x0 , ξ 0 , ξd = γ2 , τ 0 ) = b(x0 , ξ 0 + iτ 0 dx ϕ(x0 ), γ2 + iτ 0 ∂xd ϕ(x0 )) = 0. As the roots γ1 and γ2 are locally smooth with respect to (x, ξ  , τ ) and homogeneous of degree one in (ξ  , τ ) by Proposition 6.28, there exist U a

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conic open neighborhood of (x0 , ξ 0 , τ 0 ) in U 0 × Rd−1 × R+ and C, C  > 0 such that SU is compact and γ1 ( ) = γ2 ( ), Im γ2 ( ) ≥ −CλT,τ , and Im γ1 ( ) ≤ −C  λT,τ , and λT,τ ( ) ≤ 2C0 τ, and bϕ (x, ξ  , ξd = γ2 ( ), τ ) = 0, if  = (x, ξ  , τ ) ∈ U . ˜ ∈ ST0,τ be homogeneous of We let χ ∈ ST0,τ be as in the statement and χ degree zero and be such that supp(χ) ˜ ⊂ U and χ ˜ ≡ 1 on supp(χ). From the smoothness and the homogeneity of the roots, we have χγ ˜ j ∈ ST1,τ , j = 1, 2. We set ˜ 2 ) and P − = Dd − OpT (χγ ˜ 1 ). P + = Dd − OpT (χγ As in (14.4.7), we have (14.7.7) τ −1/2 OpT (χ)uτ,1,s  P + OpT (χ)uτ,0,s + τ −1/4 |OpT (χ)u|xd =0+ |τ,s+1/2 + uτ,0,−N , for τ sufficiently large, u ∈ S (Rd+ ) and s ∈ R. Let now v ∈ S (Rd+ ). We set w = OpT (χ)v. As in Case 1, we have estimates (14.7.4) and (14.7.5) since the present setting is a subcase of Case 1 for the proof of Proposition 13.10. We thus obtain (14.7.6), that is, P + w+ + τ −1/4 |w|xd =0+ |τ,1/2 + τ −1/4 |(Dd w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  τ −1/4 |B0,ϕ (w, −OpT (χ)Gd )|τ,1/2−k + τ −1 G0 + + G+ + τ −1 vτ,1 + τ −1/4 |(Dd v − iGd )|xd =0+ |τ,−3/2 . With (14.7.7) in the case s = 0, we obtain τ −1/2 OpT (χ)uτ,1 + τ −1/4 |w|xd =0+ |τ,1/2 + τ −1/4 |(Dd w − iOpT (χ)Gd )|xd =0+ |τ,−1/2  τ −1/4 |B0,ϕ (w, −OpT (χ)Gd )|τ,1/2−k + τ −1 G0 + + G+ + τ −1 vτ,1 + τ −1/4 |(Dd v − iGd )|xd =0+ |τ,−3/2 . Upon multiplication by τ , this is the sought estimate (14.7.2). This concludes the proof for Case 2.

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14.7.1.3. Case 3: Roots Are Different and Both in the Open Lower Complex Half-Plane. Here, one applies without any change the argument of Sect. 13.3.2 that gives a conic neighborhood U as in the statement and

τ OpT (χ)vτ,1 + |OpT (χ)v|xd =0+ |τ,1/2  + |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |τ,−1/2  G0 + + τ G+ + τ vτ,1,−1 , for τ > 0 chosen sufficiently large and for χ as in the statement. This estimate is stronger than (14.7.2). This concludes the proof for Case 3. 14.7.1.4. Case 4: Roots Coincide in the Open Lower Complex HalfPlane. Here, one applies without any change the argument of Sect. 13.3.3. One obtains a conic neighborhood U as in the statement and estimate (13.3.16) that we reproduce here:

τ OpT (χ)vτ,1 + |OpT (χ)v|xd =0+ |τ,1/2 + |(Dd OpT (χ)v − iOpT (χ)Gd )|xd =0+ |τ,−1/2



 G0 + + τ G+ + vτ,1 + |(Dd v − iGd )|xd =0+ |τ,−N , for τ > 0 chosen sufficiently large and for χ as in the statement. This estimate is stronger than (14.7.2). This concludes the proof for Case 4. 14.7.2. Shifted Estimate. Arguing as for the proof of Theorem 8.24, we deduce from Theorem 14.31 the following shifted estimate. Theorem 14.37. Let (M, g) be a compact Riemannian manifold with boundary, and let P = −Δg + R1 with R1 a first-order differential operator on M. Let V be an open set of M and set V∂ = V ∩ ∂M. Let ϕ ∈ C ∞ (M) be such that the pair (P, ϕ) has the sub-ellipticity property of Definition 5.1 in V . If V∂ = ∅, consider B a differential operator of order k ∈ N in V , and ˇ assume moreover that (P, B, ϕ) satisfies the Lopatinski˘ı–Sapiro condition at all points m ∈ V∂ (Definition 8.1). Then, there exist C and τ∗ > 0 such that (14.7.8) τ 1/2 eτ ϕ uτ,1 + τ 5/4 |eτ ϕ|∂M u|∂M |τ,0 + τ 5/4 |eτ ϕ|∂M (∂ν u + g(F, ν))|∂M |τ,−1   ≤ C eτ ϕ F0 L2 (M) + τ eτ ϕ F L2 V (M) + τ 5/4 |eτ ϕ|∂M B(u, g(F, ν))|τ,−k , for τ ≥ τ∗ and u ∈ H 1 (M), F0 ∈ L2 (M), F ∈ L2 V (M) such that P u = F0 + divg F, and supp(u) ∪ supp(F0 ) ∪ supp(F ) ⊂ V .

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14.8. Notes In the case of Dirichlet boundary conditions, the results of Theorems 14.1 and 14.35 were obtained in the work of O. Yu. Imanuvilov and J.-P. Puel [179]. As mentioned above, if compared to the results in Chaps. 8 and 13, such estimates yield an improvement with respect to the boundary terms. They have been used to obtain controllability results for the Navier–Stokes equations; see, for instance, the work of E. Fern´andez-Cara et al. [145]. Together with M. Yamamoto, O. Yu. Imanuvilov and J.-P. Puel further extended the result in the case of the associated parabolic operator [180]. This result is, for instance, used in the work by M. Badra [45] to obtain an estimation of a solution of the Stokes system. The proof scheme we use here has some similarity with that of [179], for instance, by writing the conjugated operator as a product of two firstorder terms. The proof of the critical estimate, that is, the estimate of Proposition 14.6 here relies however on different arguments. Here, we have chosen to rely on the strength of a Fefferman–Phong type inequality [142] as proven by J.-M. Bony [77]. For details on the proofs of such inequalities, we refer to the book by L. H¨ormander [175, Sections 18.4-6] and the book by N. Lerner [228, Chapter 2]. Here, the Carleman estimate obtained by O. Yu. Imanuvilov and J.-P. Puel [179] is generalized to any boundary condition such that the Lopatinˇ ski˘ı–Sapiro condition holds. Moreover, the optimality of the estimation is proven in Sect. 14.5. In Sect. 14.6, an estimation is also derived without prescribing any particular boundary condition. We have to assume that ∂ν ϕ = 0. To us, obtaining the optimal estimate if ∂ν ϕ is allowed to vanish is an open question, whose study is useful if one wishes to derive global estimates with boundary observation terms; see Sect. 8.5. In Chapter 3 of Volume 1 and Chaps. 5, 8, and 13 in the present volume, global estimates were derived. This could also be done in the framework of the optimal local estimations we have obtained here. Appendix 14.A. H¨ ormander Calculus Properties 14.A.1. Slowly Varying and Temperate Metrics. Here, we prove Proposition 14.7 for the metric g γ . The proof for g follows by taking γ = τ . Slow Variation. Let ε > 0 whose value will be set below, and let (x, ξ), (y, η) ∈ R2d , and τ, γ ∈ [1, +∞) such that τ ≥ γ and gγ (y, η) < ε, for  = (x, ξ, τ ). This reads (14.A.1) |y|2 +

|η|2 = |y|2 + λT,τ ( )−2 |η|2 ≤ γε/τ ≤ ε, τ 2 + |ξ  |2

with  = (x, ξ  , τ ).

¨ 14.A. HORMANDER CALCULUS PROPERTIES

369

Set ˜ = (x+y, ξ +η, τ ), and observe that it suffices to have λT,τ ()  λT,τ (˜ ) to obtain gγ˜  gγ . This reads τ + |ξ  |  τ + |ξ  + η  |.

(14.A.2) By (14.A.1), one has

τ + |ξ  |  τ + |ξ  + η  | + |η  |  τ + |ξ  + η  | + |η|  τ + |ξ  + η  | + ε1/2 (τ + |ξ  |). Thus, for ε > 0 chosen sufficiently small, one obtains (14.A.2). Temperate Property. Let (z, ζ) ∈ R2d . With  2 gγ (z, ζ) = τ γ −1 |z|2 + τ γ −1 λ−2 T,τ ( )|ζ| ,

gγ˜ (z, ζ) = τ γ −1 |z|2 + τ γ −1 λ−2  )|ζ|2 , T,τ (˜ one sees that the property described in (14.2.3) holds if one has    γ σ  ˜ N λ−2 (˜ ˜, ξ − ξ) λ−2 T,τ ( )  1 + (g ) (x − x T,τ  ), for some N with (g γ )σ given in (14.2.1). With λT,τ ( )2 = τ 2 + |ξ  |2 and  )2 = τ 2 + |ξ˜ |2 , it is then sufficient to prove that λT,τ (˜ (14.A.3)

  ˜ N (τ 2 + |ξ  |2 ). ˜, ξ − ξ) |ξ˜ |2  1 + (g γ )σ (x − x

In fact, with τ ≥ 1 and γ ≥ 1 here, one has |ξ˜ |2  |ξ  − ξ˜ |2 + |ξ  |2  (1 + τ −1 γ|ξ  − ξ˜ |2 )(τ 2 + |ξ  |2 ) ˜ 2 )(τ 2 + |ξ  |2 ).  (1 + τ −1 γ|ξ − ξ| ˜ = λ2 ( )τ −1 γ|x − x ˜ 2 , one thus finds ˜, ξ − ξ) ˜|2 + τ −1 γ|ξ − ξ| With (g γ )σ (x − x T,τ   ˜ (τ 2 + |ξ  |2 ), ˜, ξ − ξ) |ξ˜ |2  1 + (g γ )σ (x − x that is, (14.A.3) with N = 1.



14.A.2. Order Functions. Here we prove Proposition 14.10. By Remarks 14.8 and 14.9, it suffices to prove g-continuity and σ, g γ -temperateness. Note that if m1 (y) and m2 (y) are both g-continuous, then their product has also this property. Similarly, if they are both σ, g γ -temperate, then their product has also this property. We thus concentrate on the properties of λτ and λT,τ .

370

14. OPTIMAL ESTIMATES AT THE BOUNDARY

g-Continuity. For the g-continuity of λτ and λT,τ , it suffices to consider the case r = 1. See the definition of g-continuity in (14.2.4). We write the proof for λτ . The proof for λT,τ can be adapted in a straightforward manner. Let ε > 0 whose value will be set below, and let (x, ξ), (y, η) ∈ R2d and τ ∈ [1, +∞) such that g (y, η) < ε, for  = (x, ξ, τ ). This reads (14.A.4) |y|2 +

|η|2 = |y|2 + λT,τ ( )−2 |η|2 ≤ ε, τ 2 + |ξ  |2

with  = (x, ξ  , τ ).

With λτ ()2 = τ 2 +|ξ|2 and λτ (˜ )2 = τ 2 +|ξ +η|2 , where ˜ = (x+y, ξ +η, τ ), we wish to prove τ + |ξ|  τ + |ξ + η|  τ + |ξ|,

(14.A.5)

under condition (14.A.4). On the one hand, we have τ +|ξ+η| ≤ τ +|ξ|+|η| ≤ τ +|ξ|+ε1/2 (τ +|ξ  |)  τ + |ξ|, that is, one part of (14.A.5). On the other hand, by (14.A.4), one has τ + |ξ|  τ + |ξ + η| + |η|  τ + |ξ + η| + ε1/2 (τ + |ξ  |). Thus, for ε > 0 chosen sufficiently small, one obtains the second part of (14.A.5). σ, g γ -Temperate Property. For the g γ -temperate property of λτ and λT,τ , it suffices to consider the cases r = 1 and r = −1. See the definition of σ, g γ -temperateness in (14.2.5). We write the proof for λτ . The proof for λT,τ can be adapted in a straightforward manner. We write ˜ 2 + |ξ| ˜ 2  (1 + τ −1 γ|ξ − ξ| ˜ 2 )(τ 2 + |ξ| ˜ 2 ). |ξ|2  |ξ − ξ| ˜ = λ2 ( )τ −1 γ|x − x ˜ 2 , one thus finds ˜, ξ − ξ) ˜|2 + τ −1 γ|ξ − ξ| With (g γ )σ (x − x T,τ   ˜ λτ (˜ λτ ()2  1 + (g γ )σ (x − x ˜, ξ − ξ) ) 2 , that is, the result for r = 1. By symmetry, one also has ˜ 2  (1 + τ −1 γ|ξ − ξ| ˜ 2 )(τ 2 + |ξ|2 ). |ξ| This yields   ˜ λτ ()2 , λτ (˜ )2  1 + (g γ )σ (x − x ˜, ξ − ξ) that is, the result for r = −1.



14.B. SYMBOL PROPERTIES

371

14.B. Symbol Properties ˜ k Im f , k˜ Im f . Here we prove Lemma 14.B.1. Properties of k, k, 14.24. Note that ∂xα ∂ξβ k is a linear combination of terms of the form   αp βp −1   (τ /γ)p/2 ∂xα1 ∂ξβ1 (λ−1 T,τ Im f ) · · · ∂x ∂ξ  (λT,τ Im f )  (p)  × σ0 (τ /γ)1/2 λ−1 T,τ Im f , with α1 + · · · + αp = α, β1 + · · · + βp = β, and p ≤ |α| + |β|. This gives −|β |

−|β |

−|β|

−|β|

|∂xα ∂ξβ k|  (τ /γ)p/2 λT,τ 1 · · · λT,τ p  (τ /γ)p/2 λT,τ  (τ /γ)(|α|+|β|)/2 λT,τ , yielding k ∈ S(1, g γ ). Similarly, k˜ ∈ S(1, g γ ). Let α ∈ Nd and β ∈ Nd−1 . Observe now that ∂xα ∂ξβ (k˜ Im f ) is a linear   ˜ α ∂ β (Im f ), where α + α = α combination of terms of the form ∂ α ∂ β (k)∂ x

ξ

x

ξ

and β  + β  = β. On the one hand, if |α | + |β  | ≤ −1 + |α| + |β|, that is, |α | + |β  | ≥ 1, we have 

  ˜ α ∂ β (Im f )|  (τ /γ)(|α |+|β  |)/2 λ−|β | λ1−|β |∂xα ∂ξβ (k)∂ x T,τ T,τ ξ

 |

1−|β|

 (τ /γ)(−1+|α|+|β|)/2 λT,τ . On the other hand, if |α | + |β  | = 0, we have ˜  (τ /γ)(|α|+|β|−1)/2 λ1−|β| , | Im f ∂xα ∂ξβ (k)| T,τ ˜ This allows one to using that | Im f | ≤ (τ /γ)−1/2 λT,τ on the support of k.   −1/2 γ ˜ λT,τ , g . conclude that k Im f ∈ S (τ /γ) Let α ∈ Nd be such that |α| = 1. We then write ∂xα (k Im f ) = k(∂xα Im f ) + (∂xα k) Im f α ˜ ∈ S(1, g γ )S(λT,τ , g) + (τ /γ)1/2 λ−1 T,τ (∂x Im f )k Im f

⊂ S(λT,τ , g γ ) + (∂xα Im f )S(1, g γ ) ⊂ S(λT,τ , g γ ),   as S(λT,τ , g) ⊂ S(λT,τ , g γ ) and using that k˜ Im f ∈ S (τ /γ)−1/2 λT,τ , g γ as proven above. Let β ∈ Nd−1 be such that |β| = 1. Similarly, we find ∂ξβ (k Im f ) = k(∂ξβ Im f ) + (∂ξβ k) Im f ˜ ∈ S(1, g γ )S(1, g) + (τ /γ)1/2 ∂ξβ (λ−1 T,τ Im f )k Im f ˜ ⊂ S(1, g γ ) + (τ /γ)1/2 S(λ−1 T,τ , g) k Im f ⊂ S(1, g γ ), −1 γ as S(1, g) ⊂ S(1, g γ ) and S(λ−1 T,τ , g) ⊂ S(λT,τ , g ).



372

14. OPTIMAL ESTIMATES AT THE BOUNDARY

14.B.2. Applicability of the Fefferman–Phong–Bony Inequality. Here, we prove Lemma 14.26. We have a ˜ ≥ 0 if γ ≥ γ0 . By Lemma 14.24, we have k Im f ∈ S(λT,τ , g γ ) 1/2 1/2 , g γ ) implying |q |  λ ˜ and (τ /γ) λ−1 0 T,τ + T,τ {ξd − Re f, Im f }k ∈ S((τ /γ) 1/2 (τ /γ) yielding   a ˜ = γ 3/2 τ −1/2 χ ˜ q0 − C3 (τ /γ)1/2  γ 3/2 τ −1/2 λT,τ + γ  γ 3/2 τ −3/2 λ2T,τ + τ  λ2T,τ , for τ ≥ γ ≥ 1. ˜ for |α| + |β| = 2 and |α| + |β| ≥ 4. We shall now estimate ∂xα ∂ξβ a Note that ∂xα ∂ξβ (χk ˜ Im f ) is a linear combination of terms of the form 







˜ ∂xα ∂ξβ (k Im f ), (∂xα ∂ξβ χ) with α + α = α and β  + β  = β. We recall that 



|∂xα ∂ξβ (k Im f )|  (τ /γ)(|α

 |+|β  |−1)/2

1−|β  |

λT,τ

, 



˜  for |α | + |β  | ≥ 1 as stated in (14.3.3). In such a case, since |∂xα ∂ξβ χ| −|β  |

λT,τ , we find 







1−|β|

˜ ∂xα ∂ξβ (k Im f )|  (τ /γ)(|α|+|β|−1)/2 λT,τ , |(∂xα ∂ξβ χ) as τ ≥ γ. If |α | + |β  | = 0, since k Im f  λT,τ , we find 







1−|β|

˜ ∂xα ∂ξβ (k Im f )| = |(∂xα ∂ξβ χ) ˜ k Im f |  λT,τ . |(∂xα ∂ξβ χ) In both cases, we thus have 1−|β|

˜ Im f )|  (τ /γ)(|α|+|β|−1)/2 λT,τ . |∂xα ∂ξβ (χk If |α| + |β| = 2, one thus finds ˜ Im f )|  γλT,τ γ 3/2 τ −1/2 |∂xα ∂ξβ (χk

1−|β|

2−|β|

 λT,τ

using γ ≤ τ ≤ λT,τ . If |α| + |β| ≥ 4 one has ˜ Im f )|  γ 2−(|α|+|β|)/2 τ (|α|+|β|)/2−1 λT,τ γ 3/2 τ −1/2 |∂xα ∂ξβ (χk

1−|β|

(|α|−|β|)/2

 λT,τ

,

using γ ≥ 1 and 0 < τ ≤ λT,τ . ˜ d − Re f, Im f } ∈ S(γ, g γ ). We thus have from We have a1 = γλ−1 ˜k{ξ T,τ χ the properties of such symbol classes recalled in Sect. 14.2 −|β|

−|β|

(|α|−|β|)/2

|∂xα ∂ξβ a1 |  γ(τ /γ)(|α|+|β|)/2 λT,τ  τ (|α|+|β|)/2 λT,τ  λT,τ

,

if |α| + |β| ≥ 2 using γ ≥ 1 and 0 < τ ≤ λT,τ . Note that if |α| + |β| = 2, we have (|α| − |β|)/2 = 1 − |β| ≤ 2 − |β|. All the estimations written above give the conclusion of the lemma. 

14.B. SYMBOL PROPERTIES

373

14.B.3. Proof of Theorem 14.17. The proof follows that of [175, Theorem 18.5.5] that is carried out within the framework of the Weyl quantification. In this book, we chose to preferably use the so-called classical quantification, as presented in Chapter 2 of Volume 1. Here, we explain how one can deduce Theorem 14.17 from results proven in [175]. In what follows, instead of writing [175, (18.∗.∗)] or [175, Theorem 18.∗.∗], we shall instead write (H18.∗.∗) or Theorem H18.∗.∗ for concision. In this section, it is sometimes important to put forward in which space the Euclidean inner product is carried out. As opposed to the rest of this book, we shall thus write x, ξRd instead of x·ξ. If there is no ambiguity, we shall simply write x, ξ. Moreover, with this notation, the presentation is consistent with that of Chapter 18 in [175] on which all the argumentation is founded. For a1 , a2 , and a3 tangential symbols, if one writes OpT (a3 ) = OpT (a2 ) OpT (a1 ), from (H18.5.21), one finds  )| =ˆ = ei Dξ ,Dxˆ a1 ( )a2 (ˆ  )| =ˆ , (14.B.1) a3 ( ) = ei Dξ ,Dxˆ a1 ( )a2 (ˆ x, ξˆ , τ ), and x ˆ = (ˆ x , x ˆd ), using that where  = (x, ξ  , τ ), x = (x , xd ), ˆ = (ˆ the symbols are independent of ξd . To estimate a3 and its derivatives, one wishes to apply Theorems H18.4.10’ and H18.4.11. This requires however the introduction of a metric G, a quadratic form A, and the computation of the dual metric GA following the notation of Section H18.4. Observe that even if we are only interested in tangential symbols, that is, not depending on ξd , to obtain estimates on the symbols, including the behavior with respect to xd , it is conveˆ with nient to consider metrics acting on the whole R2d . For w = (w, w) ˆ ξ x ξ 2d x ˆ 2d ˆ = (w , w ) ∈ R , we set w = (w , w ) ∈ R and w G,ˆ (w) = g (w) + gγˆ (w) ˆ = |wx |2 + τ γ −1 |wxˆ |2 + λ−2 |wξ |2 T,τ,ξ  ˆ

+ τ γ −1 λ−2 ˆ |wξ |2 , T,τ,ξ

ˆ τ ) and λ2  =  , ξˆd ) = (ˆ x, ξ, where  = ( , ξd ) = (x, ξ, τ ) and ˆ = (ˆ T,τ,ξ  2 2  2 2  2 2  2 ˆ = λT,τ (ˆ  ) = τ + |ξ | , and we also set λT,τ ( ) = τ + |ξ | and λ T,τ,ξˆ

A(w) = wξ , wxˆ Rd = Aw, wR4d , with the 4d × 4d symmetric matrix A given by ⎛ 0 0 0 1⎜ 0 0 IdRd A= ⎜ 0 2 ⎝0 IdRd 0 0 0

⎞ 0 0⎟ ⎟. 0⎠ 0

The matrix A appears naturally from (14.B.1) since one has Dξ , Dxˆ Rd = A(Dx , Dξ , Dxˆ , Dξˆ), (Dx , Dξ , Dxˆ , Dξˆ)R4d .

374

14. OPTIMAL ESTIMATES AT THE BOUNDARY

Note that G,ˆ with  = ( , ξd ) and ˆ = (ˆ  , ξˆd ) is in fact independent of the values of ξd and ξˆd . The dual metric of G with respect to A is given by GA ,ˆ  (w) =

sup 4d w∈R ˜ G , ˜ ˆ(Aw) 0 and C > 0 such that (14.B.2)

G,ˆ (w) ≤ ε ⇒ Gμ,ˆμ ≤ CG,ˆ ,

ˆ ˆ τ ), μ = (x + wx , ξ + wξ , τ ), μ ˆ = (ˆ x + wxˆ , ξˆ+ wξ , τ ), for  = (x, ξ, τ ), ˆ = (ˆ x, ξ, ˆ ˆ ˆ (wx , wξ ), (wxˆ , wξ ) ∈ R2d , and τ ≥ 1. x, ξ), w = (wx , wξ , wxˆ , wξ ), with (x, ξ), (ˆ Let ε > 0 and assume G,ˆ (w) ≤ ε, that is,

(14.B.3)

g (w) + gγˆ (w) ˆ ≤ ε.

Yet, by Lemma 14.7, the metrics g and g γ are both slowly varying. Thus, for ε chosen sufficiently small, one finds gμ  g , implying (14.B.2).

gμγˆ  gγˆ ,

14.B. SYMBOL PROPERTIES

375

A-Temperate Property of G. By A-temperate, one means that for some N ∈ N,  N Gμ,ˆμ , (14.B.4) G,ˆ ≤ C 1 + GA ,ˆ  (w) for all , ˆ, μ, μ ˆ, and w as above and τ ≥ 1. Observe that we may assume ˆ ξ x w = 0 and w = 0, since otherwise GA ,ˆ  (w) = +∞. Thus, we may simply consider ˆ τ ). ˆ = (ˆ y , ηˆ , ηˆd , τ ) = (ˆ x + wxˆ , ξ, μ = (y, η  , ηd , τ ) = (x, ξ + wξ , τ ) and μ Recall now that G,ˆ and GA μ is independent ,ˆ  are independent of ξd and Gμ,ˆ of ηd . It is thus sufficient to obtain (14.B.4) in the case wx = 0,

ˆ

wξ = (wξ , 0), and wξ = 0. ˆ

For v = (v, vˆ) = (v x , v ξ , v xˆ , v ξ ) ∈ R4d , we thus need to prove that, for some N ∈ N, ˆ

|v ξ |2 + τ γ −1 λ−2 ˆ |v ξ |2 |v x |2 + τ γ −1 |v xˆ |2 + λ−2 T,τ,ξ  T,τ,ξ   −1 ξ 2 2 x ˆ 2 N  1 + τ γ|w | + λT,τ,ξ |w |  ˆ  × |v x |2 + τ γ −1 |v xˆ |2 + λ−2 |v ξ |2 + τ γ −1 λ−2 ˆ |v ξ |2 . T,τ,ξ  +w ξ T,τ,ξ

For the first, second, and fourth terms on the l.h.s., this estimation clearly holds with N = 0. It thus suffices to find some N ∈ N such that   ξ 2 −1 ξ 2 N −2 |v |  1 + τ γ|w | λT,τ,ξ +wξ |v ξ |2 , λ−2  T,τ,ξ for all v ξ ∈ Rd , that is, (14.B.5)

 N τ 2 + |ξ  + wξ |2  (τ 2 + |ξ  |2 ) 1 + τ −1 γ|wξ |2 .

As here τ, γ ∈ [1, +∞), one simply writes τ 2 + |ξ  + wξ |2  τ 2 + |ξ  |2 + |wξ |2  τ 2 + |ξ  |2 + τ γ|wξ |2 , which is less than or equal to the r.h.s. of (14.B.5), in the case N = 1. This concludes that proof of the A-temperate property of G. G-Continuity of λT,τ,ξ and λT,τ,ξˆ . For a function m(, ˆ), the Gcontinuity property means the existence of ε > 0 and C > 0 such that (14.B.6)

ˆ) ≤ Cm(, ˆ), G,ˆ (w) ≤ ε ⇒ m(, ˆ)/C ≤ m(μ, μ

ˆ ˆ τ ), μ = (x + wx , ξ + wξ , τ ), μ for  = (x, ξ, τ ), ˆ = (ˆ x, ξ, ˆ = (ˆ x + wxˆ , ξˆ+ wξ , τ ), ˆ ˆ (wx , wξ ), (wxˆ , wξˆ) ∈ R2d , and τ ≥ 1. w = (wx , wξ , wxˆ , wξ ), with (x, ξ), (ˆ x, ξ), ˆ ≤ ε. Since As seen above, having G,ˆ (w) ≤ ε reads g (w) + gγˆ (w) λT,τ,ξ is g-continuous and λT,τ,ξˆ is g γ -continuous by Proposition 14.10, we find that

λT,τ,ξ  λT,τ,μ and λT,τ,ξˆ  λT,τ,ˆμ .

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14. OPTIMAL ESTIMATES AT THE BOUNDARY

A, G-Temperate Property of λT,τ,ξ and λT,τ,ξˆ . For a function m(, ˆ), the A, G-temperate property means that, for some N ∈ N and C > 0,  N m(, ˆ) ≤ C 1 + GA m(μ, μ ˆ), ,ˆ  (w) for all , ˆ, μ, μ ˆ, and w as above and τ ≥ 1. As for the A-temperate property of G, it suffices to consider the case wx = 0,

ˆ

wξ = (wξ , 0), and wξ = 0.

As here λT,τ,ξˆ = λT,τ,ξˆ +wξˆ , the property is obvious for λT,τ,ξˆ . For λT,τ,ξ , we need to prove  N (14.B.7) λT,τ,ξ  1 + τ −1 γ|wξ |2 + λ2T,τ,ξ |wxˆ |2 λT,τ,ξ +wξ . Since γ, τ ∈ [1, +∞), we write λ2T,τ,ξ = τ 2 + |ξ  |2  τ 2 + |ξ  + wξ |2 + |wξ |2     1 + γτ −1 |wξ |2 τ 2 + |ξ  + wξ |2    1 + τ −1 γ|wξ |2 + λ2T,τ,ξ |wxˆ |2 λ2T,τ,ξ +wξ , 

and (14.B.7) follows for N = 1.

14.B.4. Proof of Fefferman–Phong–Bony Inequality. Here, we prove Theorem 14.22 from the version given in [77]. The result given by J.-M. Bony is stated in the context of the Weyl calculus. We denote by pw (x, D , τ ) or simply by pw the tangential operator associated with the symbol p(x, ξ  , τ ) in the Weyl quantification, that is, (14.B.8)

pw (x, D , τ )u 





= (2π)1−d ∫∫ ei(x −y )·ξ p((x + y  )/2, xd , ξ  , τ )u(y  , xd ) dξ  dy  , R2d−2

in the sense of oscillatory integrals (see Section 2.4 of Volume 1). Here, the tangential symbol p(x, ξ  , τ ) used may, for instance, lie in the symbol classes introduced in Section 2.10 of Volume 1 and Sect. 14.2. Theorem 14.39 (J.-M. Bony’s Theorem 3.2 in [77]). Let p(x, ξ  , τ ) ∈ × Rd−1 ) be nonnegative and such that for any α , β  ∈ Nd−1 , with  |α | + |β  | ≥ 4, there exists Cα ,β  > 0 such that

C ∞ (Rd

(14.B.9)





|∂xα ∂ξβ p(x, ξ  , τ )| ≤ Cα ,β  ,

(x, ξ  ) ∈ R2d−1 , τ ≥ 1.

Then, for some C > 0, one has (pw u, u)+ + Cu2+ ≥ 0,

u ∈ S (Rd+ ).

This result is in fact adapted to the tangential case; here, xd is considered a parameter in the symbol property of p(x, ξ  , τ ). We begin by proving that if the result holds true in the Weyl quantification, it holds also true in the classical quantification.

14.B. SYMBOL PROPERTIES

bw .

377

Following [175, Theorem 18.5.10], one can find b(x, ξ  , τ ) such that OpT (a) = More precisely, one has b(x, ξ  , τ ) = e−i Dx ,Dξ /2 a(x, ξ  , τ ),

or equivalently a(x, ξ  , τ ) = ei Dx ,Dξ /2 b(x, ξ  , τ ). One important feature of the Weyl quantification lies in having (pw )∗ = p¯w as can be readily seen formally from the form of the kernel of pw (x, D , τ ). Then, one has OpT (a) + OpT (a)∗ = 2(Re b)w . From the Taylor formula eix = 1 + ix − ∫01 (1 − t)eitx x2 dt, for a real symbol a(x, ξ  , τ ), one finds 2 Re b(x, ξ  , τ ) = e−i Dx ,Dξ /2 a(x, ξ  , τ ) + ei Dx ,Dξ /2 a(x, ξ  , τ ) yielding  1 (1 − t) t i 2 Dx ,Dξ −i 2t Dx ,Dξ   Re b(x, ξ , τ ) = a(x, ξ , τ ) − ∫ +e e 8 0 × Dx , Dξ 2 a(x, ξ  , τ )dt. We now consider the symbol a(x, ξ  , τ ) as given in the statement of  2 Theorem 14.22. One has Dx , Dξ 2 a(x, ξ  , τ ) ∈ S(1, λT,τ |dx|2 + λ−1 T,τ |dξ | ). Lemma 14.40. The operators B = e±it Dx ,Dξ /2 map S(1, λT,τ |dx|2 +  2 λ−1 T,τ |dξ | ) onto itself continuously and uniformly with respect to t ∈ [0, 1]. A proof is given in Sect. 14.B.4.1 below. With Lemma 14.40, one has  2 e±it Dx ,Dξ /2 Dx , Dξ 2 a(x, ξ  , τ ) ∈ S(1, λT,τ |dx|2 + λ−1 T,τ |dξ | ),

uniformly with respect to t ∈ [0, 1].  2 We deduce that Re b = a + c, where c ∈ S(1, λT,τ |dx|2 + λ−1 T,τ |dξ | ), yielding (Op(a) + Op(a)∗ )/2 = aw + cw , where cw is bounded on L2 (Rd+ ) from [175, Theorem 18.6.3]. Consequently, in the proof of Theorem 14.22, we may thus replace the operator OpT (a) by its Weyl quantification aw . Observe that the result of Theorem 14.39 does not apply directly to the symbol a(x, ξ  , τ ) given in the statement of Theorem 14.22. It however applies to localized versions of this symbol after the action of a symplectomorphism. Localization is performed in the (ξ  , τ ) variables. Let us use a Littlewood–Paley decomposition as introduced in the proof of Proposition 7.3. With ϕk , k ∈ N as defined in (7.4.2), one has ∞ k=0 ϕk = 1−k 1. We recall that ϕ0 is supported on [−1, 1], and ϕk (t) = ϕ(2 t), where ϕ is supported on 1 ≤ |t| ≤ 3. We set ak (x, ξ  , τ ) = a(x, ξ  , τ )ϕk (λT,τ ) and qk = ak ◦ Σk , where Σk is the following symplectomorphism in the (x , ξ  ) variables: Σk : (x, ξ  , τ ) → (2−k/2 x , xd , 2k/2 ξ  , τ ). We have the following result.

378

14. OPTIMAL ESTIMATES AT THE BOUNDARY

Proposition 14.41. The symbol qk fulfills the property (14.B.9) with constants Cα,β > 0 that are uniform with respect to k ∈ N. A proof is given in Sect. 14.B.4.2 below. We are now in a position to apply the result of J.-M. Bony given in Theorem 14.22 to the symbol qk (x, ξ  , τ ) in the x , ξ  variables with an additional integration with respect to xd on (0, +∞), yielding (qkw u, u)+ + Cu2+ ≥ 0,

u ∈ S (Rd+ ),

uniformly with respect to k ∈ N. For p a symbol in one of the classes considered here, u ∈ S (Rd+ ), and λ = 0 from (14.B.8), one finds (pw u)(λx , xd ) = (pw λ uλ )(x) with pλ (x, ξ  , τ ) = p(λx , xd , ξ  /λ, τ ) and uλ (x) = u(λx , xd ), yielding (pw (pw u, u)+ λ uλ , uλ )+ = . 2 u+ uλ 2+ One thus finds w u, u)+ + Cu2+ ≥ 0, (qk,λ

u ∈ S (Rd+ ),

for any λ > 0 with qk,λ (x, ξ  , τ ) = qk (λx , xd , ξ  /λ, τ ). For λ = 2k/2 , one has qk,λ (x, ξ  , τ ) = ak (x, ξ  , τ ). Thus, (14.B.10)

2 (aw k u, u)+ + Cu+ ≥ 0,

u ∈ S (Rd+ ),

for τ ≥ 1. As above, this inequality is uniform with respect to k ∈ N. Let ψ ∈ Cc∞ (R) be such that supp(ψ) ⊂ (1/2, 4) ψ ≡ 1 in a neighborhood of supp(ϕ). We set ψk (t) = ψ(21−k t) such that ψk ϕk = ϕk on R+ for k ≥ 1. For k = 0, let ψ0 ∈ Cc∞ (R) be such that supp(ψ) ⊂ (−2, 2) and ψ0 ≡ 1 in a neighborhood of supp(ϕ0 ). By abuse of notation, we write ψkw instead of (ψk (λT,τ ))w . Observe that ψk (λT,τ )ak (x, ξ  , τ ) = ak (x, ξ  , τ ). Applying inequality (14.B.10) to ψkw u, we find w w w 2 (aw k ψk u, ψk u)+ + Cψk u+ ≥ 0.

As ψ is real valued, we have (ψkw )∗ = ψkw yielding w w 2 (ψkw aw k ψk u, u)+ + Cψk u+ ≥ 0.

Lemma 14.42. One has ϕk (λT,τ ), ψk (λT,τ ) ∈ S(1, |dx|2 + 2−2k |dξ  |2 ) and ak ∈ S(22k , 2k |dx|2 + 2−k |dξ  |2 ). A proof is given in Sect. 14.B.4.3 below. w w w Applying [175, Theorem 18.5.5], we obtain that aw k ψk = (ak ) + ck , −kN k 2 −k  2 , 2 |dx| + 2 |dξ | ), uniformly with Respect to k ≥ 0 where ck ∈ S(2 and for all N > 0 since derivatives of ψk vanish on support of ϕk . Applying w w w the same argument a second time, one obtains ψkw aw k ψk = ak + dk , where −kN k 2 −k  2 , 2 |dx| + 2 |dξ | ), uniformly with respect to k ≥ 0 and for dk ∈ S(2

14.B. SYMBOL PROPERTIES

379

2 all N > 0. As dw k is L continuous from [175, Theorem 18.6.3] with a norm estimated by C2−N k , we deduce that w 2  −N k u2+ ≥ 0. (aw k u, u)+ + Cψk u+ + C 2

(14.B.11)

A summation with respect to k ∈ N can be carried out by the following lemma.  Lemma 14.43. There exists C > 0 such that k∈N ψkw u2+ ≤ Cu2+ ,  for u ∈ S (Rd+ ). One also has the convergence of k (aw k u, u)+ with  w ∞ (ak u, u)+ = (aw u, u)+ , u ∈ C c (Rd+ ). k∈N

A proof is given in Sect. 14.B.4.4 below. Here we recall the definition of the space of function ∞

C c (Rd+ ) = {u = v|Rd ; v ∈ Cc∞ (Rd )}. +

For some

C 

> 0, one thus obtains

(aw u, u)+ + C  u2+ ≥ 0,

∞

u ∈ C c (Rd+ ).

∞

From the density of C c (Rd+ ) in S (Rd+ ) and the continuity of aw from S (Rd+ ) into itself, we conclude the proof of Theorem 14.22.  14.B.4.1. Proof of Lemma 14.40. It suffices to apply [175, Theorem  2 18.5.10] to prove that B is continuous from S(1, λT,τ |dx|2 + λ−1 T,τ |dξ | ) onto itself. To prove the uniformity, we have to apply [175, Theorem 18.4.10]. The continuity constants depend on the constants in the estimates (18.4.2)’, (18.4.5), 18.4.13), and (18.4.14) of [175]. The estimates (18.4.2)’ and (18.4.5) involve the constants related with the metric and the order function that is the constants ε and C in the following estimates: g (y, η) < ε ⇒ C −1 g ≤ g˜ ≤ Cg ,

for ˜ = (x + y, ξ + η, τ )

and ) ≤ Cm(), g (y, η) < ε ⇒ C −1 m() ≤ m(˜

for ˜ = (x + y, ξ + η, τ ).

In our case, the metric and the order function do not depend on t. The estimates (18.4.13) and (18.4.14) involve the constants related with dual metric that is, using the previous notation,   ˜ N g˜, (14.B.12) ˜, ξ − ξ) g ≤ C 1 + gA (x − x and (14.B.13)

  ˜ N, ˜, ξ − ξ) m() ≤ Cm(˜ ) 1 + gA (x − x

where A is the quadratic form A(x , ξ  ) = tx , ξ  /2 associated with the 0 Id matrix also denoted A = (t/4) . We have to prove that C can Id 0

380

14. OPTIMAL ESTIMATES AT THE BOUNDARY

be chosen independently of t. In the lemma, the metric is g˜ = λT,τ |dx|2 +  2 λ−1 T,τ |dξ | and dual metric g˜A (w) =

sup w ∈R2(d−1) g ˜ (Aw ) 0 such that 2−|β  |



|∂xα ∂ξβ a(x, ξ  , τ )| ≤ Cα,β  λT,τ

,

(x, ξ  ) ∈ R2d−1 , τ ≥ 1.

A proof is given below. Choosing α = (α , 0) in Lemma 14.45, one obtains an interpolation of property (14.B.15). We write   ˆ  ∂xα ∂ξδ a(2−k/2 x , xd , 2k/2 ξ  , τ ) ˆ





ˆ

= 2(|δ |−|α |)k/2 (∂xα ∂ξδ a)(2−k/2 x , xd , 2k/2 ξ  , τ ). If |α | + |δˆ | ≤ 4, we have   ˆ   ˆ 2−|δˆ | |∂xα ∂ξδ a(2−k/2 x , xd , 2k/2 ξ  , τ ) |  2(|δ |−|α |)k/2 νk

14.B. SYMBOL PROPERTIES

381

  with νk2 = τ 2 + |2k/2 ξ  |2 . On supp ϕk (|(2k/2 ξ  , τ )|) , one has νk  2k . Thus, one finds ˆ



ˆ







|Iα ,δ ,δˆ |  2(|δ |−|α |)k/2 22k−|δ |k 2−k|δ |/2 = 2k(4−|α |−|β |)/2 . If |α | + |δˆ | > 4, we have   ˆ   ˆ (|α |−|δˆ |)/2  1, |∂xα ∂ξδ a(2−k/2 x , xd , 2k/2 ξ  , τ ) |  2(|δ |−|α |)k/2 νk as νk  2k . Hence, if |α | + |β  | ≥ 4, one obtains |Iα ,δ ,δˆ |  1.



Proof of Lemma 14.44. The function λT,τ is homogeneous of degree one with respect to (ξ  , τ ), with ξ  ∈ Rd−1 and τ > 0. Thus, for α ∈ Nd−1 ,  the function ∂ξα λT,τ is homogeneous of degree 1 − |α |. In particular, there exists Cα > 0 such that 1−|α |



|∂ξα λT,τ | ≤ Cα λT,τ

(14.B.16)

ξ  ∈ Rd−1 , τ > 0.

,

The result of the lemma is clear for k = 0. For k ≥ 1, we now write ϕk (|(2k/2 ξ  , τ )|) = ϕ(21−k |(2k/2 ξ  , τ )|) = ϕ(21−k/2 λT,τk ), where λ2T,τk = |ξ  |2 + τk2 with τk = 2−k/2 τ .  For δ  ∈ Nd−1 , computing ∂ξδ ϕk (|(2k/2 ξ  , τ )|) yields a linear combination of terms of the form Jm1 ,...,mj = 2−jk/2 (∂ξm 1 λT,τk ) · · · (∂ξ j λT,τk )ϕ(j) (21−k/2 λT,τk ), m

for m1 + · · · + mj = δ  . With (14.B.16), one finds 1−|m1 |

|Jm1 ,...,mj |  2−jk/2 λT,τk

1−|mj |

· · · λT,τk

|ϕ(j) (21−k/2 λT,τk )|

j−|δ  |

 2−jk/2 λT,τk |ϕ(j) (21−k/2 λT,τk )|. In ϕ(j) (21−k/2 λT,τk ), one has λT,τk  2k/2 yielding 



|Jm1 ,...,mj |  2−jk/2 2k(j−|δ |)/2  2−k|δ |/2 , 

which yields estimation (14.B.14).

Proof of Lemma 14.45. From the assumption on a(x, ξ  , τ ) in the statement of Theorem 14.22, one has 2−|β  |



|∂xα ∂ξβ a(x, ξ  , τ )|  λT,τ

if |α| + |β  | ∈ {0, 2, 4}.

With this property, one only needs to consider the cases |α| + |β  | = 1 and |α| + |β  | = 3. First, we consider the case |α| + |β  | = 1. Case |α| = 1 and |β  | = 0. Here, one estimates ∇x a(x, ξ  , τ ). To that purpose, one writes a(x + y, ξ  , τ ) = a(x, ξ  , τ ) + ∇x a(x, ξ  , τ ) · y 1

+ ∫ (1 − t)d2x a(x + ty, ξ  , τ )(y, y) dt. 0

382

14. OPTIMAL ESTIMATES AT THE BOUNDARY

One may assume ∇x a(x, ξ  , τ ) = 0, since otherwise the estimation is obvious. If one chooses y = ∇x a(x, ξ  , τ )/|∇x a(x, ξ  , τ )|, one finds |∇x a(x, ξ  , τ )|  λ2T,τ . Case |α| = 0 and |β  | = 1. Similarly, one writes a(x, ξ  + η  , τ ) = a(x, ξ  , τ ) + ∇ξ a(x, ξ  , τ ) · η  1

+ ∫ (1 − t)d2ξ a(x, ξ  + tη  , τ )(η  , η  ) dt. 0

η

We choose = δλT,τ,ξ ∇ξ a(x, ξ  , τ )/|∇ξ a(x, ξ  , τ )| with δ > 0. If δ is chosen sufficiently small, one has λT,τ,ξ  λT,τ,ξ +tη uniformly for all t ∈ [0, 1]. We then obtain λT,τ,ξ |∇ξ a(x, ξ  , τ )|  λ2T,τ,ξ , yielding the estimate in this case. Second, we consider the case |α| + |β  | = 3. If |α| ≥ 1, then arguing as in the first case treated above the result follows. The only remaining case is then |β  | = 3; one argues in the second case treated above. The details are left to the reader.  14.B.4.3. Proof of Lemma 14.42. For k ≥ 1, we set fk = ϕk (λT,τ ) =  d−1 , T,τ ). Recall that ϕ is supported in the interval [1, 3]. For β ∈ N β observe that ∂ξ fk is a linear combination of terms of the form

ϕ(21−k λ

m1 ,...,mj

fk

= 2−jk (∂ξm 1 λT,τ ) · · · (∂ξ j λT,τ )ϕ(j) (21−k λT,τ ), m

with m1 + · · · + mj = β  , and one has m1 ,...,mj

|fk

1−|m1 |

|  2−jk λT,τ

1−|mj |

· · · λT,τ

|ϕ(j) (21−k λT,τ )|

j−|β  |

using that λT,τ (14.B.17)



 2−jk λT,τ |ϕ(j) (21−k λT,τ )|  2−k|β | ,    2k in supp ϕ(j) (21−k λT,τ ) . One thus finds  β   ∂  ϕk (λT,τ )   2−k|β  | , ξ

implying the result for ϕk (λT,τ ). The same proof gives the counterpart result for ψk (λT,τ ). 

Since ak = ϕk (λT,τ )a, for α ∈ Nd , β  ∈ Nd−1 , observe that ∂xα ∂ξβ ak is a linear combination of terms of the form     ˆ  bα,δ ,δˆ = ∂xα ∂ξδ a ∂ξδ ϕk (λT,τ ) , with δ  + δˆ = β  . We consider two cases. Case |α| + |δ  | ≤ 4. In such a case, by Lemma 14.45, one has 

2−|δ  |

|∂xα ∂ξδ a(x, ξ  , τ )|  λT,τ

.

14.B. SYMBOL PROPERTIES

383

In the support of ϕk (λT,τ ), where λT,τ  2k , one thus finds 



|∂xα ∂ξδ a(x, ξ  , τ )|  22k−k|δ | . With (14.B.17), one obtains (14.B.18)



ˆ





|bα,δ ,δˆ |  22k−k|δ | 2−k|δ |  22k−k|β |  22k+k(|α|−|β |)/2 .

Case |α| + |δ  | > 4. In such a case, by the assumption of Theorem 14.22, one has (|α|−|δ  |)/2



|∂xα ∂ξδ a(x, ξ  , τ )|  λT,τ

(14.B.19)

.

Hence one finds 



|∂xα ∂ξδ a(x, ξ  , τ )|  2k(|α|−|δ |)/2 , in the support of ϕk (λT,τ ). With (14.B.17), one obtains (14.B.20)



ˆ





|bα,δ ,δˆ |  2k(|α|−|δ |)/2 2−k|δ |  2k(|α|−|β |)/2  22k+k(|α|−|β |)/2 .

Together, (14.B.18) and (14.B.20) yield 



|∂xα ∂ξβ ak |  22k+k(|α|−|β |)/2 , implying the second result of the lemma.



14.B.4.4. Proof of Lemma 14.43. One has   ψkw u2+ = (ψkw ψkw u, u)+ = (ψk2 )w u, u + = (2π)−(d−1) ∫

∫ ψk2 (λT,τ )|ˆ u(ξ  , xd )|2 dξ  dxd ,

R+ Rd−1

where u ˆ denotes the partial Fourier transform of u with respect to x , using that ψkw acts as a Fourier multiplier. With the monotone convergence theorem, one finds  2  ψkw u2+  ∫ ∫ ψk (λT,τ )|ˆ u(ξ  , xd )|2 dξ  dxd  u2+ . k∈N



R+ Rd−1 k∈N

One has a = k∈N ak pointwise. Denote by Kk (x, y) the kernel of the  operator aw k (x, D ). In the sense of oscillatory integrals, one has      Kk (x , y  ) = (2π)−(d−1) ∫ e(x −y )·ξ ak (x + y  )/2, xd , ξ  dξ  , Rd−1

with xd ≥ 0 acting as a parameter, implying that   ˇk (x + y  )/2, xd , x − y  , Kk (x , y  ) = a where a ˇk denotes the partial inverse Fourier transform of ak with respect to  ξ.    Up to change of variables, one sees that k Kk (x , y ) converges in   d−1 d−1   × R ) if and only if k a ˇk (x , xd , y ) converges in S  (Rd−1 × S (R Rd−1 ), with xdacting as a parameter. In turn, the latter property holds if and only if k ak (x , xd , ξ  ) converges in S  (Rd−1 × Rd−1 ). Since ak = ϕk (λT,τ )a, this last convergence holds because of the temperate growth of

384

14. OPTIMAL ESTIMATES AT THE BOUNDARY

 a(x, ξ) at infinity and the uniform convergence of k ak (x, ξ) to a(x, ξ) on    any compact set of R2(d−1) . Thus, k Kk (x , y ) converges to the kernel   w   d−1 d−1 × R ). Note that this convergence is K(x , y ) of a (x, D ) in S (R uniform for xd in a compact set. ∞ For u ∈ C c (Rd+ ), one obtains  ∫ Kk , u ⊗ uS  (R2(d−1) ),S (R2(d−1) ) dxd k≤n R+

→

n→+∞

∫ K, u ⊗ uS  (R2(d−1) ),S (R2(d−1) ) dxd ,

R+

yielding the second result of Lemma 14.43.



Part 5

Background Material: Geometry

CHAPTER 15

Elements of Differential Geometry Contents 15.1. 15.2. 15.2.1. 15.2.2. 15.2.3. 15.3. 15.3.1. 15.3.2. 15.3.3. 15.4. 15.4.1. 15.4.2. 15.4.3. 15.4.4. 15.5. 15.6. 15.6.1. 15.6.2. 15.6.3. 15.7. 15.7.1. 15.7.2.

Manifolds Open Coverings and Partitions of Unity Some Topological Results Partitions of Unity on Rd Partitions of Unity on a σ-Compact Manifold Tangent Space and Vector Fields Tangent Vectors Differential of a Map and Action on Tangent Vectors The Tangent Bundle, Vector Fields Cotangent Vectors and Forms Cotangent Vectors, Local Representatives Action of a Smooth Map and Differential of a Function The Cotangent Bundle One-Forms Submanifold Tensors and p-Forms Covariant Tensors p-Forms General Tensors Symplectic Structure of the Cotangent Bundle The Symplectic Two-Form Hamiltonian Vector Field and Poisson Bracket

388 390 390 392 394 396 396 399 400 402 402 403 403 404 405 406 406 407 410 411 411 412

This chapter is devoted to the exposition of facts of differential in particular to prepare for a proper presentation of integration and differential operators in the next chapter. In Chapter 9 of Volume 1, we reviewed the action of change of variables on some of the notions that are important

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 15

387

388

15. ELEMENTS OF DIFFERENTIAL GEOMETRY

for the understanding of Carleman estimates. On manifolds, the notion of change of (local) variables, through local charts, is essential for the understanding of geometrical objects such as tangent and cotangent vectors, vector fields, covariant tensors, and p-forms. We only review the essential features. An interested reader should consult a more thorough reference, such as [1, 222, 309], for analysis on manifolds 15.1. Manifolds Manifolds in the sense used here are topological spaces that have smooth properties. Local representations allow one to characterize this smoothness. We start by providing the precise definitions of manifold without and with boundary that we shall use. Definition 15.1 (Manifold Without Boundary). Let d, k ∈ N. A ddimensional C k -manifold without boundary is a topological space M equipped with a set (Oi , κi )i∈I , with, for each i ∈ I, Oi is an open subset of M and κi is a map from Oi into Rd such that (1) (2) (3) (4)

The open sets Oi , i ∈ I, cover M, that is, ∪i∈I Oi = M. κi (Oi ) is an open subset of Rd . For each i ∈ I, the map κi : Oi → κi (Oi ) is a homeomorphism. If Oij = Oi ∩ Oj = ∅, then for W  = κ (Oij ) ⊂ κ (O ),  = i or j, the map κij : W i → W j given by κj ◦ (κi )−1 is a C k -diffeomorphism.

A couple (Oi , κi ) is called a local chart. A set of local charts as above is called an atlas. If the definition holds for any k ∈ N, one says that M is a smooth or C ∞ -manifold. When in a local chart (O, κ), we shall denote by κi the map from O to R such that κi (m) is the ith coordinate of κ(m) in Rd , that is, κ(m) =  κ1 (m), . . . , κd (m)). Definition 15.2 (Manifold with Boundary). A topological space M is said to be a d-dimensional C k -manifold with boundary if it is equipped with a set (Oi , κi )i∈I , where Oi is an open subset of M and κi is a map from Oi into Rd for each i ∈ I, and such that (1) The open sets Oi , i ∈ I, cover M, that is ∪i∈I Oi = M. (2) κi (Oi ) is either an open subset of Rd or an open subset of Rd+ = {x ∈ Rd ; xd ≥ 0}. (3) For each i ∈ I, the map κi : Oi → κi (Oi ) is a homeomorphism. (4) If Oij = Oi ∩ Oj = ∅, then for W  = κ (Oij ) ⊂ κ (O ),  = i or j, the map κij : W i → W j given by κj ◦ (κi )−1 is a C k -diffeomorphism. In particular, in the case W i is an open subset of Rd+ that contains part of {xd = 0}, then W j satisfies this property as well, and the map κij : W i → W j is the restriction of a C k -map κij : W i → W j , for W i and W j open subsets of Rd such that W i ⊂ W i and W j ⊂ W j .

15.1. MANIFOLDS

389

A couple (Oi , κi ) as above is called a local chart. A point m ∈ M is said to be a boundary point of M if m ∈ O for a local chart (O, κ) with κ(O) an open set of Rd+ that meets {xd = 0} and m = κ−1 (x) with xd = 0. We denote by ∂M the boundary of M, that is, the set of all such boundary points. Observe that ∂M ⊂ M in the definition we have given. The boundary ∂M can be equipped with the structure of a manifold. It is then a manifold without boundary; we refer to Sect. 15.5. If Ω is a smooth open set of Rd , then Ω∪∂Ω is a manifold with boundary in the sense given above. We recall that an open set Ω with ∂Ω = ∅ is said to be smooth (or regular) if ∂Ω is smooth and if Ω is only located on one side of ∂Ω. Note however that Ω is not to be considered as a manifold with boundary, since ∂Ω ⊂ Ω. A manifold may not be connected. Each of its connected components can however be considered as an individual manifold. If a manifold is compact, the number of its connected components is finite. ˜ two manifolds possibly with boundaries, of respective For M and M ˜ k k ˜ It is said to be of class C and C , consider a continuous map φ : M → M.  ˜ if for every m ∈ M and every coordinate patch class C , for  ≤ min(k, k), ˜ ˜ such that m ∈ O and φ(m) ∈ O, ˜ the map (O, κ) on M and (O, κ ˜ ) on M, −1 g : κ ˜ ◦ φ ◦ κ , which is well defined in a neighborhood of κ(m) in κ(O), is of class C  . In the case M is a d-dimensional manifold with a boundary and m is a boundary point, then κ(O) is an open subset of Rd+ and C  regularity for g means such regularity up to the boundary. In particular, g is the restriction to Rd+ of a C  -map defined in a neighborhood of κ(m) in ˜ ˜ Rd into Rd , where d˜ is the dimension of M. ˜ For a function f defined on M, we define its pullback by φ, denoted by φ∗ f to be the function on M given by φ∗ f = f ◦ φ. If f is a function defined on M with values in some set E, then for every local chart C = (O, κ), f C = (κ−1 )∗ f : κ(O) → E is the local representative of f in this chart. If we now consider two local charts C  = (O , κ ),  = 1 2 1, 2, such that O = O1 ∩ O2 = ∅, then we have f C (x) = (κ12 )∗ f C (x) for x ∈ κ1 (O) and κ12 = κ2 ◦ (κ1 )−1 that is a diffeomorphism from κ1 (O) onto κ2 (O). Conversely, we have the following proposition. Proposition 15.3. Let A = {(Oi , κi ); i ∈ I} be an atlas of a manifold M. Let (f i )i∈I be such that (1) For each i ∈ I, f i : κi (Oi ) → E, for some set E. (2) For each (i, j) ∈ I 2 such that Oij = Oi ∩ Oj = ∅, we have f i (x) = (κij )∗ f j (x) for x ∈ κi (Oij ) and κij = κj ◦ (κi )−1 . Then, there exists f : M → E such that f i = f C for all i ∈ I. i

390

15. ELEMENTS OF DIFFERENTIAL GEOMETRY

15.2. Open Coverings and Partitions of Unity Let M be a C k -manifold (0 ≤ k ≤ ∞) with or without boundary. Let F be a closed subset of M, and let (V j )j∈J be an open covering of F in M, that is, for all j ∈ J , V j is an open set of M and F ⊂ ∪j∈J V j . Definition 15.4. A family of C k -functions (χj )j∈J is called a C k partition of unity of F subordinated to the open covering (V j )j∈J if we have: (1) For all j ∈ J , we have supp(χj ) ⊂ V j . (2) For all j ∈J , we have 0 ≤ χj ≤ 1.  (3) The sum j∈J χj is locally finite and j∈J χj (m) = 1 for m in a neighborhood of F . By locally finite, one means that for all m ∈ M, there exists a neighborhood Vm of m in M such that #{j ∈ J ; χj ≡ 0 in Vm } < ∞. An open covering V = (V j )j∈J is also said to be locally finite if for all m ∈ M, there exists a neighborhood Vm of m in M such that #{j ∈ J ; V j ∩Vm = ∅} < ∞. Let V = (V j )j∈J and U = (U k )k∈K be two open coverings of a closed set F in M. One says that U is finer than V if for all k ∈ K there exists j ∈ J such that U k ⊂ V j . We then observe that a partition of unity of F subordinated to U yields a partition of unity of F subordinated to V upon summing together some of the functions. Below, we shall use this argument implicitly, as in the process of constructing a partition of unity subordinated to a given open covering, we shall make use of finer open coverings. 15.2.1. Some Topological Results. A manifold M with or without boundary is σ-compact if it admits an exhaustion by compact sets: there exists a sequence of compact sets K 0 ⊂ K 1 ⊂ · · · ⊂ K n ⊂ · · · such that K n = M. K n ⊂ int K n+1 , n ∈ N, and n∈N

One also says that M is countable at infinity. In such case, for any compact set K of M, there exists n ∈ N such that K ⊂ int K n . Examples are Rd and (obviously) compact manifolds. We shall make the assumption that the considered manifold is σ-compact to build a partition of unity. The importance of this property lies in the following lemma. Lemma 15.5. Let M be a σ-compact manifold. Then, for any closed set F and for any open covering V = (V j )j∈J of F in M, there exists a second open covering U = (U k )k∈K of F in M, with K countable, that is finer than V and locally finite. In particular, this holds for F = M, precisely meaning that M is paracompact. Proof. Let (K n )n∈N be an exhaustive sequence of compact sets for M. For each n ∈ N, there exists a finite sub-family of V, (Vnk )0≤k≤kn , such that n F ∩ K n ⊂ ∪kj=0 Vnk . Note that the family (Vnk )n,k , is countable and finer than

15.2. OPEN COVERINGS AND PARTITIONS OF UNITY

391

V. We now construct a locally finite refinement of this family. Define the open sets V (0,k) = V0k ∩ int K 1 for k = 0, . . . , k0 , and V (n,k) = Vnk \ K n−1 , for n ≥ 1 and k = 0, . . . , kn . We have V (0,k) , F ∩ K n \ K n−1 ⊂ V (n,k) . F ∩ K0 ⊂ k=0,...,k0

k=0,...,kn

n ∈ N, k = 0, . . . , kn } is an open Consequently, the family U = {V (,k) n−1 ∩K = ∅ if  ≥ n, implying that the covering of F . We see that V open covering U is locally finite.  (n,k) ;

The following lemma is also useful. Lemma 15.6. Let M be a σ-compact manifold, F be a closed subset, and V = (V j )j∈J be an open covering of F . There exists an open covering U of F finer than V formed by relatively compact open sets. Moreover if V is locally finite, then U is also locally finite, and if V is countable, then U is also countable. Proof. We introduce the sequence of open sets (Ωn )n∈N given by   Ω0 = int K 1 , Ωn = int K n+1 \ K n−1 , for n ≥ 1. We then set U j,n = V j ∩ Ωn for j ∈ J and n ∈ N. The family U = (U j,n )j∈J ,n∈N is a covering of F as ∪n∈N Ωn = M and one sees that it has the required property.  In the next two sections, we shall use the previous two lemmata to construct partitions of unity, first, in Rd and, second, in a manifold. Remark 15.7. In many texts, the topological assumption made on the manifold is that of second countability. A manifold M is second-countable, if there exists a countable basis of open sets defining its topology. In fact, this implies that the manifold is σ-compact by the following lemma. Lemma 15.8. A second-countable manifold M with or without boundary is σ-compact. Proof. As M is locally homeomorphic to Rd , we find that there exists a basis of its topology made of open sets with compact closure. As M is second-countable, we may in fact choose such a basis to be countable, which we denote by (U n )n∈N . We claim that we can construct a sequence of compact sets satisfying U n ⊂ Kn

and K n ⊂ int K n+1 , n ∈ N.

We set the compact set K 0 = U 0 . Now assume that the compact sets K 0 , . . . , K n are chosen such that U j ⊂ K j , for j = 0, . . . , n, and K j−1 ⊂ int K j , for j = 1, . . . , n. As K n is compact, there exists N ≥ n + 1 such that K n ⊂ ∪0≤j≤N U j . We define the compact set K n+1 = ∪0≤j≤N U j . By the definition of N and K n+1 , we have U n+1 ⊂ K n+1 . As ∪0≤j≤N U j ⊂ K n+1 , we have ∪0≤j≤N U j ⊂ int K n+1 yielding K n ⊂ int K n+1 . Finally, as

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15. ELEMENTS OF DIFFERENTIAL GEOMETRY

M = ∪n∈N U n ⊂ ∪n∈N K n , we see that we have constructed an exhaustion by compact sets.  15.2.2. Partitions of Unity on Rd . An important lemma toward the construction of partitions of unity in Rd is the following one. Lemma 15.9. Let F be a closed subset of Rd . Let V = (V j )j∈J be a locally finite open covering of F in Rd . For all j ∈ J , there exists W j an open subset of Rd such that W j ⊂ V j and W = (W j )j∈J is a locally finite open covering of F in Rd . Proof. Let (K n )n∈N be an exhaustive sequence of compact sets for Rd . Let n ∈ N, and let Jn be the finite set of indices j ∈ J such that V j ∩K n = ∅. Let us define the continuous function dn (x) given by dn (x) = max dist(x, Rd \ V j ). j∈Jn

As a side remark, observe that dn (x) ≤ dn+1 (x) as Jn ⊂ Jn+1 . If x ∈ K n and if j ∈ J \ Jn , then x ∈ / V j and thus dist(x, Rd \ V j ) = 0. It follows that dn+ (x) = dn (x) for all  ≥ 0 and x ∈ K n . For all x ∈ Rd , the sequence dn (x) is thus nondecreasing and stationary for n sufficiently large. If x ∈ K n ∩F , then x ∈ V j for some j ∈ Jn and thus dist(x, Rd \V j ) > 0. We thus have dn (x) > 0 on K n ∩ F . As K n ∩ F is compact, we have Cn = inf x∈K n ∩F dn (x) > 0. For j ∈ Jn , we now define an open set Wnj of Rd such that Wnj ⊂ V j and given by (15.2.1)

x ∈ Wnj

⇔

x ∈ V j and dist(x, Rd \ V j ) > Cn /2.

Note that Wnj may very well be empty. We moreover set Wnj = ∅ for j ∈ J \ Jn . If x ∈ K n ∩ F , there exists j ∈ Jn such that dist(x, Rd \ V j ) = dn (x) ≥ Cn . By the definition of Wnj , we have x ∈ Wnj . Hence, (Wnj )j∈J is an open covering of K n ∩ F . We introduce the sequence of open sets (On )n∈N given by On = int Kn and we have ∪n∈N On = Rd . The sequence of compact sets (On )n∈N is also an exhaustion of Rd . We then introduce the sequence of open sets (Ωn )n∈N given by Ω0 = O0 , Ω1 = O1 ,

Ωn = On \ On−2 , for n ≥ 2.

We have ∪n∈N Ωn = Rd . We set ˜ nj = Wnj ∩ Ωn , W

j ∈ J , n ∈ N.

˜ nj ⊂ V j for j ∈ J . We define the open sets W j = ∪n∈N W Let x ∈ F . Then, for some n ∈ N, x ∈ Ωn ⊂ On ⊂ K n . As (Wnj )j∈J is an open covering of K n ∩ F , there exists j ∈ J such that x ∈ Wnj and thus ˜ nj ⊂ W j . The family of open sets (W j )j∈J thus forms an x ∈ Ωn ∩ Wnj = W open covering of F in Rd .

15.2. OPEN COVERINGS AND PARTITIONS OF UNITY

393

If j ∈ J and x ∈ W j , then there exists a sequence (xp )p∈N ⊂ W j such that xp → x in Rd . As this sequence is bounded, there exists n ∈ N such ˜ j. that {x} ∪ (xp )p∈N ⊂ On ⊂ K n , and we thus have (xp )p∈N ⊂ ∪≤n+1 W  This yields (xp )p∈N ⊂ ∪≤n+1 Wj . As a result, we have x ∈ ∪≤n+1 Wj ⊂ V j by (15.2.1). We have thus obtained that W j ⊂ V j , which concludes the proof.  Lemma 15.10. Let V be an open set of Rd and K ⊂ V a compact set. There exists ϕ ∈ Cc∞ (V ) such that 0 ≤ ϕ ≤ 1 and ϕ ≡ 1 in a neighborhood of K. Proof. Set α = dist(K, Rd \ V ). For 0 < ε < α, we define the following neighborhood of K: Kε = {x ∈ Rd ; dist(x, K) ≤ ε} ⊂ V. We let ψ be the characteristic function of Kα/2 . We also pick χ ∈ Cc∞ (Rd ) supported in the unit ball B(0, 1) such that χ ≥ 0 and ∫ χ = 1. For ε > 0, we set χε (x) = ε−d χ(x/ε). We have supp χε ⊂ B(0, ε),

∫ χε = 1.

We set ϕε = χε ∗ψ ∈ Cc∞ (Rd ). We have ϕε ≥ 0. For ε = α/4, we find ϕε ≡ 1 in Kα/4 from the definition of the convolution and supp(ϕε ) ⊂ K3α/4 ⊂ V by the support theorem for the convolution of functions.  We can now prove the following theorem that states the existence of partitions of unity in Rd . Theorem 15.11. Let F be a closed subset of Rd and let (V j )j∈J be an open covering of F in Rd . There exists a C ∞ -partition of unity of F subordinated to this open covering. Proof. By Lemmata 15.5 and 15.6, we first obtain a finer open covering (On )n∈N of F in Rd that is countable, locally finite, and formed by relatively compact open sets. By Lemma 15.9, there exists a second locally finite open covering W = (W n )n∈N of F in Rd such that W n ⊂ On for all n ∈ N. The open set W = ∪n∈N W n is a neighborhood of F . By Lemma 15.10 for all n ∈ N, there exists φn ∈ Cc∞ (On ) such that 0 ≤ φn ≤ 1 and φn ≡ 1 in a neighborhood of W n . We then define the following sequence of functions: ϕ0 = φ 0 ,

ϕn = φ n

n−1

(1 − φj ), for n ≥ 1.

j=0

and, as the open covering (On )n∈N of F We have ϕn ≥ 0, supp(ϕn ) ⊂ On ,  is locally finite, we see that ϕ = n∈N ϕn is a well defined C ∞ function. Let x0 ∈ W . With the same argument, there exists k ∈ N such that a

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15. ELEMENTS OF DIFFERENTIAL GEOMETRY

neighborhood U0 ⊂ W of x0 does not intersect On for n > k. We thus have  ϕ= ϕn in U0 . 0≤n≤k

We claim that ϕ ≡ 1 in U0 , which concludes the proof. In fact, define the functions ρk = φ k ,

k 

ρ = φ  +

n=+1

In U0 , we have ϕ =

ρ0 .

φn

n−1

(1 − φj ), for 0 ≤  < k.

j=

We then write, for  = 0, . . . , k − 1,

ρ − 1 = φ − 1 + φ+1 (1 − φ ) +

k  n=+2

= (1 − φ ) − 1 + φ+1 +

k  n=+2

= (1 − φ )(ρ 

φn

+1

φn

n−1

(1 − φj )

j= n−1

(1 − φj )



j=+1

− 1).

By induction, we find, for  = 0, . . . , k − 1,



(15.2.2) (1 − φn )(ρ+1 − 1) = − (1 − φn ). ρ0 − 1 = 0≤n≤

0≤n≤k

If x ∈ U0 ⊂ W , then there exists n ∈ {1, . . . , k} such that x ∈ W n implying  φn (x) = 1. We thus have ϕ(x) − 1 = ρ0 (x) − 1 = 0 by (15.2.2). 15.2.3. Partitions of Unity on a σ-Compact Manifold. Let M be a σ-compact C k -manifold. Let A be an atlas of M with A = (Oi , κi )i∈I . Let F be a closed subset of M, and let V = (V j )j∈J be an open covering of F in M. Upon refining both open coverings, according to Lemmata 15.5 and 15.6, we may assume that both the coverings (Oi )i∈I and V are locally finite and (at most) countable and formed by relatively compact open sets. We can thus use I = J = N to index the two families of open sets. Setting now V i,j = Oi ∩ V j , i, j ∈ N, we see that we can furthermore assume that the covering V of F is finer than the covering of M associated with the atlas. Remark 15.12. Observe that if F is compact, then the manifold M need not be assumed σ-compact as the results of Lemmata 15.5 and 15.6 are clear in this case. The following lemma is the counterpart to Lemma 15.9 for manifolds. Lemma 15.13. Let V = (V j )j∈N be a countable locally finite open covering of F in M made of relatively compact open sets. For all j ∈ N, there exists a relatively compact open subset W j of M such that W j  V j and W = (W j )j∈N is a locally finite open covering of F in M. Proof. We first consider the open set V 0 . It is contained in a relatively compact open set Oi0 associated with a local chart (Oi0 , κi0 ) of the atlas for some i0 ∈ N. Note that different choices of i0 may be possible. We set

15.2. OPEN COVERINGS AND PARTITIONS OF UNITY

395

Vˇ 0 = ∪j≥1 V j . As F ⊂ V 0 ∪ Vˇ 0 , the closed set L0 = F \ Vˇ 0 satisfies L0 ⊂ V 0 ⊂ Oi0 . As Oi0 is compact, L0 is itself compact. ˜ = κi0 (Oi0 ) an ˜ 0 = κi0 (L0 ) and V˜ 0 = κi0 (V 0 ) both subsets of O We set L ˜ 0 is a compact set of Rd . open subset of Rd . As κ is a homeomorphism, L 0 ˜ Thus there exists an open set W that is relatively compact and such that ˜ 0  V˜ 0 . We set W 0 = (κi0 )−1 (W ˜ 0 ), and we have ˜0 ⊂ W L L0 ⊂ W 0  V 0 , implying that W 0 = {W 0 } ∪



{V j } is an open covering of F in M.

j≥1

Next starting from W 0 , setting L1 = F \ (W 0 ∪ ∪j≥2 V j ), we construct W 1 such that L1 ⊂ W 1  V 1 and such that {V j } is an open covering of F in M. W 1 = {W 0 , W 1 } ∪ j≥2

By induction, we construct the relatively compact open sets W j , j ∈ N, such that W j  V j and {W j } ∪ {V j } is an open covering of F in M. ∀k ∈ N, W k = 0≤j≤k

j≥k+1

Let x ∈ F , then there exists Jx = {j1 , . . . , jn } ⊂ N such that x ∈ V j if and only if j ∈ Jx as the covering is locally finite. We thus see that there exists j ∈ Jx such that x ∈ W j using that W k is an open covering of F in M for k > max Jx . Hence W = (W j )j∈N is an open covering of F in M. As W j ⊂ V j , this covering is necessarily locally finite.  With this lemma, we may then conclude to the existence of a partition of unity subordinated to a given open covering. Theorem 15.14. Let M be a σ-compact C k -manifold. Let F be a closed subset of M, and let V = (V j )j∈J be an open covering of F in M. There exists a C k -partition of unity of F in M subordinated to this covering. Proof. Upon choosing refinements of the atlas and the covering as in the beginning of this section, we may assume that both are at most countable (and indexed by N), locally finite, and made of relatively compact open sets. We may also assume that the covering is finer than the covering given by the atlas. We can then apply Lemma 15.13 and have a second locally finite open covering W = (W n )n∈N of F in M such that W n  V n , for all n ∈ N. For each n ∈ N, there exists φn ∈ Cck (V n ) such that φn ≡ 1 on a neighborhood of W n . To see this, we consider a chart C = (O, κ) such that V n ⊂ O, and ˜ n = κ(W n ). Then, by Lemma 15.10, there exists we set V˜ n = κ(V n ) and W ˜ n . Then φn = κ∗ φ˜n has the required φ˜n ∈ Cc∞ (V˜ n ) such that φ˜n ≡ 1 on W properties.

396

15. ELEMENTS OF DIFFERENTIAL GEOMETRY

We then define the following sequence of C k -functions on M: ϕ0 = φ 0 ,

ϕn = φ n

n−1

(1 − φj ), for n ≥ 1.

j=0

 Then, the sum n∈N ϕn is locally finite and forms a partition of unity of F in M subordinated to the covering (V n )n∈N , by the same argument as in the end of proof of Theorem 15.11.  We shall see below that partitions of unity can be very important to define proper geometrical objects on a manifold. Remark 15.15. Proposition 15.3 characterizes functions on a manifold. In fact, if f : M → E is a C k -function and if A = (On , κn )n∈N is an atlas chosen countable and the covering (On )n∈N of M locally finite if M is assumed σ-compact. The local representative of f in a local chart C = (O, κ) is (κ−1 )∗ f . This function is C k in κ(O) but not compactly supported. In some arguments, it can be useful to have some compactly supported representative of f . A partition of unity comes in handy. Assume indeed that  n of unity subordinated to the covering (On )n∈N . We n∈N ϕ is a partition  n then write f = n∈N f with f n = f ϕn compactly supported in On . Upon application of the pullback by (k n )−1 , we then obtain some sort of compactly n n supported representative  n −1 of f in κ (O ). Naturally, this representation is n only exact in κ (ϕ ) ({1}) . We conclude this section by the following useful lemma that is the counterpart on manifolds of Lemma 15.10. Lemma 15.16. Let M be a d-dimensional manifold. Let V be an open set of M and K ⊂ V be a compact set. There exists ϕ ∈ Cc∞ (V ) such that 0 ≤ ϕ ≤ 1 and ϕ ≡ 1 in a neighborhood of K. Proof. By Lemma 15.6, there exists an open set V  ⊂ V such that V is relatively compact in M and K ⊂ V  . Then, the partition of unity provided by Theorem 15.14 yields a single function in Cc∞ (V ) that fulfills the sought properties.  

15.3. Tangent Space and Vector Fields Let M be a C 1 d-dimensional manifold and m0 ∈ M. We shall use equivalent classes of C 1 -curves to define tangent vectors at m0 . 15.3.1. Tangent Vectors. We call a curve γ passing through m0 on M any map R ⊃ I → M of class C 1 for I an open interval such that γ(t0 ) = m0 for some t0 ∈ I. Without any loss of generality, we may choose I = (−ε, ε), with ε > 0, and t0 = 0. If m0 is a boundary point, we choose I = (−ε, 0] or [0, ε) and derivatives below need to be replaced by one-sided derivatives.

15.3. TANGENT SPACE AND VECTOR FIELDS

397

Let C 1 = (O1 , κ1 ) be a local chart with m0 ∈ O1 . We set v(1) = Fm0 ,C 1 (γ) :=

d 1 κ (γ(t))|t=0 . dt

Note that this derivative exists because of the C 1 -regularity of γ. This vector in Rd is the tangent vector of the curve t → κ1 (γ(t)) at κ1 (m0 ) (for t = 0) in Rd . Observe that for any v ∈ Rd , the local curve γ(t) = (κ1 )−1 (κ1 (m0 ) + tv) passes through m0 and is such that Fm0 ,C 1 (γ) = v. The map Fm0 ,C 1 is thus onto Rd . If C 2 = (O2 , κ2 ) is a second local chart with m0 ∈ O2 , then we may also d 2 κ (γ(t))|t=0 . Setting κ12 = κ2 ◦ (κ1 )−1 , which is set v(2) = Fm0 ,C 2 (γ) := dt 1 well defined near κ (m0 ) in κ1 (O1 ), we write, for t near 0,  κ2 (γ(t)) = κ12 κ1 (γ(t))). As κ12 is a C 1 -diffeomorphism, this yields the following relation between v(2) and v(1) :   (15.3.1) v(2) = dκ12 (κ1 (m0 )) v(1) . We then observe that if two curves, γ and γ˜ , both passing through m0 , γ ), that is, they share the same tangent are such that Fm0 ,C 1 (γ) = Fm0 ,C 1 (˜ 1 vector v(1) in the local chart C , then we have Fm0 ,C 2 (γ) = Fm0 ,C 2 (˜ γ ) in the second local chart C 2 . We may thus define the following equivalence relation on the set of local curves passing through m0 , independently of the picked local chart. One says that γ ∼ γ˜ if, in any chart C = (O, κ), they share the same tangent vector v. We denote by [γ] the equivalence class. We call [γ] a tangent vector. The vectorial property of such an object is the subject of the next paragraph. We define the tangent set of M at m0 as the set of those equivalence classes, and we denote it by Tm0 M. For any chart C = (O, κ), we may define the bijective map F˜m0 ,C : Tm0 M → Rd , by F˜m0 ,C ([γ]) = Fm0 ,C (γ). This induces a linear structure on Tm0 M, which is independent of the considered local chart, as we have the linear correspondence (15.3.1) for the local representative of the tangent vectors. The tangent set Tm0 M is then a vector space of dimension d. It is called the tangent vector space of M at m0 . Naturally, this vectorial structure on Tm0 M makes the bijection F˜m0 ,C an isomorphism. Observe that this definition of the tangent space only requires the manifold M to be C 1 . Let f be a C 1 real valued function defined in a neighborhood of m0 in M. For v ∈ Tm0 M, we consider γ a representative of v, that is, v = [γ]. For a local chart C = (O, κ), we define its local representative, that is, the C 1 -function f C : κ(O) → R by f C = (κ−1 )∗ f . The map t → κ(γ(t)) is a C 1 -curve in Rd that goes through x0 = κ(m0 ) at t = 0. As we have

398

15. ELEMENTS OF DIFFERENTIAL GEOMETRY

f (γ(t)) = f C (κ(γ(t))) and (15.3.2)

d dt κ(γ(t))|t=0

= Fm0 ,C (γ) = F˜m0 ,C (v), we obtain

d f (γ(t))|t=0 = df C (x0 )(F˜m0 ,C (v)). t→0 dt lim

If we consider a second representative γ˜ of v, that is, γ ∼ γ˜ , then we also have d d d γ (t))|t=0 = F˜m0 ,C (v). This implies that dt f (γ(t))|t=0 = dt f (˜ γ (t))|t=0 . dt κ(˜ We may thus define a linear form f → v(f ) by this common value: (15.3.3)

v(f ) :=

d d f (γ(t))|t=0 = f (˜ γ (t))|t=0 . dt dt

Observe that this map is linear. Moreover, from (15.3.2), we find that v(f h) = f (m0 )v(h) + v(f )h(m0 ),

f, h ∈ C 1 .

The map f → v(f ) is thus a derivation on the C 1 -functions defined locally near m0 . Note in particular that v vanishes on constant functions. Remark 15.17. In the case of a C ∞ -manifold, the tangent vector space at m0 can be alternatively defined as the vector space of derivations on C ∞ functions defined in a neighborhood of m0 , as is often done in the literature. Note, however, that the vector space of derivations on C k -functions defined in a neighborhood of m0 , with k ∈ N, does not coincide with Tm0 M as the former is in fact infinite dimensional [263, 316]. In particular, this approach cannot be used for C k -manifolds. This can however be repaired by additional requirements on the derivations [132]. Let us further express the local representative of a tangent vector v at m0 . Consider a local chart C = (O, κ). Let v C = (v 1 , . . . , v d ) be the coordinates of the local representative of v in C, that is, v C = F˜m0 ,C (v) in the notation used above. We denote by Xi the map from Rd to R such that Xi (x) = xi . We then set κi = κ∗ Xi . We have, as seen above, for γ a d κ(γ(t))|t=0 , meaning that representative of v, v C = dt (15.3.4)

vi =

d κi (γ(t))|t=0 = v(κi ). dt

This allows one to obtain the action of v on C 1 -functions as follows. As above, for a C 1 real valued function f given in a neighborhood of m0 , we set f C = (κ−1 )∗ f from a neighborhood in Rd of x0 = κ(m0 ) into R. Setting x = κ(m) ∈ Rd , yielding xi = κi (m), we then write  f (m) = f C (x) = f C (x0 ) + ∂xj f C (x0 )(xi − x0 ) + ε(x), 1≤j≤d

with ε(x) = w(x)|x − x0 |Rd of class C 1 , with lim w(x) = 0 as x → x0 . Observe that we have t−1 ε(κ(γ(t))) = t−1 |κ(γ(t)) − x0 |Rd w(κ(γ(t))) ∼ |v| w(κ(γ(t))) ∼ 0,

15.3. TANGENT SPACE AND VECTOR FIELDS

399

as t → 0. This implies that v(ε ◦ κ) = limt→0 t−1 ε(κ(γ(t))) = 0. By (15.3.4), we then obtain  i  (15.3.5) ∂xi f C (x0 )v(κi ) = v ∂xi f C (x0 ). v(f ) = 1≤i≤d

1≤i≤d

Let us consider as above two local charts C  = (O , κ ),  = 1, 2, with 1 2 m0 ∈ O = O1 ∩O2 and κ12 = κ2 ◦(κ1 )−1 . For v ∈ Tm0 M, we let v C , v C ∈ Rd be its local representative in the two local charts. Formula (15.3.1) can also (1) be written in the form, with x0 = κ1 (m0 ), (15.3.6) V C = (κ12 ) (x0 ) V C , 2

(1)

1

with V C = t (v C 

 ,1

, . . . , vC

 ,d

),  = 1, 2,

where (κ12 ) (x0 ) is the Jacobian matrix of the map κ12 : κ1 (O) → κ2 (O) (1) at x0 , that is,   (1)  2 1 (15.3.7) ∂xj κ12 i (x0 ) v C ,j , v C ,i = (1)

1≤j≤d



 where κ12 i = (κ12 )∗ Xi , with the function Xi defined as above. 15.3.2. Differential of a Map and Action on Tangent Vectors. ˜ of respective dimensions d and d˜ We consider two C 1 -manifolds M and M 1 ˜ and a C -map φ from M into M defined in a neighborhood of m0 . We let v ∈ Tm0 M and γ be such that v = [γ]. Then, the function t → φ(γ(t)) is ˜ a curve that goes through m ˜ 0 = φ(m0 ). We set v˜ = [φ ◦ γ] ∈ Tm ˜ 0 M. We denote this tangent vector by v˜ = T φ(m0 )(v), and the map T φ(m0 ) is called the differential of φ at m0 or sometimes the tangent of φ at m0 . As both v and v˜ are defined independently of local charts, it is sufficient to consider the action of T φ(m0 ) in local charts. Let thus C = (O, κ) be a ˜ κ ˜ with m ˜ ˜ ) be a chart on M ˜ 0 ∈ O. chart on M with m0 ∈ O and C˜ = (O, −1 d We then set ψ = κ ˜ ◦ φ ◦ κ that maps a neighborhood in R of x0 = κ(m0 ) ˜ into a neighborhood in Rd of x ˜0 = κ ˜ (m˜0 ). This function ψ is the local representative of the function φ with respect to the two chosen local charts. Let v C = (v 1 , . . . , v d ) be the coordinates of the local representative of v in C, that is, v i = v(κi ) in the notation used above. Similarly we let ˜ ˜ ˜ v 1 , . . . , v˜d ) be the coordinates of the local representative of v˜ in C, v˜(C) = (˜ i κi ). Then, from the standard differential calculus, we have that is, v˜ = v˜(˜ (15.3.8)

˜

v˜C = dψ(x0 )v C .

˜ or t v˜(C) = ψ  (x0 ) t v C , where ψ  (x0 ) is the d˜ × d Jacobian matrix of ψ at x0 . From this local representative, we see that T φ(m0 ) is a linear map from ˜ Tm0 M into Tm ˜ 0 (M ).

400

15. ELEMENTS OF DIFFERENTIAL GEOMETRY

Observe that in the case φ = IdM , we recover the change of variables formula (15.3.1). Note also that we have for the chart C: (15.3.9) T κ(m0 )(v) = F˜m0 ,C (v) = v C ∈ Tx0 Rd ∼ = Rd , v ∈ Tm0 M. We recall that T κ(m0 ) is an isomorphism. This gives T κ(m0 )(v) = (v(κ1 ), . . . , v(κd )). ˜ defined in a neighborhood of m0 ∈ M If we have two maps φ : M → M ˜ then Φ = ϕ ◦ φ : ˜ →M ˆ defined in a neighborhood of φ(m0 ) ∈ M, and ϕ : M ˆ M → M is well defined in a neighborhood of m0 and, from (15.3.8), using local charts, we find (15.3.10)

T Φ(m0 ) = T ϕ(φ(m0 )) ◦ T φ(m0 ).

15.3.3. The Tangent Bundle, Vector Fields. With the tangent vector space defined at any point m of M, we then set {m} × Tm M TM = m∈M

to be the tangent bundle associated with M. We denote by π the natural projection π : T M → M given by π(m, v) = m. We have π −1 (m) = {m} × Tm M, and we refer to Tm M or to π −1 (m) as the fiber above m in the tangent bundle. If C = (O, κ) is a local chart on M with m0 ∈ O, we set T O = ∪m∈O {m} × Tm M and set the map, using (15.3.9), T κ : T O  (m, v) → (κ(m), T κ(m)(v)) ∈ Rd × Rd . We have T κ(m, v) = (κ1 (m), . . . , κd (m), v(κ1 ), . . . , v(κd )). We see that T κ is a bijection onto κ(O) × Rd . If C j = (Oj , κj ), j = 1, 2, are two local charts with O = O1 ∩ O2 = ∅, then T κ2 ◦ (T κ1 )−1 is a diffeomorphism from κ1 (O) × Rd onto κ2 (O)Rd of class C k−1 if M is a C k -manifold (k ≥ 2). In fact, with κ12 = κ2 ◦ (κ1 )−1 , we have   T κ2 ◦ (T κ1 )−1 (x, w) = κ12 (x), dκ12 (x)(w) , (x, w) ∈ κ1 (O) × Rd . If (Oi , κi )i∈I is an atlas for M and if we equip T M with a topology that makes the maps T κi as defined above continuous on T Oi into R2d , we thus obtain that T M is a C k−1 -manifold of dimension 2d. An atlas is then given by the set of charts (T Oi , T κi )i∈I . ˜ a C k˜ -manifold. Let φ be a C  -map, Let M be a C k -manifold and M ˜ from M into M ˜ defined on an open set U of M. If  ≥ 1, we  ≤ min(k, k), ˜ by can define the map T φ : T U → T M   T φ(m, v) = φ(m), T φ(m)(v) ,

15.3. TANGENT SPACE AND VECTOR FIELDS

401

which is called the differential or the tangent map of φ on T U . As T U is an ˜ open set of T M, T φ is a C −1 -map from T U into T M. Let U be an open set of M, a C k -manifold. For  ≤ k − 1, a C  vector field v on U is a C  -map from U into T U that satisfies the property πv(m) = m, that is, v(m) is in the fiber above m in the tangent bundle. A vector field is also called a section of the tangent vector bundle. We denote by C  V (U ) the space of C  -vector fields on U . On a C ∞ -manifold, we set C ∞ V (U ) = ∩≥0 C  V (U ). For v ∈ C  V (M), its action on a C +1 -function f on M yields a C  -function defined by v(f )(m) = vm (f ),

with v(m) = (m, vm ).

In a local chart C = (O, κ), where v is defined, the local representative of v is (x, vxC ) with vxC = T κ(m)(vm ) if v(m) = (m, vm ) and x = κ(m). As vxC is a function of x, we have vxC = (v 1 (x), . . . , v d (x)), with v i (x) = vm (κi ) from (15.3.4), and considering the action given in (15.3.5), we write  i v (x)∂xi . vxC = 1≤i≤d

If we denote by f C the representative of f in the local chart, that is, f C (x) = f (κ−1 (x)), we have  i v (x)∂xi f C (x), x = κ(m). v(f )(m) = vm (f ) = vxC (f C ) = 1≤i≤d

˜ a C k˜ -manifold, and let the two maniLet M be a C k -manifold and M folds be of the same dimension. If a map φ is a C 1 -diffeomorphism from U onto φ(U ) with U an open set of M and if v is a vector field on U , we can define the image of v by φ, called the push-forward of v by φ, denoted by φ∗ (v), φ∗ (v)(m) ˜ = T φ(v(m)),

for m ˜ ∈ φ(U ), with m = φ−1 (m). ˜

This means that φ∗ (v)(m) ˜ = T φ(m, vm ) = (m, ˜ T φ(m)(vm )) with (m, vm ) = v(m) ∈ T M. We finish with the following consequence of Proposition 15.3. Corollary 15.18. Let A = (T Oi , T κi )i∈I be an atlas for T M as dei fined above, and let (f C )i∈I be such that: (1) For each i ∈ I, f C : T κi (T Oi ) → E, for some set E. (2) For each (i, j) ∈ I 2 such that Oij = Oi ∩ Oj = ∅, we have i j j f C (x, v) = (T κij )∗ f C (x, v) = f C (T κij (x, v)) for x ∈ κi (Oij ), v ∈ Rd , and T κij = T κj ◦ (T κi )−1 . i

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15. ELEMENTS OF DIFFERENTIAL GEOMETRY

Then, there exists f : T M → E such that for all i ∈ I, f C is the local i representative of f in the chart C i = (T Oi , T κi ), that is, f C = ((T κi )−1 )∗ f . i

Recall that T κij (x, v) = (κij (x), dκij (x)v) for (x, v) ∈ κi (Oij ) × Rd . 15.4. Cotangent Vectors and Forms For m ∈ M, a C k d-dimensional manifold, we define the cotangent ∗ M. vector space at m as the dual space of Tm M. It is denoted by Tm 15.4.1. Cotangent Vectors, Local Representatives. Let m0 ∈ M ∗ M and C = (O, κ) be a local chart with m0 ∈ O and x0 = κ(m0 ). If ω ∈ Tm 0 C 1 d and v ∈ Tm0 M, and if v = (v , . . . , v ) = (v(κ1 ), . . . , v(κd )) is the local representative of v, that is, v C = T κ(m0 )v, we write −1 −1   ω(v) = ω, v = ω, T κ(m0 ) v C  = t T κ(m0 ) ω, v C , −1  which yields the representative ξ = ω C = t T κ(m0 ) ω of ω in the local d ) the dual base of (∂ , . . . , ∂ ) of chart. If we denote by (dx1 , . . . , dx x1 xd d d i . Since ξ, ∂  = ξ , we ∼ ξ dx Tκ(m0 ) R = R , then we have ξ = i x i i 1≤i≤d find with the above duality identity  −1 ξi = ω, (∂xOi )m0 , with (∂xOi )m0 = T κ(m0 ) (∂xi )x0 . Let us now consider two local charts C  = (O , κ ),  = 1, 2, with m0 ∈ O =  () ∗ M, let ξ () = ω C  = i O1 ∩O2 and κ12 = κ2 ◦(κ1 )−1 . If ω ∈ Tm 1≤i≤d ξi dx 0

be its local representative in (O , κ ). Let now v ∈ Tm0 M, with v C its local representative,  = 1, 2. We have 

ω, v = ξ (1) , v C  = ξ (2) , v C . 1

2

(1)

From (15.3.1), this yields, with x0 = κ1 (m0 ), (1)

ξ (1) = t dκ12 (x0 )ξ (2) .

(15.4.1)

In terms of coordinates, this reads (15.4.2) Ξ(1) = t (κ12 ) (x0 ) Ξ(2) , (1)

()

()

with Ξ() = t (ξ1 , . . . , ξd ),  = 1, 2,

that is, (15.4.3)

(1)

ξi

=

 1≤j≤d

  (1) (2) ∂xi κ12 j (x0 ) ξj ,

which is the counterpart of (15.3.6) and (15.3.7) for cotangent vectors.

15.4. COTANGENT VECTORS AND FORMS

403

15.4.2. Action of a Smooth Map and Differential of a Function. ˜ of respective dimensions We consider now two C 1 -manifolds M and M 1 ˜ ˜ d and d and a C -map φ from M into M defined in a neighborhood of ˜ the transpose map t T φ(m0 ) : m0 . As T φ(m0 ) maps Tm0 M into Tφ(m0 ) M, ∗ ∗ M is given by ˜ → Tm Tφ(m M 0 0) t T φ(m0 )˜ ω , v = ˜ ω , T φ(m0 )v,

∗ ˜ v ∈ Tm M. ω ˜ ∈ Tφ(m M, 0 0)

In particular, observe that if C = (O, κ) is a local chart with m0 ∈ O, then −1  (15.4.4) ξ = ω C = t T κ(m0 ) ω = t T κ−1 (x0 )ω, with x0 = κ(m0 ). We recall that if f is a C 1 -function on M, m0 ∈ M, and v ∈ Tm0 M, then, from (15.3.5), the map v → v(f ) is well defined and linear. It thus defines a cotangent vector at m0 which we denote by df (m0 ). In fact, by (15.3.5), we have  i v ∂xi g(x0 ), df (m0 ), v = v(f ) = 1≤i≤d

with g = (κ−1 )∗ f and x0 = κ(m0 ). Observe that if M = Rd , then f = g, and we recover the usual definition of the differential of a function at one point. In fact, in the local chart (O, κ), the above identity precisely  means that the local representative of the cotangent vector df (m0 ) is 1≤i≤d ∂xi g(x0 )dxi .  By abuse of notation, one often writes df (m0 ) = 1≤i≤d ∂xi f (m0 )dxi . 15.4.3. The Cotangent Bundle. With the cotangent vector space defined at any point m of M, we set ∗ (15.4.5) {m} × Tm M T ∗M = m∈M

to be the cotangent bundle associated with M. We denote by π ˜ the natural ˜ (m, ω) = m. We have (˜ π )−1 (m) = projection π ˜ : T ∗ M → M given by π ∗ M, and we refer to T ∗ M or to (˜ π )−1 (m) as the fiber above m in {m} × Tm m the cotangent bundle. ∗ M, If C = (O, κ) is a local chart on M, we set T ∗ O = ∪m∈O {m} × Tm and we define the map, using (15.4.4), T ∗ κ : T ∗ O  (m, ω) → (κ(m), t (T κ(m))−1 (ω)) ∈ Rd × Tκ(m) Rd ∼ = Rd × Rd . We have O T ∗ κ(m, ω) = (κ1 (m), . . . , κd (m), ω((eO 1 )m ), . . . , ω((ed )m )),  −1 (∂xj )x if x = κ(m). We see that T ∗ κ is a bijection for (eO j )m = T κ(m) onto κ(O) × Rd . If C j = (Oj , κj ), j = 1, 2, are two local charts with O = O1 ∩ O2 = ∅, then T ∗ κ2 ◦ (T ∗ κ1 )−1 is a diffeomorphism from κ1 (O) × Rb

404

15. ELEMENTS OF DIFFERENTIAL GEOMETRY

onto κ2 (O) × Rb of class C k−1 if M is a C k -manifold (k ≥ 2). In fact, with κ12 = κ2 ◦ (κ1 )−1 , we have   T ∗ κ2 ◦ (T ∗ κ1 )−1 (x, ξ) = κ12 (x), t d(κ12 )−1 (x)ξ , (x, ξ) ∈ κ1 (O) × Rd . If (Oi , κi )i∈I is an atlas for M and if we equip T ∗ M with a topology that makes the maps T ∗ κi as defined above continuous on T ∗ Oi into R2d , we thus obtain that T ∗ M is a C k−1 -manifold of dimension 2d. An atlas is then given by the set of charts (T ∗ Oi , T ∗ κi )i∈I . 15.4.4. One-Forms. Let U be an open set of M, a C k -manifold. For  ≤ k − 1, a C  one-form ω on U is a C  -map from U into T ∗ U that satisfies the property π ˜ ω(m) = m, that is, ω(m) is in the fiber above m in the cotangent bundle. A one-form is also called a section of the cotangent vector bundle. A one-form on U is also called a cotangent vector field on U . A one-form ω acts on a vector field v as follows: ω, v(m) = ωm , vm Tm∗ M,Tm M ,

m ∈ M,

for ω(m) = (m, ωm ), and v(m) = (m, vm ). One then obtains a function on M. If both ω and v are C  , then f = ω, v is C  . We denote by C  Λ(U ) the space of C  -one-forms on U . On a C ∞ manifold, we set C ∞ Λ(U ) = ∩≥0 C  Λ(U ). Recalling that if f is a C +1 -function on M,  ≤ k − 1, then df (m) ∈ ∗ Tm M, we see that the map ω : m → (m, df (m)) is in fact a C  one-form: ω, v(m) = df (m), vm Tm∗ M,Tm M = vm (f ), for a vector field v. One often identifies ω(m) and df (m) and uses the notation df for ω given above. We call df the differential of f on M. ˜ a C k˜ -manifold, and let d and d˜ be their Let M be a C k -manifold and M 1 ˜ with U an open set respective dimensions. If φ is C -map from U into M ˜ of M and if ω is a one-form on M, we can define the pullback of ω by φ, denoted by φ∗ (ω), with φ∗ (ω)(m) = (m, φ∗ (ω)m ), by φ∗ (ω)m , vTm∗ M,Tm M = ωφ(m) , T φ(m)vT ∗

φ(m)

˜ φ(m) M ˜, M,T

for v ∈ Tm M, m ∈ M. Note that as opposed to the definition of pushforwards of vector fields at the end of Sect. 15.3, we need not require the map φ to be a diffeomorphism to define pullback of one-forms. If φ is a C 1 -diffeomorphism, we however have φ∗ (ω), v(m) = φ∗ (ω)m , vm Tm∗ M,Tm M = ωφ(m) , T φ(m)vm T ∗

φ(m)

= ωφ(m) , φ∗ vφ(m) T ∗

φ(m)

˜ φ(m) M ˜ M,T

˜ φ(m) M ˜ M,T

= ω, φ∗ v(φ(m)),

˜ and v a vector field on M. for ω a one-form on M

15.5. SUBMANIFOLD

405

We finish with the following consequence of Proposition 15.3. Corollary 15.19. Let A = (T ∗ Oi , T ∗ κi )i∈I be an atlas for T ∗ M as i defined above, and let (f C )i∈I be such that (1) For each i ∈ I, f C : T ∗ κi (T ∗ Oi ) → E, for some set E. (2) For each (i, j) ∈ I 2 such that Oij = Oi ∩ Oj = ∅, we have i j f C (x, ξ) = (T ∗ κij )∗ f C (x, ξ) for x ∈ κi (Oij ), ξ ∈ Rd , and T ∗ κij = T ∗ κj ◦ (T ∗ κi )−1 . i

Then, there exists f : T ∗ M → E such that for all i ∈ I, f C is the local repi resentative of f in the chart C i = (T ∗ Oi , T ∗ κi ), that is, f C = ((T ∗ κi )−1 )∗ f . i

Remark 15.20. If we have (y, η) = T ∗ κij (x, ξ) = T ∗ κj ◦ (T ∗ κi )−1 (x, ξ) = (κij (x), t d(κij )−1 (x)ξ) for (x, ξ) ∈ κi (Oij ) × Rd , then ξ = t dκij (x)η. Thus, property (2) in Corollary 15.19 also reads f C (x, t dκij (x)η) = f C (κij (x), η), i

j

x ∈ κi (Oij ),

η ∈ Rd .

15.5. Submanifold Let M be a d-dimensional smooth manifold with or without boundary. Definition 15.21 (Submanifold). A subset N such that N ∩ ∂M = ∅ is called a smooth submanifold (without boundary) of codimension n ∈ N if for every m0 ∈ N , there exists a neighborhood V0 of m0 in M and n real valued smooth functions f1 , . . . , fn such that the rank of (df1 (m0 ), . . . , dfn (m0 )) in ∗ M is n, and we have N ∩V = {m ∈ M∩V ; f (m) = · · · = f (m) = 0}. Tm 0 0 1 n 0 Such a submanifold is also called an embedded submanifold. Upon choosing an open set O ⊂ V0 , we can furthermore enforce the rank of (df1 (m), . . . , dfn (m)) to be constant and equal to n in O. We provide N with the induced subset topology. For a point m0 of N and in a neighborhood O as above, the n functions f1 , . . . , fn can be used to generate a diffeomorphism κ : O → κ(O) ⊂ Rd , with a possibly reduced open set O, such that κ(m) = (˜ κ(m), f1 (m), . . . , fn (m)) with κ ˜ : O → Rd−n smooth. The couple (O, κ) can then be used as a local chart if added to any atlas. In such a local chart, the submanifold is given by xd = · · · = xd−n+1 = 0. Let (Oi , κi )i∈I be a family of such local charts that covers N , that is, κi (m), 0Rn ) if and only if N ⊂ ∪i∈I Oi . For m ∈ Oi , we then have κi (m) = (˜ ˜ i, κ ˜ i = Oi ∩ N , then C˜i = (O ˜ i ) is a local chart for N and m ∈ N . If we set O i ˜ (C )i∈I forms an atlas that yields a smooth (d − n)-dimensional manifold structure for N . If we consider the inclusion map φ : N → M, we see that for any m ∈ N , the map T φ(m) : Tm N → Tm M is an injection. Then, any tangent vector v ∈ Tm N , m ∈ N , is identified with its image, T φ(m)v. This allows

406

15. ELEMENTS OF DIFFERENTIAL GEOMETRY

one to consider Tm N as a linear subspace of Tm M and, in turn, T N is itself a submanifold of T M. If v ∈ Tm M, then v ∈ Tm N if and only if, in a local chart C = (O, κ), where N is given by xd = · · · = xd−n+1 = 0, its v C is of the form v C = (v 1 , . . . , v d−n , 0, . . . , 0) =  local representative i 1≤i≤d−n v ∂xi . This above identification of Tm N as a subspace of Tm M allows cotangent ∗ M to act on tangent vectors in T N . This can also be done vectors in Tm m ∗ M to T ∗ N by the inclusion by pulling back any cotangent vector ω ∈ Tm m map φ. Observe that the pullback φ∗ is not injective. In fact, we denote by ∗ N the cotangent vectors in T ∗ M that vanish on T N . Those vectors Nm m m ∗M = are called conormal. With the identification made above, we have Tm ∗ ∗ ∗ ∗ Tm N ⊕Nm N , with dim Tm N = d−n and dim Nm N = n. We define N ∗ N = ∗ N , called the conormal bundle of N . It is a submanifold of ∪m∈N {m} × Nm ∗ T M of codimension d (and thus of dimension d). In the particular case of ∗ N is of dimension one. a submanifold N of codimension one, Nm If M is a manifold with boundary, then in every chart C = (O, κ) such that O ∩ ∂M = ∅, we have κ(m) = (κ1 (m), . . . , κd (m)) with κd (m) ≥ 0, ˜ (m) = and the boundary ∂M is locally given by κd (m) = 0. Setting κ (κ1 (m), . . . , κd−1 (m)), we see that the above analysis applies and allows one to give a manifold structure to ∂M. We also say that ∂M is a submanifold of M of codimension 1. Then, if Ω is a smooth open subset of M, its boundary ∂Ω can be seen as a submanifold of codimension one of either M or Ω. Note in particular that ∂M does not have a boundary. Let N be a submanifold of M, and let U be an open subset of N . We define a C  -vector field on M along the open set U of the submanifold N to be a C  -map v : U → T M that satisfies the property πv(m) = m. This is not a vector field on N as v(m) = (m, vm ) and vm may not be in Tm N , but rather vm ∈ Tm M. Similarly, we define that a C  -one-form on M along the ˜ ω(m) = m. open set U is C  -map ω : U → T ∗ M that satisfies the property π In particular, if U is an open set of ∂M, a C  -one-form ω on M along U ∗ ∂M is called a conormal vector field on U . such that ωm ∈ Nm

15.6. Tensors and p-Forms 15.6.1. Covariant Tensors. Let M be a smooth d-dimensional manifold and m ∈ M. For r ∈ N, the vector space of multilinear forms from ∗ M. We have dim ⊗r T ∗ M = dr . Such (Tm M)r into C is denoted by ⊗r Tm m forms are called r-covariant tensors at m. If ω and ω ˜ are r- and s-covariant tensor, respectively, we define their tensor product ω ⊗ ω ˜ as the following p-covariant tensor, with p = r + s, ω⊗ω ˜ (v, v˜) = ω(v)˜ ω (˜ v ),

v ∈ (Tm M)r , v˜ ∈ (Tm M)s .

15.6. TENSORS AND p-FORMS

407

˜ for m ∈ M, we define ˜ is a smooth map and ω ∈ ⊗r T ∗ M, If φ : M → M φ(m) ∗ r ∗ its pullback φ ω ∈ ⊗ Tm M as (15.6.1) φ∗ ω(v1 , . . . , vr ) = ω(T φ(m)(v1 ), . . . , T φ(m)(vr )),

v1 , . . . , vr ∈ Tm M.

∗ ˜ M Observe then that we have φ∗ (ω1 ⊗ ω2 ) = φ∗ ω1 ⊗ φ∗ ω2 for ω1 ∈ ⊗r Tφ(m) ∗ ˜ This allows us to analyze how local representatives M. and ω2 ∈ ⊗s Tφ(m) ∗ M, in local charts C j = change from one chart to the other. If ω ∈ ⊗r Tm j j (O , κ ), j = 1, 2, its representatives are of the form  j ωC = ωij1 ...ir dxi1 ⊗ · · · ⊗ dxir . 1≤i1 ,...,ir ≤d

If we set κ = = ◦ (κ1 )−1 on κ1 (O) with m ∈ O = O1 ∩ O2 , we then 1 2 have ω C = κ∗ ω C , which reads  ωi11 ...ir = (15.6.2) ∂i1 (κ)j1 (x(1) ) · · · ∂ir (κ)jr (x(1) )ωj21 ...jr , κ12

κ2

1≤j1 ,...,jr ≤d

x(1)

κ1 (m).

= ∗ M, which can be given a manWe define ⊗r T ∗ M = ∪m∈M {m} × ⊗r Tm ∗ ifold structure similar to that of T M . It is called the r-covariant tensor bundle. If U is an open set of M, a C  -r-covariant tensor field on U is a C  -section of ⊗r T ∗ M, that is, a C  -map ω : U → ⊗r T ∗ M such ∗ M. We denote by C  Λr (U ) the that ω(m) = (m, ωm ) with ωm ∈ ⊗r Tm  space of C -r-covariant tensor fields on U . On a C ∞ -manifold, we set C ∞ Λr (U ) = ∩≥0 C  Λr (U ). For r = 1, we recover one-forms as defined in Sect. 15.4, that is, C  Λ1 (U ) = C  Λ(U ). ˜ is a smooth map and ω If φ : M → M ˜ is an r-covariant tensor field on ˜ we define its pullback as (φ∗ ω ˜ )m = φ∗ (˜ ωφ(m) ), which by (15.6.1) gives, M, for v1 , . . . , vr ∈ Tm M, for

(15.6.3)

(φ∗ ω ˜ )m (v1 , . . . , vr ) = ω ˜ φ(m) (T φ(m)(v1 ), . . . , T φ(m)(vr )).

15.6.2. p-Forms. Above we defined one-forms as sections of the cotangent bundle. We now consider p ≥ 1, and we denote by Amp M the vector space of p-covariant tensors at m that are alternating, that is, if ω ∈ Amp M, then for all (v1 , . . . , vp ) ∈ (Tm M)p , we have ω(v1 , . . . , vp ) = 0 if vi = vj for some i, j ∈ {1, . . . , p}, i = j. Consequently, we have ω(vσ(1) , . . . , vσ(p) ) = (σ)ω(v1 , . . . , vp ), with σ ∈ Sp , that is, a permutation of {1, . . . , p} and (σ) its signature. In particular, we have  (15.6.4) (σ) ω(vσ(1) , . . . , vσ(p) ) = p! ω(v1 , . . . , vp ). σ∈Sr

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15. ELEMENTS OF DIFFERENTIAL GEOMETRY

∗ M and A p M = {0} if p > d. By Observe that we have Am1 M = Tm m 0 no argument. convention, we also set Am M = C, that is, forms that  take p d d! For 0 ≤ p ≤ d, the dimension of Am M is given by p = (d−p)!p! . ˜ −ω ˜ ⊗ ω ∈ Am2 M. More generally, If ω, ω ˜ ∈ Am1 M, we see that ω ⊗ ω p p˜ ˜ ∈ Am M, with p, p˜ ∈ N, we define ω ∧ ω ˜ ∈ Amr M, with if ω ∈ Am M and ω r = p + p˜, by (15.6.5) 1  (σ) ω(vσ(1) , . . . , vσ(p) ) ω ˜ (vσ(p+1) , . . . , vσ(r) ), ω∧ω ˜ (v1 , . . . , vr ) = p!˜ p! σ∈Sr

which we call the wedge or exterior product of ω and ω ˜. Proposition 15.22. Let ω ∈ Amp M and ω ˜ ∈ Amp˜ M, with p, p˜ ∈ N. The wedge product has the following properties: (1) The wedge product is bilinear and associative. (2) If p = 0, that is, ω = λ ∈ C, one has ω ∧ ω ˜ = λ˜ ω. p

For ωj ∈ Amj M, j = 1, . . . , N , one finds by induction, with r = p1 + · · · + pN , ω1 ∧ · · · ∧ ωN (v1 , . . . , vr ) =

N

 1 (σ) ωj (vσ(Pj−1 +1) , . . . , vσ(Pj−1 +pj ) ), p1 ! · · · pN ! σ∈Sr j=1

with P0 = 0 and Pj+1 = Pj + pj+1 . For p1 = · · · = pN = 1, this gives in particular  (σ) ω1 (vσ(1) ) · · · ωN (vσ(N ) ). ω1 ∧ · · · ∧ ωN (v1 , . . . , vN ) = σ∈SN

This allows us to write, for s ∈ SN , (15.6.6)  (σ) ωs(1) (vσ(1) ) · · · ωs(N ) (vσ(N ) ) ωs(1) ∧ · · · ∧ ωs(N ) (v1 , . . . , vN ) = σ∈SN

=



(σ) ω1 (vσ◦s−1 (1) ) · · · ωN (vσ◦s−1 (N ) )

σ∈SN

= (s)ω1 ∧ · · · ∧ ωN (v1 , . . . , vN ). ˜ is a smooth map and ω ∈ A p M, ˜ for m ∈ M, we If φ : M → M φ(m) p ∗ define its pullback φ ω ∈ Am M as in (15.6.1). Observe then that we have p ˜ and ω2 ∈ A q M. ˜ This allows M φ∗ (ω1 ∧ ω2 ) = φ∗ ω1 ∧ φ∗ ω2 for ω1 ∈ Aφ(m) φ(m) us to analyze how local representatives change from one chart to the other. In a local chart, a basis of Amp M is given by dxi1 ∧ · · · ∧ dxip ,

with 1 ≤ i1 < · · · < ip ≤ d.

With the local representation this basis provides, one deduces the anticommutativity property of the wedge product, which could also be obtained from the computations above.

15.6. TENSORS AND p-FORMS

409

Proposition 15.23. The wedge product is anticommutative in the sense that ˜ ∧ ω, ω∧ω ˜ = (−1)p˜p ω ˜ ∈ Amp˜ M. for ω ∈ Amp M and ω Note that for p = d, one has Amd M = span(dx1 ∧· · ·∧dxd ) and moreover dx1 ∧ · · · ∧ dxd (v1 , . . . , vd ) = det(v1 , . . . , vd ). If ω ∈ A p M, in local charts C j = (Oj , κj ), j = 1, 2, its representatives are of the form  j ωij1 ...ip dxi1 ∧ · · · ∧ dxip . ωC = 1≤i1 0, is said to be differential with a large parameter, with smooth coefficients and of order k, if (1) Pτ is local. (2) In every chart C = (O, κ), the representative of the operator is in Dτk , as introduced in Section 2.3 of Volume 1. We denote by Dτk (M) the set of all such operators. We then have  k−j τ Pj , Pτ = 0≤j≤k

where Pj , j = 0, . . . , k, is a differential operator of j on M. The principal symbol of Pτ is defined as the function T ∗ M with τ as a parameter  k−j σ(Pτ )(m, ω) = (16.3.3) τ σ(Pj )(m, ω), 0≤j≤k

where σ(Pj ) denotes the principal symbol of Pj in the sense of the differential operators of order j on M.4 The following proposition follows from Proposition 16.11 and yields a coordinate-free characterization of the principal symbol of Pτ . Proposition 16.16. Let Pτ be a differential operator with a large parameter of order k on M. Let (m0 , ω0 ) ∈ T ∗ M. If f ∈ C ∞ (M; R) is such that df (m0 ) = ω0 , we then have   σ(Pτ )(m0 , ω0 ) = lim λ−k e−iλf (m0 ) Pλτ eiλf (m) m=m0 . λ→∞

In each local chart, the representative of σ(Pτ ) coincides with the principal symbol of the representative of Pτ as defined in Sections 2.2–2.3, in the case of differential operators, and as given in (2.2.3). 1 Consider two local charts C  = (O , κ ),  = 1, 2, and let P C (x, D, τ ) and 2 P C (x, D, τ ) be the representatives of the operator Pτ in these two charts, using the notation of Section 2.3 of Volume 1. The principal symbols of 1 2 P C (x, D, τ ) and P C (x, D, τ ) are then the representative of σ(Pτ ). They thus satisfy the identity (16.3.4) σ(P C )(x, t dκ12 (x)η, τ ) = σ(P C )(κ12 (x), η, τ ), 1

2

x ∈ O1 , η ∈ Rd ,

with κ12 = κ2 ◦ (κ1 )−1 .

4If P is in fact of order less than or equal to j − 1, then σ(P ) = 0. j j

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16. INTEGRATION AND DIFFERENTIAL OPERATORS ON MANIFOLDS

An important example for us here is the case of a conjugated differential operator on M. Proposition 16.17. Let P be a differential operator of order k on M and ϕ be a smooth function on M. Let Pϕ = eτ ϕ P e−τ ϕ . It is a differential operator with a large parameter of order k on M, and its principal symbol is given by (16.3.5)

σ(Pϕ )(m, ω, τ ) = σ(P )(m, ω + iτ dϕ(m)),

where σ(P ) is the principal symbol of P as defined in Section 16.3.1. Note that (16.3.5) makes sense as t → σ(P )(m, ω+tdϕ(m)) is polynomial by Proposition 16.13. Proof. In a local chart C = (O, κ), if P C (x, D), PϕC (x, D, τ ), and ϕC C C are the representatives of P , Pϕ , and ϕ, we have PϕC = eτ ϕ P C e−τ ϕ . The discussion in the beginning of Section 9.3.1 of Volume 1 shows that PϕC is a differential operator with a large parameter of order k on κ(O). Its principal symbol is given by σ(PϕC )(x, ξ, τ ) = σ(P C )(x, ξ + iτ dϕC (x)). As Pϕ is local, this precisely means that it is a differential operator with a large parameter of order k on M and that we have (16.3.5).  Note that we could have also used either the formula of change of variables for conjugated operators as provided in Section 9.3.1 or Proposition 16.16.

CHAPTER 17

Elements of Riemannian Geometry Contents 17.1. 17.1.1. 17.1.2. 17.1.3. 17.1.4. 17.2. 17.3. 17.4. 17.4.1. 17.4.2. 17.4.3. 17.4.4. 17.5. 17.5.1. 17.5.2. 17.5.3. 17.6. 17.7.

Riemannian Structure on a Manifold Metric and Distance Representation in Local Charts Musical Isomorphisms Volume Form and Integration Gradient, Divergence, and Laplace–Beltrami Operators Canonical Positive Density Function and Divergence Formula Linear Connection and Covariant Derivatives Linear Connection Differentiation Along a Curve Extension to Tensors of All Orders The Levi-Civita Connection Geodesics and Geodesic Flows Tangent Geodesic Flow Length Minimization The Hamiltonian Geodesic Flow Normal Geodesic Coordinates at the Boundary Higher-Order Covariant Derivatives

438 438 438 439 440 442 445 447 448 448 449 450 451 452 454 454 455 460

This chapter is devoted to the exposition of some facts of Riemannian geometry. Basic aspects of differential geometry are exposed in Chap. 15. One goal is to properly define the Laplace–Beltrami operator. As second goal is to give the necessary tools to prepare for Chap. 18 where Sobolev spaces on a Riemannian manifold are presented and where the elliptic properties of the Laplace–Beltrami operator are studied.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 17

437

438

17. ELEMENTS OF RIEMANNIAN GEOMETRY

An interested reader should consult a more thorough reference, such as [120, 189, 221] for Riemannian geometry. 17.1. Riemannian Structure on a Manifold We consider a smooth manifold M of dimension d. Having a Riemannian structure on M means having a way to measure distances on the manifold. 17.1.1. Metric and Distance. On Rd , the Euclidean distance or anisotropic versions of it can be expressed by a quadratic form. For manifolds, this is generalized by providing a quadratic form at every point m ∈ M, with some smoothness requirement. A quadratic form in Rd can be represented by its associated bilinear form that acts on vectors of Rd . The same is done on M with an action on tangent vectors at m. Definition 17.1. A Riemannian metric is a smooth 2-covariant tensor field g on M that satisfies the following properties, for any m ∈ M and v, w ∈ Tm M, (1) symmetry: we have gm (v, w) = gm (w, v); (2) positivity: we have gm (v, v) > 0, for v = 0. A couple (M, g) is called a Riemannian manifold. The metric g thus yields for each m ∈ M an inner product on Tm M and the following associated norm |v|gm = gm (v, v)1/2 ,

v ∈ Tm M.

Having such a metric allows one to define the length of a C 1 -path γ : [a, b] → M: b

(17.1.1)

1/2 ˙ γ(t)) ˙ dt, (γ) = ∫ gγ(t) (γ(t),

where γ˙ =

a

d γ, dt

that is independent of the chosen (injective) parametrization γ. For M connected, which we shall always assume here, we define the distance (17.1.2)

distg (x, y) =

inf

γ∈P(x,y)

(γ),

x, y ∈ M,

where P(x, y) is the set of all the C 1 -paths γ : [0, 1] → M such that γ(0) = x and γ(1) = y. This distance yields a metric space structure for (M, g) and the induced topology is equivalent to the manifold topology. Connected to the notion of distance is that of geodesics. They are covered in Sect. 17.5. 17.1.2. Representation in Local Charts. In a local chart C = (O, κ), for every m ∈ M and x = κ(m), the local representative of gm, that is, gxC = C (x) (κ−1 )∗ gm identifies with a symmetric positive definite matrix gij 1≤i,j≤d and  C gm (v, w) = gij (x)v i wj , 1≤i,j≤d

17.1. RIEMANNIAN STRUCTURE ON A MANIFOLD

439

where (v i )1≤i≤d and (wi )1≤i≤d are the coordinates of the representatives of v and w in the local chart, that is,  C gij (x)dxi ⊗ dxj . g C (x) = gxC = 1≤i,j≤d

We shall most often use the notation gxC close to the notation we use for covariant tensors but at places we shall use g C (x) if the symbol of the metric has a subscript. In the case of the Euclidean space Rd , the metric is simply  dxi ⊗ dxi , 1≤i≤d

often called the flat metric. Let C k = (Ok , κk ), k = 1, 2, be two local charts such that m ∈ O1 ∩  C k (k)  k 2 (x ) 1≤i,j≤d be the local matrix O . For x(k) = κk (m) let gxC(k) = gij k representatives of g in C . The formula expressing the action of a change of coordinates on 2-covariant tensor fields (15.6.2) gives    C 1 (1) C 2 (2) (x ) = (x ), ∂xi κ ∂xj κq (x(1) ) gq gij 1≤,q≤d

with κ = κ2 ◦ (κ1 )−1 . In matrix form this reads gxC(1) = t κ (x(1) ) gxC(2) κ (x(1) ). 1

2

In particular, this gives   1 1/2 2 (17.1.3) = | det κ (x(1) )| det gxC(2) )1/2 . det gxC(1) ∗ M we define ω  ∈ T M 17.1.3. Musical Isomorphisms. For ω ∈ Tm m

by ω, v = gm (ω  , v),

v ∈ Tm M.

∗ M → T M and its inverse  : T M → T ∗ M are called The map  : Tm m m m musical isomorphisms. In a local chart C = (O, κ) with m ∈ O and x = κ(m), this reads  C,ij g (x)ωj , (ω  )i = 1≤j≤d

(g C,ij (x))

C where 1≤i,j≤d is the inverse of the matrix (gij (x))1≤i,j≤d . Similarly, for v ∈ Tm M we have  C (v )i = gij (x)v j . 1≤j≤d ∗M We then define the following bilinear form and associated norm on Tm ∗ (and more generally on T M)

(17.1.4)

(ω, ω  )gm = gm (ω  , ω  ) = ω, ω  ,

|ω|2gm = (ω, ω)gm ,

440

17. ELEMENTS OF RIEMANNIAN GEOMETRY

∗ M. In local coordinates, we have for ω, ω  ∈ Tm  g C,ij (x)ωi ωj . (ω, ω  )gm = 1≤i,j≤d

The musical isomorphism  and  are introduced here for cotangent and tangent vectors, respectively; their action is naturally extended to one-forms and vector fields. 17.1.4. Volume Form and Integration. The Riemannian structure on M allows one to choose a particular volume form if M is orientable. Let us consider an atlas A = (Oi , κi )i∈I for which M is oriented. In any local chart C i = (Oi , κi ) of this atlas, we consider (17.1.5)

dv i (x) = (det gxC )1/2 dx1 ∧ · · · ∧ dxd . i

By (15.6.7) and (17.1.3) we find that (17.1.6)

(κij )∗ dv j = dv i ,

κij = κj ◦ (κi )−1 ,

as det(dκij (x)) > 0 here. For m ∈ M, we can thus set (17.1.7)

(dvg )m = (κi )∗ dv i (x) ∈ Amd M,

x = κi (m),

if m ∈ Oi , as this d-form at m is independent of the choice of i ∈ I by (17.1.6). The d-form dvg : m → (m, (dvg )m ) has the properties of a volume form (see Sect. 16.1). We call it the canonical Riemannian volume form. If N is a submanifold of M, the metric g induces a smooth 2-covariant tensor field gN on N since Tm N naturally injects in Tm M (see Sect. 15.5). One can readily check that the properties of Definition 17.1 are fulfilled, meaning that gN is a Riemannian metric on N . If N is orientable this metric can yield a canonical volume form once an orientation is chosen. The particular case of the boundary ∂M of M in the case M is oriented is of interest. In fact, once a smooth outward pointing vector field ν is chosen along ∂M, the canonical volume form dvg on M (associated with the orientation of M, see above) yields a volume form (dvg )∂ on ∂M and thus an orientation (see Sect. 16.1). However, the induced metric g∂ along with this orientation yields a canonical volume form dvg∂ on ∂M. These two volume forms coincide for the following natural choice of the vector field ν along ∂M. Proposition 17.2. Let ν be the unique outward pointing vector field along ∂M such that, for all m ∈ ∂M, gm (νm , νm ) = 1 and gm (νm , u) = 0 for all u ∈ Tm ∂M, that is, ν is unitary and orthogonal to Tm ∂M in the sense of g. Let the volume form (dvg )∂ on ∂M be given at m by (dvg )∂ (u1 , . . . , ud−1 ) = dvg (ν, u1 , . . . , ud−1 ),

u1 , . . . , ud−1 ∈ Tm ∂M.

Denote by dvg∂ the canonical Riemannian volume form associated with the induced metric g∂ on ∂M and with the orientation of ∂M given by the volume form (dvg )∂ .

17.1. RIEMANNIAN STRUCTURE ON A MANIFOLD

441

(1) We have (dvg )∂ = dvg∂ . Moreover, if C = (O, κ) is a local chart such that O∩∂M = ∅ and U+ = κ(O) is an open set of Rd+ , we have, writing κ(m) = x = (x , xd ),  1/2  1/2 C,d  C (2) det g(x = det g∂C (x ) /|ν (x , 0)|;  ,0) C,d C,d  (3) gm (u, ν) = −u /|ν (x , 0)| if u ∈ Tm M and m ∈ ∂M. Proof. To prove the first item, it suffices to prove that |(dvg )∂ | = |dvg∂ | since both volume forms share the same orientation. We choose a local chart C = (O, κ) near m ∈ ∂M where M is given by {xd ≥ 0}, that is U+ = κ(O) is an open set of Rd+ . We set (x , 0) = κ(m). We set ∂U+ = {xd = 0} ∩ U+ . We have T(x ,0) U+ ∼ = Rd and let (ej )1≤j≤d be the canonical basis of Rd . d−1 (canonically) injects in T(x ,0) U+ ∼ As T(x ,0) ∂U+ ∼ =R = Rd , a vector u ∈ T(x ,0) ∂U+ viewed as a vector u of Rd , that is u = (u1 , . . . , ud ) is characterized by ud = 0. Then, (ej )1≤j≤d−1 is a basis of T(x ,0) ∂U+ . For such a vector u = (u1 , . . . , ud−1 , 0) ∈ T(x ,0) ∂U+ we shall write u = (u1 , . . . , ud−1 ) the associated vector in Rd−1 . In what follows we simply write ν C in place of ν C (x , 0). The tangent C C C vector ν C = (ν 1 , . . . , ν d ) is such that g(x  ,0) (ν , ν ) = 1 and C C g(x  ,0) (ν , u) = 0,

u = (u , 0).

C The matrix (Gi,j )1≤i,j≤d of the bilinear form g(x  ,0) (., .) in this basis C  C ˜ i,j )1≤i,j≤d−1 given by Gi,j = g (x , 0) = g  (ei , ej ). We denote by (G ij

(x ,0)

˜ i,j = Gi,j , 1 ≤ i, j ≤ d − 1, that is, the submatrix given by G ⎛ ⎜ ⎜ G=⎜ ⎜ ⎝

˜ G G1,d

· · · Gd−1,d

⎞ G1,d ⎟ .. ⎟ . ⎟. Gd−1,d ⎟ ⎠ Gd,d

(17.1.8)

 At (x , 0), we have dvgC = det(G)dx1 ∧ · · · ∧ dxd . Then, on the one hand, at (x , 0) we have, for u1 , . . . , ud−1 ∈ T(x ,0) ∂U+ , from Sect. 16.1,  (dvg )C∂ (u1 , . . . , ud−1 ) = dvgC (ν C , u1 , . . . , ud−1 ) = det(G) det(ν C , u1 , . . . , ud−1 )  = (−1)d+1 ν d det(G) det(u1 , . . . , ud−1 ), ˜ is that of the since ud1 = · · · = udd−1 = 0. On the other hand, the matrix G induced metric g∂ on ∂M at m. We thus " find that the associated volume C 1 ∧ · · · ∧ dxd−1 | at (x , 0), ˜ form dvg on ∂M is such that |dvg | = det(G)|dx ∂

∂

implying, for u1 , . . . , ud−1 ∈ T(x ,0) ∂U+ , " C   ˜ | det(u , . . . , u )|. |dvg∂ |(u1 , . . . , ud−1 ) = det(G) 1 d−1

442

17. ELEMENTS OF RIEMANNIAN GEOMETRY

It thus remains to prove that

  ˜ = ν d 2 det(G), det(G)

(17.1.9)

which is precisely the second item of the proposition. To that purpose, we consider the basis (e1 , . . . , ed−1 , ν) of Rd . The maC trix (Hi,j )1≤i,j≤d of the bilinear form g(x  ,0) (., .) in this second basis and the invertible matrix P representing the change of basis are

⎛ ⎜ ⎜ H =⎜ ⎜ ⎝

˜ G 0 ··· 0

0 .. . 0 1

⎞

⎛

⎟ ⎟ ⎟, ⎟ ⎠

⎜ ⎜ P =⎜ ⎜ ⎝

⎞

Idd−1 0

··· 0

ν1 ⎟ .. ⎟ . ⎟. ⎟ ν d−1 ⎠ νd

(17.1.10)

  ˜ = ν d 2 det(G). We have H = t P G P yielding det(G) We now prove the last item of the proposition. We note that |ν d | = −ν d as the vector field ν is outward pointing (see Sect. 16.1.3). We thus need to prove that ud /ν d = g C (ν C , u) for any u ∈ Rd . Let u be the column vector formed by the coordinates of u in the second basis (e1 , . . . , ed−1 , ν C ). We have t u = P u yielding ud = ν d ud . Next, using the second basis we find  g C (ν C , u) = (0, . . . , 0, 1)Hu = ud . Remark 17.3. Note that this proof can be simplified by using normal geodesic coordinates as introduced in Section 9.4 of Volume 1. However, the proof of the existence of such coordinates is more involved than the above argument that is only based on linear algebra. 17.2. Gradient, Divergence, and Laplace–Beltrami Operators In the Euclidean space we have ∇f (m) · v = df (m)(v) = v(f ), for any vector v at m, yielding a definition of the gradient vector. On (M, g), this definition remains unchanged, yet replacing the inner product by the bilinear form g. Hence the gradient at m of a C k -function f : M → C at m is defined as the unique ∇g fm ∈ Tm M such that (17.2.1)

gm (∇g fm , v) = v(f ) = df (m)(v),

∀v ∈ Tm M.

Letting m vary we obtain a C k−1 -vector field ∇g f : m → ∇g fm on M. In a local chart C = (O, κ), the representative of ∇g fm is given by, with x = κ(m),  C,ij (17.2.2) g (x)∂xj ((κ−1 )∗ f )(x), (∇g fm )C,i = 1≤j≤d C (x)) where (g C,ij (x))1≤i,j≤d is the inverse of the matrix (gij 1≤i,j≤d of the local representative of the metric. With the musical notation (see Sect. 17.1.3) we have ∇g fm = df (m) . For two smooth functions f1 and f2 one has

(17.2.3)

∇g (f1 f2 ) = f1 ∇g f2 + f2 ∇g f1 .

17.2. GRADIENT, DIVERGENCE, AND LAPLACE–BELTRAMI OPERATORS

443

For a smooth vector field u on an oriented Riemannian manifold M we define its divergence divg u as the unique function that satisfies ∫ ϕ divg u dvg = − ∫ g(∇g ϕ, u)dvg ,

(17.2.4)

M

M

D ∞ (M),

for any ϕ ∈ with the volume form dvg defined above. In a local chart C = (O, κ), if ui (x) are the coordinates of the local representative of u, uC = κ∗ u, we have, by (17.1.5) and (17.1.7),    κ∗ divg u(x) = (det gxC )−1/2 ∂xi (det gxC )1/2 ui (x). (17.2.5) 1≤i≤d

From (17.2.3) and (17.2.4) we find, for a smooth function f and a smooth vector field u, (17.2.6)

divg (f u) = f divg u + g(∇g f, u) = f divg u + u(f ).

The Laplace–Beltrami operator acting on C 2 -functions is then defined as the composition of the gradient and the divergence operators: Δg f = divg ∇g f, yielding a function on M. In particular, if f1 and f2 are two C 2 -functions supported away from the boundary of M, from (17.2.4) one finds (17.2.7)

∫ f1 Δg f2 dvg = − ∫ g(∇g f1 , ∇g f2 )dvg = ∫ (Δg f1 )f2 dvg .

M

M

M

In a local chart C = (O, κ), from (17.2.2) and (17.2.5), we have (17.2.8) (κ−1 )∗ (Δg f )(x)

  ∂xi (det gxC )1/2 g C,ij (x)∂xj ((κ−1 )∗ f ) (x).



= (det gxC )−1/2

1≤i,j≤d

One sees that the Laplace–Beltrami operator fulfills the definition of differential operator on M in the sense of Sect. 16.3. The principal symbol of Δg is a function on T ∗ M by Proposition 16.10. It is given by σ(Δg )(m, ω) = −|ω|2gm , with |.|gm as defined in (17.1.4). In the local chart C, it is given by  g C,ij (x)ξi ξj . − 1≤i,j≤d

Note that the local representative of the Laplace–Beltrami operator is often written    ∂xi (det g)1/2 g ij ∂xj f , Δg f = (det g)−1/2 1≤i,j≤d

by abuse of notation (compare with (17.2.8)). With a similar abuse of notation for the gradient operator one writes  ij g ∂xj f, i = 1, . . . , d. (∇g f )i = 1≤j≤d

(See, for instance, Chap 5).

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17. ELEMENTS OF RIEMANNIAN GEOMETRY

When considering smooth vector fields, we obtain the classical divergence theorem. Proposition 17.4. Let (M, g) be an oriented Riemannian manifold with boundary and let u be a smooth compactly supported vector field on M. Let ν be the unique outward pointing vector field along ∂M such that, for all m ∈ ∂M, gm (νm , νm ) = 1 and gm (νm , u) = 0 for all u ∈ Tm ∂M, that is, ν is unitary and orthogonal to Tm ∂M in the sense of g. We have ∫ divg (u)dvg = ∫ g(u, ν)(dvg )∂ .

M

∂M

Compare with (16.1.6) that states this result in the flat case. Proof. Arguing as in the beginning of the proof of the Stokes formula (Theorem 16.2), it suffices the consider the vector field u supported in a single local chart C = (O, κ) that meets the boundary in an atlas compatible with the chosen orientation of M. We set U+ = κ(O). It is an open set of Rd+ , that is, we have U+ = Rd+ ∩ U with U an open set of Rd . By abuse of notation, we write u, g, and dvg to denote the local representatives of the vector field, the metric and the volume form. We then have ∫ divg (u)dvg = ∫ divg (u)(det g)1/2 dx = ∫ div((det g)1/2 uj )dx.

M

U+

U+

Applying the divergence formula in Rd , see (16.1.6), we obtain ∫ divg (u)dvg = ∫ (−1)d−1 (det g)1/2 ud dx1 ∧ · · · ∧ dxd−1

M

∂U+

= − ∫ ud (det g)1/2 dx . ∂U+

We have (det g)1/2 (x , 0) = (det g∂ (x ))1/2 /|ν d (x , 0)| where g∂ is the restriction of the metric to ∂U+ by the second item of Proposition 17.2, yielding ∫ divg (u)dvg = − ∫

M

∂U+

ud (det g∂ )1/2 dx . |ν d |

The third item of Proposition 17.2 states −ud /|ν d | = g(u, ν) implying ∫ divg (u)dvg = ∫ g(u, ν)(det g∂ )1/2 dx = ∫ g(u, ν)(dvg∂ )

M

∂U+

∂M

= ∫ g(u, ν)(dvg )∂ , ∂M

using the first item of Proposition 17.2 for the two equivalent formulations  of the volume form (dvg )∂ , which is the sought result. We finish this section by the observation that the gradient, the divergence, and the Laplace–Beltrami operators are truly of geometrical nature observing how they transform through the action of a diffeomorphism.

17.3. CANONICAL POSITIVE DENSITY FUNCTION. . .

445

Proposition 17.5. Let (M1 , g1 ) and (M2 , g2 ) be two oriented Riemannian manifolds and φ : M1 → M2 a diffeomorphism such that φ∗ g2 = g1 . Then, we have κ∗ ∇g1 κ∗ = ∇g2 ,

divg1 = κ∗ divg2 κ∗ ,

Δg1 κ∗ = κ∗ Δg2 .

17.3. Canonical Positive Density Function and Divergence Formula Consider an atlas A = (Oi , κi )i∈I . In any local chart C i = (Oi , κi ), we define the function f C (x) = (det g C )1/2 on κi (Oi ). i

i

By (17.1.3), we see that the change of variable formula (16.2.4) for density functions is satisfied, thus yielding by (16.2.6) a smooth positive density i function μg such that its local representative is precisely the functions f C , i that is, μCg = Tf Ci (using the notation of Chapter 8 of Volume 1 for L1loc functions identified with distributions). Such a positive density function allows one to identify functions and density functions on M through the bijective map φ → φμg in a canonical way. The canonical density function allows to define L1 -functions (resp. L1loc ) on M as the functions f such that f μg is an L1 -density functions (resp. L1loc ). We denote by L1 (M) (resp. L1loc (M)) the space of such functions. Above, the divergence of a vector field was defined through the integration of d-forms on an oriented Riemannian manifold. On a nonoriented Riemannian manifold, the integration of densities allows one to have a similar definition that yields the same form in local coordinates. Indeed, for u a C 1 -vector field, the map Su : φ → − ∫ u(φ)μg = − ∫ g(∇g φ, u)μg , M

M

φ ∈ 0 Dc∞ (M),

is a density distribution. A computation shows that Su is in fact a density function with Su = divg u μg with divg u ∈ C 0 (M) given in local charts by (17.2.5). This yields the following identity defining divg u on a nonoriented manifold ∫ φ divg u μg = − ∫ u(φ)μg ,

M

M

φ ∈ 0 Dc∞ (M).

With the canonical positive density μg and this definition of the divergence we can state the following form of the divergence theorem in the context of integration of densities on nonoriented manifolds. Proposition 17.6 (Divergence Formula). Let (M, g) be a Riemannian manifold with boundary and let u be a smooth compactly supported vector field on M. Let ν be the unique outward pointing vector field along ∂M such that, for all m ∈ ∂M, gm (νm , νm ) = 1 and gm (νm , v) = 0 for all v ∈ Tm ∂M, that is, ν is unitary and orthogonal to Tm ∂M in the sense of g. We have

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17. ELEMENTS OF RIEMANNIAN GEOMETRY

∫ divg (u)μg = ∫ g(u, ν)μg∂ ,

M

∂M

where g∂ is the induced Riemannian metric on ∂M and μg∂ the resulting canonical positive density function on ∂M. We insist on the fact that here, as opposed to Proposition 17.4, M and ∂M need not be oriented. Proof. Arguing as in the beginning of the proof of Theorem 16.2, it suffices the consider the vector field u supported in a single local chart C = (O, κ) that meets the boundary. In this single chart, an orientation can be chosen, allowing to identify density functions and d-forms (see the last paragraph of Sect. 16.2.4). Then the result of Proposition 17.4 applies.  If X ∈ C ∞ V (M), that is, X is a smooth vector field it can be viewed as a first-order differential operator on M and one can compute t X, the associated transpose operator as defined in Sect. 16.3.3. Proposition 17.7 (Transposition of a Vector Field). Let X ∈ C ∞ V (M). Then t X = −X − divg X. Moreover, we have ∫ (Xf1 )f2 μg = − ∫ f1 (Xf2 + f2 divg X)μg + ∫ f1 f2 g(X, ν)μg∂ ,

M

M

∂M

for f1 and f2 smooth compactly supported functions on M. Proof. With (17.2.6) we write divg (f1 f2 X) = f1 divg (f2 X) + f2 (Xf1 )   = f1 f2 divg X + X(f2 ) + f2 (Xf1 ). 

With Proposition 17.6 the result follows.

Proposition 17.8 (Green Formula). We have t Δg = Δg . Moreover, we have the Green formula ∫ f1 Δg f2 μg + ∫ g(∇g f1 , ν)f2 μg∂ = ∫ (Δg f1 )f2 μg + ∫ f1 g(∇g f2 , ν)μg∂ ,

M

M

∂M

∂M

for f1 and f2 smooth compactly supported functions on M. Proof. We have divg (f1 ∇g f2 ) = f1 Δg f2 + g(∇g f1 , ∇g f2 ) by (17.2.6). From Proposition 17.6 we deduce (17.3.1)

∫ f1 Δg f2 μg = − ∫ g(∇g f1 , ∇g f2 )μg + ∫ f1 g(∇g f2 , ν)μg∂ .

M

M

∂M

Writing the same identity after interchanging the rˆ oles of f1 and f2 and subtracting the two equations yields the Green formula. For two functions f1 , f2 ∈ 0 Dc∞ (M) we naturally recover (17.2.7), (17.3.2)

∫ f1 Δg f2 μg = ∫ (Δg f1 )f2 μg .

M

M

Yet, here integration concerns densities instead of d-forms. From the definition of the transpose of an operator in Sect. 16.3.3 we have t Δg = Δg . 

17.4. LINEAR CONNECTION AND COVARIANT DERIVATIVES

447

Anticipating upon the definition of the L2 -inner product recalled in Sect. 18.1, changing f2 into f2 the Green formula reads (f1 , Δg f2 )L2 (M) + (∂ν f1 |∂M , f2 |∂M )L2 (∂M) = (Δg f1 , f2 )L2 (M) + (f1 |∂M , ∂ν f2 |∂M )L2 (∂M) , with ∂ν w|∂M = g(∇g w, ν)|∂M . A formula expressing the transposition of the Laplace–Beltrami operator in the sense of distribution is given in Sect. 18.3. An extension of the Green formula for functions in H 2 (M) is given in Proposition 18.30 at the end of Sect. 18.5 after traces of Sobolev functions are properly defined. A Green formula for a function f that is only L2 is given in Lemma 18.34 under the requirement that Δg f ∈ L2 (M). We say that a vector field u is L1 (resp. L1loc ) if g(u, u)1/2 is a L1 (resp. function on M. For such nonsmooth functions with support away from the boundary we have the following result that will be of use in what follows.

L1loc )

Proposition 17.9. Let (M, g) be a Riemannian manifold with boundary and let u be a compactly supported L1 -vector field on M such that supp u ∩ ∂M = ∅ and divg u ∈ L1 (M). We have ∫M divg (u)μg = 0. Proof. Arguing as in the beginning of the proof of Theorem 16.2, it suffices to consider the vector field u supported in a single local chart C = (O, κ). In this chart, using the local form of the divergence given by (17.2.5), we thus have to prove that    ∂xi (det g C )1/2 ui (x) dx J= ∫ κ(O) 1≤i≤d

vanishes if for each i = 1, . . . , d, supp(ui ) ⊂ int κ(O), ui ∈ L1 (int κ(O)), and  moreover 1≤i≤d ∂xi wi ∈ L1 (int κ(O)), with wi = (det g C )1/2 ui . For χ ∈ Cc∞ (int κ(O)) such that χ ≡ 1 in a neighborhood of supp(ui ), i = 1, . . . , d we write   ∂xi wi , χD  (Rd ),Cc∞ (Rd ) = − wi , ∂xi χD  (Rd ),Cc∞ (Rd ) = 0, J= 1≤i≤d

1≤i≤d

as ∂xi χ = 0 in a neighborhood of supp(wi ).



17.4. Linear Connection and Covariant Derivatives A linear connection is a natural way to formalize the directional differentiation of a vector field. In particular, it allows one to formulate the differentiation of a vector field along a curve. Recall that C ∞ V (M) is the space of smooth vector fields over M.

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17. ELEMENTS OF RIEMANNIAN GEOMETRY

17.4.1. Linear Connection. A linear connection is a map D : C ∞ V (M) × C ∞ V (M) → C ∞ V (M), (u, v) → D(u, v) = Du v that satisfies the following properties: (1) D is linear over C ∞ (M) w.r.t. its first argument, that is, Df u+gu˜ v = f Du v + g Du˜ v, C ∞ (M)

and u, u ˜, v ∈ C ∞ V (M). for f, g ∈ (2) D is linear over R w.r.t. its second argument, that is, v ) = λ Du v + μ Du v˜, Du (λv + μ˜ for λ, μ ∈ R and u, v, v˜ ∈ C ∞ V (M). (3) D acts as a derivation w.r.t. its second argument, that is, Du (f v) = f Du v + u(f ) v, for f ∈ C ∞ (M) and u, v ∈ C ∞ V (M). For such a connection, in a local chart C = (O, κ), setting in O the vector fields ei , i = 1, . . . , d, such that their local representatives are ∂xi , for i, j ∈ {1, . . . , d}, there exists Γkij , k = 1, . . . , d, such that  k (17.4.1) Γij (m)ek (m). Dei (ej )(m) = 1≤k≤d

The real functions Γkij , for i, j, k ∈ {1, . . . , d}, defined on O are the socalled Christoffel symbols associated with the connection D. The Christoffel symbols fully characterize the connection D. For two vector fields u and v, the coordinates of the representative of Du v are given by  C,i  (Du v)C,k = (17.4.2) u ∂xi v C,k + uC,i v C,j Γkij , 1≤i≤d

1≤i,j≤d

at x = κ(m) with Γkij = Γkij (m). We observe that for m ∈ M, the tangent vector (Du v)m only depends on um . This appears quite clearly in (17.4.2). We may in fact define a connection as a map Tm M × C ∞ V (M) → Tm M for all m ∈ M. For u ∈ Tm M, one calls Du v the covariant derivative of v with respect u (at m). 17.4.2. Differentiation Along a Curve. As mentioned above a primary use of a connection is to differentiate vector fields along curves. Let γ : I → M is a smooth curve, I an interval of R. Consider v : I → T M such that v(t) ∈ Tγ(t) M, that is, a vector field along γ. For t0 ∈ I, there exist ε > 0, a neighborhood V of γ(t0 ) in M and a vector field v˜ on V such that v(t) = v˜γ(t) for t ∈ (t0 − ε, t0 + ε). If given a connection one introduces the following differentiation of v along γ: (17.4.3)

Dv = D dγ (t0 ) v˜ at the point γ(t0 ). dt dt

17.4. LINEAR CONNECTION AND COVARIANT DERIVATIVES

With (17.4.2) one observes that this definition of choice of v˜.

Dv dt

449

is independent of the

17.4.3. Extension to Tensors of All Orders. The action of Du on C ∞ V (M) can be extended to tensors of all orders. For a tensor of order zero, that is a smooth function on M, for u ∈ Tm M, we set (17.4.4)

Du f (m) = um (f ) = df (m)(u).

For a one-form ω we set (Du ω)(v) = u(ω(v)) − ω(Du v),

v ∈ C ∞ V (M),

meaning that we have the product law u(ω(v)) = (Du ω)(v) + ω(Du v). We observe that Du ω is a one-form, meaning that its action on v at m only depends on vm . This can be seen by computing the (Du ω)(v) in a local chart C for instance, with uC = ui ∂xi , v C = v j ∂xj and ω C = ωk dxk , (17.4.5) (Du ω)(v) =

 1≤i,j≤d

=



uj ∂xj (ωi v i ) −



1≤i,j≤d

v u ∂ x j ωi − i j

1≤i,k≤d





ωi (uj ∂xj v i +

uj v k Γijk )

1≤k≤d

ωi v k uj Γijk .

1≤i,j,k≤d

In particular, we may define (Du ω)(v) at m for u, v ∈ Tm M without requiring u and v to be vector fields. As a generalization, for θ a r-covariant,s-contravariant tensor field we set (17.4.6)   (Du θ)(v1 , . . . , vr , ω1 , . . . , ωs ) = u θ(v1 , . . . , vr , ω1 , . . . , ωs )  − θ(v1 , . . . , Du vi , . . . , vr , ω1 , . . . , ωs ) 1≤i≤r

−



θ(v1 , . . . , vr , ω1 , . . . , Du ωi , . . . , ωs ),

1≤j≤s

for v1 , . . . , vr ∈ C ∞ V (M) and ω1 , . . . , ωs smooth one-forms. With a computation similar to (17.4.5), we see that (Du θ)(v1 , . . . , vr , ω1 , . . . , ωs ) evaluated at m ∈ M is only a multilinear function of um , (v1 )m , . . . , (vr )m and (ω1 )m , . . . , (ωs )m . This shows that Du θ is a r-covariant,s-contravariant tensor at m and (u, v1 , . . . , vr , ω1 , . . . , ωs ) → (Du θ)(v1 , . . . , vr , ω1 , . . . , ωs ) is a (r + 1)-covariant,s-contravariant tensor at m. For a tangent vector field v, we set D v to be the 1-covariant,1-contravariant tensor field given by, at a point m, D v(u, ω) = (Du v)(ω) = ω, Du v, ∗ M. In a local chart, this yields for u ∈ Tm M and ω ∈ Tm  j k (D v)ki = (Dei v)k = ∂xi v k + (17.4.7) v Γij . 1≤j≤d

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17. ELEMENTS OF RIEMANNIAN GEOMETRY

Similarly, for a one-form ω, we set D ω to be the 2-covariant tensor field by, at a point m, D ω(v, u) = (Du ω)(v) = Du ω, v, for u, v ∈ Tm M. In a local chart, this yields (17.4.8)



(D ω)ij = (Dej ω)i = ∂xj ωi −

ωk Γkji .

1≤k≤d

More, generally for θ a r-covariant,s-contravariant tensor field, we set D θ to be the (r + 1)-covariant,s-contravariant tensor field given by, D θ(v1 , . . . , vr , u, ω1 , . . . , ωs ) = (Du θ)(v1 , . . . , vr , ω1 , . . . , ωs ), for v1 , . . . , vr ∈ C ∞ V (M) and ω1 , . . . , ωs smooth one-forms. In a local chart, this yields (17.4.9)

...js ...js = (De θ)ji11...i (D θ)ji11...i r r ...js )− = ∂x (θij11...i r

+

 1≤n≤s 1≤k≤d

 1≤n≤r 1≤k≤d

j ...j

Γjkn θi11...irn−1

...js Γkin θij11...i n−1 k in+1 ...ir k jn+1 ...js

,

which is a generalization of (17.4.7) and (17.4.8). For a function f we find D f = df . 17.4.4. The Levi-Civita Connection. For a connection D on M its torsion is defined as the map T : C ∞ V (M) × C ∞ V (M) → C ∞ V (M) given by T (u, v) = Du v − Dv u − [u, v], where [., .] denotes the Lie bracket of two vector fields. A connection is said to be torsion free or symmetric if T vanishes identically. In the case of a Riemannian manifold, one says that the connection D is compatible with the metric g if one has     u g(v, w) = Du g(v, w) = g(Du v, w) + g(v, Du w), (17.4.10) for u, v, w ∈ C ∞ V (M). With (17.4.6) this means that Du g vanishes for any vector field u. The following result is fundamental in Riemannian geometry. Theorem 17.10 (Levi-Civita Connection). If (M, g) is a smooth Riemannian manifold, there exists a unique torsion free linear connection D that is compatible with the metric g. This connection is called the LeviCivita connection.

17.5. GEODESICS AND GEODESIC FLOWS

451

We refer to [221] for a proof. In particular, one can find therein a clear explanation of the reason for this choice of connection. In each chart C = (O, κ) the Christoffel symbols associated with the Levi-Civita connection are given by 1  C,k C C C (17.4.11) g (∂xi gj + ∂xj gi − ∂x gij ). Γkij = 2 1≤≤d On a Riemannian manifold (M, g), if D is the Levi-Civita connection, the compatibility between D and g simply reads D g ≡ 0. We also have ∇g f = (df ) = (D f ) . Let ν be the unique outward pointing vector field along ∂M such that, for all m ∈ ∂M, gm (νm , νm ) = 1 and gm (νm , u) = 0 for all u ∈ Tm ∂M. In (5.3.1) in Sect. 5.3, for m ∈ ∂M we have defined ∂ν f (m) to be νm (f ) = df (m)(νm ). Note that we have ∂ν f (m) = gm (∇g (f )m , νm ). We see that we may also write the normal derivative by means of the connection: ∂ν f (m) = D fm (νm ) = Dνm f (m).

(17.4.12)

17.5. Geodesics and Geodesic Flows Geodesics can be defined once provided a connection. Here, we consider the Levi-Civita connection given by Theorem 17.10. Some results are stated without proofs and we refer the interested reader to [120, 189, 221]. For a curve t → x(t), as is done classically, we denote d2 x dx and v(t) ˙ =x ¨(t) = 2 . dt dt Recall that T M is a smooth manifold; see Sect. 15.3.3. In a local chart C = (O, κ), T M is locally diffeomorphic to κ(O) × Rd . Then t → x(t), v(t) is a curve on T M. One deduces that   v(t), v(t) ˙ ∈ T(x(t),v(t)) T M. v(t) = x(t) ˙ =

The definition of a geodesic is based on the differentiation of vector fields along curves given in (17.4.3). In a local chart, from (17.4.2) and (17.4.3), we have  D x˙ =x ¨+ (17.5.1) Γjk (x)x˙ k x˙  ∂j . dt 1≤j,k,≤d Definition 17.11. A curve γ : I → M, with I an interval of R, is called a geodesic if D γ˙ =0 dt for γ(t) ˙ =

d dt γ(t)

∈ Tγ(t) M.

t ∈ I,

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17. ELEMENTS OF RIEMANNIAN GEOMETRY

In other words geodesics are curves with vanishing ‘acceleration’. If t → γ(t) is a geodesic, then t → |γ(t)| ˙ g is constant since, by (17.4.4) and (17.4.10), we have     d d 2 ˙ γ(t) ˙ = Dγ(t) ˙ γ(t) ˙ |γ(t)| ˙ gγ(t) γ(t), gγ(t) γ(t), ˙ g = dt dt   = 2gγ(t) Dγ(t) γ(t), ˙ γ(t) ˙ ˙  D γ(t), ˙ γ(t) ˙ = 0. = 2gγ(t) dt Setting c = |γ(t)| ˙ g , with the length of a curve given as in (17.1.1), the length of the geodesic joining γ(t0 ) and γ(t1 ) is then simply c|t1 − t0 |. Remark 17.12. In Definition 17.11 where a geodesic is characterized as a curve with vanishing ‘acceleration’ one sees that the parametrization of the curve is essential. Hence a geodesic is not defined here as the set of points visited by γ, Gγ = {γ(t); t ∈ I}, but the map itself. If fact, a change of parametrization will not affect Gγ yet can yield Ddtγ˙ = 0. Having Ddtγ˙ = 0 gives a second-order system; cast into a first-order system, it takes the form  d d γ = v, v=− (17.5.2) Γjk γ˙ k γ˙  ∂j . dt dt 1≤j,k,≤d From the Cauchy–Lipschitz theorem, given (m0 , v 0 ) ∈ T M we see that there exists a unique maximal geodesic that goes through (m0 , v 0 ). If t → γ(t) is a geodesic we saw above that     v(t), v(t) ˙ = γ(t), ˙ γ¨ (t) ∈ T(γ(t),v(t)) T M,  with v(t) ˙ = − 1≤j,k,≤d Γjk (γ(t))v k (t)v  (t)∂j . Conversely, if m ∈ M and v ∈ Tm M, then    Γjk (m)v k v  ∂j . v, w ∈ T(m,v) T M, with w = − 1≤j,k,≤d

In fact, consider γ(t) theunique (maximal) geodesic such that γ(0) = m   and γ(0) ˙ = v. Then v, w = γ(0), ˙ γ¨ (0) ∈ T(m,v) T M. 17.5.1. Tangent Geodesic Flow. For (m, v) ∈ T M, we saw above  for wv with − 1≤j,k,≤d Γjk (x)v k v  ∂j as local representative, then (v, wv ) ∈ T(m,v) T M. Setting G(m,v) = (v, wv ), one thus defines a vector field on T M. It is called the (tangent) geodesic vector field. From (17.5.2) we find that a curve t → x(t) is a geodesic if and  only if t → x(t), x(t) ˙ is integral curve of G on T M. The flow χt associated

17.5. GEODESICS AND GEODESIC FLOWS

453

with G is called the (tangent) geodesic flow. A geodesic t → γ(t) is thus solution to  d . γ(t), γ(t) ˙ = G(γ(t),γ(t)) ˙ dt For (x, v) ∈ T M, t → γ(t) = χt (m, v) yields the maximal unique geodesic such that γ(0) = m and γ(0) ˙ = v. Proposition 17.13 (Local Geodesic Flow Box). Let m0 ∈ M. There exist an open neighborhood V of m0 , C > 0, and δ > 0 such that χt (m, v) is well defined for t ∈ (−δ, δ), m ∈ V and v ∈ Tm M such that |v|g ≤ C. Observing that the following holds χt (m, λv) = χλt (m, v),

λ > 0,

if one side makes sense, Proposition 17.13 thus yields the following result. Proposition 17.14. Let m0 ∈ M and T > 0. There exist an open neighborhood VT of m0 , CT > 0 such that χt (m, v) is well defined for t ∈ (−T, T ), m ∈ VT and v ∈ Tm M such that |v|g ≤ CT . For m0 ∈ M and T > 1 one can then define the so-called exponential map by exp(m, v) = χ1 (m, v) = χ|v|g (m, v/|v|g ), for (m, v) ∈ Tm M with m ∈ V and v ∈ Tm M such that |v|g ≤ CT , where V and CT > 0 are given by Proposition 17.14. For m ∈ V one sets expm (v) = exp(m, v). Proposition 17.15. Let m ∈ M. Define the open ball Bg (0, ε) = {v ∈ Tm M; |v|g < ε}. There exists ε > 0 such that expm maps Bg (0, ε) diffeomorphically onto an open subset of M. The proof is in fact that of Lemma 4.36 of Volume 1 once connection is made with the Hamiltonian geodesic flow; see Sect. 17.5.3 and Proposition 17.20. If the exponential map at m, expm , is a diffeomorphism from an open neighborhood V of 0 in Tm M onto an open subset U of M one says that U is a normal open neighborhood of m. Any neighborhood U  of m such that U  ⊂ U is also called a normal neighborhood of m. In the previous proposition m is kept fixed. The following theorem provides a locally uniform version of the previous result. 0 U Theorem 17.16. Let m0 ∈ M. There exists a neighborhood   of m and δ > 0 such that for all m ∈ U the geodesic ball expm Bg (0, δ) is well defined and contains U .

454

17. ELEMENTS OF RIEMANNIAN GEOMETRY

  For ε > 0 as in Proposition 17.15, one calls expm Bg (0, ε) a geodesic ball of radius ε centered at m. For 0 ≤ δ < ε, if we set Sg (0, δ) = {v ∈ Tm M; |v|g = δ},   then one calls expm S(0, δ) the geodesic sphere of radius δ centered at m. It is a submanifold of M of codimension one. Lemma 17.17 (Gauss’ Lemma).  Let m ∈ M and let δ > 0 be such that the geodesic sphere Sm,δ = expm Sg (0, δ) is well defined. If v ∈ Sg (0, δ) and if γ(t) = χt (m, v), then γ(1) ˙ is orthogonal to Tγ(1) Sm,δ in the sense of g. 17.5.2. Length Minimization. The following results characterize geodesics locally as the curves that minimize the Riemannian distance.   Proposition 17.18. Let m ∈ M and let B = expm Bg (0, ε) be a geodesic ball centered at m. Let γ(t) = χt (m, v) with |v|g < ε. Let ρ : [0, 1] → M be a piecewise differential curve such that ρ(0) = γ(0) = m and ρ(1) = γ(1) ∈ B . Then (γ) ≤ (ρ) and if equality holds then ρ([0, 1]) = γ([0, 1]). Proposition 17.19. Let γ : R ⊃  [a, b] →  M be a piecewise differentiable curve such that, for some c > 0,  γ([a, t]) = c|t − a| for any t ∈ [a, b]. If for any other piecewise differentiable curve ρ joining γ(a) to γ(b) one has (γ) ≤ (ρ), then γ is a geodesic. In particular γ is a smooth curve. The first proposition states that locally a geodesic is a length minimizing curve. The second proposition is a converse. Moreover, in this second proposition, the result holds globally: a length minimizing curve (with parameter proportional to arc length) is a geodesic. However, considering the sphere Sd−1 ⊂ Rd one sees that geodesics may not minimize the Riemannian distance globally. 17.5.3. The Hamiltonian Geodesic Flow. Recall that T ∗ M is a manifold; see Sect. 15.4.3. For a function f : T ∗ M → R one associates the Hamiltonian vector field Hf on T ∗ M; see Sect. 15.7.2. Here, we consider f (m, ω) = gm (ω, ω)/2, that is, half of the principal symbol of the Laplace– Beltrami operator. In a local chart one has  1  g ij (x)ξj ∂xi − ∂xk g ij (x)ξi ξj ∂ξk . Hf (x, ξ) = 2 1≤i,j≤d 1≤i,j,k≤d Maximal integral curves of Hf are unique by the Cauchy–Lipschitz theorem and they provide an alternative way to describe geodesics. We call the flow associated with Hf the Hamiltonian geodesic flow. Proposition 17.20. Let (m0 , ξ 0 ) ∈ T ∗ M and ϕ(t) = (m(t), ξ(t)) be the unique maximal integral curve of Hf such ϕ(0) = (m0 , ξ 0 ). Set v 0 = ξ 0 . If γ(t) = χt (m0 , v 0 ) is the unique maximal integral curve of the geodesic vector field G such that γ(0) = m0 and γ(0) ˙ = v 0 , then γ(t) = (m(t), ξ  (t)).

17.6. NORMAL GEODESIC COORDINATES AT THE BOUNDARY

455

Remark 17.21. Because of the uniqueness of the two families of integral curves considered here, the converse follows, that is, if (m0 , v 0 ) ∈ T M and if γ(t) = (m(t), v(t)) = χt (m0 , v 0 ) is the unique maximal integral curve of the geodesic vector field G such that γ(0) = m0 and γ(0) ˙ = v 0 , then

ϕ(t) = (m(t), v (t)) is the unique maximal integral curve of Hf such ϕ(0) = (m0 , v 0 ). Proof. It is sufficient to consider a local chart. An integral curve t → ˙ ˙ ξ(t)) = Hf (x(t), ξ(t)), that is, (x(t), ξ(t)) of Hf on T ∗ M is solution to (x(t),  i 1  g (x(t))ξ (t), ξ˙i (t) = − ∂x g k (x(t))ξk (t)ξ (t). x˙ i (t) = 2 1≤k,≤d i 1≤≤d The first equation reads x(t) ˙ = ξ  (t). We thus set v(t) = ξ  (t). For the sake of concision we omit sums in the proof and use the Einstein summation convention on repeated indices. We also write g ij , v, x and ξ in place of g ij (x(t)), v(t), x(t) and ξ(t). We have d  ij  1   v˙ j = g ξi = ∂xk g ij v k ξi + g ij ξ˙i = ∂xk g ij v k ξi − g ij ∂xi g k  ξk ξ dt 2 

  1    = ∂xk g ij gi − g ij ∂xi g k  gkk g v k v  . 2   j ij Since we have g gi = δ we find ∂xk g ij gi = −g ij ∂xk gi . We thus obtain   1  1     ∂xk g ij gi − g ij ∂xi g k  gkk g = −g ij ∂xk gi + g ij g k  gkk ∂xi g 2 2 1 ij ij = −g ∂xk gi + g ∂xi gk . 2 We thus have   1 1 v˙ j = − g ij 2∂xk gi − ∂xi gk v k v  = − g ij ∂xk gi + ∂x gik − ∂xi gk v k v  . 2 2 With the formula for the Christoffel symbols given in (17.4.11), we thus find  v˙ j = −Γjkl v k v  . By (17.5.2), the proof is complete. 17.6. Normal Geodesic Coordinates at the Boundary The Laplace–Beltrami operator Δg is canonically defined through the metric g. Let m ∈ ∂M. By Theorem 9.7 of Volume 1 we can design a local chart C = (O, κ) with m ∈ O, such that the principal symbol of the representative of P = −Δg in C has the form  pij (x)ξi ξj . ξd2 + 1≤i,j≤d−1

˜ κ ˜ the represenIn fact, if one considers any local chart C˜ = (O, ˜ ) with m ∈ O tative of P in this local chart is a second-order elliptic operator with smooth ˜ an open set of Rd . One can apply Theorem 9.7 to that coefficients in κ ˜ (O), + operator and this generates new coordinates in a possibly smaller open set. This new open set and the new coordinates yield the local chart C = (O, κ).

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17. ELEMENTS OF RIEMANNIAN GEOMETRY

Since P = −Δg the principal symbol also reads  g C,ij (x)ξi ξj , pC (x, ξ) = 1≤i,j≤d

implying that in the local chart C the representative of the metric is such that C (x) = δdj , j = 1, . . . , d, x ∈ κ(O), gdj

meaning that (17.6.1)

gxC = gxC  + dxd ⊗ dxd ,



with gxC  =

C gij (x)dxi ⊗ dxj ,

1≤i,j≤d−1

or rather, in matrix form,

⎛

⎞ 0 ⎜ .. ⎟ gx . ⎟. gxC = ⎜ ⎝ 0 ⎠ 0 ··· 0 1 (17.6.2) C Note that g(x  ,x ) is a metric on κ(O ∩ ∂M) with xd acting as a parameter. d C For xd = 0, g(x  ,0) is the representative of the induced metric g∂ on the boundary ∂M. In local coordinates, the representative of Δg is given by (17.2.8). Here, this gives   P C = (det gxC  )−1/2 Dd (det gxC  )1/2 Dd    Dxi (det gxC  )1/2 g C ,ij (x)Dxj . + (det gxC  )−1/2 1≤i,j≤d−1

If we set R(x, Dx ) = (det gxC  )−1/2



  Dxi (det gxC  )1/2 g C ,ij (x)Dxj ,

1≤i,j≤d−1

we then have (17.6.3)

P C = Dd2 + R(x, Dx ) + Q1 (x, D),

with   i Q1 (x, D) = − (det gxC  )−1 ∂d det gxC  Dd . 2 The first-order operator Q1 (x, D) thus acts only in the normal direction to ∂M given by the variable xd in the local chart. (17.6.4)

Normal geodesic coordinates as describes above are defined locally. In a bounded part of a Riemannian manifold they can be used to provide a chart that describes an open set near the boundary in a very natural manner.

17.6. NORMAL GEODESIC COORDINATES AT THE BOUNDARY

457

Theorem 17.22. Let (M, g) be a Riemannian manifold and let Γ be a bounded open subset of ∂M. There exist O an open set of M, z0 > 0, and a diffeomorphism Φ, such that Γ = O ∩ ∂M and Φ : Γ × [0, z0 ) → O (m , z) → Φ(m , z), and the following additional properties hold. (1) For any m ∈ O with m = Φ(m , z) we have distg (m, ∂M) = distg (m, Γ) = distg (m, m ) = z. (2) There exists a Riemannian metric g  (z) on Γ that smoothly depends on z such that  (Φ∗ g)(m ,z) = gm  (z) ⊗ 1z + dz ⊗ dz ,

and g  (0) = g∂ on Γ. A point in the open set O is thus diffeomorphically parameterized by its Riemannian distance z to ∂M and the point m of ∂M where this distance is realized. In particular the map O  m = Φ(m , z) → m provides a projection onto Γ naturally associated with the Riemannian structure. As seen in the proof below, for m = Φ(m , z) ∈ O the curve t → Φ(m , tz), t ∈ [0, 1], is a geodesic that joins m and m. This geodesic leaves ∂M in a normal manner and it minimizes the distance between the two points. This result further justifies the name normal geodesic coordinates that one also uses for the “chart” (O, Φ−1 ). The open set O is illustrated in Fig. 17.1. In the particular case where M is a compact manifold, one can choose Γ = ∂M and thus obtain a global parameterization of an open neighborhood of ∂M by means of normal geodesic coordinates. Proof. Let m0 be an arbitrary point of Γ. From the discussion at the ˆ κ beginning of the section there exists a local chart Cˆ = (O, ˆ ) that yields z0 m

z m

Γ

O

M

∂M

Figure 17.1. The open subset Γ of ∂M and the open subset O of M

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17. ELEMENTS OF RIEMANNIAN GEOMETRY

normal geodesic coordinates. Without any loss of generality we may assume ˆ  an open set of Rd−1 and ˆ takes the form κ ˆ =O ˆ  × [0, zˆ0 ] for O that κ ˆ (O) ˆ (O)  ˆ ˆ ˆ ˆ (O ∩ ∂M) = O . We choose zˆ0 and O sufficiently small so that zˆ0 > 0 and κ ˆ lies in a neighborhood U of m0 given by Theorem 17.16. It then follows O ˆ minimizes distances between any pair of its that any geodesic that lies in O points by Proposition 17.18. ˆ  so that for m ∈ O ˆ the distance We further reduce the size of zˆ0 and O ˆ distg (m, ∂M) is only realized by points in O ∩ ∂M. ˆ κ Consider m ∈ O, ˆ (m) = (x , z). The curve γ(t) = (x , tz) is a geodesic. In fact, γ(t) ˙ = (0, z), γ¨ (t) = (0, 0), and with (17.5.1) we compute  j D ˙ = Γdd (x)z 2 ∂j = 0, γ(t) ˙ = Dγ(t) ˙ γ(t) dt 1≤j≤d since Γjdd = 0 by (17.4.11) and the form of the metric in the chosen local chart. ˆ κ We now claim that for m ∈ O, ˆ (m) = (x , z) then distg (m, ∂M) = z. In ˜ m) = distg (m, ∂M) and fact, assume that m ˜ ∈ ∂M is such that δ = distg (m, ˆ is a normal neighborhood of m there exists a m ˜ = m = κ ˆ −1 (x , 0). Since O unique geodesic joining m and m ˜ by the Gauss lemma (Lemma 17.17), it is  orthogonal to the geodesic sphere expm Sg (m, δ) . On the one hand, if this geodesic is orthogonal to ∂M, then the above analysis and the uniqueness of geodesics show that m ˜ = m . On theother hand,  if this geodesic is not orthogonal to ∂M, then points of expm Sg (m, δ) lie outside M and yield points different from m ˜ that are closer to m. As Γ is compact, there exists a finite number of such charts (Cˆi )1≤i≤n , i ˆ i . For each 1 ≤ i ≤ n, κ ˆ ˆ i, κ ˆ i) = C = (O ˆ i ) such that Γ ⊂ ∪1≤i≤n O ˆ i (O ˆ i an open set of Rd−1 and zˆi > 0. Set z0 = min1≤i≤n zˆi ˆ i × [0, zˆi ) with O O 0 0 0 and   i ˆ i , i = 1, . . . , n, ˆ × [0, z0 ) ⊂ O κi )−1 O O = ∪1≤i≤n Oi . Oi = (ˆ We have Γ ⊂ O. We denote by C i the modified chart C i = (κi , Oi ) with ˆ i × [0, z0 ) κi : Oi → O m → κ ˆ i (m). In the chart C i , the representative of the metric g has the form  i i i Ci (17.6.5) gxC = gxC  + dxd ⊗ dxd with gxC  = gij (x)dxi ⊗ dxj , 1≤i,j≤d−1 Ci (x ,x

is a metric on κ(Oi ∩ ∂M) with xd acting as a parameter. where g d) For each chart C i , we denote by κid (m) the last coordinate associated with the diffeomorphism κi , that is, κid (m) = xd if x = κi (m) for m ∈ Oi . We also define smooth maps ψ i : m → (κi )−1 (x , 0) where x ∈ Rd−1 is such that κi (m) = (x , xd ). From the form of the representative of the metric

17.6. NORMAL GEODESIC COORDINATES AT THE BOUNDARY

459

gxC given above, the point m = ψ i (m) is the unique point in ∂M such that distg (m, ∂M) = distg (m, m ). Moreover, the path t → (x , t) for t ∈ [0, xd ] yields a parameterization of the unique geodesic joining m and m . If m ∈ Oi ∩ Oj , we have i

(17.6.6)

κid (m) = κjd (m) = distg (m, ∂M),

and ψ i (m) = ψ j (m),

from the discussion above. We introduce the functions ψ : O → O ∩∂M = Γ and ϕ : O → [0, z0 ) by ψ(m) = ψ i (m),

ϕ(m) = κid (m),

if m ∈ Oi ,

and we see that they are both well defined and smooth as the functions ψ i coincide where the charts overlap.   If we set Ψ : O → Γ×[0, z0 ) with Ψ(m) = ψ(m), ϕ(m) we see that Ψ is a diffeomorphism and the sought diffeomorphism is Φ = Ψ−1 : Γ × [0, z0 ) → O. Finally, for two overlapping charts C i and C j , setting κ = κj ◦ (κi )−1 , denote by κ∂ its restriction to ∂M. It is a diffeomorphism from κi (Oi ∩∂M) onto κj (Oj ∩ ∂M) and we have κ(x , xd ) = (κ∂ (x ), xd ),

(x , xd ) ∈ κi (Oi ∩ Oj ).

The formula expressing the action of a change of coordinates on 2-covariant i j tensor fields (15.6.2) relating g C and g C and (17.6.5) then yields g C  (x , xd ) = κ∗∂ g C  (κ∂ (x ), xd ). i

j

Here z = xd is simply a parameter and this formula shows that there exists a family of metrics g  (z) on Γ, smoothly depending on z, such that its i representative in the local chart C i is given by g C  (x , z). Then, g  (z)⊗1z is 2covariant tensor field on Γ × [0, z0 ). Let m = Φ(m , z) ∈ O and u, v ∈ Tm M. We set (u , uz ) = Φ∗ u and (v  , vz ) = Φ∗ v both in Tm ∂M × R. From (17.6.5) we conclude that    z z gm (u, v) = gm  (z)(u , v ) + u v ,  (z) ⊗ 1 + d ⊗ d . meaning that Φ∗ gm = gm  z z z



The following corollary generalizes the local result obtained in (17.6.3)– (17.6.4). It is a consequence of the definition of the Laplace–Beltrami operator. Corollary 17.23. Let Γ, O and Φ : Γ × [0, z0 ) → O be as given by Theorem 17.22. Set g˜ = Φ∗ g, that is,  g˜(m ,z) = gm  (z) ⊗ 1z + dz ⊗ dz .

The Laplace–Beltrami operator on Γ × [0, z0 ) associated with this metric satisfies Δg˜ = Φ∗ Δg (Φ−1 )∗ and is given by   1 Δg˜ = ∂z2 + Δg (z) + (det g  (z))−1 ∂z det g  (z) ∂z , 2

460

17. ELEMENTS OF RIEMANNIAN GEOMETRY

where Δg (z) is the Laplace–Beltrami operator on Γ associated with the metric g  (z), with z as a parameter. Note that the formula Δg˜ = Φ∗ Δg (Φ−1 )∗ follows from Proposition 17.5. The following result can be easily proven by means of the normal geodesic coordinates obtained in Theorem 17.22. Proposition 17.24. Let (M, g) be a Riemannian manifold. Let Ω be a connected open set of M. Let Γ be a bounded open subset of ∂M with Γ ⊂ Ω. Let also V be an open neighborhood of Γ in M. There exists an open neighborhood W of Γ in M such that W  V and Ω \ W is connected. Proof. Let Γ be a bounded open neighborhood of Γ in Ω ∩ ∂M such that Γ ⊂ V ∩ ∂M. By Theorem 17.22 there exist O an open set of M, z0 > 0, and a diffeomorphism Ψ, such that Γ = O ∩ ∂M and Ψ : Γ × [0, z0 ) → O (m , z) → Ψ(m , z).   We claim that there exists z1 > 0 sucht that Ψ−1 Γ × [0, 2z1 ) ⊂ Ω. Othertoward a point wise, we can find a sequence (mn )n ⊂ M \ Ω that converges  of Γ ; a contradiction. We set O1 = Ψ−1 Γ × [0, z1 ) . We now consider a function h ∈ Cc∞ (Γ ) with 0 ≤ h ≤ 1 such that h ≡ 1 on a neighborhood of Γ. We then define the following map H : Ω → Ω,  m if m ∈ / O1 ; H(m) =    Ψ(m , (1 − h(m ))z + h(m )z1 ) if m ∈ O1 and (m , z) = Ψ−1 (m). This map is continuous and thus H(Ω) is connected. If we set W = Ω\H(Ω) we see that W is bounded and fulfills the required properties.  17.7. Higher-Order Covariant Derivatives For θ a r-covariant,s-contravariant tensor field, setting D0 θ = θ, for k ∈ N∗ , one defines by induction, the (r+k)-covariant,s-contravariant tensor field given by, (17.7.1)

Dk θ = D(Dk−1 θ).

For a function f we saw above that D f = df . The 2-covariant tensor H f = D2 f is called the Hessian of f . In  a local chartC C i= (O, jκ) we find C the representative of H f to be (H f ) = 1≤i,j≤j (H f )ij dx ⊗ dx , with  k (17.7.2) Γij ∂xk f C . (H f )Cij = ∂x2i xj f C − 1≤k≤d

For  ∈ N, following (17.4.12), we define accordingly (17.7.3)

∂ν f (m) = D fm (νm , . . . , νm ). # $% & 

17.7. HIGHER-ORDER COVARIANT DERIVATIVES

461

We have the following proposition that shows that this definition is consistent with what one expects from the Euclidean case. Proposition 17.25. Let m0 ∈ ∂M and C = (O, κ) be local chart such that m0 ∈ O and such that κ provides normal geodesic coordinates in O. If x = (x1 , . . . , xd ) = κ(m) for m ∈ O, we have ∂ν f (m) = (−∂xd ) f C (x),

x = κ(m), m ∈ ∂M.

For normal geodesic coordinates we refer to Sections 9.4 and 17.6. We ˜ is an open recall that a local chart at a boundary point is such that κ ˜ (O) d ˜ = {xd = 0} ∩ κ ˜ ˜ (∂M ∩ O) ˜ (O). set of R+ and κ Proof. In the normal geodesic coordinates provided by κ, the representative of ν in C is ν C = (0, . . . , 0, −1). We have ∂ν f (m) = D fm (νm , . . . , νm ) = (−1) (D fxC )d . . . d . #$%& # $% & 



For concision we shall write f in place of f C here. It now suffices to prove that (D fx )d . . . d = ∂x d f (x) #$%&

for all x ∈ κ(O).



The proof goes by induction. The result is clear for  = 1. We assume that the results hold for . With (17.4.9), we have  k Γ (D fx )d . . . d k d...d . (D+1 fx )d . . . d = ∂xd (D fx )d . . . d − #$%& #$%& 1≤n≤ dd #$%& +1



1≤k≤d

n−1

For normal geodesic coordinates we have Γkdd (x) = 0 by (17.4.11). The conclusion follows. 

CHAPTER 18

Sobolev Spaces and Laplace Problems on a Riemannian Manifold Contents L2 and H 1 -Spaces Sobolev Spaces L2 -Based Spaces Lp -Based Spaces Transposition of the Laplace–Beltrami Operator and Action on H 1 Functions 18.4. The Laplace Problem on a Compact Manifold Without Boundary 18.5. Continuous Sobolev Scale and Traces 18.6. The Dirichlet-Laplace Problem 18.6.1. Homogeneous Dirichlet Boundary Conditions 18.6.2. Spaces with Remarkable Trace Properties 18.6.3. The Dirichlet Lifting Map 18.6.4. More Spaces with Remarkable Trace Properties 18.7. The Neumann-Laplace Problem 18.7.1. The Neumann Lifting Map 18.8. The Laplace Problem with Mixed Neumann-Dirichlet Boundary Conditions 18.8.1. Homogeneous Case 18.8.2. The Mixed Dirichlet–Neumann Lifting Map 18.9. Second-Order Elliptic Operators in the Euclidean Space Appendix 18.A. Traces Extension: Technical Aspects 18.A.1. L2 Functions with L2 ‘Laplacian’ 18.A.2. Green-Like Formula in WP (M)

18.1. 18.2. 18.2.1. 18.2.2. 18.3.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7 18

464 467 467 470 471 472 474 482 482 484 486 488 489 495 497 497 498 501 506 506 506 509

463

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18. SOBOLEV SPACES AND LAPLACE PROBLEMS

18.A.3. L2 Functions with H −1 Laplacian 18.A.4. H 1 Functions with L2 Laplacian 18.A.5. Normal Traces for Vector Fields in H(divg , M)

512 512 513

With this chapter, our first goal is to give a geometrical definition of Sobolev spaces on a Riemannian manifold, as the norms associated with those spaces are used in some of the Carleman estimates proven in Chapter 3 of Volume 1 and Chap. 8. A second goal is the study of the elliptic properties of the Laplace–Beltrami operator and diverse boundary value problems associated with that operator. 18.1. L2 and H 1 -Spaces Let (M, g) be a smooth d-dimensional Riemannian manifold and let μg be the canonical positive density function defined in Sect. 17.3. It allows one to identify functions and density functions, that is, 0-density functions and 1-density functions, 0-density distributions and 1-density distributions, etc. An L2 -function f on M (resp. L2loc ) is then a complex valued function f such that |f |2 μg is an L1 -density function (resp. L1loc ). On L2 (M), the vector space of L2 -functions, we define the inner product and associated norm (f, g)L2 (M) = ∫ f gμg , M

f 2L2 (M) = ∫ |f |2 μg , M

which yields a Hilbert space structure. Remark 18.1. If f ∈ L2 (M), then for every local chart C = (O, κ) its representative is such that f C ∈ L2 (κ(O)). Conversely, if M is compact, if f C ∈ L2 (κ(O)) for every chart (using charts in a finite atlas suffices), then f ∈ L2 (M). Proposition 18.2. Let M be σ-compact. The space 0 Dc∞ (M) is dense in L2 (M). Proof. Let f ∈ L2 (M) and ε > 0. If (Kn )n is an exhaustive sequence of compact sets for M, we set fn = 1Kn f . The Lebesgue dominatedconvergence theorem yield n0 ∈ N such that fn0 − f L2 (M) ≤ ε. Then as supp(fn0 ) ⊂ Kn0 , working in a finite number of local charts that cover Kn0 , there exists a sequence of smooth functions h supported in Kn0 such  that fn0 − hL2 (M) ≤ ε, which yields the result. We recall the notation C ∞ V (M) for the space of smooth vector fields. We define the following inner product and norm, for w, v ∈ C ∞ V (M), (u, v)L2 V (M) = ∫ g(u, v)μg , and u2L2 V (M) = ∫ g(u, u)μg . M

We denote by L2 V (M) the completion of the space   u ∈ C ∞ V (M); uL2 V (M) < ∞

M

18.1. L2 AND H 1 -SPACES

465

respect to this norm. If v ∈ L2 V (M) we say that v is an L2 -vector field. For a smooth function f, h on M we define the H 1 -inner product and norm (f, h)H 1 (M) = (f, h)L2 (M) + (∇g f, ∇g h)L2 V (M) , f 2H 1 (M) = f 2L2 (M) + ∇g f 2L2 V (M) . We then define the space H 1 (M) as the completion of   f ∈ C ∞ (M); f H 1 (M) < ∞ with respect to this norm. We then obtain a Hilbert space. The map f → f|∂M on C ∞ (M) extends uniquely as a map from H 1 (M) into L2loc (∂M) and is referred to as the trace map as in Euclidean spaces. For H 1 -functions we use the classical notation f|∂M for its trace on ∂M. Remark 18.3. Similarly to Remark 18.1 we see that if f ∈ H 1 (M), then for every local chart C = (O, κ) its representative is such that f C ∈ H 1 (κ(O)). Conversely, if M is compact, if f C ∈ H 1 (κ(O)) for every chart (using charts in a finite atlas suffices), then f ∈ H 1 (M). The space H01 (M) is defined as the closure for the norm .H 1 (M) of 0 ∞ Dc (M). Proposition 18.4. Let (M, g) be a complete Riemannian manifold without boundary. Then H01 (M) = H 1 (M). A Riemannian manifold is said to be complete if it is complete for its natural distance as given in (17.1.2). This result is based on the exhaustion of M by compact balls with respect to the Riemannian distance and a Lipschitz cutoff of H 1 -functions that reduces the analysis to compactly supported H 1 functions. In the case of a compact manifold without boundary the proof is simply based on the use a finite partition of unity subordinated to an atlas covering, allowing one to use the result known for bounded regular open sets in Rd . A similar argument, in the case of a compact Riemannian manifold with boundary yields the following density result (anticipating the notion of traces that we present in Sect. 18.5 below). Proposition 18.5. Let (M, g) be a compact Riemannian manifold with boundary. Then H01 (M) is the space of H 1 -functions f such that f|∂M = 0. Remark 18.6. Note that the property of Proposition 18.4 is open for Sobolev spaces of higher order if additional assumptions are not made on the manifold [167]. Theorem 18.7 (Rellich-Kondrachov). Given χ continuous with compact support on M the map u → χu is compact from H 1 (M) into L2 (M).

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This can be proven using a finite number of local charts that cover the support of χ and by using the counterpart result on Rd (see, for instance, [90, Theorem 9.16]). We have the following Poincar´e inequalities. Lemma 18.8 (Poincar´e Inequality). Let (M, g) be a connected Riemannian manifold. Let U be an open subset of M such that U is a compact of M and U = M. There exists C > 0 such that (18.1.1)

f L2 (M) ≤ C∇g f L2 V (M) ,

for f ∈ H 1 (M) supported in U . Proposition 18.9 (Poincar´e Inequalities). Let (M, g) be a connected compact Riemannian manifold. If Γ is an open subset of ∂M, there exists C > 0 such that (18.1.2)

f L2 (M) ≤ C∇g f L2 V (M) ,

for f ∈ H 1 (M) such that f|Γ = 0. A similar inequality holds for H 1 functions with zero mean value, that is, ∫M f μg = 0. If Γ = ∂M, we call Poincar´e constant Cg the largest positive constant such that Cg1/2 f L2 (M) ≤ ∇g f L2 V (M) for f ∈ H 1 (M). If A ⊂ U is of positive measure, there exists also C > 0 such that   f L2 (M) ≤ C f L2 (A) + ∇g f L2 V (M) , (18.1.3) for f ∈ H 1 (M). Proofs of Lemma 18.8 and Proposition 18.9. Assume that inequality (18.1.1) does not hold. Then, there exists a sequence (f n )n∈N ⊂ H 1 (M) with supp(f n ) ⊂ U and such that (18.1.4)

f n L2 (M) = 1, ∇g f n L2 V (M) → 0.

In particular, f n is bounded in H 1 (M) and, thus, there exists f ∈ H 1 (M) such that f n  f in H 1 (M) up to a subsequence, since the unit ball of a Hilbert space is weakly compact by the Banach Alaoglu theorem [90, Theorem 3.16]. Choose χ ∈ Cc∞ (M) be such that χ = 1 in a neighborhood of U using Lemma 15.16. Then, f n = χf n converges strongly to f = χf in L2 (M) by the Rellich–Kondrachov theorem (see Theorem 18.7). As we have ∇g f n  ∇g f in L2 V (M), we find that ∇g f = 0. Because of the form of ∇g given in (17.2.2) this implies that f ≡ Cst in every chart and it thus constant on the whole M as M is connected. As supp(f ) ⊂ U = M we have f ≡ 0. We have thus found that χf n strongly converges to 0 in L2 (M), which yields a contradiction as χf n L2 (M) = f n L2 (M) = 1. This concludes the proof of Lemma 18.8.

18.2. SOBOLEV SPACES

467

For the proof of Proposition 18.9 the manifold M is compact and we can use the same setting with U = M and χ ≡ 1. Considering a sequence (f n )n∈N ⊂ H 1 (M) that contradicts (18.1.2) by fulfilling (18.1.4) we find as above that, up to a subsequence, f n  f in H 1 (M) and f n → f in L2 (M) with f a constant function. n = 0 since the map h → h 1 2 If f|Γ |Γ is continuous on H (M) into Lloc (Γ) n  f . Since f n = 0 then f = 0, which implies that f ≡ 0 we find that f|Γ |Γ |Γ |Γ on M. If ∫M f n μg = 0 we have 0 = ∫ f n μg = (f n , 1)L2 (M) → (f, 1)L2 (M) = ∫ f μg . M

M

As f ≡ Cst, we find f ≡ 0. We now consider a sequence (f n )n∈N ⊂ H 1 (M) that contradicts (18.1.3), that is, f n L2 (M) = 1, f n L2 (A) → 0, ∇g f n L2 V (M) → 0. As above we find that, up to a subsequence, f n  f in H 1 (M) and f n → f in L2 (M) with f a constant function. We also find gives 0 = f n L2 (A) → f L2 (A) implying that f ≡ 0 since |A| > 0. In all cases the same contradiction as above is reached.  s (M) as the space of (0-density) For s ∈ R, we also define the space Hloc s in every chart. distributions u on M such its local representative is Hloc s (M) if for every chart More precisely, for u ∈ 0 D  (M), we say that u ∈ Hloc C s C = (O, κ) we have u ∈ Hloc (int κ(O)). Recall that int κ(O) here denotes that interior of κ(O) in Rd .

Proposition 18.10. Let u ∈ H 1 (M) and ϕ ∈ 0 Dc∞ (M). One has ∫ g(∇g u, ∇g ϕ) μg + ∫ uΔg ϕ μg = 0.

M

M

Proof. One simply applies the divergence formula of Proposition 17.9  to the L1 -vector field u∇g ϕ whose divergence is also L1 . 18.2. Sobolev Spaces 18.2.1. L2 -Based Spaces. Let s ∈ R. We recall that given a (0s (M), if for each chart density) distribution u on M, we say that u ∈ Hloc s C = (O, κ) its representative is in Hloc (κ(O)). We see immediately that if Q s (M) into H s−m (M) is a differential operator of order m ∈ N it maps Hloc loc for all s ∈ R. In contrast, to introduce the Sobolev space H s (M), a proper norm needs to be defined. On a compact manifold, given a finite atlas and a subordinated partition of unity, one can define Sobolev norms. However, such norms

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18. SOBOLEV SPACES AND LAPLACE PROBLEMS

are not intrinsic. On Riemannian manifolds the notion of covariant derivative allows one to define intrinsic Sobolev norms that appear as natural extensions of those used on Rd . On a Riemannian manifold (M, g), for m ∈ M, as seen above in Sect. 17.1, the metric g yields a natural norm on tangent vectors at m, that is, |v|2gm = gm (v, v¯) if v ∈ Tm M and a natural norm on cotangent vectors at ∗ M. In a local chart C = (O, κ), for ¯  ) if ω ∈ Tm m, that is |ω|2gm = gm (ω  , ω x = κ(m), it reads    C C,j C,j  gjj v¯ , and |ω|2gm = g C,ii (x)ωiC ω ¯ iC . |v|2gm =  (x)v 1≤j,j  ≤d

1≤i,i ≤d

A generalization of these norms to r-covariant,s-contravariant tensors θ at m is the following norm (18.2.1) |θ|2gm =

 1≤i1 ,i1 ,...,ir ,ir ,≤d  ,...,j ,j  ≤d 1≤j1 ,j1 s s

   C,i1 i  j ...js j1 ...js  1 · · · g C,ir ir g C  · · · g C  θ 1 θ g (x).   j j j1 j s s i1 ...ir i ...i 1

r

1

This norm is independent of the chosen local chart. The associated inner product is  C,i1 i   j ...js ¯j1 ...js  1 · · · g C,ir ir g C  · · · g C  θ 1 g (θ, ϑ)gm = js js i1 ...ir ϑi ...i (x). j1 j 1

1≤i1 ,i1 ,...,ir ,ir ,≤d  ,...,j ,j  ≤d 1≤j1 ,j1 s s

1

r

A r-covariant, s-contravariant tensor fields θ is said to be L2 if |θ|gm ∈ L2 (M). We denote by L2 Trs (M) the space of such tensor fields. Equipped with the norm θ2L2 Trs (M) = ∫ |θm |2gm μg , M

and the associated inner product (θ, ϑ)L2 Trs (M) = ∫ (θm , ϑm )gm μg , M

one obtains a Hilbert space. We recall the notation C ∞ Trs (M) introduced in Sect. 15.6.3 for the space of smooth r-covariant, s-contravariant tensor fields on M. The space L2 Trs (M) is in fact the completion of {θ ∈ C ∞ Trs (M); θL2 Trs (M) < ∞} with respect to .L2 Trs (M) . s (M). For k ∈ N If θ ∈ C ∞ Trs (M), and j ∈ N, one has Dj θ ∈ C ∞ Tr+j we define the norm k  2 θ2H k Trs (M) =  Dj θL2 T s (M) , j=0

r+j

18.2. SOBOLEV SPACES

469

and the Sobolev space H k Trs of H k r-covariant,s-contravariant tensor fields as the completion of {θ ∈ C ∞ Trs (M); θH k Trs (M) < ∞}, with respect to this norm. Then, for θ ∈ H k Trs (M) we have Dj θ ∈ s (M) for j = 0, . . . , k. Here, the action of Dj can be understood H k−j Tr+j as a unique extension or in a weak sense (the latter can be defined in local charts). The inner product associated with .H k Trs (M) is given by (θ, ϑ)H k Trs (M) =

k 

s (M) , (Dj θ, Dj ϑ)L2 Tr+j

j=0

for θ, ϑ ∈ H k Trs (M). With this inner product H k Trs (M) is a Hilbert space. In the case r = s = 0 we simply write L2 (M) and H k (M), and we recover the L2 - and H 1 -functions introduced in Sect. 18.1. Given a function f on M, Dk f is a k-covariant tensor field. As (D f ) = (df ) = ∇g f we see that the H 1 -norm defined in Sect. 18.1 coincides with the H 1 -norm given here. Similarly to Remarks 18.1 and 18.3 we see that if f ∈ H k (M), then for every local chart C = (O, κ) its representative is such that f C ∈ H k (κ(O)). Conversely, if M is compact, if f C ∈ H k (κ(O)) for every chart, then f ∈ H k (M). In fact, we can prove the following result. Proposition 18.11. Let (M, g) be a compact manifold (with or without boundary). Let A = (C i )1≤i≤N be a finite atlas with C i = (Oi , κi ). There exists C > 0 such that  i (18.2.2) f C H k (κi (Oi )) ≤ Cf H k (M) . C −1 f H k (M) ≤ 1≤i≤N

The constant C depends on the choice of the atlas. Moreover, if i ∈ {1, . . . , N }, for some Ci > 0, we have (18.2.3)

Ci−1 f H k (M) ≤ f C H k (κi (Oi )) ≤ Ci f H k (M) i

if supp(f ) ⊂ Oi . With the atlas used in this proposition and with the density of smooth functions in each local chart on see that smooth functions are dense in H k (M) if M is compact with or without boundary. In the case of vector fields, we used the notation L2 V (M) in Sect. 18.1. It coincides with L2 T01 (M) here and similarly we use H k V (M) to denote H k T01 (M). In the case of one-forms, we use the notation L2 Λ (M) instead of L2 T10 (M) and similarly we use H k Λ(M) to denote H k T10 (M). In the case of r-covariant tensor fields, we use the notation L2 Λr (M) instead of L2 Tr0 (M) and similarly we use H k Λr (M) to denote H k Tr0 (M).

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18. SOBOLEV SPACES AND LAPLACE PROBLEMS

The Sobolev spaces defined above coincide with the usual spaces if M\∂M is an open set1 of Rd . The action of differential operator on Sobolev space is consistent with the results on Rd . For simplicity, to avoid boundedness assumptions on the coefficients of the operator we consider compact manifold in the following proposition. Proposition 18.12. Let (M, g) be a smooth Riemannian manifold such that M is compact (with or without boundary). Let Q be a differential operator of order k with smooth coefficients on M. Then, for  ∈ N, the operator Q maps H +k (M) into H  (M) continuously. Finally, for k ∈ N, we define H0k (M) as the closure for the norm . m (M) of 0 Dc∞ (M). Here, H0k (M) is a space 0-densities. Its dual space   H H0k (M) is a space of 1-density distributions. Using the canonical 1-density   μg on (M, g) we can identify H0k (M) with a space of 0-density distributions that we denote by H −k (M). Using the space L2 (M) as a pivot space, we find that v μg , u, v¯H −k (M),H k (M) = (u, v)L2 (M) = ∫ u¯ 0

M

and u ∈ Note that using 1/2-density spaces instead if v ∈ is an alternative approach. The norm .H −k (M) is naturally defined as the norm of a dual space of normed vector space and yields a Hilbert space structure on H −k (M). H0k (M)

L2 (M).

18.2.2. Lp -Based Spaces. Above we considered Sobolev spaces that are based on the L2 -structure. In the present book we need very few aspects of Sobolev spaces based on a Lp -structure, with 1 ≤ p ≤ ∞. First, we treat the case 1 ≤ p < ∞. One says that f is a Lp function if p |f | is a L1 -function and one sets f pLp (M) = ∫ |f |p μg . M

With (18.2.1) defining the function |θm |gm for a r-covariant,s-contravariant tensor field θ, one says that θ is Lp on M if |θm |gm is a Lp function on M. One writes θ ∈ Lp Trs (M) and one defines the norm θpLp T s (M) = ∫ |θm |pgm μg . r

If θ ∈

C ∞ Trs (M)

M

and k ∈ N we define the norm θpW k,p T s (M) = r

k  j=0

p

 Dj θLp T s

r+j (M)

,

1Recall that here manifolds with boundary are defined such that ∂M ⊂ M; see Definition 15.2.

18.3. TRANSPOSITION OF THE LAPLACE–BELTRAMI OPERATOR

471

and the Sobolev space W k,p Trs of W k,p r-covariant,s-contravariant tensor fields as the completion of {θ ∈ C ∞ Trs (M); θW k,p Trs (M) < ∞}, with respect to this norm. One finds that if θ ∈ W k,p Trs (M), then Dj θ ∈ s (M) for 1 ≤ j ≤ k. For k ∈ N and 1 ≤ p < ∞, one can check W k−j,p Tr+j that W k,p Trs (M) is a Banach space if equipped with the above norm. Ci

Second, we treat the case p = ∞. Given an atlas A = (C i )i∈I , with = (Oi , κi ), for a function f defined on M, a L∞ -norm can be defined as f L∞ (M) = sup f C L∞ (κi (Oi )) , i

i∈I

Ci

where f is the representative of f in C i . One can check that this norm is in fact independent of the chosen atlas. For θ a r-covariant,s-contravariant tensor field, one says that θ is L∞ on M if |θm |gm ∈ L∞ (M). One writes θ ∈ L∞ Trs (M) and one defines the norm θL∞ Trs (M) = |θm |gm L∞ (M) . s (M) for 1 ≤ j ≤ k one says that If θ ∈ L∞ Trs (M) and if Dj θ ∈ L∞ Tr+j θ ∈ W k,∞ Trs (M). Here, the action of Dj is to be understood in a weak sense, which can be defined in local charts. One sets

θW k,∞ Trs (M) = max  Dj θL∞ T s 0≤j≤k

r+j (M)

.

For k ∈ N one can check that W k,∞ Trs (M) is a Banach space if equipped with the above norm. Finally, in the case of functions, that is r = s = 0, for 1 ≤ p ≤ ∞, one write W k,p (M) in place of W k,p T00 (M). Note also that one has W k,2 Tsr (M) = H k Tsr (M). 18.3. Transposition of the Laplace–Beltrami Operator and Action on H 1 Functions tΔ

In Sect. 17.3 we saw that the transpose of Δg (for the density μg ) is g = Δg . If u ∈ 0 D  (M) and ϕ ∈ 0 Dc∞ (M) we thus have

(18.3.1)

Δg u, ϕμg 0 D  (M),1 Dc∞ (M) = u, (Δg ϕ)μg 0 D  (M),1 Dc∞ (M) .

Let now v ∈ L2 V (M). For ϕ ∈ 0 Dc∞ (M), the map Sv : ϕ → − ∫M v(ϕ)μg is a 1-density distribution and we define the 0-density distribution divg v such that Sv = divg v μg following Sect. 17.3. If u ∈ H 1 (M), then ∇g u ∈ L2 V (M) ˜ g u = divg ∇g u. Observe that for ϕ ∈ 0 Dc∞ (M) we have and we set Δ ˜ g u)μg , ϕ1  = ∫ ∇g u(ϕ)μg = ∫ g(∇g u, ∇g ϕ)μg . (−Δ 0 ∞ D (M), Dc (M)

M

M

472

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

By Proposition 18.10 and (18.3.1) this yields ˜ g u)μg , ϕ1  (−Δ D (M),0 Dc∞ (M) = − ∫ uΔg ϕ μg M

= uμg , −Δg ϕ1 D  (M),0 Dc∞ (M) = −Δg uμg , ϕ1 D  (M),0 Dc∞ (M) , ˜ g u = Δg u, for u ∈ H 1 (M) with the latter defined as the meaning that Δ action of Δg on a distribution. We then find |(Δg u)μg , ϕ1 D  (M),0 Dc∞ (M) |  uH 1 (M) ϕH 1 (M) , and Δg u ∈ H −1 (M). By density one obtains the following proposition. Proposition 18.13. Let u ∈ H 1 (M). Then, Δg u ∈ H −1 (M) and −Δg u, wH −1 (M),H01 (M) = ∫ g(∇g u, ∇g w)μg , M

w ∈ H01 (M).

18.4. The Laplace Problem on a Compact Manifold Without Boundary In the case of a compact Riemannian manifold (M, g) without boundary, we have H k (M) = H0k (M). This property is not clear for noncompact manifold; see the discussion below Proposition 18.4. However, in the case of a compact manifold without boundary, as in the case k = 1, the proof is simply based on the use a finite partition of unity subordinated to an atlas covering, allowing one to use the result known for bounded regular open sets in Rd . We consider the Laplace–Beltrami problem −Δg u + qu = f, where q ∈ L∞ (M; R) and f ∈ H −1 (M), with the definition of H −1 (M) given at the end of Sect. 18.2. If q = 0 we have a kernel formed by constant functions on M. As is classical in this case, we thus choose a nonnegative potential function q that exhibits some positivity in a domain of positive measure. Proposition 18.14 (Laplace Problem on a Manifold without Boundary). Let (M, g) be a compact connected Riemannian d-dimensional manifold without boundary. Let q ∈ L∞ (M; R) be such that q ≥ 0 and q ≥ C0 > 0 on some subset A of M with |A| > 0. There exists C > 0 such that, for any f ∈ H −1 (M), there exists a unique u ∈ H 1 (M) such that (18.4.1) ∫ g(∇g u, ∇g ϕ)μg + ∫ quϕμg = f, ϕH −1 (M),H 1 (M) , M

M

ϕ ∈ H 1 (M),

with C −1 f H −1 (M) ≤ uH 1 (M) ≤ Cf H −1 (M) .

18.4. THE LAPLACE–BELTRAMI PROBLEM

473

Moreover, the (0-density) distribution Δg u is in H −1 (M) and −Δg u + qu = f. Let k ∈ N and assume that in addition q ∈ W k,∞ (M; R). There exists Ck > 0, such that if f ∈ H k (M), then moreover u ∈ H k+2 (M) and Ck−1 f H k (M) ≤ uH k+2 (M) ≤ Ck f H k (M) . and the equation −Δg u + qu = f holds in H k (M). Proof. On H 1 (M), we consider the continuous bilinear form v μg . a(u, v) = ∫ g(∇g u, ∇g v¯)μg + ∫ qu¯ M

M

It is coercive as a(v, v)  (∇g v, ∇g v)L2 V (M) + (v, v)L2 (A)  v2H 1 (M) , by the Poincar´e inequality (18.1.3) given in Proposition 18.9. As v → f, v¯H −1 (M),H 1 (M) is continuous, the Lax–Milgram theorem then gives a unique u ∈ H 1 (M) such that (18.4.1) holds. It also yields the existence of C > 0 such that C −1 f H −1 (M) ≤ uH 1 (M) ≤ Cf H −1 (M) . For a function in H 1 (M) the action of Δg yields a 0-distribution in H −1 (M) by Proposition 18.13. With this proposition we moreover find that −Δg u + qu = f holds in H −1 (M). If now f ∈ H k (M) with k ∈ N, the regularity result can be obtained by using Proposition 18.11, a finite atlas, and the counterpart of the result in  open sets of Rd ; see [161, Section 8.4] and [90, Section 9.6]. We define P0 as an operator on L2 (M) with domain D(P0 ) = {u ∈ H 1 (M); Δg u ∈ L2 (M)}, given by P0 u = −Δg u. As 0 Dc∞ (M) ⊂ D(P0 ) we see by Proposition 18.2 that D(P0 ) is dense in L2 (M). We set Lq = P0 + q with q ≥ 0 and q ≥ C0 > 0 on some subset A of M with |A| > 0, in agreement with the properties required in Proposition 18.14. If u ∈ D(P0 ), then we set f = Lq u ∈ L2 (M). Then, the function u is precisely the solution of the variational problems of the above propositions since the operator Lq : D(P0 ) → L2 (M) is injective by the following lemma. Lemma 18.15. Let u ∈ D(P0 ) be such that Lq u = 0. Then u = 0. Proof. Consider ϕ ∈ 0 Dc∞ (M). By the definition of Δg for a H 1 function we find that ¯ g + ∫ quϕμ ¯ g. 0 = ∫ g(∇g u, ∇g ϕ)μ

(18.4.2)

M

M

density of Dc∞ (M) in H 1 (M) by Proposition 18.4, we see that holds for all ϕ ∈ H 1 (M). The function u is thus the unique solution 0

By the (18.4.2) of the corresponding homogeneous variational problem. This yields u = 0. 

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18. SOBOLEV SPACES AND LAPLACE PROBLEMS

We thus conclude that D(P0 ) = H 2 (M). The analysis recalled in Section 10.1 of Volume 1 can be adapted to the case of the operator −Δg on M. We denote by Sq the map L2 (M) into H 2 (M) such that Sq (f ) provides the unique solution u ∈ H 2 (M) of Lq u = f . If ι is the natural injection of H 2 (M) into L2 (M), we find that the resolvent map Rq = ι ◦ Sq : L2 (M) → L2 (M) is compact by the Rellich– Kondrachov theorem (see Theorem 18.7). Let f1 , f2 ∈ L2 (M) and u1 , u2 ∈ D(P0 ) the associated solutions, that is, uj = Sq (fj ), j = 1, 2. The variational form of elliptic problem gives in particular (18.4.3) (f1 , u2 )L2 (M) = ∫ g(∇g u1 , ∇g u2 )μg + ∫ qu1 u2 μg = (u1 , f2 )L2 (M) , M

M

which reads (f1 , Rq f2 )L2 (M) = (Rq f1 , f2 )L2 (M) . The resolvent map is thus selfadjoint. As Rq is closed we immediately have that Lq and thus P0 are closed operators. Moreover, (18.4.3) also reads (Lq u1 , u2 )L2 (M) = (u1 , Lq u2 ) L2 (M) , that is, (P0 u1 , u2 )L2 (M) = (u1 , P0 u2 )L2 (M) , using the properties of the solution of the elliptic problem given by Proposition 18.14. The operator P0 = (−Δg , D(P0 )) is thus symmetric. Observe also that P0 is monotone and maximal (see Definitions 12.22 and 12.10 of volume 1) as can be seen by choosing q = 1 in the variational problem. By Proposition 12.28 of Volume 1 we obtain that P0 is selfadjoint. Let u ∈ ker P0 . We then have 0 = (−Δg u, u)L2 (M) = ∇g u2L2 V (M) , meaning that u ≡ Cst as M is connected. The kernel of P0 is thus precisely the one dimensional space of constant functions. With the features listed above for P0 , following the lines of Section 10.1 of Volume 1 in the case of a bounded open set of Rd , we obtain the following properties: Hilbert basis of eigenfunctions: There exists a nondecreasing sequence of real eigenvalues 0 = μ0 < μ1 ≤ μ2 ≤ · · · ≤ μn (counted with their multiplicity) and an associated sequence of eigenfunctions of P0 , denoted by (φj )j∈N , that forms a Hilbert basis of L2 (M). Weyl Law: The eigenvalues satisfy the Weyl law of Theorem 10.3 of Volume 1. 18.5. Continuous Sobolev Scale and Traces With the Sobolev spaces H k (M) for k ∈ N defined above on a Riemannian manifold the spaces H s (M) for s > 0 can be defined by interpolation [74, 236].

18.5. CONTINUOUS SOBOLEV SCALE AND TRACES

475

If the manifold M does not have a boundary we may use the spectral decomposition of the Laplace–Beltrami operator on M to define H s (M) for all s ∈ R. Let us denote the eigenvalues of Id +P0 by 1 = 1+μ0 < 1+μ1 ≤ 1+μ2 ≤ · · · ≤ 1 + μn as in Sect. 18.4 with associated eigenfunctions φj that form a Hilbert basis of L2 (M). Following Section 10.1.3, for s ≥ 0 we define K s (M) = {u ∈ L2 (M); (1 + μj )s/2 uj ∈ 2 (C)}, where uj = (u, φj )L2 (M) and we have  uj φj , u= j∈N

with convergence in L2 (M). We set  (1 + μj )s |uj |2 . u2K s (M) = j∈N

With this norm K s (M) is a Hilbert space. As φj is smooth by the elliptic regularity result of Proposition 18.14, we see that smooth functions are dense in K s (M). For k ∈ N, we have K 2k (M) = D((Id +P0 )k ) and uK 2k (M) = (Id +P0 )k uL2 (M) . Proposition 18.16. Let M be a compact Riemannian manifold without boundary. For k ∈ N we have H k (M) = K k (M). Moreover, the norms .H k (M) and .K k (M) are equivalent. Proof. The equivalence is obvious for k = 0. For k = 1, for u smooth, using formula (17.3.1), we write u2K 1 (M) = (u, (Id +P0 )u) = u2L2 (M) + ∇g u2L2 V (M) = u2L2 (M) + Du2L2 Λ1 (M) = u2H 1 (M) . As smooth functions are dense in both H 1 (M) and K 1 (M) we obtain the equality of the two norms and that H 1 (M) = K 1 (M). We next proceed by induction. Assuming the result holds for k ∈ {0, . . . , N }, N ≥ 1, we write, for u smooth, uK N +1 (M) = (Id +P0 )uK N −1 (M)  (Id +P0 )uH N −1 (M) , and we conclude that uK N +1 (M)  uH N +1 (M) using the elliptic regularity result of Proposition 18.14. We then conclude by density as in the case k = 1.  Because of the previous proposition, we are led to define the Sobolev as follows.

476

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

Definition 18.17. Let M be a compact Riemannian manifold without boundary. For s ≥ 0, we define the space H s (M) to be the Hilbert space K s (M) equipped with the norm .K s (M) , that is,  u2H s (M) = (1 + μj )s |uj |2 . j∈N

In particular note that smooth functions are dense in H s (M). With this definition we recover the spaces we could also obtain by interpolation starting from the Sobolev spaces defined in Sect. 18.2 as K s (M) is the natural interpolation of K k (M) and K k+1 (M) if s ∈ (k, k + 1). Sobolev space  of negative orders can be defined too. For s ≥ 0, the dual  space H s (M) is a space of 1-density distributions since smooth functions are dense in H s (M). Using the canonical density function μg on (M, g)   we can identify H s (M) with a space of 0-density distributions that we denote by H −s (M). Definition 18.18. Let M be a compact Riemannian manifold without boundary. For s ≥ 0, we define the space H −s (M) as   H −s (M) = {u ∈ 0 D  (M); μg u ∈ H s (M) }.

We define uH −s (M) = μg u(H s (M)) . Equipped with this norm H −s (M) is a Banach space. The space H −s (M) can be characterized in a very natural manner.  Proposition 18.19. Let s > 0 and (uj )j ⊂ C be such that (1 +   μj )−s/2 uj j ∈ 2 (C). Then, the series j uj φj converges in H −s (M). Con  versely, let u ∈ H −s (M) and v = μg u ∈ H s (M) . Set uj = v, φj    −s/2 u 2 j j ∈  (C) and (H s (M)) ,H s (M) , j ∈ N. Then, one has (1 + μj )  −s u = j uj φj where convergence is understood in H (M), with respect to the norm .H −s (M) . Moreover,  u2H −s (M)  (1 + μj )−s |uj |2 . j

The proof can be adapted from what is done in Section 10.1.3 of Volume 1 in the case of an open set of Rd , the only difficulty being the switching between 1-densities and 0-densities by means of μg . For u ∈ H −s (M) and v ∈ H s (M), we define u, vH −s (M),H s (M) = μg u, v(H s (M)) ,H s (M) .

18.5. CONTINUOUS SOBOLEV SCALE AND TRACES

477

Apart from the case s = 0, we do not identify H −s (M) and H s (M) through the inner product of H s (M) and the Riesz theorem but rather use L2 (M) as a pivot space. We then have, for s ≥ 0, u ∈ L2 (M), v ∈ H s (M).

u, v¯H −s (M),H s (M) = (u, v)L2 (M) ,

Remark 18.20. Note that in the case of a compact manifold with ∂M = ∅ along with homogeneous Dirichlet boundary conditions, the its/2 erated domains K s (M) = D(−Δg ) form a scale of Sobolev-like spaces. As in the case of a bounded regular open set in Rd , K s (M) do not coincide with the natural Sobolev spaces as given in Sect. 18.2 above. Proposition 18.21. Let M be a compact Riemannian manifold without boundary. Let s ∈ R+ and let u ∈ L2 (M). We have u ∈ H s (M) if and only if for every local chart C = (O, κ) its representative is such that uC ∈ H s (κ(O)). If A = (C i )1≤i≤N is a finite atlas with C i = (Oi , κi ), there exists C = CA > 0 such that  i C −1 uH s (M) ≤ uC H s (κi (Oi )) ≤ CuH s (M) , 1≤i≤N

if u ∈

H s (M).

Moreover, if i ∈ {1, . . . , N }, for some Ci > 0, we have Ci−1 uH s (M) ≤ uC H s (κi (Oi )) ≤ Ci uH s (M) i

(18.5.1) if supp(u) ⊂ Oi .

This result follows from Proposition 18.11 and an interpolation argument. The following proposition treats the case s < 0. Proposition 18.22. Let M be a compact Riemannian manifold without boundary, let A = (C i )1≤i≤N be a finite atlas with C i = (Oi , κi ), and let (ψi )1≤i≤N be a smooth partition of unity subordinated to the open covering (Oi )1≤i≤N . Let s < 0 and let u ∈ 0 D  (M). We have u ∈ H s (M) if and only if for i all i ∈ {1, . . . , N }, (ψi u)C ∈ H s (Rd ). Moreover, there exists C = CA > 0 such that  i (ψi u)C H s (Rd ) ≤ CuH s (M) , C −1 uH s (M) ≤ 1≤i≤N

if u ∈

H s (M).

Moreover, if i ∈ {1, . . . , N }, for some Ci > 0, we have Ci−1 uH s (M) ≤ uC H s (Rd ) ≤ Ci uH s (M) i

(18.5.2)

if supp(u) ⊂ Oi . Proof. One first proves (18.5.2). We thus let u ∈ H s (M) with supp(u) i ⊂ Oi . One has u ∈ 0 D  (Oi ) and thus uC ∈ D  (κi (Oi )). If v i ∈ Cc∞ (κi (Oi )) we write uC , v i D  (Rd ),Cc∞ (Rd ) = μCg uC , (μCg )−1 v i D  (Rd ),Cc∞ (Rd ) . i

i

i

i

478

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

 i  With v = (κi )∗ (μCg )−1 v i ∈ Cc∞ (Oi ), understood as a 0-density function on M, one has uC , v i D  (Rd ),Cc∞ (Rd ) = μg u, v1 D  (Oi ),0 Dc∞ (Oi ) = μg u, v(H −s (M)) ,H −s (M) i

= u, vH s (M),H −s (M) . We thus find   Ci i Ci u , v D  (Rd ),C ∞ (Rd )  ≤ u s H (M) vH −s (M)  uH s (M) v H −s (Rd ) c  uH s (M) v i H −s (Rd ) , implying that uC ∈ H s (Rd ) and i

uC H s (Rd )  uH s (M) , i

from the density of Cc∞ (κi (Oi )) in the set of function in H −s (Rd ) with compact support in κi (Oi ). Similarly, for v ∈ Cc∞ (Oi ) one has u, vH s (M),H −s (M) = uC , v i D  (Rd ),Cc∞ (Rd ) = uC , v i H s (Rd ),H −s (Rd ) i

i

with v i = μCg v C . One thus finds i

i

u, vH s (M),H −s (M)  ≤ uC H s (Rd ) v i H −s (Rd )  uC H s (Rd ) vH −s (M) , i

i

yielding uH s (M)  uC H s (Rd ) , i

from the density of Cc∞ (Oi ) in the set of function in H −s (M) with compact support in Oi . The first part of the proof follows by using a partition of unity associated with the chosen atlas.  This property is used in Chap. 8, allowing us to use H s -norms in local charts that can be obtained easily by means of Fourier multipliers. There, the manifold without boundary under consideration is the boundary of a compact manifold. With an interpolation argument we also prove the following continuity result for differential operators. Proposition 18.23. Let (M, g) be a smooth Riemannian manifold such that M is compact without boundary. Let s ∈ R and let Q be a differential operator of order k with smooth coefficients on M. Then, the operator Q maps H s (M) into H s−k (M) continuously. We conclude this section with the following trace inequality.

18.5. CONTINUOUS SOBOLEV SCALE AND TRACES

479

Proposition 18.24. Let (M, g) be a compact Riemannian manifold. The trace u → u|∂M defined for smooth functions can be uniquely extended to functions in H 1 (M). Moreover, there exists C > 0 such that u ∈ C ∞ (M).

|u|∂M |H 1/2 (∂M) ≤ CuH 1 (M) ,

Proof. On Rd+ arguing as in the proof of Proposition 6.9 (taking τ = 1, for instance) we obtain, for s > 0, |u|xd =0+ |H 1/2 (Rd−1 )  uH 1 (Rd ) ,

(18.5.3)

u ∈ S (Rd+ ).

+

Let A = (C i )1≤i≤N , C i = (Oi , κi ) be a finite atlas of M. Choose a partition of unity (χi )1≤i≤N of M subordinated to the open covering (Oi )1≤i≤N and i set ui = χi u. We denote by uCi the representative of ui in the chart C i . We set J = {1 ≤ i ≤ N ; Oi ∩ ∂M = ∅}. For j ∈ J, we have by (18.5.3), with a density argument, |uCj

j

|

|xd =0+ H 1/2 (κj (Oj ))

=|uCj

j

 uCj H 1 (Rd ) =uCj H 1 (κj (Oj )) . j

|

|xd =0+ H 1/2 (Rd−1 )

j

+

By (18.2.3) in Proposition 18.11 and (18.5.1) in Proposition 18.21 we thus have |χj u|∂M |H 1/2 (∂M)  χj uH 1 (M) . We then write |u|∂M |H 1/2 (∂M) 

 j∈J

|χj u|∂M |H 1/2 (∂M) 

 j∈J

χj uH 1 (M)  uH 1 (M) ,

using Proposition 18.23 in the last inequality.



More generally, we have the following result. Theorem 18.25 (Trace Theorem). Let k ∈ N, k ≥ 1. The map T defined by T : H k (M) →

k−1

H k−j−1/2 (∂M)

j=0

u → (u|∂M , . . . , ∂νk−1 u|∂M ) is well defined, continuous, and surjective. Moreover, T has a linear and continuous right-inverse. Recall that ∂ν is defined in (17.7.3). We refer to [236, Theorem 8.3 in Chapter 1] and [2, Theorem 7.39 and Lemma 7.41]. In fact, the following lemmata provide explicit forms of right inverses in the case of a half-space. A proof of Theorem 18.25 in the cases k = 1, 2 can then be deduced by working locally at the boundary. Lemma 18.26. Let χ ∈ Cc∞ (R) be even and such that ∫R χ = 2π. Let v ∈ S (Rd−1 ) and denote by vˆ its Fourier transform. The function ξ = (ξ  , ξd ) → ξ  −1 χ(ξd /ξ  )ˆ v (ξ  )

480

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

is in S (Rd ) and if we define u as its inverse Fourier transform in Rd , then u ∈ S (Rd ) and we have u|xd =0 = v,

∂xd u|xd =0 = 0,

and uH s+1/2 (Rd )  |v|H s (Rd−1 ) ,

for s ∈ R. / supp(χ) and Lemma 18.27. Let χ ∈ Cc∞ (R) be even, such that 0 ∈ d−1 ∫R χ = 2π. Let v ∈ S (R ) and denote by vˆ its Fourier transform. The function v (ξ  ), ξ = (ξ  , ξd ) → (iξd ξ  )−1 χ(ξd /ξ  )ˆ is in S (Rd ) and if we define u as its inverse Fourier transform in Rd , then u ∈ S (Rd ) and we have u|xd =0 = 0,

∂xd u|xd =0 = v,

and uH s+3/2 (Rd )  |v|H s (Rd−1 ) ,

for s ∈ R. v (ξ  ). As χ is Proof of Lemma 18.26. Set w(ξ) = ξ  −1 χ(ξd /ξ  )ˆ  compactly supported we have |ξd |  ξ  in supp(w) and thus ξ  ξ  ,

(18.5.4)

ξ ∈ supp(w).

We then deduce that w ∈ S (Rd ) since v ∈ S (Rd−1 ). For s ∈ R, we have, 2

v (ξ  )|2 dξ ξs+1/2 wL2 (Rd ) = ∫ ξ2s+1 ξ  −2 χ2 (ξd /ξ  )|ˆ Rd

   ∫ ξ  2s |ˆ v (ξ  )|2 ∫ ξ  −1 χ2 (ξd /ξ  )dξd dξ  Rd−1



= ∫ χ (t)dt 2

R



R  2s

∫ ξ  |ˆ v (ξ  )|2 dξ 

Rd−1

 |ξ  s v|L2 (Rd−1 )  |v|2H s (Rd−1 ) . 2

If u is the inverse Fourier transform of w, that is, u(x) =

1 (2π)d

∫Rd eix·ξ w(ξ)dξ,

we then have uH s+1/2 (Rd )  ξs+1/2 wL2 (Rd )  |v|H s (Rd−1 ) . Now, since we have 1   ∫ eix ·ξ w(ξ  , ξd ) dξ  dξd , u(x , 0) = d (2π) Rd we compute vˆ(ξ  ) vˆ(ξ  ) 1 ∫ w(ξ  , ξd )dξd = ∫ ξ  −1 χ(ξd /ξ  )dξd = ∫ χ(t)dt = vˆ(ξ  ). 2π R 2π R 2π R This implies that u(x , 0) = v(x ). We also have i   ∫ eix ·ξ ξd w(ξ  , ξd ) dξ  dξd . ∂xd u(x , 0) = d (2π) Rd We compute v (ξ  ) ∫ tχ(t)dt = 0, ∫ ξd w(ξ  , ξd ) dξd = vˆ(ξ  ) ∫ (ξd /ξ  )χ(ξd /ξ  )dξd = ξ  ˆ

R

R

R

18.5. CONTINUOUS SOBOLEV SCALE AND TRACES

481

since χ is even. This implies ∂xd u|xd =0 = 0, which concludes the proof.



Proof of Lemma 18.27. Set w(ξ) = (iξd ξ  )−1 χ(ξd /ξ  )ˆ v (ξ  ). Arguing as in the proof of Lemma 18.26 we find that w ∈ S (Rd ). For s ∈ R, we have, 2

v (ξ  )|2 dξ ξs+3/2 wL2 (Rd ) = ∫ ξ2s+3 (|ξd |ξ  )−2 χ2 (ξd /ξ  )|ˆ Rd

   ∫ ξ  2s |ˆ v (ξ  )|2 ∫ ξ  −1 |ξd /ξ  |−2 χ2 (ξd /ξ  )dξd dξ  Rd−1



= ∫ |t| R

−2 2

χ (t)dt

R



∫ ξ  2s |ˆ v (ξ  )|2 dξ 

Rd−1

 |ξ  s v|L2 (Rd−1 )  |v|2H s (Rd−1 ) . 2

If u is the inverse Fourier transform of w, we then have uH s+3/2 (Rd )  ξs+3/2 wL2 (Rd )  |v|H s (Rd−1 ) . Now, since we have u(x , 0) =

1   ∫ eix ·ξ w(ξ  , ξd ) dξ  dξd , (2π)d Rd

we compute vˆ(ξ  ) 1 ∫ w(ξ  , ξd )dξd = −i ∫ ξ  −1 (ξd /ξ  )−1 χ(ξd /ξ  )dξd 2π R 2πξ   R vˆ(ξ  ) −1 ∫ t χ(t)dt = 0, = −i 2π R as χ is even. This implies that u(x , 0) = 0. We also have 1   ∫ eix ·ξ iξd w(ξ  , ξd ) dξ  dξd . ∂xd u(x , 0) = d (2π) Rd We compute vˆ(ξ  ) vˆ(ξ  ) 1 ∫ iξd w(ξ  , ξd )dξd = ∫ ξ  −1 χ(ξd /ξ  )dξd = ∫ χ(t)dt = vˆ(ξ  ). 2π R 2π R 2π R This implies that ∂xd u(x , 0) = v(x ), which concludes the proof.



Observe that the divergence formula stated in Proposition 17.6 for smooth vector fields extends to H 1 -vector fields by density. Proposition 18.28 (Divergence Formula for H 1 -Vector Fields). Let (M, g) be a Riemannian manifold with boundary and let u ∈ H 1 V (M) be compactly supported on M. Let ν be the unique outward pointing vector field along ∂M such that, for all m ∈ ∂M, gm (νm , νm ) = 1 and gm (νm , v) = 0 for all v ∈ Tm ∂M, that is, ν is unitary and orthogonal to Tm ∂M in the sense of g. We have ∫ divg (u)μg = ∫ g(u|∂M , ν) μg∂ ,

M

∂M

482

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

where g∂ is the induced Riemannian metric on ∂M and μg∂ the resulting canonical positive density function on ∂M. ¯ g u) = wΔ ¯ g u + g(∇g w, ¯ ∇g u), for u ∈ H 2 (M) and w ∈ Since divg (w∇ we obtain as an application

C ∞ (M)

(Δg u, w)L2 (M) = −(∇g u, ∇g w)L2 V (M) + (∂ν u|∂M , w|∂M )L2 (∂M) . By density we find (18.5.5)

(Δg u, w)L2 (M) = −(∇g u, ∇g w)L2 V (M) + (∂ν u|∂M , w|∂M )L2 (∂M) ,

for u ∈ H 2 (M) and w ∈ H 1 (M), both with compact supports. By density of C ∞ (M) ∩ H 1 (M) in H 1 (M) we obtain the following ‘integration by parts’ formula from Proposition 17.7. Proposition 18.29. Let X ∈ C ∞ V (M) and let f1 , f2 ∈ H 1 (M) with compact supports. Then ¯ 2 +f2 divg X) ¯ L2 (M) +(f1 g(X, ν)|∂M , f2 |∂M )L2 (∂M) . (Xf1 , f2 )L2 (M) = − (f1 , Xf From (18.5.5), we can extend the Green formula to compactly supported H 2 -functions. Proposition 18.30. Let f1 , f2 ∈ H 2 (M) have compact supports. We have (f1 , Δg f2 )L2 (M) + (∂ν f1 |∂M , f2 |∂M )L2 (∂M) = (Δg f1 , f2 )L2 (M) + (f1 |∂M , ∂ν f2 |∂M )L2 (∂M) .

18.6. The Dirichlet-Laplace Problem If (M, g) is a compact Riemannian manifold with boundary we are interested in solving the following problem −Δg u + qu = f in M,

u|∂M = h on ∂M,

that is, an elliptic problem with Dirichlet boundary condition. The potential function q will be assumed bounded in what follows. 18.6.1. Homogeneous Dirichlet Boundary Conditions. We denote by P0 the differential operator P0 = −Δg . We start by considering the case of homogeneous Dirichlet boundary conditions, that is, P0 u + qu = f,

u|∂M = 0,

where q ∈ L∞ (M; R) and f ∈ H −1 (M), with the definition of H −1 (M) given at the end of Sect. 18.2.

18.6. THE LAPLACE–DIRICHLET PROBLEM

483

Proposition 18.31 (Dirichlet-Laplace Problem). Let (M, g) be a compact connected Riemannian d-dimensional manifold with boundary. Let q ∈ L∞ (M; R). Let Cg > 0 be the Poincar´e constant. There exists C > 0 such that, if q > −Cg , for any f ∈ H −1 (M), there exists a unique u ∈ H01 (M) such that (18.6.1) ¯)μg + ∫ qϕ¯ u μg = f¯, ϕH −1 (M),H01 (M) , ϕ ∈ H01 (M), ∫ g(∇g ϕ, ∇g u M

M

with C −1 f H −1 (M) ≤ uH 1 (M) ≤ Cf H −1 (M) . Moreover, the (0-density) distribution P0 u is in H −1 (M) and P0 u + qu = f holds in H −1 (M). Conversely, if f ∈ H −1 (M) and if u ∈ H01 (M) is such that P0 u + qu = f holds in H −1 (M), then u is the unique solution of the variational form of the homogeneous Dirichlet problem (18.6.1). Let k ∈ N. For q ∈ W k,∞ (M; R), there exists Ck > 0, such that if f ∈ H k (M), then moreover u ∈ H k+2 (M) and Ck−1 f H k (M) ≤ uH k+2 (M) ≤ Ck f H k (M) and the equation P0 u + qu = f holds in H k (M). Proof. On H01 (M), we consider the continuous Hermitian form ¯)μg + ∫ qv u ¯μg = (∇g v, ∇g u)L2 V (M) + (qv, u)L2 (M) . a(v, u) = ∫ g(∇g v, ∇g u M

M

If Cg > 0 is the Poincar´e constant, then Cg1/2 vL2 (M) ≤ ∇g vL2 V (M) ,

v ∈ H01 (M).

by Proposition 18.9. Thus, if q > −Cg we obtain that a(v, v)  v2H 1 (M) and consequently a(., .) is coercive on H01 (M). As v → (v, f¯)L2 (M) is continuous, the Lax–Milgram theorem then gives a unique u ∈ H01 (M) such that (18.6.1) holds. It also yields the existence of C > 0 such that C −1 f H −1 (M) ≤ uH 1 (M) ≤ Cf H −1 (M) . We then find that P0 u + qu = f holds in H −1 (M) by Proposition 18.13. Conversely, if f ∈ H −1 (M) and if u ∈ H01 (M) is such that P0 u + qu = f holds in H −1 (M), then by applying (18.5.5), consequence of the divergence formula of Proposition 18.28, one finds that u is the unique solution of (18.6.1). If now f ∈ H k (M) with k ∈ N, the regularity result can be obtained by using Proposition 18.11, a finite atlas, and the counterpart of the result in  open sets of Rd ; see [161, Section 8.4] and [90, Section 9.6]. Following Sects. 10.1 and 18.4, define P0 as an unbounded operator on with domain

L2 (M)

(18.6.2)

D(P0 ) = {u ∈ H01 (M); P0 u ∈ L2 (M)},

484

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

given by P0 u = P0 u. We have D(P0 ) = H 2 (M) ∩ H01 (M). We find that P0 is closed, selfadjoint and maximal monotone. Moreover, ker P0 = {0}. Proposition 18.31 thus shows that the map, with k ∈ N, (18.6.3)

Lq : H k+2 (M) ∩ H01 (M) → H k (M) u → P0 u + qu

is an isomorphism if q ∈ W k,∞ (M; R) and q > −Cg . We define the resolvent map R0 : L2 (M) → H 2 (M) ∩ H01 (M), as the inverse of the above map in the case q = 0. It is compact map if considered as a map from L2 (M) into itself. We then obtain the following properties: Hilbert basis of eigenfunctions: There exists a nondecreasing sequence of real eigenvalues 0 < μ0 < μ1 ≤ μ2 ≤ · · · ≤ μn (counted with their multiplicity) and an associated sequence of eigenfunctions of P0 , denoted by (φj )j∈N , that forms a Hilbert basis of L2 (M). Weyl Law: The eigenvalues satisfy the Weyl law of Theorem 10.3 of Volume 1. Second, we wish to consider the nonhomogeneous Dirichlet problem on a compact Riemannian manifold. To obtain case of Dirichlet data with low regularity we first consider some subspaces of L2 (M). 18.6.2. Spaces with Remarkable Trace Properties. Let P = P0 + R1 with P0 = −Δg and R1 a first-order differential operator with smooth coefficients. We introduce the following subspace of L2 (M) associated with P: (18.6.4)

WP (M) = {u ∈ L2 (M); P u ∈ L2 (M)}.

In the case P = P0 we simply write Wg (M). Here, the action of P on u is to be understood in the sense of distributions, as expressed in (18.3.1), and not as that of the unbounded operator (P0 + R1 , D(P0 )) on L2 (M). One can readily check that WP (M) is a Hilbert space if equipped with the inner product (u, v)WP (M) = (u, v)L2 (M) + (P u, P v)L2 (M) ,

u, v ∈ WP (M),

and the associated norm u2WP (M) = u2L2 (M) + P u2L2 (M) ,

u ∈ WP (M).

We find that both the Dirichlet and Neumann trace operators extend to functions in WP (M). Lemma 18.32. The Dirichlet trace map γ D : u → u|∂M and the Neumann trace map γ N : u → ∂ν u|∂M , both well defined on H 2 (M), admit a unique

18.6. THE LAPLACE–DIRICHLET PROBLEM

485

extension to WP (M), that we still denote by γ D and γ N . If u ∈ WP (M) we have γ D (u) ∈ H −1/2 (∂M)

and γ N (u) ∈ H −3/2 (∂M),

and the linear maps γ D : WP (M) → H −1/2 (∂M) and γ N : WP (M) → H −3/2 (∂M) are both bounded. Finally, the Leibniz rule is compatible with these trace extensions, that is, for a ∈ C ∞ (M), we have γ N (au) = γ D (a)γ N (u) + γ N (a)γ D (u). A proof is given in Appendix 18.A.1. If R1 = X + f with X a vector field and f a function, the transpose operator of P is given by t P = −Δg − X − divg X + f ; see Sect. 16.3.3, Propositions 17.7 and 17.8. For functions in WP (M), we have a Green-like formula. Lemma 18.33. Let R1 = X + f with X a smooth vector field and f a smooth function and P = −Δg + R1 . Let u ∈ WP (M) and w ∈ H 2 (M). We have ¯ H −3/2 (∂M),H 3/2 (∂M) (P u, w)L2 (M) + γ N (u), γ D (w) = (u, t P¯ w)L2 (M) + γ D (u), γ N (w) ¯ H −1/2 (∂M),H 1/2 (∂M) + γ D (g(X, ν)u), γ D (w) ¯ H −1/2 (∂M),H 1/2 (∂M) . A proof is given in Appendix 18.A.2. For functions in Wg (M), that is, in the case P = P0 = −Δg , this gives the following Green formula. Lemma 18.34. Let u ∈ Wg (M) and w ∈ H 2 (M). We have (P0 u, w)L2 (M) + γ N (u), γ D (w) ¯ H −3/2 (∂M),H 3/2 (∂M) = (u, P0 w)L2 (M) + γ D (u), γ N (w) ¯ H −1/2 (∂M),H 1/2 (∂M) . We also consider the following subspace of L2 (M): (18.6.5)

Wg,−1 (M) = {u ∈ L2 (M); P u ∈ H −1 (M)}.

Note that this space is independent of the first-order term R1 in P . We have Wg (M) ∪ H 1 (M) ⊂ Wg,−1 (M). The inner product (u, v)Wg,−1 (M) = (u, v)L2 (M) + (P0 u, P0 v)H −1 (M) ,

u, v ∈ Wg,−1 (M),

and the associated norm, u2Wg,−1 (M) = u2L2 (M) + P0 u2H −1 (M) ,

u ∈ Wg,−1 (M),

yield a Hilbert space structure on Wg,−1 (M). As the Neumann trace of a H 1 -function makes no sense in general the same is true for a function in Wg,−1 (M). However, the Dirichlet trace makes sense in H −1/2 (∂M) like we found above for functions in Wg (M).

486

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

Lemma 18.35. The Dirichlet trace map γ D : u → u|∂M well defined on H 1 (M) admits a unique extension to Wg,−1 (M), that we still denote by γ D . If u ∈ Wg,−1 (M) we have γ D (u) ∈ H −1/2 (∂M) and the linear maps γ D : Wg,−1 (M) → H −1/2 (∂M) is bounded. For any w ∈ H 2 (M) ∩ H01 (M) we have ¯ H −1 (M),H01 (M) = (u, P0 w)L2 (M) + γ D (u), γ N (w) ¯ H −1/2 (∂M),H 1/2 (∂M) . P0 u, w A proof is given in Appendix 18.A.3. 18.6.3. The Dirichlet Lifting Map. We are interested in defining a map that acts as a solution operator for the following elliptic problem P0 u = 0 in M

and γ D (u) = h on ∂M,

for h ∈ H −1/2 (∂M). The Hilbert space Wg (M) appears as a natural space for solutions of this problem. In fact, we have the following result. Proposition 18.36. Let h ∈ H −1/2 (∂M). There exists a unique u ∈ Wg (M) such that P0 u = 0 and γ D (u) = h. Moreover, the map h → u from H −1/2 (∂M) into Wg (M) is bounded. Definition 18.37 (Dirichlet Lifting Map). The bounded map D : h → u from H −1/2 (∂M) into Wg (M) given by Proposition 18.36 is called the Dirichlet lifting map. Proof of Proposition 18.36. The proof makes use of the resolvent map R0 : L2 (M) → D(P0 ) introduced below (18.6.3). First, we address uniqueness. By linearity, we consider u ∈ Wg (M) is such that P0 u = 0 and γ D (u) = 0. Then, by Lemma 18.34 we have (u, P0 v)L2 (M) = 0 for all v ∈ H 2 (M) ∩ H01 (M). Let f ∈ L2 (M); we choose v = R0 f . We obtain (u, f )L2 (M) = 0. As f ∈ L2 (M) is arbitrary we conclude that u = 0. Second, we address existence. We consider the following form: U : L2 (M) → C ¯ 1/2 f → −γ N R0 f, h H (∂M),H −1/2 (∂M) . By the trace formula of Theorem 18.25, this form is bounded and we have |U (f )| ≤ |h|H −1/2 (∂M) R0 f H 2 (M)  |h|H −1/2 (∂M) f L2 (M) . By the Riesz theorem, there exists u ∈ L2 (M) such that U (f ) = (f, u)L2 (M) for all f ∈ L2 (M). Moreover, uL2 (M)  |h|H −1/2 (∂M) . If ϕ ∈ 0 Dc∞ (M) and f = P0 ϕ, then ϕ = R0 f and we find ¯ 1/2 U (f ) = −γ N (ϕ), h = 0. −1/2 H

(∂M),H

(∂M)

18.6. THE LAPLACE–DIRICHLET PROBLEM

487

Consequently u, P0 ϕμg 0 D  (M),1 Dc∞ (M) P0 u, ϕμg 0 D  (M),1 Dc∞ (M) = ¯ = (P0 ϕ, u)L2 (M) = U (f ) = 0. Thus, in the sense of distributions we have P0 u = 0. Thus u ∈ Wg (M) and uWg (M)  |h|H −1/2 (∂M) . Lemma 18.38. There exists a bounded map N0 : H 1/2 (∂M) → H 2 (M) ∩ such that γ N ◦ N0 = IdH 1/2 (∂M) .

H01 (M)

This is a particular case of Theorem 18.25. A proof follows from Lemma 18.27 by working locally at the boundary. Let ψ ∈ H 1/2 (∂M). For v = N0 ψ ∈ H 2 (M) ∩ H01 (M) as given by Lemma 18.38, we have u)H 1/2 (∂M),H −1/2 (∂M) = 0, (P0 v, u)L2 (M) + ψ, γ D (¯ by Lemma 18.34. We have ¯ 1/2 (P0 v, u)L2 (M) = (P0 v, u)L2 (M) = U (P0 v) = −γ N R0 P0 v, h H (M),H −1/2 (M) ¯ 1/2 ¯ = −γ N v, h H (M),H −1/2 (M) = −ψ, hH 1/2 (M),H −1/2 (M) . ¯ − γ D (¯ u)H 1/2 (∂M),H −1/2 (∂M) = 0, For all ψ ∈ H 1/2 (∂M) we thus find ψ, h which concludes the existence part of the proof.  If h is more regular than H −1/2 , then additional regularity is transferred to D(h), with consistency with the trace formula of Theorem 18.25. Proposition 18.39. Let m ∈ N. If h ∈ H m−1/2 (∂M), then D(h) ∈ Moreover, for some C > 0, we have

H m (M). (18.6.6)

D(h)H m (M) ≤ C|h|H m−1/2 (∂M) ,

h ∈ H m−1/2 (∂M).

Proof. The case m = 0 was treated above. Let m ≥ 1 and h ∈ H m−1/2 (M). By Theorem 18.25, there exists v ∈ H m (M) such that γ D (v) = h and vH m (M)  |h|H m−1/2 (∂M) . We have f = P0 v ∈ H m−2 (M) with f H m−2 (M)  |h|H m−1/2 (∂M) , and there exists a unique v˜ ∈ H m (M) ∩ H01 (M) such that P0 v˜ = f by Proposition 18.31. We have ˜ v H m (M)  |h|H m−1/2 (∂M) . Consider next w = v − v˜ ∈ H m (M) . We have P0 w = 0 and γ D (w) = γ D (v) = h. By the uniqueness part of Proposition 18.36 we have w = D(h) and (18.6.6) holds.  The above results allow one to fully treat the nonhomogeneous DirichletLaplace problem on a compact manifold with boundary exploiting the linearity of the problem.

488

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

Theorem 18.40. Let (M, g) be a compact connected Riemannian ddimensional manifold with boundary and let q ∈ L∞ (M; R) with q > −Cg , where Cg is as given in Proposition 18.31. The map Wg,−1 (M) → H −1 (M) ⊕ H −1/2 (∂M)   u → P0 u + qu, γ D (u) is an isomorphism. Moreover, for k ∈ N, if q ∈ W k,∞ (M; R), the map L : H k+1 (M) → H k−1 (M) ⊕ H k+1/2 (∂M)   u → P0 u + qu, γ D (u)

D

is an isomorphism. We also have the following regularity results. Corollary 18.41. Let u ∈ L2 (M). (1) Assume that P0 u ∈ H −1 (M), that is, u ∈ Wg,−1 (M) and assume yet that γ D (u) ∈ H 1/2 (∂M). Then u ∈ H 1 (M). (2) More generally, let m, m ∈ N and assume that P0 u ∈ H m−1 (M)   and γ D (u) ∈ H m +1/2 (∂M). Then u ∈ H min(m,m )+1 (M). Proof. Let u ∈ Wg,−1 (M) with γ D (u) ∈ H 1/2 (M). Set f = P0 u ∈ and h = γ D (u) ∈ H 1/2 (M). By Theorem 18.40 there exists a unique v ∈ H 1 (M) such that P0 v = f and γ D (v) = h. Then w = u − v ∈ L2 (M) is such that P0 w = 0, that is w ∈ Wg (M) and moreover γ D (w) = 0. From the uniqueness part of Proposition 18.36 we conclude that w = 0. Let now u ∈ L2 (M) such that f = P0 u ∈ H m−1 (M) and h = γ D (u) ∈  H m +1/2 (∂M). Set n = min(m, m ). We have f ∈ H n−1 (M) and h ∈ H n+1/2 (∂M). By Theorem 18.40 there exists a unique v ∈ H n+1 (M) such  that P0 v = f and γ D (v) = h. The conclusion then follows as above.

H −1 (M)

In particular, with this last corollary, we see that the domain of the operator P0 with domain defined in (18.6.2) can equivalently be defined by (18.6.7)

D(P0 ) = {u ∈ L2 (M); Δg u ∈ L2 (M) and γ D (u) = 0},

which is a more natural definition as the operator P0 is an unbounded operator on L2 (M). This, however, requires the knowledge of the existence of a Dirichlet trace for functions in Wg (M). 18.6.4. More Spaces with Remarkable Trace Properties. By Lemma 18.32 if u ∈ Wg (M), then γ D (u) ∈ H −1/2 (∂M)

and γ N (u) ∈ H −3/2 (∂M).

If u is moreover in H 1 (M) we naturally have γ D (u) ∈ H 1/2 (∂M) by the trace formula of Theorem 18.25. In fact, there is also an improvement of regularity for the Neumann trace.

18.7. THE NEUMANN-LAPLACE PROBLEM

489

Lemma 18.42. Let u ∈ Wg (M) ∩ H 1 (M). Then γ N (u) ∈ H −1/2 (∂M) and

  |γ N (u)|H −1/2 (∂M) ≤ C uH 1 (M) + P0 uL2 (M) ,

for some C > 0 (independent of u). A proof based on the Dirichlet lifting map introduced in Sect. 18.6.3 is given in Appendix 18.A.4. Lemma 18.42 is also a consequence of a normal trace result on the following Hilbert space of vector fields H(divg , M) = {U ∈ L2 V (M); divg U ∈ L2 (M)}, equipped with the inner product (U, V )H(divg ,M) = (U, V )L2 V (M) + (divg U, divg V )L2 (M) . In the definition of H(divg , M) the action divg is to be understood in the sense of distributions. Lemma 18.43. The trace map U → g(U, ν)|∂M , well defined on the space of H 1 -vector fields, admit a unique extension to H(divg , M). If U ∈ H(divg , M) one has g(U, ν)|∂M ∈ H −1/2 (∂M) and for some C > 0, g(U, ν)|∂M H −1/2 (∂M) ≤ CU H(divg ,M) ,

U ∈ H(divg , M).

A proof is given in Appendix 18.A.5. 18.7. The Neumann-Laplace Problem Here, for q ∈ L∞ (M; R) such that q ≥ 0, we consider the elliptic problem (18.7.1)

(P0 + q)u = f in M,

γ N (u) = h on ∂M.

If f ∈ L2 (M) and if we seek u ∈ H 2 (M), then the trace formula of Theorem 18.25 indicates that the proper space for h is H 1/2 (∂M). Next, with the divergence formula of Proposition 18.28, we observe that (18.7.2)

∫ f μg + ∫ γ N (u) μg∂ = ∫ qu μg .

M

∂M

M

We refer to Sects. 16.2.4 and 17.3 for integration on Riemannian manifolds. In particular, μg (resp. μg∂ ) is the canonical positive density function on M (resp. ∂M). If q = 0 we see that the integral of qu is immediately known. On H 1 (M) we define the following Hermitian form: (18.7.3)

a0 (v, u) = (D v, D u)L2 Λ1 (M) = (∇g v, ∇g u)L2 V (M) ¯(m)) μg , = ∫ gm (∇g v(m), ∇g u M

and (18.7.4)

aq (v, u) = a0 (v, u) + (v, qu)L2 (M)

490

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

Proposition 18.44. Let q ∈ L∞ (M; R) be such that q ≥ 0 and q ≥ C > 0 on A ⊂ U of positive measure. If f ∈ L2 (M) and h ∈ H 1/2 (∂M), there exists a unique u ∈ H 1 (M) such that (18.7.5)

aq (v, u) = (v, f )L2 (M) + (v|∂M , h)L2 (∂M) ,

v ∈ H 1 (M).

Moreover, u ∈ H 2 (M) and we have (P0 + q)u = f in M,

γ N (u) = h on ∂M.

and there exists C > 0 such that   uH 2 (M) ≤ C f L2 (M) + |h|H 1/2 (∂M) . If k ∈ N, if q ∈ W k,∞ (M; R) in addition to the above requirements, there exists C > 0 such that if f ∈ H k (M) and h ∈ H k+1/2 (M), then u ∈ H k+2 (M) and   uH k+2 (M) ≤ C f H k (M) + |h|H k+1/2 (∂M) . One says that (18.7.5) is the variational form of the Neumann problem (18.7.1). Proof. By the trace formula of Theorem 18.25 there exists h ∈ H 2 (M) such that γ N (h) = h and hH 2 (M)  |h|H 1/2 (∂M) . With (18.5.5) consequence of the divergence formula of Proposition 18.28 we then have, for all v ∈ H 1 (M) aq (v, h) = (v, H)L2 (M) + (v|∂M , h)L2 (∂M) , where H = (q + P0 )h ∈ L2 (M), with HL2 (M)  |h|H 1/2 (∂M) . By linearity, it thus suffices to solve Problem (18.7.5) for h = 0 and f replaced by f − H. One then observes that aq (., .) is coercive on H 1 (M) by the third Poincar´e inequality given in Proposition 18.9. As v → (v, f )L2 (M) is continuous and linear on H 1 (M), the Lax–Milgram theorem yields the existence of u ∈ H 1 (M) solution to (18.7.5). The remainder of the proof is classical. We refer, for instance, to [90, Section 9.5, Example 4] that can be adapted to the manifold case.  Corollary 18.45 (Uniqueness of Strong Solutions). Let f ∈ L2 (M) and h ∈ H 1/2 (∂M). Let u ∈ H 2 (M) be such that P0 u + qu = f in M and γ N (w) = h on ∂M. Then u is the unique solution of the variational form of the Neumann problem (18.7.5). Proof. It suffices to observe that u is itself a solution of (18.7.5) by (18.5.5) consequence of the divergence formula of Proposition 18.28.  We now consider the case q = 0 in (18.7.1). Then, we see from (18.7.2) that a necessary condition for the solvability of the elliptic problem is (18.7.6)

∫ f μg + ∫ h μg∂ = 0.

M

∂M

18.7. THE NEUMANN-LAPLACE PROBLEM

491

Moreover, in the case q = 0, uniqueness cannot be obtained in the space H 2 (M) since u + Cst is a solution if u solves the problem. We shall recall below that this is the only obstruction to well-posedness under condition (18.7.6). Let k ∈ N and set C = {u ∈ H k (M); u ≡ Cst} ∼ = C. With the above observations, we define the equivalence relation on H k (M) u ∼ u ⇔ u − u ∈ C,

(18.7.7)

and we denote by H k (M)/C or rather H k (M)/ C the quotient space H k (M)/ ∼, and by Φ : H k (M) → H k (M)/C the surjective linear application that maps u to its class [u] in H k (M)/C. The subspace C is precisely the kernel of Φ. We have a Banach space structure on H k (M)/C (see e.g. [320, Chapter 11]) with the norm given by uH k (M)/C =

inf

u∈H k (M) [u]=u

uH k (M) ,

that makes the map Φ continuous. We define the following linear continuous map Π : H k (M) → H k (M) ffl u → u − u μg , where

ffl

M

M u μg

=

1 |M| ∫M u μg

is the average of u on M. We observe that for u ∈ H k (M).

[u] = [Πu],

We set H k (M) = Ran(Π) and note that (18.7.8)

H k (M) = {u ∈ H k (M);

ffl M

u μg = 0}.

In particular, H k (M) is a closed linear subspace of H k (M). For k = 0, we denote H 0 (M) = L2 (M). Lemma 18.46. We have C ⊕ H k (M) = H k (M) and Π is the orthogonal projector onto H k (M). Proof. As H k (M) is closed, it suffices to prove that C = H k (M)⊥ to obtain the first part of the result. The second part follows as Π vanishes on C and coincides with the identity on H k (M). Let u be in H k (M)⊥ . We then have 0 = (u, Πu)H k (M) yielding u2L2 (M) ≤ u2H k (M) = ( ≤

ffl M

u μg )(u, 1)H k (M) =

2 1  ∫ |u| μg . |M| M

2 1  ∫ u μg  |M| M

492

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

With the Cauchy–Schwarz inequality, we have 2 1  ∫ |u| μg ≤ u2L2 (M) . |M| M As a result ∫M |u| μg = |M|1/2 uL2 (M) , meaning equality in the Cauchy– Schwarz inequality, which implies that u ≡ Cst. Conversely, if u ≡ Cst, we  immediately see that u ∈ H k (M)⊥ . With this lemma, we see that H k (M)/C is to be identified with H k (M). In fact, if we denote by Φ the map (18.7.9)

Φ : H k (M) → H k (M)/C u → Φ(u) = [u],

we have the following result formalizing this identification. Lemma 18.47. The map Φ is an isometry of H k (M) onto H k (M)/C. In particular, this yields a Hilbert space structure on H k (M)/C with the inner product (18.7.10)   [u], [v] H k (M)/C = ((Φ)−1 [u], (Φ)−1 [v])H k (M) ,

for [u], [v] ∈ H k (M)/C.

This is summarized in the following diagram. H k (M) Φ

Π

H k (M)

Φ (Φ)−1

H k (M)/C

For [u] ∈ H k (M)/C, we shall denote by u its unique representative in H k (M), that is, u = (Φ)−1 [u], and we have (18.7.11)

[u]H k (M)/C = uH k (M) .

For u ∈ H k (M), since [u] = [Πu], we note that we have Πu = (Φ)−1 [u] = u. We may now state how the Neumann problem can be solved in the case q = 0. Proposition 18.48. Let f ∈ L2 (M) and h ∈ H 1/2 (∂M) be such that (18.7.12)

∫ f μg + ∫ h μg∂ = 0.

M

∂M

There exists a unique [u] ∈ H 1 (M)/C such that, (18.7.13) a0 (v, u) = (v, f )L2 (M) + (v|∂M , h)L2 (∂M) ,

u ∈ [u], v ∈ H 1 (M),

18.7. THE NEUMANN-LAPLACE PROBLEM

493

with the Hermitian form a0 (., , .) given in (18.7.3). Moreover, [u] ∈ H 2 (M) /C and we have, for u ∈ [u], P0 u = f in M,

γ N (u) = h on ∂M.

Finally, there exists C > 0 such that (18.7.14)

  uH 2 (M) = [u]H 2 (M)/C ≤ C f L2 (M) + |h|H 1/2 (∂M) .

If k ∈ N there exists Ck > 0 such that if moreover f ∈ H k (M) and h ∈ H k+1/2 (∂M), then [u] ∈ H k+2 (M)/C and   uH k+2 (M) = [u]H k+2 (M)/C ≤ C f H k (M) + |h|H k+1/2 (∂M) . Proof. By the trace formula of Theorem 18.25 there exists h ∈ H 2 (M) such that γ N (h) = h and hH 2 (M)  |h|H 1/2 (∂M) . With (18.5.5) consequence of the divergence formula of Proposition 18.28 we then have, for all v ∈ H 1 (M), a0 (v, h) = (v, H)L2 (M) + (v|∂M , h)L2 (∂M) , where H = P0 h ∈ L2 (M), with HL2 (M)  |h|H 1/2 (∂M) . Choosing v = 1 yields, with (18.7.12), (18.7.15)

∫ Hμg + ∫ h μg∂ = 0, that is, ∫ f μg = ∫ Hμg .

M

M

∂M

M

By linearity, it thus suffices to solve Problem (18.7.13) for h = 0 and f replaced by f − H. By (18.7.15), we then assume that (18.7.16)

∫ f μg = 0,

M

which is consistent with choosing h = 0 in (18.7.12). The map  : v → (v, f )L2 (M) is continuous on H 1 (M) and (v) = (v  ) if v ∼ v  (in the sense of (18.7.7)) as ∫M f μg = 0. Thus, there exists a continuous form ˜ on H 1 (M)/C such that  = ˜ ◦ Φ. On H 1 (M)/C we also consider the well defined continuous Hermitian form b0 ([v], [u]) = a0 (v, u),

[u], [v] ∈ H 1 (M)/C.

By the Poincar´e inequality of Proposition 18.9, we have b0 ([v], [v]) =  D vL2 Λ1 (M)  vH 1 (M) = [v]H 1 (M)/C , [v] ∈ H 1 (M)/C, implying that b0 (., .) is coercive on H 1 (M)/C. As it is Hermitian symmetric, by the Riesz representation theorem, there exists a unique [u] ∈ H 1 (M)/C ˜ for all [v] ∈ H 1 (M)/C. This reads such that b0 ([v], [u]) = ([v]) a0 (v, u) = (v, f )L2 (M) ,

v ∈ H 1 (M).

We may then prove that u ∈ H 2 (M), γ N (u) = 0, and prove inequality (18.7.14) following, for instance, [90] that can be adapted to the manifold case. 

494

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

Corollary 18.49 (Uniqueness of Strong Solutions Modulo Constants). Let f ∈ L2 (M) and h ∈ H 1/2 (∂M) be such that ∫ f μg + ∫ h μg∂ = 0.

(18.7.17)

M

∂M

Let w ∈ be such that P0 w = f in M and γ N (w) = h on ∂M. If [u] ∈ H 2 (M)/C is the unique solution of the variational form of the Neumann problem (18.7.13), then w ∈ [u], that is, w = u + Cst. H 2 (M)

Proof. It suffices to observe that w is itself a solution of (18.7.13) by (18.5.5) consequence of the divergence formula of Proposition 18.28.  The case of a homogeneous Neumann boundary condition, h ≡ 0, naturally leads to defining the unbounded operator PN on L2 (M), with domain D(PN ) = {u ∈ H 2 (M); γ N (u) = 0}, by PN u = P0 = −Δg u. For λ > 0, by Proposition 18.48 we have uH 2 (M)  (P0 + λ Id)uL2 (M) ,

(18.7.18)

u ∈ D(PN ).

Let [u] ∈ H 2 (M)/C be such that γ N (u) = 0 (recall that u = (Φ)−1 [u]). Then, by (18.7.6) we have ∫M P0 u μg = 0 meaning that by (18.7.8) P0 u is in the space Φ

L2 (M) = ΠL2 (M) ∼ = L2 (M)/C. We can then define the unbounded operator PN,∗ on L2 (M)/C, with domain (18.7.19)   D(PN,∗ ) = [u] ∈ H 2 (M)/C; γ N (u) = 0

and PN,∗ [u] = Φ(P0 u).

By Proposition 18.48, if [u] ∈ D(PN,∗ ), we have (18.7.20)

[u]H 2 (M)/C = uH 2 (M)  P0 uL2 (M)  [P0 u]L2 (M)/C .

Moreover, the same proposition shows that PN,∗ : H 2 (M)/C → L2 (M)/C is surjective. With the continuity of the trace u → γ N (u) as given in Theorem 18.25 for u ∈ H 2 (M) and (18.7.18) (resp. (18.7.20)) we have the following result. Proposition 18.50. The unbounded operators (PN , D(PN )) and (PN,∗ , D(PN,∗ )) are closed operators on L2 (M) and L2 (M)/C, respectively. We conclude this section with the following lemma that is of use on several occasions in what follows. Lemma 18.51. Let u ∈ L2 (M) be such that, for all v ∈ H 2 (M) with = 0,

γ N (v)

(u, P0 v)L2 (M) = 0. Then, u ≡ Cst.

18.7. THE NEUMANN-LAPLACE PROBLEM

495

Proof. Let f ∈ L2 (M) = ΠL2 (M), that is f ∈ L2 (M) such that 2 M f μg = 0; see (18.7.8). By Proposition 18.48, there exists [v] ∈ H (M)/C such that P0 v = f and γ N (v) = 0 for v ∈ [v]. We thus obtain, by the assumption made on u, ffl

(u, f )L2 (M) = (u, P0 v)L2 (M) = 0. As f is arbitrary in L2 (M), the conclusion follows by Lemma 18.46.



18.7.1. The Neumann Lifting Map. We introduce here a map that is the counterpart of the Dirichlet lifting map for the Neumann problem. We are interested in defining a map that acts as a solution operator for the following elliptic problem P0 u + u = 0 in M

and γ N (u) = h on ∂M,

for h ∈ H −3/2 (∂M). Note that we consider P0 + Id instead of P0 to enforce uniqueness without having to deal with classes of functions modulo constants (see the previous section). Here also the Hilbert space Wg (M) introduced in Sect. 18.6.2 plays an essential rˆ ole. Proposition 18.52. Let h ∈ H −3/2 (∂M). There exists a unique u ∈ Wg (M) such that P0 u + u = 0 and γ N (u) = h. Moreover, the map h → u from H −3/2 (∂M) into Wg (M) is bounded. With appropriate changes the proof is an adaptation of that of Proposition 18.36. In fact, it is a particular case of the proof of Proposition 18.59 in Sect. 18.8. We thus omit it. Definition 18.53 (Neumann Lifting Map). The bounded map N : h → u from H −3/2 (∂M) into Wg (M) given by Proposition 18.52 is called the Neumann lifting map. If h is more regular than H −3/2 , then additional regularity is transferred to N(h), with consistency with the trace formula of Theorem 18.25. Proposition 18.54. Let m ∈ N. If h ∈ H m−3/2 (∂M), then N(h) ∈ H m (M). Moreover, for some C > 0, we have N(h)H m (M) ≤ C|h|H m−3/2 (∂M) ,

h ∈ H m−3/2 (∂M).

Proof. The case m = 0 was treated above. The proof can be adapted from that of Proposition 18.39 for m ≥ 2. The case m = 1 follows by an interpolation argument [74, 236].  The above results allow one to fully treat the nonhomogeneous NeumannLaplace problem on a compact manifold with boundary, using the linearity of the equation.

496

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

Theorem 18.55. Let (M, g) be a compact connected Riemannian ddimensional manifold with boundary and let q ∈ L∞ (M; R) be such that q ≥ 0 and q ≥ C > 0 on A ⊂ M of positive measure. Then, the maps Wg (M) → L2 (M) ⊕ H −3/2 (∂M)   u → P0 u + qu, γ N (u) and Wg (M) ∩ H 1 (M) → L2 (M) ⊕ H −1/2 (∂M)   u → P0 u + qu, γ N (u) are isomorphisms. Let k ∈ N, if moreover q ∈ W k,∞ (M; R), the map L : H k+2 (M) → H k (M) ⊕ H k+1/2 (∂M)   u → P0 u + qu, γ N (u)

N

is also an isomorphism. Note that for the definition of the second map and its properties we use Lemma 18.42. In the case q = 0 both existence and uniqueness are lost in the Neumann problem. One, however, finds the following result with the above results, again exploiting the linearity of the problem. Theorem 18.56. Let (M, g) be a compact connected Riemannian ddimensional manifold with boundary. For k ∈ N, the map L : H k+2 (M) → H k (M) ⊕ H k+1/2 (∂M)   u → P0 u, γ N (u)

N

has a one dimensional kernel formed by the set of constant functions and its range is given by the set of (f, h) ∈ H k (M) × H k+1/2 (∂M) such that ∫ f μg + ∫ hμg∂ = 0.

M

∂M

The same holds for the maps Wg (M) → L2 (M) ⊕ H −3/2 (∂M)   u → P0 u, γ N (u) and Wg (M) ∩ H 1 (M) → L2 (M) ⊕ H −1/2 (∂M)   u → P0 u, γ N (u) , with the range condition replaced by ∫ f μg + h, 1H −3/2 (∂M),H 3/2 (∂M) = 0

M

18.8. THE LAPLACE PROBLEM WITH MIXED BOUNDARY CONDITIONS

497

and ∫ f μg + h, 1H −1/2 (∂M),H 1/2 (∂M) = 0,

M

respectively. 18.8. The Laplace Problem with Mixed Neumann-Dirichlet Boundary Conditions If (M, g) is compact and has a boundary ∂M, the latter may be composed of a finite number of connected components. We are interested in solving an elliptic problem with Dirichlet boundary conditions on some of the connected components of ∂M and Neumann boundary conditions on others. We exclude the case of pure Dirichlet or Neumann boundary conditions here: there is a least one connected component of ∂M where we impose Dirichlet boundary conditions and one connected component of ∂M where we impose Neumann boundary conditions. The treatment of the Dirichlet problem and the Neumann problem can be found in Sects. 18.6 and 18.7, respectively. We denote by D∂M the union of the connected components of ∂M where we impose Dirichlet boundary conditions and by N∂M the union of the other connected components of ∂M where we impose Neumann boundary conditions. As already mentioned we assume D∂M = ∅ and N ∂M = ∅. The problem under consideration then reads, with P0 = −Δg , P0 u + qu = f,

u|D∂M = Dh,

∂ν u|N∂M = Nh.

The potential function q will be assumed bounded in what follows. If f ∈ L2 (M) and if we seek u ∈ H 2 (M), then the trace formula of Theorem 18.25 indicates that the proper space for Dh is H 3/2 (D∂M) and that for Nh is H 1/2 (N∂M). We first treat the homogeneous case. 18.8.1. Homogeneous Case. We define the following closed subset of H 1 (M) HD1 (M) = {u ∈ H 1 (M); u|D∂M = 0}. If q ∈ L∞ (M; R), on H 1 (M) we define the Hermitian form: aq (., .) as in (18.7.4). Proposition 18.57 (Homogeneous Mixed Dirichlet–Neumann Laplace Problem). Let q ∈ L∞ (M; R). There exist C0 > 0 and C > 0 such that, if q > −C0 , for any f ∈ L2 (M), there exists a unique u ∈ HD1 (M) such that (18.8.1)

aq (ϕ, u) = (ϕ, f )L2 (M) ,

ϕ ∈ HD1 (M).

Moreover, u ∈ H 2 (M), ∂ν u|N∂M = 0 and C −1 f L2 (M) ≤ uH 2 (M) ≤ Cf L2 (M) . Moreover, the equation P0 u + qu = f holds in L2 (M).

498

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

Let k ∈ N. For q ∈ W k,∞ (M; R), there exists Ck > 0, such that if f ∈ H k (M), then moreover u ∈ H k+2 (M) and Ck−1 f H k (M) ≤ uH k+2 (M) ≤ Ck f H k (M) and the equation P0 u + qu = f holds in H k (M). Note that the constant C0 depends on the metric g but also on the choice of connected components of ∂M where Dirichlet boundary conditions are imposed. Proof. Let C0 > 0 be the best possible constant in the first Poincar´e inequality of Proposition 18.9, that is, 1/2

C0 vL2 (M) ≤ ∇g vL2 V (M) ,

v ∈ HD1 (M).

Thus, if q > −Cg we obtain that aq (v, v)  v2H 1 (M) and consequently aq (., .) is coercive on HD1 (M). As v → (v, f¯)L2 (M) is continuous on HD1 (M), the Lax–Milgram theorem gives a unique u ∈ HD1 (M) such that (18.8.1) holds. It also yields the existence of C > 0 such that C −1 f L2 (M) ≤ uH 1 (M) ≤ Cf L2 (M) . The remainder of the proof is classical.



Corollary 18.58 (Uniqueness of Strong Solutions). Let f ∈ L2 (M) and let u ∈ H 2 (M)∩HD1 (M) be such that P0 u+qu = f in M and ∂ν u|N∂M = 0. Then u is the unique solution of the variational form of the homogeneous mixed Dirichlet–Neumann Laplace problem (18.7.5). Proof. It suffices to observe that u is itself a solution of (18.8.1) by (18.5.5) consequence of the divergence formula of Proposition 18.28.  18.8.2. The Mixed Dirichlet–Neumann Lifting Map. We introduce here a map that is the counterpart of the Dirichlet lifting map for the present mixed boundary problem. The Hilbert space Wg (M) introduced in Sect. 18.6.2 plays once again an essential rˆ ole. For r, s ∈ R we set (18.8.2)

r,s (∂M) = H r (D∂M) ⊕ H s (N∂M). HDN

For a function w ∈ H 2 (M), we set (18.8.3)

γ(w) = (w|D∂M , ∂ν w|N∂M ), 3/2,1/2

and we have γ(w) ∈ HDN (∂M). By Lemma 18.42 this trace map extends 1/2,−1/2 to a function w ∈ Wg (M) ∩ H 1 and γ(w) ∈ HDN (∂M); and finally, by −1/2,−3/2 Lemma 18.32, for a function w ∈ Wg (M) we have γ(w) ∈ HDN (∂M).

18.8. THE LAPLACE PROBLEM WITH MIXED BOUNDARY CONDITIONS

499

−1/2,−3/2

Proposition 18.59. Let h ∈ HDN (∂M). There exists a unique u ∈ Wg (M) such that P0 u = 0 and γ(u) = h. Moreover, the map −1/2,−3/2

HDN

(∂M) → Wg (M) h → u

is bounded. Definition 18.60 (Mixed Dirichlet–Neumann Lifting Map). We call the −1/2,−3/2 bounded map M : HDN (∂M) → Wg (M) given by Proposition 18.59 the mixed Dirichlet–Neumann lifting map. Proof of Proposition 18.59. With appropriate changes the proof is an adaptation of that of Proposition 18.36. Denote by R : L2 (M) → H 2 (M) the resolvent map associated with the homogeneous problem of Proposition 18.57. First, we address uniqueness. By linearity, we consider u ∈ Wg (M) such that P0 u = 0 and γ(u) = 0. Let f ∈ L2 (M) and choose w = Rf . We have w ∈ H 2 (M) and γ(w) = 0. By Lemma 18.34 we obtain (u, f )L2 (M) = (u, P0 w)L2 (M) = 0. As f ∈ L2 (M) is arbitrary we conclude that u = 0. Second, we address existence. To ease −1/2,−3/2 (∂M). For a function w ∈ H 2 (M) set H = HDN γ  (w) = (∂ν w|D∂M , w|N∂M ) ∈ H  = HDN

1/2,3/2

notation

we

set

(∂M).

For h = (Dh, Nh) ∈ H and k = (Dk, Nk) ∈ H  , we set k, hH  ,H = −Dk, DhH 1/2 (D∂M),H −1/2 (D∂M) +Nk, NhH 3/2 (N∂M),H −3/2 (N∂M) . For h ∈ H, consider then the map U : L2 (M) → C ¯ H  ,H . f → γ  Rf, h By the trace formula of Theorem 18.25, this form is bounded and we have |U (f )| ≤ |h|H Rf H 2 (M)  |h|H f L2 (M) . By the Riesz theorem, there exists u ∈ L2 (M) such that U (f ) = (f, u)L2 (M) for all f ∈ L2 (M). Moreover, uL2 (M)  |h|H . If ϕ ∈ 0 Dc∞ (M) and f = P0 ϕ, then ϕ = Rf and we find ¯ H  ,H = 0. U (f ) = γ(ϕ), h Consequently P0 u, ϕμg 0 D  (M),1 Dc∞ (M) = ¯ u, P0 ϕμg 0 D  (M),1 Dc∞ (M) = (P0 ϕ, u)L2 (M) = U (f ) = 0. Thus, in the sense of distributions we have P0 u = 0. Thus u ∈ Wg (M) and uWg (M)  |h|H .

500

γ

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

Lemma 18.61. There exists a bounded map M0 : H  → H 2 (M) such that ◦ M0 = IdH  and γ ◦ M0 = 0.

This is a particular case of Theorem 18.25. A proof follows from Lemmata 18.26 and 18.27 by working locally at the boundary. Let k ∈ H  . For v = M0 k ∈ H 2 (M) as given by Lemma 18.61, we have u)H  ,H , (P0 v, u)L2 (M) = k, γ(¯ by Lemma 18.34. We also have ¯ H  ,H (P0 v, u)L2 (M) = U (P0 v) = γ  R(P0 v), h ¯ H  ,H = k, h ¯ H  ,H . = γ  (v), h ¯ − γ(¯ For all k ∈ H  we thus find k, h u)H  ,H = 0, implying γ(u) = h, which concludes the existence part of the proof.  k−1/2,k−3/2

Proposition 18.62. Let k ∈ N. If h ∈ HDN H k (M). Moreover, for some C > 0, we have

(∂M), then M(h) ∈

M(h)H k (M) ≤ C|h|H k−1/2,k−3/2 (∂M) . DN

Proof. The case m = 0 was treated above. The proof can be adapted from that of Proposition 18.39 for m ≥ 2. The case m = 1 follows by an interpolation argument [74, 236].  With the mixed Dirichlet–Neumann lifting map and Proposition 18.57 we obtain the following theorem. Theorem 18.63. Let (M, g) be a compact connected Riemannian ddimensional manifold with boundary and let q ∈ L∞ (M; R) with q > −C0 , where C0 is as given in Proposition 18.31. Then, the maps −1/2,−3/2

Wg (M) → L2 (M) ⊕ HDN   u → P0 u + qu, γ(u)

(∂M)

and 1/2,−1/2

Wg (M) ∩ H 1 (M) → L2 (M) ⊕ HDN   u → P0 u + qu, γ(u)

(∂M)

are isomorphisms. Let k ∈ N, if moreover q ∈ W k,∞ (M; R), the map k+3/2,k+1/2

L : H k+2 (M) → H k (M) ⊕ HDN   u → P0 u + qu, γ(u)

M

(∂M)

is also an isomorphism. Recall that the trace map γ is given in (18.8.3). Note that for the definition of the second map and its properties we use Lemma 18.42.

18.9. SECOND-ORDER ELLIPTIC OPERATORS IN THE EUCLIDEAN SPACE

501

18.9. Second-Order Elliptic Operators in the Euclidean Space In this section, we explain how the analysis of a second-order elliptic operator in an open set of Rd can be made with the point of view of Riemannian geometry. As a typical example, we consider below the definition of the Dirichlet map D. On Ω a regular open set of Rd , consider the second-order elliptic operator   Di (q ij (x)Dj ), with q ij (x)ξi ξj ≥ C|ξ|2 , Q0 (x, D) = 1≤i,j≤d

C ∞ (Ω; R)

1≤i,j≤d

∈ such that = 1≤ where  i, j ≤ d. At each point x ∈ Ω, define by gx = gij (x) the inverse matrix of  ij  q (x) . If M = Ω is equipped with the metric g this yields the structure of Riemannian manifold, with boundary if Ω = Rd . In what follows we shall use the notation M, ∂M, etc. for the notions associated with the Riemannian structure and Ω, ∂Ω for those associated with the Euclidean structure and the operator Q0 (x, D). Note that the operator Q0 (x, D) is not the Laplace–Beltrami operator in general. In fact, we have the following result. q ij

q ij

q ji ,

Proposition 18.64. Define a(x) = (det gx )1/4 . Then, a−1 ◦ Q0 (x, D) ◦ a = P0 + λ,

P0 = −Δg ,

for λ ∈ C ∞ (Ω) given by λ(x) = a−1 (x)Q0 (x, D)a(x). On Rd the Lebesgue measure identifies with the volume form dv = dx1 ∧ · · · ∧ dxd or the Riemannian density μg . The L2 -inner product then reads (18.9.1)

v (x)dx = ∫ u¯ v dv. (u, v)L2 (Ω) = ∫ u(x)¯ Ω

Ω

Note that Q0 (x, D) is symmetric for this inner product. The Laplace–Beltrami operator Δg is, however, symmetric for the L2 inner product associated with the volume form dvg = (det gx )1/2 dv = a2 dv, that is, (18.9.2)

(u, v)L2 (M) = ∫ u¯ v dvg . Ω

These observations lead to the following proof. Proof of Proposition 18.64. The operators a−1 ◦Q0 (x, D)◦a and P0 share the same principal symbol. Their difference a−1 ◦ Q0 (x, D) ◦ a − P0 = R1 (x, D) is thus a first-order operator with real coefficients. For u, v ∈ Cc∞ (Ω) = 0 Dc∞ (M) we write (a−1 Q0 (x, D)(au), v)L2 (M) = ∫ Q0 (x, D)(au)¯ v a(x)dx Ω

= (Q0 (x, D)(au), av)L2 (Ω) = (au, Q0 (x, D)av)L2 (Ω) = (u, a−1 Q0 (x, D)(av))L2 (M) .

502

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

One sees that a−1 ◦ Q0 (x, D) ◦ a is symmetric for (., .)L2 (M) just as P0 and thus, so is R1 (x, D). This shows that R1 (x, D) is in fact a function λ(x) as its coefficients are real. Then, we simply have λ = a−1 Q0 (x, D)a since P0 vanishes on constant functions.  We now wish to understand the connection between the two volume forms dv∂ and dvg∂ on ∂Ω = ∂M that are associated with dv and dvg , respectively. Let ν be the unique outward pointing vector field along ∂Ω such that, for all x ∈ ∂Ω, νx · νx = 1 and νx · t = 0 for all t ∈ Tx ∂Ω. As the Euclidean metric is characterized by  the identity matrix at every point we may identify the tangent vector ν = 1≤i≤d ν i ∂i with its covariant  version n = 1≤i≤d ni ∂ i , that is ni = ν i , i = 1, . . . , d. Let also νg be the unique outward pointing vector field along ∂M such that, for all x ∈ ∂M, gx ((νg )x , νx ) = 1 and gx ((νg )x , t) = 0 for all t ∈ Tx ∂Ω. For w ∈ H 2 (Ω) we set γgN (w) = ∂νg w|∂M = g(νg , ∇g w)|∂M , and γ N (w) = ∂ν w|∂Ω =

 1≤i,j≤d

g ij ni ∂j w|∂Ω .

In this section, we use the notation γgN to avoid any possible confusion. The maps γgN and γ N are related yet different in general. Proposition 18.65. The volume forms dvg∂ and (dv)∂ are related according to 1/2

dvg∂ = (dvg )∂ = |n|g (det g)|∂M (dv)∂ .  N −1 N We also have νg = |n|−1 g n and γg = |n|g γ .

In this proposition, the musical isomorphism is related to the Riemannian structure associated with the metric g. Using the coordinates (that are global here) we have   −1/2  ij g kl (nx )k (nx )l g (x)(nx )i , (νg )ix = 1≤k,l≤d

1≤j≤d

for x ∈ ∂M. Proof. If we set ng = n/|n|g , then as (ng )x is conormal to Tx ∂M and unitary, then ng fulfills the characterizing properties of νg . Next, we write |n|g γgN (w) = |n|g g(νg , ∇g w)|∂M = g(n , ∇g w) = n, ∇g w|∂M = ∂ν w|∂M . Since dvg = (det g)1/2 dv, we find (dv)∂ (w1 , . . . , wd−1 ) = dv(ν, w1 , . . . , wd−1 ),

18.9. SECOND-ORDER ELLIPTIC OPERATORS IN THE EUCLIDEAN SPACE

503

and (dvg )∂ (w1 , . . . , wd−1 ) = dvg (νg , w1 , . . . , wd−1 ) 1/2

= (det g)|∂M dv(νg , w1 , . . . , wd−1 ), for w1 , . . . , wd−1 ∈ C 1 V (∂M). As ν is unitary for the Euclidean metric we have νg = (νg · νx )ν + ν  with  νx ∈ (νx )⊥ = (nx )⊥ = Tx ∂M. Observing that dv(ν  , w1 , . . . , wd−1 ) = 0, if w1 , . . . , wd−1 ∈ C 1 V (∂M) we find 1/2

(dvg )∂ (w1 , . . . , wd−1 ) = (det g)|∂M (νg · ν)dv(ν, w1 , . . . , wd−1 ), 1/2

that is, (dvg )∂ = (det g)|∂M (νg · ν)(dv)∂ . Finally, we write  (νg · ν)(x) = nx , (νg )x  = |nx |−1 g nx , nx  = |nx |g ,



which yields the result.

If u, v ∈ H 2 (Ω), the following Green type formula for the operator Q0 (x, D) follows from two applications of the divergence formula given in (16.1.6) for the Euclidean case: (18.9.3)

(Q0 (x, D)u, v)L2 (Ω) + ∫ γ N (u)¯ v|∂Ω (dv)∂ ∂Ω

v ) (dv)∂ . = (u, Q0 (x, D)v)L2 (Ω) + ∫ u|∂Ω γ N (¯ ∂Ω

In what follows, we shall use this formula to show how the correspondences we put forward above can be used to transfer properties obtained for P0 = −Δg to Q0 (x, D) and vice versa. Let u, v ∈ H 2 (Ω) and with a(x) = (det gx )1/4 as above let u ˜, v˜ be such that u = a˜ u and v = a˜ v . With Proposition 18.65 and the Green formula of Proposition 18.30 we have u, v˜)L2 (M) (Q0 (x, D)u, v)L2 (Ω) = ((P0 + λ)˜ = (˜ u, (P0 + λ)˜ v )L2 (M) − (γgN (˜ u), γ D (˜ v ))L2 (∂M) + (γ D (˜ u), γgN (˜ v ))L2 (∂M) = (u, Q0 (x, D)v)L2 (Ω) − (γgN (˜ u), γ D (˜ v ))L2 (∂M) + (γ D (˜ u), γgN (˜ v ))L2 (∂M) . With the next lemma we see that the Green formula of Proposition 18.30 and (18.9.3) can be deduced from one another. Lemma 18.66. One has u), γgN (˜ v ))L2 (∂M) − (γgN (˜ u), γ D (˜ v ))L2 (∂M) (γ D (˜ where γ N (w) = ∂ν w|∂Ω

= (γ D (u), γ N (v))L2 (∂Ω) − (γ N (u), γ D (v))L2 (∂Ω) ,  = 1≤i,j≤d g ij ni ∂j w|∂Ω .

504

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

Proof. We write u), γgN (˜ v ))L2 (∂M) − (γgN (˜ u), γ D (˜ v ))L2 (∂M) I = (γ D (˜  −1 D −1 = ∫ a γ (u) g(νg , ∇g (a v¯))|∂M ∂M  − a−1 γ D (¯ v ) g(νg , ∇g (a−1 u))|∂M dvg∂ . Due to some cancellation, one obtains

  u g(νg , ∇g (a−1 v¯)) − v¯ g(νg , ∇g (a−1 u)) = a−1 u g(νg , ∇g v¯) − v¯ g(νg , ∇g u) ,

yielding I= ∫ ∂M

 D  γ (u) γgN (¯ v ) − γ D (¯ v ) γgN (u) a−2 |∂M dvg∂ .

N −1 N By proposition 18.65, we have a−2 |∂M dvg∂ = |n|g (dv)∂ and γg = |n|g γ . We thus obtain   v ) − γ D (¯ v ) γ N (u) (dv)∂ , I = ∫ γ D (u) γ N (¯ ∂M



which is the result.

We now show how the above observation can be used to see how the extended Green formula of Proposition 18.34 can be transposed to the operator Q0 (x, D) on Ω. First we need to consider the duality bracket for Sobolev spaces on ∂M. Let s ∈ R. Recall that on ∂M equipped with the metric g∂ the duality between H s (∂M) and H −s (∂M) is understood using L2 (∂M) as a pivot space with the inner product (ϕ, ψ)L2 (∂M) = ∫ ϕψ¯ (dvg )∂ . ∂M

The same can be done on ∂Ω for the inner product (ϕ, ψ)L2 (∂Ω) = ∫ ϕψ¯ (dv)∂ , ∂Ω

and H −s (∂Ω). In fact, these spaces and we denote these spaces by can be identified: if ϕ ∈ H s (∂Ω) and ψ ∈ H −s (∂Ω), then ϕ˜ = b−1 ϕ ∈ 1/2 1/4 1/2 H s (∂M) and ψ˜ = b−1 ψ ∈ H −s (∂M), with b = |n|g (det g)∂M = |n|g γ D (a), and ˜ H s (∂M),H −s (∂M) . (18.9.4) ˜ ψ ϕ, ψH s (∂Ω),H −s (∂Ω) = ϕ, H s (∂Ω)

Second, we recall the space Wg (Ω) = {u ∈ L2 (Ω); Q0 (x, D)u ∈ L2 (Ω)} introduced in Section 10.5 of Volume 1. With the counterpart space Wg (M) introduced in (18.6.4), with Proposition 18.64 we see that ˜ = a−1 u ∈ Wg (M). u ∈ Wg (Ω) ⇔ u

18.9. SECOND-ORDER ELLIPTIC OPERATORS IN THE EUCLIDEAN SPACE

505

As a is smooth, from Lemma 18.32 we obtain the result of Lemma 10.54 of Volume 1, that is the existence of both the Dirichlet and Neumann traces of functions in Wg (Ω). With Lemma 18.34 we may then obtain its counterpart on Wg (Ω). Lemma 18.67. Let u ∈ Wg (Ω) and w ∈ H 2 (Ω). We have (Q0 (x, D)u, w)L2 (Ω) + γ N (u), γ D (w) ¯ H −3/2 (∂Ω),H 3/2 (∂Ω) = (u, Q0 (x, D)w)L2 (Ω) + γ D (u), γ N (w) ¯ H −1/2 (∂Ω),H 1/2 (∂Ω) . Proof. We set u ˜ = a−1 u and v˜ = a−1 v with a(x) = (det gx )1/4 . By Proposition 18.64 we have (Q0 (x, D)u, w)L2 (Ω) − (u, Q0 (x, D)w)L2 (Ω) = (P0 u ˜, v˜)L2 (M) − (˜ u, P0 v˜)L2 (M) . With Lemma 18.34 it thus suffices to prove that ¯ H −3/2 (∂Ω),H 3/2 (∂Ω) − γ D (u), γ N (w) ¯ H −1/2 (∂Ω),H 1/2 (∂Ω) γ N (u), γ D (w) = γgN (˜ u), γ D (w) ˜ H −3/2 (∂M),H 3/2 (∂M) − γ D (˜ u), γgN (w) ˜ H −1/2 (∂M),H 1/2 (∂M) . One cannot apply Lemma 18.66 since there is no density argument to invoke here. One rather reproduces the argument of the proof of Lemma 18.66 yet, here, on the level of the duality bracket and using the trace properties of the functions in Wg (Ω) and in Wg (M). In particular, using Proposition 18.65 one has u) + γ N (a)γ D (˜ u), H −3/2 (∂Ω)  γ N (u) = |n|g γ D (a)γgN (˜ by the Leibniz-like formula for the traces given in Lemma 18.32, where the first term on the r.h.s. is in H −3/2 (∂M) and the second term in H −1/2 (∂M). As we also have ˜ + γ N (a)γ D (w), ˜ H 1/2 (∂Ω)  γ N (w) = |n|g γ D (a)γgN (w) where the first term on the r.h.s. is in H 1/2 (∂M) and the second term in H 3/2 (∂M), we obtain I = γ N (u), γ D (w) ¯ H −3/2 (∂Ω),H 3/2 (∂Ω) − γ D (u), γ N (w) ¯ H −1/2 (∂Ω),H 1/2 (∂Ω) = |n|g γ D (a)γgN (˜ u), γ D (aw) ˜ H −3/2 (∂Ω),H 3/2 (∂Ω) − γ D (a˜ u), |n|g γ D (a)γgN (w) ˜ H −1/2 (∂Ω),H 1/2 (∂Ω) since u), γ D (aw) ˜ H −3/2 (∂Ω),H 3/2 (∂Ω) γ N (a)γ D (˜ = γ D (a˜ u), γ N (a)γ D (w) ˜ H −1/2 (∂Ω),H 1/2 (∂Ω) .

506

18. SOBOLEV SPACES AND LAPLACE PROBLEMS 1/2

With b = |n|g γ D (a) as above, we have u), bγ D (w) ˜ H −3/2 (∂Ω),H 3/2 (∂Ω) I = bγgN (˜ − bγ D (˜ u), bγgN (w) ˜ H −1/2 (∂Ω),H 1/2 (∂Ω) , which gives I = γgN (˜ u), γ D (w) ˜ H −3/2 (∂M),H 3/2 (∂M) − γ D (˜ u), γgN (w) ˜ H −1/2 (∂M),H 1/2 (∂M) 

by (18.9.4).

With these results we can then extend mutatis mutandis the results of Section 18.6.3 and prove results of Section 10.5 of Volume 1: Proposition 10.55 that allows one to define the Dirichlet map as in Definition 10.56 therein; similarly, the proof of Proposition 10.57 in Volume 1 is identical to that of Proposition 18.39. Appendix 18.A. Traces Extension: Technical Aspects 18.A.1. L2 Functions with L2 ‘Laplacian’. Here, we provide a proof of Lemma 18.32. We use normal geodesic coordinates as given by Theorem 17.22: there exist O an open set of M neighborhood of ∂M, z0 > 0, and a diffeomorphism Φ, such that (18.A.1)

Φ : N = ∂M × [0, z0 ) → O (m , z) → Φ(m , z).

The pullback of the metric takes the form  Φ∗ g(m ,z) = gm  (z) ⊗ 1z + dz ⊗ dz ,

where g  (z) is a Riemannian metric on ∂M that smoothly depends on z and such that g  (0) = g∂ . By Corollary 17.23 the pullback of the Laplace– Beltrami operator on N is given by (18.A.2)

P0 = (det g  (z))−1/2 Dz (det g  (z))1/2 Dz ) − Δg (z) .

The operator we consider P = P0 + R1 with R1 a first-order differential operator with smooth coefficients. 2 (M), that is, H 2 If u ∈ WP (M), by standard elliptic theory u ∈ Hloc ∞ away from the boundary. Let χ ∈ Cc (−z0 , z0 ) such that χ ≡ 1 in a neighborhood of 0. Observe then that v = χ(z)Φ∗ u is such that v ∈ L2 (N ) and (P0 + R1 )v ∈ L2 (N ), and we have vWP (N )  uWP (M) , where v2WP (N ) =v2L2 (N ) +P v2L2 (N ) . It now suffices to prove the result for v supported in N .

18.A. TRACES EXTENSION: TECHNICAL ASPECTS

507

Lemma 18.68. We have v ∈ H 2 (R+ ; H −2 (∂M)) ∩ H 1 (R+ ; H −1 (∂M)). Moreover, there exists C > 0 (independent of v) such that vH 2 (R+ ;H −2 (∂M)) + vH 1 (R+ ;H −1 (∂M)) ≤ CvWP (N ) . A proof is given below. Consequently, the traces v|z=0+ and ∂z v|z=0+ are both uniquely defined v|z=0+ = lim v(z, .) ∈ H −1 (∂M),

(18.A.3)

z→0+

and ∂z v|z=0+ = lim ∂z v(z, .) ∈ H −2 (∂M).

(18.A.4)

z→0+

We shall now improve upon these two results. Let ψ ∈ C ∞ (∂M). According to the trace formula of Theorem 18.25 we can pick w ∈ C ∞ (N ) be such that w|z=0+ = ψ and |ψ|H 1/2 (∂M)  wH 1 (N ) . With the regularity of v obtained in Lemma 18.68 we can write v|z=0+ , ψH −1 (∂M),H 1 (∂M) +∞   = − ∫ ∂z v, wH −1 (∂M),H 1 (∂M) + v, ∂z wH −1 (∂M),H 1 (∂M) dz 0 +∞

= − ∫ ∂z v, wH −1 (∂M),H 1 (∂M) dz − v, ∂z wL2 (N ),L2 (N ) , 0

which gives

  |v|z=0+ , ψH −1 (∂M),H 1 (∂M) |  vH 1 (R+ ;H −1 (∂M)) + vL2 (N ) wH 1 (N )  vWP (N ) ψH 1/2 (∂M) .

As ψ is arbitrary, it follows that v|z=0+ ∈ H −1/2 (∂M) and moreover we have |v|z=0+ |H −1/2 (∂M)  vWP (N ) . Again with ψ ∈ C ∞ (∂M), according to the trace formula of Theorem 18.25 we can pick w ∈ C ∞ (N ) be such that w|z=0+ = ψ, ∂z w|z=0+ = 0, and |ψ|H 3/2 (∂M)  wH 2 (N ). With the regularity of v obtained in Lemma 18.68 we can write ∂z v|z=0+ , ψH −2 (∂M),H 2 (∂M) +∞   = − ∫ ∂z2 v, wH −2 (∂M),H 2 (∂M) + ∂z v, ∂z wH −2 (∂M),H 2 (∂M) dz 0 +∞ 

=− ∫ 0

 ∂z2 v, wH −2 (∂M),H 2 (∂M) + ∂z v, ∂z wH −1 (∂M),H 1 (∂M) dz,

which gives

  | ∂z v|z=0+ , ψ H −2 (∂M),H 2 (∂M) |  vH 2 (R+ ;H −2 (∂M)) + vH 1 (R+ ;H −1 (∂M)) wH 2 (N )  vWP (N ) ψH 3/2 (∂M) .

As ψ is arbitrary, it follows that v|z=0+ ∈ H −3/2 (∂M) and moreover we have |v|z=0+ |H −3/2 (∂M)  vWP (N ) .

508

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

Finally, since v ∈ H 2 (R+ ; H −2 (∂M)), if a is a smooth function, then ∂z (av) = v∂z a + a∂z v ∈ H 1 (R+ ; H −2 (∂M)) and we find that γ N (av) is also well defined in H −2 (∂M) and moreover ∂z (av)|z=0+ = (v∂z a)|z=0+ + (a∂z v)|z=0+ . In fact, (v∂z a)|z=0+ ∈ H −1/2 (∂M) and (a∂z v)|z=0+ ∈ H −3/2 (∂M), implying that ∂z (av)|z=0+ ∈ H −3/2 (∂M). This concludes the proof of Lemma 18.32.  Proof of Lemma 18.68. The operator R1 reads R1 = α(m , z)Dz + where α is a smooth function and R1 (z) is a family of first-order differential operators on ∂M smoothly parameterized by z. With β(m , z) = ∫0z α(m , σ)dσ, we write     e−iβ Dz eiβ (det g  (z))1/2 Dz v = Dz (det g  (z))1/2 Dz v + (det g  (z))1/2 αDz v   = (det g  (z))1/2 P v + Δg (z) v − R1 (z)v R1 (z),

∈ L2 (R+ ; H −2 (∂M)),   and Dz eiβ (det g  (z))1/2 Dz v L2 (R+ ;H −2 (∂M))  vWP (N ) . With Lemma 18.69 below, for r = 0 and s = −2, we find that Dz v ∈ H 1 (R+ ; H −2 (∂M)) and Dz vH 1 (R+ ;H −2 (∂M))  vWP (N ) . Applying Lemma 18.69 a second time for r = 1 and s = −2, we find that v ∈ H 2 (R+ ; H −2 (∂M)) and vH 2 (R+ ;H −2 (∂M))  vWP (N ) . Finally, we conclude that v ∈ H 1 (R+ ; H −1 (∂M)) and vH 1 (R+ ;H −1 (∂M))  vWP (N ) , with an interpolation argument [74, 236].



Lemma 18.69. Let V be Rn or some compact Riemannian manifold.  that Dz w ∈ Let also r ∈ N and s ∈ R and w ∈ 0 D  (R+ × V )s be such  r s r+1 (a, +∞); H (V ) for some a > 0. H R+ ; H (V ) and w|(a,+∞)×V ∈ H   Then, w ∈ H r+1 R+ ; H s (V ) and   wH r+1 (R+ ;H s (V )) ≤ C Dz wH r (R+ ;H s (V )) + wH r+1 ((a,+∞);H s (V )) , for some C > 0. Proof. We pick ϕ ∈ Cc∞ (R) such that ϕ ≡ 1 in a neighborhood of [0, a] with supp(ϕ) ⊂ [−b, b] with b > a. We set θ(x) = ϕ(z). We compute Dz (θw) = θDz w + (Dz θ)w. As Dz θ is supported in (a, +∞) × V we have (Dz θ)w ∈ H r+1 (R+ ; H s (V )) ⊂ H r (R+ ; H s (V )).

18.A. TRACES EXTENSION: TECHNICAL ASPECTS

509

  We thus have Dz (θw) ∈ H r R+ ; H s (V ) and, because of its support, we have θw ∈ H r+1 R+ ; H s (V ) and b

θw(., z) = − ∫ ∂d (θw)(., s) ds. z

Moreover, we have θwH r+1 (R+ ;H s (V ))  θDz wH r (R+ ;H s (V )) + (Dz θ)wH r (R+ ;H s (V ))  Dz wH r (R+ ;H s (V )) + wH r+1 ((a,+∞);H s (V )) . Since (1 − θ)w ∈ H r+1 (R+ ; H s (V )) the result follows.



18.A.2. Green-Like Formula in WP (M). Here we prove Lemma 18.33. We use the diffeomorphism Φ defined in (18.A.1). Let ψ ∈ Cc∞ (−z0 , z0 ) ˜  , z) = ψ(z) and such that ψ ≡ 1 in a neighborhood of 0 and set ψ(m −1 ∗ ˜ This function χ ≡ 1 in a neighborhood of ∂M. χ = (Φ ) ψ.   Let u ∈ WP (M). We have P (χu) = P (χ − 1)u + P u. Since u ∈ 2 (M), that is, H 2 away from the boundary by standard elliptic theory, Hloc then χu ∈ WP (M). Observe that   (P (1 − χ)u , w)L2 (M) = ((1 − χ)u, t P w)L2 (M) by Propositions 18.29 and 18.30. It thus suffices to prove the result with u replace by χu. In turn, if v = Φ∗ (χu) it suffices to prove that (18.A.5)

¯ H −3/2 (∂M),H 3/2 (∂M) (P v, w)L2 (N ) + γ N (v), γ D (w) ¯ H −1/2 (∂M),H 1/2 (∂M) = (v, t P¯ w)L2 (N ) + γ D (v), γ N (w) + γ D (g(X, ν)v), γ D (w) ¯ H −1/2 (∂M),H 1/2 (∂M) ,

for w ∈ C ∞ (N ). By abuse of notation we also denote by P0 the Laplace– Beltrami operator of N associated with the pulled-back metric. Similarly, the pullbacks of the vector field X and the function f are also denoted by X and f , respectively. With the form of the metric we have μg = μg (z) dz and μg (0) = μg∂ . Observe that (det g  (z))−1/2 μg (z) = (det g  (0))−1/2 μg (0) = (det g∂ )−1/2 μg∂ . This can be simply seen in a local chart for ∂M. In particular ρz = (det g∂ )−1/2 (det g  (z))1/2 is to be understood as a function on ∂M with z acting as a parameter. We write ¯ g (z) dz = ∫ ∫ ρz (P v)wμ ¯ g∂ dz. (P v, w)L2 (N ) = ∫ (P v)wμ N

R+ ∂M

As v ∈ H 2 (R+ ; H −2 (∂M)) ∩ H 1 (R+ ; H −1 (∂M)) ∩ L2 (R+ × ∂M) by Lemma 18.68 and P v = P0 v + Xv + f v ∈ L2 (M), we may write P v as a sum one term in L2 (R+ ; H −2 (∂M)), that is, P0 v, one term in L2 (R+ ; H −1 (∂M)), that is, Xv, and one term in L2 (R+ × ∂M), that is, f v. Recalling that

510

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

the duality bracket between H −k (∂M) and H k (∂M) is understood using L2 (∂M) as a pivot space with the inner product on L2 (∂M) based on the density μg∂ , we may then write (P v, w)L2 (N ) = I1 + I2 + I3 , with ¯ H −2 (∂M),H 2 (∂M) dz, I1 = ∫ ρz P0 v, w R+

I2 = ∫ ρz Xv, w ¯ H −1 (∂M),H 1 (∂M) dz, R+

¯ L2 (∂M),L2 (∂M) dz. I3 = ∫ ρz f v, w R+

We write I1 = I1a − I1b , with ¯ H −2 (∂M),H 2 (∂M) dz, I1a = ∫ (det g∂ )−1/2 Dz (det g  (z))1/2 Dz v), w R+

¯ H −2 (∂M),H 2 (∂M) dz. I1b = ∫ ρz Δg (z) v, w R+

Integrations by parts with respect to z yield ¯ H −2 (∂M),H 2 (∂M) dz I1a = ∫ (det g∂ )−1/2 Dz (det g  (z))1/2 Dz v), w R+

= ∫ (det g∂ )−1/2 Dz v, (det g  (z))1/2 Dz wH −2 (∂M),H 2 (∂M) dz R+

¯|z=0+ H −2 (∂M),H 2 (∂M) + ∂z v|z=0+ , w = ∫ (det g∂ )−1/2 v, Dz ((det g  (z))1/2 Dz w)H −2 (∂M),H 2 (∂M) dz R+

¯|z=0+ H −2 (∂M),H 2 (∂M) + ∂z v|z=0+ , w − v|z=0+ , ∂z w ¯|z=0+ H −2 (∂M),H 2 (∂M) . Using that v ∈ L2 (R+ × ∂M) and using Lemma 18.32 we may then write ¯ H −3/2 (∂M),H 3/2 (∂M) − γ D (v), γ N (w) ¯ H −1/2 (∂M),H 1/2 (∂M) I1a + γ N (v), γ D (w) = ∫ (det g∂ )−1/2 v Dz ((det g  (z))1/2 Dz w)μg∂ dz N

= (v, (det g  (z))−1/2 Dz ((det g  (z))1/2 Dz w))L2 (N ) . Almost everywhere on R+ , we have, using (18.3.1), ¯ H −2 (∂M),H 2 (∂M) = ρz Δg (z) v, wμ ¯ g∂ 0 D  (∂M),1 Dc∞ (∂M) ρz Δg (z) v, w = Δg (z) v, wμ ¯ g (z) 0 D  (∂M),1 Dc∞ (∂M) = v, (Δg (z) w)μ ¯ g (z) 0 D  (∂M),1 Dc∞ (∂M) = ∫ v(Δg (z) w)μ ¯ g (z) . ∂M

18.A. TRACES EXTENSION: TECHNICAL ASPECTS

511

yielding I1b = (v, Δg (z) w)L2 (N ) . Combining the computations of I1a and I1b we then obtain ¯ H −3/2 (∂M),H 3/2 (∂M) I1 = −γ N (v), γ D (w)

(18.A.6)

¯ H −1/2 (∂M),H 1/2 (∂M) + (v, P0 w)L2 (N ) . + γ D (v), γ N (w)  a b Setting X = X  + X z ∂z with X(m  ,z) ∈ Tm ∂M, we find I2 = I2 + I2 with

I2a = ∫ ρz X z ∂z v, w ¯ H −1 (∂M),H 1 (∂M) dz, R+

I2b

= ∫ ρz X  v, w ¯ H −1 (∂M),H 1 (∂M) dz. R+

We write ¯ H −1 (∂M),H 1 (∂M) dz I2a = ∫ ρz X z ∂z v, w R+

¯ H −1 (∂M),H 1 (∂M) = −γ D (v), γ D (X z w) − ∫ v, ∂z (ρz X z w) ¯ L2 (∂M),L2 (∂M) dz. R+

¯ H −1/2 (∂M),H 1/2 (∂M) = γ D (g(X, ν)v), γ D (w) − ∫ v, ∂z (ρz X z w) ¯ L2 (∂M),L2 (∂M) dz, R+

using that ρz = 1 for z = 0. Observe that z ¯ L2 (∂M),L2 (∂M) = ∫ vρ−1 ¯ g ∫ v, ∂z (ρz X z w) z ∂z (ρz X w)μ N

R+

and

  z ¯ = X z ∂z w ¯ + (det g  (z))−1/2 ∂z (det g  (z))1/2 X z w. ρ−1 ¯ z ∂z (ρz X w)

Observe now that ¯ H −1 (∂M),H 1 (∂M) dz I2b = ∫ ρz X  v, w R+

¯ g∂ 0 D  (∂M),1 Dc∞ (∂M) dz = ∫ ρz X  v, wμ R+

¯ g (z) 0 D  (∂M),1 Dc∞ (∂M) dz = ∫ X  v, wμ R+

= ∫ v, t X  (w)μ ¯ g (z) 0 D  (∂M),1 Dc∞ (∂M) dz R+

= (v, t X  w)L2 (N ) . We have t X  = −X  − divg (z) X  . As we have   divg (z) X  + (det g  (z))−1/2 ∂z (det g  (z))1/2 X z = divg X,

512

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

by Proposition 17.7, with (17.2.5) we thus obtain (18.A.7) ¯ + divg X)w) ¯ I2 = γ D (g(X, ν)v), γ D (w) ¯ H −1/2 (∂M),H 1/2 (∂M) − (v, (X L2 (N ) . Finally, we have I3 = (v, f¯w)L2 (M) . Together with (18.A.6) and (18.A.7)  this yields the result as t X = −X − divg X by Proposition 17.7. 18.A.3. L2 Functions with H −1 Laplacian. Here, we provide a proof of Lemma 18.35. Let u ∈ Wg,−1 (M). Set f = P0 u ∈ H −1 (M). By Proposition 18.31 ˜ = f and there exists u ˜ ∈ H01 (M) such that P0 u ˜ uH 1 (M)  f H −1 (M)  uWg,−1 (M) . Then v = u − u ˜ ∈ L2 (M) and P0 v = 0 meaning that v ∈ Wg (M). As a result v admit a Dirichlet trace γ D (u) in H −1/2 (∂M) and uL2 (M)  uWg,−1 (M) . |γ D (v)|H −1/2 (∂M)  vWg (M)  uL2 (M) + ˜ u) = 0 we find that the Dirichlet trace of u makes sense and Since γ D (˜ D γ (u) = γ D (v) ∈ H −1/2 (∂M). From Lemma 18.34, if w ∈ H 2 (M) ∩ H01 (M) we find ¯ H −1/2 (∂M),H 1/2 (∂M) . 0 = (v, P0 w)L2 (M) + γ D (v), γ N (w) which we write (˜ u, P0 w)L2 (M) = (u, P0 w)L2 (M) + γ D (v), γ N (w) ¯ H −1/2 (∂M),H 1/2 (∂M) . By Proposition 18.13 applied twice we have ˜, ∇g w)μg = (P0 u ˜, w)L2 (M) , (˜ u, P0 w)L2 (M) = ∫ g(∇g u M

which concludes the proof.



18.A.4. H 1 Functions with L2 Laplacian. Here, we prove Lemma 18.42. By Lemma 18.32 we have γ N (u) ∈ H −3/2 (∂M). Let w ∈ H 2 (M). Lemma 18.34 gives ¯ H −3/2 (∂M),H 3/2 (∂M) (P0 u, w)L2 (M) + γ N (u), γ D (w) = (u, P0 w)L2 (M) + γ D (u), γ N (w) ¯ H −1/2 (∂M),H 1/2 (∂M) . The formula (18.5.5) consequence of the divergence formula of Proposition 18.28 yields ¯ H −1/2 (∂M),H 1/2 (∂M) = (∇g u, ∇g w)L2 V (M) . (u, P0 w)L2 (M) + γ D (u), γ N (w) We thus obtain γ N (u), γ D (w) ¯ H −3/2 (∂M),H 3/2 (∂M) = −(P0 u, w)L2 (M) + (∇g u, ∇g w)L2 V (M) .

18.A. TRACES EXTENSION: TECHNICAL ASPECTS

513

Let now ψ ∈ H 3/2 (∂M) and set w = D(ψ). We have w ∈ H 2 (M) by Proposition 18.39 and wH 2 (M)  |ψ|H 3/2 (∂M) . Moreover, wH 1 (M)  |ψ|H 1/2 (∂M) according to the same proposition. We thus find   ¯ −3/2 |γ N (u), ψ H (∂M),H 3/2 (∂M) |  P0 uL2 (M) + uH 1 (M) wH 1 (M)    P0 uL2 (M) + uH 1 (M) |ψ|H 1/2 (∂M) . This gives the result. 18.A.5. Normal Traces for Vector Fields in H(divg , M). Here, we prove Lemma 18.43. We use normal geodesic coordinates as given by Theorem 17.22: there exist O an open set of M neighborhood of ∂M, z0 > 0, and a diffeomorphism Φ, such that Φ : N = ∂M × [0, z0 ) → O (m , z) → Φ(m , z). The pullback of the metric takes the form  g˜ = Φ∗ g(m ,z) = gm  (z) ⊗ 1z + dz ⊗ dz ,

where g  (z) is a Riemannian metric on ∂M that smoothly depends on z and such that g  (0) = g∂ . If V is a smooth vector field on N it can be written as V = (V  , V z ); the divergence operator takes the form   divg˜ V = (det g  (z))−1/2 ∂z (det g  (z))1/2 V z + divg (z) V    1 = ∂z V z + divg (z) V  + (det g  (z))−1 ∂z det g  (z) V z , 2 where divg (z) is the divergence operator on ∂M associated with the metric g  (z), with z as a parameter. For such a vector field V one has Z0   V 2L2 V (N ) = V z 2L2 (N ) + ∫ ∫ g  (z) V  (m , z), V  (m , z) μg (z) dz 0 ∂M

= V

z 2 L2 (N )

+ (0, V  )L2 V (N ) . 2

Let χ ∈ Cc∞ (−z0 , z0 ) such that χ ≡ 1 in a neighborhood of 0. Let U be as in the statement of the lemma. Observe that it suffices to prove the result for V = χ(z)Φ∗ U . As above the vector field has the form V = (V  , V z ). It is in L2 V (N ) and divg˜ V ∈ L2 (N ). One has divg (z) V  ∈ L2 (R+ ; H −1 (∂M)) and  divg (z) V  L2 (R Since

+ ;H

−1 (∂M))

 (0, V  )L2 V (N )  V L2 V (N ) .

  1 ∂z V z = divg˜ V − divg (z) V  − (det g  (z))−1 ∂z det g  (z) V z , 2

514

18. SOBOLEV SPACES AND LAPLACE PROBLEMS

One finds ∂z V z ∈ L2 (R+ ; H −1 (∂M)) and ∂z V z L2 (R+ ;H −1 (∂M))   divg˜ V L2 (N ) +  divg (z) V  L2 (R

+ ;H

−1 (∂M))

+ V z L2 (N )

  divg˜ V L2 (N ) + V L2 V (N )  V H(divg˜ ,N ) . Since V z ∈ L2 (N ) one has V z ∈ H 1 (R+ ; H −1 (∂M)). In particular, the z z −1 (∂M). We g (ν, V ) is uniquely defined and V|z=0 trace V|z=0 + = −˜ + ∈ H shall now improve upon this result. Let ψ ∈ C ∞ (∂M). According to the trace formula of Theorem 18.25 we can pick w ∈ C ∞ (N ) be such that w|z=0+ = ψ and |ψ|H 1/2 (∂M)  wH 1 (N ) . With the regularity of V z obtained above we can write z V|z=0 + , ψH −1 (∂M),H 1 (∂M) +∞ 

=− ∫

0 +∞

 ∂z V z , wH −1 (∂M),H 1 (∂M) + V z , ∂z wH −1 (∂M),H 1 (∂M) dz

= − ∫ ∂z V z , wH −1 (∂M),H 1 (∂M) dz − V z , ∂z wL2 (N ),L2 (N ) , 0

which gives

 z  z z |V|z=0 + , ψH −1 (∂M),H 1 (∂M) |  V H 1 (R ;H −1 (∂M)) + V L2 (N ) wH 1 (N ) +  V H(divg˜ ,N ) ψH 1/2 (∂M) . z −1/2 (∂M) and moreover we As ψ is arbitrary, it follows that V|z=0 + ∈ H z  V H(divg˜ ,N ) .  have |V|z=0 + | −1/2 H

(∂M)

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Index N.B. A page number in normal type refers to Volume 1, in italic type to Volume 2. B Boundary-damped wave equation, 221, 250 damping operator, 222, 251 generator, 223, 252 semigroup formalism, 223, 252 C Calder´on projector, 39 Canonical transform, 305 Carleman estimate characteristic set, 65 conjugated operator, 65 local estimate, 69 local estimate at the boundary, 71 patching local estimates together, 91 sub-ellipticity condition, 66, 121, 146, 337 weight function, 64 global, 92, 101, 126, 127 limiting, 150 Change of variables action on a differential, 303 action on a vector field, 303 action on conjugated differential operators, 308 action on cotangent space, 303

action on differential operators, 304 invariance of sub-ellipticity condition, 309 symplectomorphism, 305 action on Hamiltonian vector field, 306 action on Poisson bracket, 307 Conic set, 16, 30, 40 Constants, 16 Controllability controllability to trajectories, 256 exact controllability, 253 null-controllability, 253, 256, 269 observability, 257, 270 reachable set, 273 D Damped wave equation strong solution, 217 strong solution (boundary damping), 222, 251 weak solution, 217 weak solution (boundary damping), 227 Density L1 , 428 L1loc , 425 a-density, 430 distribution, 429 Radon measure, 423 Differential geometry canonical one-form, 411

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7

537

538

cotangent vector bundle, 403 cotangent vector space, 402 derivation, 398 differential, 404 differential of a composition, 400 differential of a map, 302 local representation, 399 differential of a map, tangent map, 399, 401 divergence formula, 422, 482 exterior derivative, 410 Hamiltonian vector field, 412 Lie bracket, 413 one-form, 302, 404 p-forms, 410 Poisson bracket, 412 pullback of a one-form, 404 pullbacks, 302 push forward, 401 push-forwards, 302 r-covariant tensor field, 407 r-covariant,s-contravariant tensor field, 411 Stokes’ formula, 419 symplectic two-form, 412 tangent vector, 397 tangent vector bundle, 400 tangent vector space, 397 vector field, 302, 401 Distribution, 288 compact support, 291 convergence, 289, 294 convolution, 297 derivation, 288 dilation, 289 finite order, 290 Fourier transform, 295 positive, 290 product, 289 pullbacks, 292 restriction, 289 support, 291 temperate, 294 tensor product, 297 translation, 289 Dual space, 358 norm, 358 strong topology, 358

Index

Duhamel formula, 340, 376 E Elliptic map, 35 Dirichlet boundary condition, 48 Energy boundary-damped wave, 225, 252 damped wave, 217 Euclidean inner product, 14 F Fourier transformation, 15, 293, 295 inverse, 15 partial, 15 G Green formula, 99, 443, 446, 482, 485, 503, 505 G˚ arding inequality, 30 on a half space, 139 microlocal version, 30 standard operators, 43 for a system, 31 microlocal version, 31 tangential operator, 39 microlocal version, 39 H Hamiltonian vector field, 67, 306, 412 Hardy inequality, 213 Homogeneity, 16 I Integer part, 325 Interior quadratic form, 136 L Lifting map Dirichlet lifting map, 349, 486 mixed Dirichlet-Neumann lifting map, 499 Neumann lifting map, 495 Linear operator accretive, 372, 376 adjoint, 358, 364 bound, 356 bounded operators, 356 continuous, 356 domain, 355 domain of the adjoint, 358, 364

539

Index

Fredholm operator, 358 graph, 356 graph norm, 356 index, 358 kernel, 356 maximal monotone, 372 monotone, 372, 376 range, 355 resolvent operator, 357 resolvent set, 357 selfadjoint, 364 spectrum, 357 symmetric, 364 Littlewood-Paley decomposition, 156 Locally finite sum, 390 ˇ Lopatinski˘ı-Sapiro conditions, 19 for conjugated operator, 166 M Manifold, 388 H k -tensor, 469 H k -vector field, 469 L1 , L1loc vector field, 447 L1 -density, 428 L1loc -density, 425 L2 , H k one-form, 469 L2 , H k -function, 469 L2 -vector field, 465, 469 L∞ , W k,∞ -function, 471 Lp , W k,p -function, 470 W k,∞ -tensor, 471 W k,p -tensor, 471 C k -map, 389 σ-compact, 390 atlas, 388 direct frame, 417 exhausting sequence of compact, 390 fiber, 400, 403 local chart, 388 manifold with boundary, 389 normal geodesic coordinates, 457 oriented manifold, 416 outward-pointing vector, 418 paracompact, 390 partition of unity, 390 pullback of a function, 389 submanifold, 405 tangent vector, 397

Multi-index, 14 N Normal derivative, 14, 72 first order, 123, 451 higher-order, 461 Normal geodesic coordinates, 74, 310, 457 O Observability, 254, 257, 270, 271 approximate, 271 partial, 267 Oscillatory integrals, 22 P Parabolic equation strong solution, 343 weak solution (boundary data), 344, 351 Partition of unity, 290, 390 Phase space, 16 Plancherel equality, 295 Poincar´e constant, 466 Poisson bracket, 28, 37, 307, 412 Pseudo-differential operator adjoint, 26, 36, 131, 341 commutator, 28, 37 composition, 27, 37, 341 with a large parameter, 21 on a half-space, 131 adjoint, 133 composition, 133 principal symbol, 21, 36, 42 semi-classical operator, 22, 41 Sobolev bound, 29 standard, 42 tangential, 36 Pullbacks, 302 Push-forwards, 302 R Radon measure, 290 Resolvent estimate, 220, 221, 231, 256 Riemannian geometry Christoffel symbols, 448, 451 connection, 448 covariant derivative, 448 distance, 438

540

divergence, 443 divergence formula, 446, 482 exponential map, 453 geodesic, 452 geodesic ball, 454 geodesic sphere, 454 gradient, 442 Green formula, 443, 446, 482, 485 Hessian, 460 Laplace-Beltrami operator, 443 Levi-Civita connection, 451 metric, 438 musical isomorphisms, 439 normal neighborhood, 454 normal geodesic coordinates, 457 Riemannian volume form, 440 S Schwartz space, 15, 294 on a half-space, 15 Second-order elliptic operator, 315 associated unbounded operator, 88, 317 coercivity, 316 Hilbert basis of eigenfunctions, 100, 318 maximal monotone, 317 parabolic kernel, 336 parabolic semigroup, 327, 328 selfadjointness, 90, 318 Sobolev scale, 324 variational form, 316 Weyl law, 319 Semigroup, 367 analytic, 375 bounded, 368 differentiable, 374 strongly continuous, 368 Smooth open set, 14 Sobolev norms and spaces classical norms and spaces, 43 action of differential operators, 470 on a manifold, 467, 469, 470, 476 negative order, 470, 476 norm with a large parameter, 18, 29 boundary norm on a manifold, 154

Index

on a half space, 133 inner norm on a manifold, 153 negative order, 154 trace norm, 134 trace norm on a manifold, 156 spaces on Rd with a large parameter, 28 Solution classical, 375 mild, 340, 376 parabolic equation weak, 253, 276 Spectral inequality, 260 Stabilization, 220 boundary-damped wave equation, 231 damped wave equation, 221 exponential, 222 resolvent estimate, 220, 221, 224, 231, 256 weak solution, 223, 232 Sub-ellipticity condition, 66, 121, 146, 337 invariance, 308 necessity, 146 sufficiency, 69 Symbol adjoint, 26, 36, 341 asymptotic series, 20, 36 calculus, 26 characteristic set, 65 commutator, 28, 37 composition, 27, 37, 341 with a large parameter, 19 on a half-space, 130 adjoint, 133 asymptotic series, 131 composition, 133 polyhomogeneous symbol, 131 polyhomogeneous symbol, 21 principal symbol, 19, 20, 35, 42 standard, 42 tangential, 35 Symplectic structure canonical one-form, 411 Hamiltonian vector field, 306, 412 Poisson bracket, 412 symplectic two-form, 305, 412

541

Index

symplectomorphism, 305 T Theorem Hille–Yosida, 371 kernel theorem, 298 Lumer–Philips, 372, 377 Poincar´e inequalities, 466 Rellich-Kondrachov, 465 Rouch´e, 148 Trace inequality, 134, 135, 479 Transpose of a differential operator, 434 U Unbounded operator, 355 graph norm, 218, 223, 247, 252, 268 resolvent set, 221 See also Linear operator, spectrum, 221 See also Linear operator,

Unique continuation global, 186 global quantification away form boundary, 190, 208 up to boundary, 216 boundary to boundary, 201, 219 up to boundary, 197, 207 local, 184 local quantification at a boundary, 194, 199, 202, 210, 218 away from boundary, 187 Unit sphere and cosphere bundle cosphere bundle, 16 half-unit sphere, 16 Unitary outward-pointing normal, 14, 72 W Wedge product, 408 Weight function, 64 See also Carleman estimate,

Index of notation N.B. A page number in normal type refers to Volume 1, in italic type to Volume 2. B Boundary operator and condition B∂ , 222, 251 L∂ , 42 M∂ , 43 CD,D , CD,N , CN,D , CN,N , 39 C, 39 Q, 38 QD , QN , 39 (j) (j) QD , QN , 39 (j) Q , 38 β (operator order), 34 k ∂M, 34, 89 ˇb(m, ω  , z), 19 pˇ(m, ω  , z), 19 pˇ+ ϕ , 165 pˇϕ , 165 ν, 72 b0,ϕ , bϕ , 179 B0,ϕ , 179 trace γ D , 349 D γN , 90 γ N , 37, 349 N γN , 90 ∂ν , 14, 72, 123, 451 ∂ν , 461 γ˜ N , 38 tr(.), 134, 156

C Carleman estimate notation B(.), 77 τ˜, 124 symbol and operator B0,ϕ , 179 P2 , P1 , 65, 120 Pϕ , 65, 120, 164 P0,ϕ , 179 Char, 65 α(m, ω  , τ ), 165 α(x, ξ  , τ ), 179 pˇϕ , 165 p˜, 64, 75 b0,ϕ , bϕ , 179 p2 , p1 , 65 p2 , p1 , 120 pϕ , 65, 120 p0,ϕ , 179 pˇ+ ϕ , 165 D Differential geometry Tx∗ X, 302 Tx X, 302 Differentiation D, Dx , 14 D , Dx , 74 Dα , 14 ∂, 14 Distribution and distribution space Hλ , 289 T ⊗ S, 297

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Le Rousseau et al., Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II, PNLDE Subseries in Control 98, https://doi.org/10.1007/978-3-030-88670-7

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544

T|U , 289 E  (K), 291 E  (Ω), 291 ∗, 297 κ∗ T , 292 0 0 D (U ), 427 0  D (M), 292 0 0 D (M), 292 1  D (U ), 429 1 0 D (U ), 423 1  D (M), 293 a 0 D (U ), 430 a  D (U ), 430 a 0 D (M), 293 ⊗, 296 τh , 289 D  (Ω), 288 D N (Ω), 290 S  (Rd ), 294 1 0 D (M), 293 a  D (M), 293 Dual space X  , 358 .X  , 358 E Elliptic map L, L(m) , 35 D LP , 48 D MP , 48 Energy E(.)(t), E(., .), 217, 225, 252 Euclidean space and distance B(x, r), 14 B d (x, r), 14 Ωε , 193 dist(., .), 14 (., .)x , 64 |.|x , 64 x · y, 14 ˜ε , 198 U F Fourier transformation F , 293, 295 F −1 , 295 Tˆ, 295 ϕ, ˆ 293 Function space

Index of notation

L1 -density, 428 L1 (M), 445 L1loc -density, 425 L2 (M), 464 L∞ (M), 471 Lp (M), 470 L1loc (M), 445 L2loc (M), 464 ∞ C c (U+ ), 75, 124, 178 ∞ C c (Rd+ ), 75, 379 Ω

Cc∞ (W ), 71 C ∞ (Ω), 92 Cc∞ (Ω), 287 Dc0 (U ), 423 Dck (U ), 429 W(Ω), 348 WP (M), 98, 484 Wg (M), 98, 484 Wg (Ω), 504 W−1 (Ω), 350 Wg,−1 (M), 485 ∗, 297 0 k Dc (U ), 429 0 0 Dc (M), 292 0 ∞ Dc (M), 292 1 0 Dc (U ), 426 1 ∞ Dc (U ), 429 1 0 Dc (M), 293 1 ∞ Dc (M), 293 a k Dc (U ), 430 a 0 Dc (M), 293 a ∞ Dc (M), 293 ⊗, 296 S (Rd+ ), 15 S (Rd ), 15, 294 vector fields and tensors C  V (M), 401 C  Λ(M), 404 C  Λr (M), 407 C  Ωp (M), 410 C  Trs M, 411 G Geometry differential geometry ∧, 408 differential geometry T M, 400

Index of notation

T φ, 401 T φ(m), 399 T ∗ M, 403 ∗ M, 402 Tm Tm M, 397 [., .], 413 d, 410 φ∗ f , 389 φ∗ (ω), 404 φ∗ (v), 401 π, 400 π ˜ , 403 {., .}, 412 df , 404 df (m), 403 differential operator D k (M), 431 Dτk (M), 435 Riemannian geometry Δg , 443 Γkij , 448, 451 D, 448 Du , 448 distg (., .), 438 divg , 443 exp, expm , 453 Uˆε , 210 H, 460 ∇g , 442 |.|gm , 438, 439 , , 439 g C,ij , g ij , 439 C , gij , 439 gij gm (., .), 438 g∂ , 440 H Half-space notation U+ , 75 Rd+ , 14 x , 15, 35 L Lifting map M, 499 D, 349, 486 N, 495 Linear operator A∗ , 358, 364

D(A), 355 D(A∗ ), 358, 364 F B(X, Y ), 359 G(.), 356 Rλ (.), 357 .D(A) , 218, 223, 252, 356 .L (X,Y ) , 356 def A, 358 ind(A), 358 ker(.), 356 B(X, Y ), 357 nul A, 358 Ran(.), 355 ρ(.), 221, 357 sp(.), 221, 357 L (X, Y ), 355 M Miscellaneous E[.], 325 , 16 , 16 , 16 Multi-index Dα , 14 β ≤ α, 15 |α|, 15 ξ α , 14 α!, 15 N Norm and inner product .τ,k,s , 133 .τ,k , 133, 153 .τ,s , 28, 154 .+ , 39, 76, 133 .H k (M)/C , 491 .H k (M) , 469 .H s (Rd ) , 43 .H −s (M) , 476 .H s (M) , 476 .L2 (Rd ) , 28 ϕK,N , 287 (., .)+ , 39, 76, 133 at a boundary (., .)∂ , 76, 133 |.|τ,s , 154 | tr(u)|τ,m,s , 134, 156

545

546

Index of notation

|.|∂ , 76, 133 O Oscillatory integrals Iτ (a, u), 23 P Pseudo-differential operator classical DTm , 43 Λsτ , 43 Op(.), 42 Ψm , 42 Ψm T , 43 D m , 22, 42 with a large parameter Dτm , 21 m DT,τ , 36 s Λτ , 28 ΛsT,τ , 39 Op(.), 21, 131 OpT (.), 36, 131 Ψm τ , 21 Ψm,r τ , 131 Ψm,r τ,ph , 131 Ψm T,τ , 36 Ψm T,τ,ph , 131 m ST,τ , 130 λT,τ , 130 Pseudo-differential symbol calculus ◦, 37, 341 ∼, 20, 36, 130 {., .}, 28 a∗ , 26, 36, 131, 341 characteristic set Char, 65 classical Am ρ , 25 m Nρ,k (.), 25 m S , 42 S −∞ , 42 S ∞ , 42 STm , 43 ., 23, 42 principal symbol σ(.), 19–21, 35, 36, 42 with a large parameter Sτm , 19

Sτ−∞ , 19 Sτ∞ , 19 Sτm,r , 130 m Sτ,ph , 21, 131 m,r Sτ,ph , 131 m ST,τ , 35 m ST,τ,ph , 131 λτ , 19 λT,τ , 35 S Second-order elliptic operator D(P0 ), 317 D(P−1 ), 322 K s (Ω), 324 P0 , 315 S(t), 327, 328 P0 , 317 P−1 , 322 kt (x, x ), 336 Sobolev space on Rd Hτs (Rd ), 28 on a manifold H(divg , M), 489 H k (M), 469 H k V (M), 469 H k Λ(M), 469 H k Λr (M), 469 H k Trs (M), 469 H k (M)/C, 491 H0k (M), 470 H s (M), 476 H −k (M), 470 H −s (M), 476 s Hloc (M), 467 2 L (M), 464, 469 L2 V (M), 469 L2 Λ(M), 469 L2 Λp (M), 469 L2 Trs (M), 468 L∞ Trs (M), 471 L∞ (M), 471 Lp Trs (M), 470 Lp (M), 470 L2loc (M), 464 W k,∞ (M), 471 W k,∞ Trs (M), 471

Index of notation

W k,p (M), 471 W k,p Trs (M), 471 HD1 (M), 251, 497 HD1 (M), 106 HD2 (M), 251 H k (M), 491 on boundary Hτs (N ), 155 (r)

HB (∂M), 34

r,s (∂M), 498 HDN H 3/2,1/2 (∂M), 101 H r,s (∂M), 51, 113 Sphere and cosphere bundle SW , 16 Sd−1 + , 16 Symplectic structure Hf , 67, 306, 412 σ(., .), 305, 412 {., .}, 28, 37, 307, 412

547