Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I: Dirichlet Boundary Conditions on Euclidean Space 3030886735, 9783030886738

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Progress in Nonlinear Differential Equations and Their Applications PNLDE Subseries in Control 97

Jérôme Le Rousseau Gilles Lebeau Luc Robbiano 

Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I Dirichlet Boundary Conditions on Euclidean Space

Progress in Nonlinear Differential Equations and Their Applications PNLDE Subseries in Control Volume 97

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 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 I Dirichlet Boundary Conditions on Euclidean Space

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 978-3-030-88673-8 ISBN 978-3-030-88674-5 (eBook) https://doi.org/10.1007/978-3-030-88674-5 Mathematics Subject Classification: 35, 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 Part 1. Calculus with a Large Parameter, Carleman Estimates Derivation Chapter 1. Introduction 1.1. Some Aspects of Unique Continuation 1.2. Form of Carleman Estimates and Quantification of Unique Continuation 1.3. Application to Stabilization and Controllability 1.4. Outline 1.5. Missing Subjects 1.6. Acknowledgement 1.7. Some Notation

1 3 4 4 6 8 12 14 14

Chapter 2. (Pseudo-)Differential Operators with a Large Parameter 17 2.1. Introduction 18 2.2. Classes of Symbols 19 2.3. Classes of Pseudo-Differential Operators 21 2.4. Oscillatory Integrals 22 2.5. Symbol Calculus 26 2.6. Sobolev Spaces and Operator Bound 28 2.7. Positivity Inequalities of G˚ arding Type 30 2.8. Parametrices 32 2.9. Action of Change of Variables 34 2.10. Tangential Operators 35 2.11. Semi-Classical Operators 41 2.12. Standard Pseudo-Differential Operators 42 2.13. Notes 44 Appendix 44 2.A. Technical Proofs for Pseudo-Differential Calculus 44

V

VI

CONTENTS

Chapter 3. Carleman Estimate for a Second-Order Elliptic Operator 63 3.1. Setting 64 3.2. Weight Function and Conjugated Operator 64 3.3. Local Estimate Away from Boundaries 69 3.4. Local Estimates at the Boundary 71 3.5. Patching Estimates 89 3.6. Global Estimates with Observation Terms 91 3.7. Alternative Approach 106 3.8. Notes 118 Appendices 120 3.A. Poisson Bracket and Weight Function 120 3.B. Symbol Positivity 125 3.C. An Explicit Computation 128 Chapter 4. Optimality Aspects of Carleman Estimates 4.1. On the Necessity of the Sub-ellipticity Property 4.2. Limiting Weights and Limiting Carleman Estimates 4.3. Carleman Weight Behavior at a Boundary 4.4. Notes Appendix 4.A. Some Technical Results

131 132 150 159 170 171 171

Part 2.

181

Applications of Carleman Estimates

Chapter 5. Unique Continuation 5.1. Introduction 5.2. Local and Global Unique Continuation 5.3. Quantification of Unique Continuation 5.4. Unique Continuation Initiated at the Boundary 5.5. Unique Continuation Without Any Prescribed Boundary Condition 5.6. Notes Appendix 5.A. A Hardy Inequality Chapter 6. Stabilization of the Wave Equation with an Inner Damping 6.1. Introduction and Setting 6.2. Preliminaries on the Damped Wave Equation 6.3. Stabilization and Resolvent Estimate 6.4. Remarks and Non-Quantified Stabilization Results 6.5. Resolvent Estimate for the Damped Wave Generator 6.6. Alternative Proof Scheme of the Resolvent Estimate 6.7. Notes Appendices 6.A. The Generator of the Damped-Wave Semigroup

183 183 184 186 198 202 210 213 213 215 216 216 220 222 224 229 232 235 235

CONTENTS

VII

6.B. Well-Posedness of the Damped Wave Equation 6.C. From a Resolvent to a Semigroup Stabilization Estimate 6.D. Proofs of Non-Quantified Stabilization Results

239 243 247

Chapter 7. Controllability of Parabolic Equations 7.1. Introduction and Setting 7.2. Exact Controllability for a Parabolic Equation 7.3. Null-Controllability for Semigroup Operators 7.4. Observability for the Semigroup Parabolic Equation 7.5. A Spectral Inequality 7.6. Partial Observability and Partial Control 7.7. Construction of a Control Function for a Parabolic Equation 7.8. Dual Approach for Observability and Control Cost 7.9. Properties of the Reachable Set and Generalizations 7.10. Boundary Null-Controllability for Parabolic Equations 7.11. Notes

251 251 254 256 258 260 267 268 270 273 276 280

Part 3. Background Material: Analysis and Evolution Equations

285

Chapter 8. A Short Review of Distribution Theory 8.1. Distributions on an Open Set of Rd and on a Manifold 8.2. Temperate Distributions on Rd and Fourier Transformation 8.3. Distributions on a Product Space 8.4. Notes

287 287 293 296 298

Chapter 9. Invariance Under Change of Variables 9.1. A Review of the Actions of Change of Variables 9.2. Action on Symplectic Structures 9.3. Invariance of the Sub-ellipticity Condition 9.4. Normal Geodesic Coordinates

301 302 304 308 309

Chapter 10. Elliptic Operator with Dirichlet Data and Associated Semigroup 10.1. Resolvent and Spectral Properties of Elliptic Operators 10.2. The Parabolic Semigroup 10.3. The Nonhomogeneous Parabolic Cauchy Problem 10.4. Elementary Form of the Maximum Principle 10.5. The Dirichlet Lifting Map 10.6. Parabolic Equation with Dirichlet Boundary Data

315 315 327 340 347 348 351

Chapter 11.1. 11.2. 11.3. 11.4.

355 355 356 357 358

11. Some Elements of Functional Analysis Linear Operators in Banach Spaces Continuous and Bounded Operators Spectrum of a Linear Operator in a Banach Space Adjoint Operator

VIII

CONTENTS

11.5. Fredholm Operators 11.6. Linear Operators in Hilbert Spaces Chapter 12.1. 12.2. 12.3. 12.4.

12. Some Elements of Semigroup Theory Strongly Continuous Semigroups Differentiable and Analytic Semigroups Mild Solution of the Inhomogeneous Cauchy Problem The Case of a Hilbert Space

358 363 367 368 374 375 376

Bibliography

379

Index

401

Index of notation

407

Part 1

Calculus with a Large Parameter, Carleman Estimates Derivation

CHAPTER 1

Introduction Contents 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.7.1. 1.7.2. 1.7.3. 1.7.4. 1.7.5. 1.7.6. 1.7.7.

Some Aspects of Unique Continuation Form of Carleman Estimates and Quantification of Unique Continuation Application to Stabilization and Controllability Outline Missing Subjects Acknowledgement Some Notation Open Sets Euclidean Inner Products and Norms Differential Operators Fourier Transformation Function Norms Homogeneity and Conic Sets Miscellaneous

4 4 6 8 12 14 14 14 14 14 15 15 16 16

In 1939, T. Carleman introduced some weighted estimates to achieve uniqueness properties for the Cauchy problem of an elliptic operator in two dimensions [105]. Estimates of this type now bear his name. In the late 50s A.-P. Calder´ on and L. H¨ormander further developed Carleman’s method [100, 171]. To this day, the method based on Carleman estimates remains essential to prove unique continuation properties. In more recent years, the field of applications of Carleman estimates has gone beyond the original domain; they are also used in the study of stabilization and controllability properties of partial differential equations, two applications we shall consider in this book. Inverse problems are also a field of applications for Carleman estimate; we shall however not touch upon that subject. © 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 I, PNLDE Subseries in Control 97, https://doi.org/10.1007/978-3-030-88674-5 1

3

4

1. INTRODUCTION

1.1. Some Aspects of Unique Continuation On Ω a smooth connected open set of Rd , let u ∈ H 2 (Ω) be the solution to the following Laplace–Dirichlet problem  −Δu + qu = 0 in Ω, (1.1.1) on ∂Ω. u|∂Ω = 0 If the potential function q is such that qL∞ < C02 , where C0 is the optimal constant in the Poincar´e inequality, that is, C0 wL2 (Ω) ≤ ∇wL2 (Ω) for w ∈ H01 (Ω), the variational formulation of this elliptic problem in H01 (Ω) yields u = 0. To the contrary, one can find a nonvanishing solution associated with some well chosen q ∈ L∞ (Ω) such that qL∞ ≥ C02 . As an example setting q = −C02 , there exists u0 ∈ H 2 (Ω), u0 = 0, such that (1.1.1) holds. The function u0 is in fact an eigenfunction associated with the lowest eigenvalue of −Δ. However, if one makes the additional requirement of having ∂ν u|Γ = 0 for Γ an open subset of ∂Ω, one finds u = 0. Here, ∂ν is the normal derivative at the boundary. More generally, if one considers u ∈ H 2 (Ω) solution to  −Δu + b · ∇u + qu = 0 in Ω, (1.1.2) on Γ, u|Γ = ∂ν u|Γ = 0 where b ∈ L∞ (Ω; Rd ), one has necessarily u = 0. This result cannot be deduced from the variational formulation of the elliptic problem (1.1.1). Note that the connectedness of Ω is important at this stage. A problem for which such a result holds is said to have the unique continuation property. Indeed, if u1 and u2 both satisfy the elliptic equation in Ω and have both their Dirichlet and Neumann traces that coincide on Γ, that is, u1 |Γ = u2 |Γ and ∂ν u1 |Γ = ∂ν u2 |Γ , their difference is the solution to (1.1.2) and one concludes that u1 = u2 . Similarly, if ω is an open set of Ω, and if one considers u ∈ H 2 (Ω) solution to  −Δu + b · ∇u + qu = 0 in Ω, (1.1.3) u=0 in ω, then one can conclude that u = 0 in Ω. The two previous results can be proven by means of Carleman estimates. In fact, as of today, Carleman estimates remain one of the most efficient tools to achieve such results. Moreover, as we shall see, Carleman estimates imply quantitative versions of these two unique continuation properties. 1.2. Form of Carleman Estimates and Quantification of Unique Continuation The Carleman estimates we consider in this book are weighted energy estimates associated with L2 -norms. The weights are of exponential form,

1.2. FORM OF CARLEMAN ESTIMATES AND QUANTIFICATION . . .

5

viz., exp(τ ϕ), with τ > 0 and ϕ a smooth function defined in (part of) Ω. A typical form is (1.2.1)



τ α eτ ϕ vL2 (Ω) + τ α eτ ϕ ∇vL2 (Ω) ≤ Ceτ ϕ P vL2 (Ω) ,

with P = −Δ+b·∇+q as above. This inequality holds uniformly for τ ≥ τ∗ , for some τ∗ > 0 chosen sufficiently large, and v with compact support near a region of interest in Ω. The values of α, α ∈ R can be of importance. For the operator P , with a proper choice of the function ϕ, one can have α = 3/2 and α = 1/2. The value of C in (1.2.1) is independent of τ and v, yet it depends on the geometry, in particular the dimension, of the region in Ω where v is supported and on the choice made for the weight function ϕ. Observe that one cannot directly apply such an estimate to the function u solution to, say (1.1.2) or (1.1.3), because of the support requirement. One uses a cut-off function χ with the proper support and one sets v = χu. Estimate (1.2.1) then applies to the function v. The interesting feature in such an estimate is the ability to have τ ≥ τ∗ be as large as needed. As a result, regions where ϕ is large will be predominant in the estimate and have influence on v, and thus u, elsewhere. This leads to the idea that if u vanishes in an open subset ω of Ω as in (1.1.3), choosing ϕ large over ω and smaller outside ω, one can use (1.2.1) to prove that u vanishes in a larger region. A connection argument can then be used to find that u vanishes everywhere. The unique continuation property can be described quite finely by means of the Carleman estimate (1.2.1). In fact, if x(0) ∈ Ω and r > 0 is such that the open ball B(x(0) , 4r) is contained in Ω, one can prove that for some C > 0 and δ ∈ (0, 1) one has  δ (1.2.2) uH 1 (B(x(0) ,3r)) ≤ Cu1−δ H 1 (Ω) f L2 (Ω) + uH 1 (B(x(0) ,r)) , for u ∈ H 2 (Ω) with f = P u ∈ L2 (Ω). Hence if f is small in Ω and u is small in the ball B(x(0) , r), one obtains that u is also small in the bigger ball B(x(0) , 3r). In particular, if f = 0 in Ω and u = 0 in the small ball, then u vanishes in the bigger ball. Above, we have described estimates away from the boundary. With prescribed boundary conditions and a well chosen weight function ϕ, one can also obtain a Carleman estimate in the neighborhood of a point y of ∂Ω. It takes the form (1.2.3) τ 3/2 eτ ϕ vL2 (Ω) + τ 1/2 eτ ϕ ∇vL2 (Ω) + τ 1/2 |eτ ϕ ∂ν v|∂Ω |L2 (∂Ω)   ≤ C eτ ϕ P vL2 (Ω) + τ 3/2 |eτ ϕ v|∂Ω |H 1 (∂Ω) , for v supported near y and τ ≥ τ∗ , for some τ∗ > 0 chosen sufficiently large. Here, the Dirichlet trace v|∂Ω appears on the right-hand side of the estimate. We thus say that the Carleman estimate is derived for this boundary condition. In fact, this estimate holds if one has ∂ν ϕ|∂Ω < 0 near y, with ν

6

1. INTRODUCTION

the outward-pointing normal unit vector field. This means that, locally, ϕ is larger inside Ω than on the boundary. This indicates, as exposed above, that one can control the solution u up to the boundary from the interior of Ω, once the Dirichlet trace is given. Indeed, from (1.2.3), one can deduce that for some neighborhood W of y, C > 0, and δ ∈ (0, 1), one has  δ f  (1.2.4) + u , uH 1 (W ∩Ω) ≤ Cu1−δ L2 (Ω) H 1 (Ωε ) H 1 (Ω) for u ∈ H 2 (Ω) satisfying u|∂Ω = 0, with f = P u ∈ L2 (Ω), and where Ωε = {x ∈ Ω; dist(x, ∂Ω) > ε}. For general references on Carleman estimates and their application to unique continuation we refer the reader to the seminal work of L. H¨ormander [172, Chapter 8]; see also [174, Chapter 28]. We also refer to the book of C. Zuily [330] for many results and to the recent book of N. Lerner [229]. 1.3. Application to Stabilization and Controllability In this book we present two other applications of Carleman estimates for second-order elliptic operators. One concerns the stabilization property of the damped wave equation. Another one concerns the null-controllability of the heat equation. First, consider the following wave equation:  ∂t2 y − Δy + α(x)∂t y = 0 in (0, +∞) × Ω, (1.3.1) y|t=0 = y 0 , ∂t y|t=0 = y 1 in Ω, where α(x) is a nonnegative bounded function. With initial conditions satisfying y 0 ∈ H 2 (Ω) ∩ H01 (Ω) and y 1 ∈ H01 (Ω), one obtains a solution       y ∈ C 2 [0, +∞); L2 (Ω) ∩C 1 [0, +∞); H01 (Ω) ∩C 0 [0, +∞); H 2 (Ω)∩H01 (Ω) . Introducing the energy E(y)(t) =

 1 ∂t y(t)2L2 (Ω) + ∇y(t)2L2 (Ω) , 2

one finds d 2 E(y)(t) = −α1/2 ∂t y(t)L2 (Ω) ≤ 0. dt One calls α(x)∂t y the damping term; it is responsible for this decay of the energy. One refers to (1.3.1) as the damped wave equation. If one assumes that α > 0 in some nonempty open subset ω of Ω, then, with the Carleman estimates we derive in this book, one can obtain the following decay of the energy:   C 0 2 1 2 (1.3.2) E(y)(t) ≤  2 Δy L2 (Ω) + ∇y L2 (Ω) . log(2 + t)

1.3. APPLICATION TO STABILIZATION AND CONTROLLABILITY

Introducing (1.3.3)

A=



 0 −1 , −Δ α(x)

 Y (t) =

 y(t) , ∂t y(t)

7

 0 y Y = , y1 0

the damped wave equation (1.3.1) reads, in a semigroup form, d Y (t) + AY (t) = 0, Y|t=0 = Y 0 . dt In fact, by means of Carleman estimates, for example through the quantification of unique continuation, one proves a resolvent estimate for A on the imaginary axis that takes the form, for some K > 0, (iσ Id −A)−1 L (H ) ≤ KeK|σ| ,

σ ∈ R,

where H = H01 (Ω) ⊕ L2 (Ω). Then, a functional analysis result for semigroups yields the decay estimate in (1.3.2). Note that the stabilization of the damped wave equation expressed in (1.3.2) does not rely on any particular geometrical assumption on the nonempty open set ω. Second, for T > 0, we consider the following heat equation: ⎧ ⎪ in (0, T ) × Ω, ⎨∂t y − Δy = 1ω v (1.3.4) y=0 on (0, T ) × ∂Ω, ⎪ ⎩ 0 in Ω, y(0) = y where ω is a nonempty open subset of Ω. The null-controllability property stands for the ability to choose the source term v, depending on the initial condition y 0 , to obtain a vanishing solution at time t = T . The functional framework is as follows: y 0 ∈ L2 (Ω) and v ∈ L2 ((0, T ) × Ω) yielding a solution y ∈ C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)). Thus, considering the value of y in L2 (Ω) at time t = T is sensible. Observe that the source term v, called the control function, only acts on the solution in the (possibly small) region ω. A functional analysis argument shows that the null-controllability property is equivalent to having the following observability inequality, for some Cobs > 0, (1.3.5)

z(T, .)L2 (Ω) ≤ Cobs zL2 ((0,T )×ω) ,

where z is the solution to ⎧ ⎪ ⎨∂t z − Δz = 0 z=0 ⎪ ⎩ z(0) = z 0

in (0, T ) × Ω, on (0, T ) × ∂Ω, in Ω,

with z 0 ∈ L2 (Ω). Let us denote by (φj )j∈N a Hilbert basis of L2 (Ω) formed by eigenfunctions of the Dirichlet–Laplace operator, with 0 < μ0 ≤ μ1 ≤ · · · the associated real and positive eigenvalues. By means of Carleman

8

1. INTRODUCTION

estimates, for example through the quantification of unique continuation, one proves the following spectral inequality, for some K > 0, (1.3.6)

wL2 (Ω) ≤ eK

√

μ

wL2 (ω) ,

w ∈ span(φj ; μj ≤ μ).

In turn, exploiting the decay properties of the heat semigroup, one can prove that the spectral inequality (1.3.6) implies the observability inequality (1.3.5). This allows one to conclude to the null-controllability of the heat equation for both an arbitrary time T > 0 and an arbitrary open subset ω. For general references on control theory and stabilization for partial differential equations we refer the reader to the book of J.-L. Lions [235] and the more recent books of J.-M. Coron [109] and of M. Tucsnak and G. Weiss [321]. 1.4. Outline Chapters 2 to 4 form the first part of the book where Carleman estimates are derived and fundamental aspects of these estimates are put forward. For the derivation of Carleman estimates we have chosen to rely on pseudo-differential calculus with a large parameter. On the one hand, this imposes some smoothness constraints on the coefficients in the principal part of the elliptic operators we consider and on the open sets where the elliptic problems are written. On the other hand, it provides us with powerful tools to achieve inequalities such as G˚ arding type inequalities, in particular near the boundary. These tools allow to control the dependency on the large parameter τ that is key to obtain an estimate as in (1.2.1) where the constant C is independent of τ . Moreover, these tools allow us to shade some light on the sub-ellipticity property that is at the heart of Carleman estimates. Note also that smoothness of the coefficients and of the boundary allows one to perform smooth change of variables that turn out very useful to concentrate near a point or a region of interest. Pseudo-differential calculus is presented in Chap. 2 in the presence of a large parameter. This large parameter will naturally be τ > 0 as appearing in the Carleman estimate (1.2.1). For completeness, all proofs are provided in Chap. 2, either in the main text or in an appendix. Chapter 3 is devoted to the derivation of Carleman estimates for secondorder elliptic operators, first away from a boundary, second at a boundary. In particular, we point out a sufficient condition on the weight function and the elliptic operator, namely the sub-ellipticity property, for the Carleman estimate to hold. At the boundary, we either do not consider any prescribed boundary condition or we consider Dirichlet boundary conditions. For the derivation of Carleman estimates in the case of other boundary conditions ˇ (Neumann, Robin, or more general Lopatinski˘ı-Sapiro conditions) the reader is referred to Volume 2. In all proofs we introduce the conjugated operator Pϕ = eτ ϕ P e−τ ϕ , that is a second-order differential operator, with τ > 0 as a large parameter.

1.4. OUTLINE

9

This conjugated operator fits the framework introduced in Chap. 2. If p(x, ξ) denotes the principal symbol of P , then, p(x, ξ + iτ dϕ(x)) is the principal symbol of Pϕ . The sub-ellipticity property reads |dϕ(x)| > 0 and 1 {pϕ , pϕ }(x, ξ, τ ) > 0, 2i where {., .} denotes the Poisson bracket. In Chap. 3 we insist on the local nature of Carleman estimates. Thus, for an estimate near a point of the boundary, we first prove a Carleman estimate for functions supported in a small neighborhood of this point. This allows one to use local coordinates that simplify the arguments and computations used along the derivation. We also show how local estimates can be patched together to, first, form a local estimate without having to shrink the neighborhood near the point of interest and, second, form a global estimate in Ω. A global estimate requires the construction of a suitable weight function on the whole Ω. We show how this construction can be done in various settings: global estimates with an inner observation term or with a boundary observation term. Finally, we also show how Carleman estimates can be derived when less regularity is assumed on the coefficients of the elliptic operator, namely Lipschitz regularity. (1.4.1)

pϕ (x, ξ, τ ) = 0 ⇒

Chapter 4 is devoted to optimality aspects of Carleman estimates. We prove some necessity aspects of the sub-ellipticity condition. First, having (1.4.2)

τ α eτ ϕ vL2 (Ω) ≤ Ceτ ϕ P vL2 (Ω) ,

for some α ∈ R implies the following nonnegativity property: 1 {pϕ , pϕ }(x, ξ, τ ) ≥ 0. 2i We also prove that α ≤ 2 if dimension d = 1 and α ≤ 3/2 if d ≥ 2. In the case d ≥ 2 we prove that if the Carleman estimate holds with α = 3/2, then the sub-ellipticity property (1.4.1) holds. We also introduce the so-called limiting Carleman weights. They are weights that satisfy the condition pϕ (x, ξ, τ ) = 0 ⇒

1 {pϕ , pϕ }(x, ξ, τ ) = 0. 2i With such a weight a Carleman estimate holds as in (1.2.1) with α = 1 and with α = 0. Finally, at the boundary, we also prove that the weight function needs to satisfy ∂ν ϕ ≤ 0 if an estimation of the type of (1.4.2) holds for some α ∈ R and for functions supported near the boundary. pϕ (x, ξ, τ ) = 0 ⇒

Chapters 5 to 7 form the second part of the book that is devoted to some applications of Carleman estimates.

10

1. INTRODUCTION

In Chap. 5 we address unique continuation issues. First, we prove a non-quantified unique continuation result in the whole domain Ω, without assuming any boundary condition in the elliptic problem under consideration and (therefore) only relying on Carleman estimates derived away from boundaries in Chap. 3. Second, with the Carleman estimates of Chap. 3 both away and at the boundary, we derive local quantifications of the unique continuation property, away from the boundary as in (1.2.2), or at the boundary as in (1.2.4). We show how such inequalities can be propagated across Ω yielding an estimate of the form  δ (1.4.3) uH 1 (U ) ≤ Cu1−δ H 1 (Ω) f L2 (Ω) + uL2 (ω) , where U and ω are two nonempty open subsets of Ω, for u ∈ H 2 (Ω) satisfying u|∂Ω = 0 in a neighborhood of U and f = P u, with P = −Δ + b · ∇ + q as in Sects. 1.1 and 1.2. Using a Carleman estimate of Chap. 3 on part of the boundary, one can replace the inner observation term uL2 (ω) by boundary observation terms, that is, (1.4.4)  δ uH 1 (U ) ≤ Cu1−δ f  + |u | + |∂ u | , 2 1 ν |∂Ω |∂Ω L (Ω) H (Ω) H 1 (V ∩∂Ω) L2 (V ∩∂Ω) for some V neighborhood of a point of the boundary, and U , u, and f as above, having in particular that u|∂Ω = 0 in a neighborhood of U . Third, we provide a quantification of the unique continuation property up to the boundary without prescribing any boundary condition: for U and ω two nonempty open subsets of Ω one has   uH 2 (Ω) , uH 1 (U ) ≤ CuH 2 (Ω) h f L2 (Ω) + uL2 (ω) with

  log 2 + log(2 + r) , h(r) = log(2 + r)

for u ∈ H 2 (Ω) and f = P u. In Chap. 6, we prove the logarithmic stabilization of the damped wave equation as stated in (1.3.2). In particular, we show how a resolvent estimate of the form σ ∈ R, (iσ Id −B)−1 L (H) ≤ KeK|σ| , for the generator B of a bounded C0 -semigroup S(t) on a Hilbert space H implies the estimate C S(t)B −k L (H) ≤  k , t > 0, log(2 + t) for all k ∈ N. Applied to the semigroup associated with the damped wave equation, the logarithmic decay in (1.3.2) follows. This result only concerns

1.4. OUTLINE

11

strong solutions of the damped wave equation. However, as a corollary, one can prove the stabilization of weak solutions, yet without any estimation of the decay rate. We provide two proofs of the resolvent estimate for the generator of the damped wave semigroup A given in (1.3.3). One is based on the quantification of unique continuation given by (1.4.3); another one is based on the derivation of a global Carleman estimate. In Chap. 7, we address the null-controllability of the heat equation (1.3.4). First, we show that, due to the regularizing properties of the heat semigroup, exact controllability cannot hold unless ω = Ω. Second, we state the equivalence between null-controllability and observability. Third, we provide two proofs of the spectral inequality (1.3.6). One is based on the quantification of unique continuation initiated at the boundary given by (1.4.4); another one is based on the derivation of a global Carleman estimate. Fourth, we show how the spectral inequality allows one to either construct a control function or prove the observability inequality. We also address the case of a heat equation with a source term acting in an open set Γ of the boundary: ⎧ ⎪ in (0, T ) × Ω, ⎨∂t y − Δy = 0 on (0, T ) × ∂Ω, y = 1Γ v ⎪ ⎩ 0 in Ω. y(0) = y The functional framework is the following: for y 0 ∈ H −1 (Ω) and v ∈ L2 ((0, T ) × ∂Ω) there exists a unique weak solution     y ∈ C 0 [0, T ]; H −1 (Ω) ∩ L2 0, T ; L2 (Ω) . Null-controllability at time T > 0 means in this case that for any y 0 as above there exists v ∈ L2 (0, T ; ∂Ω) such that y(T ) = 0. We show that this result can be deduced from the null-controllability of (1.3.4). Chapters 8 to 12 form the third part of the book that is devoted to the exposition of background material. In Chap. 8, we provide a short review of the theory of distributions. Some of these results are useful for the exposition of pseudo-differential operators in Chap. 2. In Chap. 9, we review the action of change of variables on differential operators and their principal symbols. Basic symplectic structure is also considered for the understanding of the action of change of variables on Poisson brackets. This allows one to prove that the sub-ellipticity property used in the derivation of Carleman estimates is of geometrical nature and thus independent of the chosen local variables. Finally, we expose how normal geodesic coordinates, used to prove Carleman estimates at a boundary, can be constructed.

12

1. INTRODUCTION

In Chap. 10, we review some basic aspects of a second-order elliptic operator including its spectral properties if associated with homogeneous Dirichlet boundary conditions. A Sobolev-like scale is provided by means of a Hilbert basis of L2 (Ω) composed of eigenfunctions of the operator. We recall the properties of the semigroup generated by this elliptic operator and of solutions of the associated homogeneous parabolic equation. The nonhomogeneous parabolic equation is also considered and strong and weak solutions are presented. Finally, parabolic equations with nonhomogeneous boundary conditions are introduced, also via a semigroup formulation by means of the Dirichlet lifting map. In Chap. 11, we review some elements of functional analysis with regard to bounded and unbounded operators on Banach spaces: spectrum, adjoint, Fredholm property. A particular focus is placed on the case of Hilbert spaces. In Chap. 12 elementary aspects of semigroup theory on a Banach space are exposed including the Hille–Yosida and the Lumer–Phillips theorems. The associated homogeneous and nonhomogeneous Cauchy problems are presented. This background material is of importance for the study of the damped wave equation in Chap. 6 and the parabolic semigroup and equations in Chaps. 7 and 10. Figure 1.1 describes the interrelation of the different parts and chapters in Volume 1. In Volume 1, all the analysis is performed in the case of open sets in the Euclidean space. However, extension to Riemannian manifolds can be easily done from the results we obtain. This is done in Volume 2 for the derivation of Carleman estimates and some of the applications we give. In Parts 1 and 2, we have avoided bibliographical references in the main text to enhance readability. Yet, chapters end with a section entitled “Notes” where we have gathered some historical perspective and many references. 1.5. Missing Subjects This book could have contained other chapters on Carleman estimates and others on some different applications. For such missing topics we mention references in the notes at the end of some of the chapters. Below, we give few examples. The unique continuation properties we consider are connected with the size of the solution: u is small in the L2 -norm or even vanishes in some open set of an interior point, or on one side of an hypersurface, or the Dirichlet and Neumann traces of u are small or vanish in some neighborhood of a point of the boundary. We however do not consider strong unique continuation, meaning the flatness of the solution near a point implying that the solution vanishes locally. Also, as mentioned above, we only consider Carleman estimates based on L2 -norms. Carleman estimates can also be

1.5. MISSING SUBJECTS

Part 1

13

Part 3 Chapter 2

Chapter 8

Chapter 3

Chapter 9 Chapter 4

Part 2 Chapter 5

Chapter 6

Chapter 7

Part 3 Chapter 10

Chapter 11

Chapter 12

Figure 1.1. Interrelation of the parts and chapters in Volume 1

obtained with more general Lp -norms that can be used to prove unique continuation properties in the case of unbounded potential functions. For references see the notes of Chap. 5. Here, we only consider Carleman estimates for elliptic operators. Other types of estimates could be treated. The derivation of estimates for parabolic operators of the form of the heat operator ∂t − Δ can be made, by many aspects, very similar to that of estimates for the Laplace operator. Estimates for the heat operator allow one to also deduce the null-controllability of the heat equation. Moreover, they allow one to consider controllability issues for semilinear heat equations and other nonlinear equations like the Navier–Stokes equations; for a reference see the work of A. Fursikov and O. Yu. Imanuvilov [156] and more references in the notes of Chap. 7. A notable application of elliptic Carleman estimates of the form we derive here is the study of the local decay of the energy of the solution of the wave equation outside a compact obstacle; we refer to the work of N. Burq [94]. Inverse problems were mentioned above. In fact, we provide Carleman estimates with the so-called limiting weights in Sect. 4.2. These estimates can be used for instance to address uniqueness in the Carlder´on inverse problem, that is, the identification of conductivity c in the operator A =

14

1. INTRODUCTION

− div c∇, by means of boundary measurement of solutions; see for instance the work of C. Kenig, J. Sj¨ ostrand, and G. Uhlmann [195]. 1.6. 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. They also wish the two reviewers for their remarks and corrections. In particular one of the reviewers permitted an improvement in the description of the reachable set of the controlled heat equation. 1.7. Some Notation Here, we set some basic notation to be used throughout this book. Other notations are introduced along the different chapters. Many are gathered in the index of notation. 1.7.1. Open Sets. We shall say that Ω is a smooth (or regular) open set of Rd if it is open and in the case ∂Ω = ∅, if ∂Ω is smooth and if Ω is only located on one side of ∂Ω. At the boundary of such an open set we shall denote by ∂ν the normal derivative where ν is the outward-pointing normal unit vector at the boundary. Most of the analysis in this book will concern differential operators on open sets in Rd . Yet by localization and the use of a local charts, we shall very often be able to pullback the analysis to Rd or the half-space Rd+ = {x = (x1 , . . . , xd ) ∈ Rd ; xd > 0}, which will ease the direct use of Fourier analysis. In Rd+ , we shall write x = (x , xd ) with x = (x1 , . . . , xd−1 ). 1.7.2. Euclidean Inner Products and Norms. In Rd , the Euclidean inner product is denoted by

x·y = x i yi , 1≤i≤d

and by |x| the associated norm. The Euclidean distance between two points of Rd is denoted by dist(x, y) = |x − y|. For x ∈ Rd and r ≥ 0, we denote by B(x, r) the Euclidean ball centered at x with radius r. If dimension needs to be made precise, we shall use the notation B d (x, r). 1.7.3. Differential Operators. We shall use the notation Dj for Dxj = −i∂xj and D will stand for (D1 , . . . , Dd ). If differentiations with respect to other variables occur, we may write Dx in place of D to ensure that no ambiguity is created. Similarly, we shall use both ∂j and ∂xj . For a multi-index α, that is α = (α1 , . . . , αd ) ∈ Nd , we shall write ξ α = ξ1α1 · · · ξdαd ,

∂ α = ∂1α1 · · · ∂dαd ,

Dα = D1α1 · · · Ddαd ,

1.7. SOME NOTATION

15

and |α| = α1 + · · · + αd ,

α! = α1 ! · · · αd !.

For two multi-indices α and β we shall write β ≤ α to actually mean βj ≤ αj for all j ∈ {1, . . . , d}. 1.7.4. Fourier Transformation. We recall the definition of the Schwartz space of smooth functions on Rd : S (Rd ) = {u ∈ C ∞ (Rd ); ∀α, β ∈ Nd , sup |xα Dβ u(x)| < ∞}. x∈Rd

If we work on the half-space Rd+ , it can be convenient to use the following space of rapidly decreasing smooth functions: S (Rd+ ) = {u = v|Rd ; v ∈ S (Rd )}. +

As a set of semi-norms on S (Rd ) is given by (1.7.1) pn,k (ϕ) := sup{xn |Dα ϕ(x)|; α ∈ Nd , |α| ≤ k, x ∈ Rd },

x2 = 1 + |x|2 ,

n, k ∈ N, endowing S (Rd ) with a Fr´echet space topology. For a function u in the proper space, e.g. u ∈ S (Rd ), we shall denote by u ˆ the Fourier transform of u (resp. its partial Fourier transform with respect to x = (x1 , . . . , xd−1 )):     u ˆ(ξ) = ∫ e−ix·ξ u(x) dx, resp. u ˆ(ξ  , xd ) = ∫ e−ix ·ξ u(x , xd ) dx , Rd

Rd−1

and the inverse Fourier transform gives     ˆ(ξ) dξ, resp. = (2π)1−d ∫ eix ·ξ u ˆ(ξ  , xd ) dξ  . u(x) = (2π)−d ∫ eix·ξ u Rd

Rd−1

More on Fourier transformation can be found in Chap. 8. 1.7.5. Function Norms. When considering function norms in the interior of the considered domain, say Ω, Rd+ , or the whole Rd , we shall use the notation ., that is, .L2 (Ω) or .L2 (Rd ) . To the contrary, when func+

tion norms computed at the boundary of the domain, ∂Ω or ∂Rd+  Rd−1 , we shall use the notation |.|, that is, |.|L2 (∂Ω) and |.|L2 (Rd−1 ) . Note that no confusion should arise with the Euclidean norm introduced above. For a function v defined in Rd+ we shall often write v+ for vL2 (Rd ) + and |v|xd =0+ |∂ for |v|xd =0+ |L2 (Rd−1 ) . To avoid some cumbersome notation for norms of vector fields, we shall use the norm notation DuL2 (Ω) (resp. ∇uL2 (Ω) ) instead of DuL2 (Ω)d (resp. ∇uL2 (Ω)d ). This was done above.

16

1. INTRODUCTION

1.7.6. Homogeneity and Conic Sets. In phase space the variables will be (x, ξ, τ ) with x ∈ Rp , ξ ∈ Rd and τ ∈ [0, +∞), with often p = d. Here, we shall have an additional parameter τ as compared to the usual Fourier analysis. It will be meant to be large. We shall say that a set W in phase space is conic in the (ξ, τ ) variables if (x, ξ, τ ) ∈ W implies (x, tξ, tτ ) ∈ W for all t > 0. We shall often simply say conic instead of conic in the (ξ, τ ) variables if there is no possible ambiguity. On such a conic set we shall say that a function a(x, ξ, τ ) is homogeneous of degree m if one has a(x, tξ, tτ ) = tm a(x, ξ, τ )

for (x, ξ, τ ) ∈ W, t > 0.

Homogeneity and smoothness are often not compatible, as a singularity may arise at the origin. We shall thus often refer to functions that are smooth and homogeneous for |(ξ, τ )| sufficiently large, say |ξ|2 + τ 2 ≥ 1. In fact, the analysis tools originating from microlocal analysis that we shall use in this book, in particular the calculus of pseudo-differential operators, the value of the function a(x, ξ, τ ) near (ξ, τ ) = 0 will not have much impact on the estimation that we shall perform. Hence, we often only need to know that the function is smooth there. The behavior of a(x, ξ, τ ) for |(ξ, τ )| large will be the important feature. To exploit homogeneity in a conic set W ⊂ Rd × Rd × [0, ∞) we shall often bring the analysis to the cosphere bundle, (1.7.2)

SW = {(x, ξ, τ ) ∈ W ; |ξ|2 + τ 2 = 1}.

For d ≥ 1, we also introduce the following notation for the half-unit sphere (1.7.3)

Sd+ = {(ξ, τ ) ∈ Rd+1 ; ξ ∈ Rd , τ ≥ 0, and |ξ|2 + τ 2 = 1}.

Some of our analysis will often be based on tangential operators. Then, if x = (x , xd ) and ξ = (ξ  , ξd ) with x ∈ Rd−1 and ξ  ∈ Rd−1 as introduced above, we shall say that a set W ∈ Rd × Rd−1 × R+ , with the associated variables (x, ξ  , τ ), is conic if it is conic with respect to the variables (ξ  , τ ). Homogeneity of a function a(x, ξ  , τ ) on such a set is to be understood accordingly. 1.7.7. Miscellaneous. Throughout the book, the letter C will denote a constant whose value may change from one line to another. If we wish to keep track of the precise value of a constant, we shall use another letter. Often, to avoid the introduction of such a generic constant, especially in the course of proofs, we shall use the standard notation A  B to be read as A ≤ CB for some C > 0. In particular, the constant C will be independent of the large parameter τ that is used throughout this book. We shall also write A  B to actually mean A  B  A. For two sets in a topological space, following a standard notation, we shall write U  V when U is compact and a subset of V .

CHAPTER 2

(Pseudo-)Differential Operators with a Large Parameter Contents 2.1. Introduction 18 2.2. Classes of Symbols 19 2.2.1. Homogeneous and Polyhomogeneous Symbols 20 2.3. Classes of Pseudo-Differential Operators 21 2.4. Oscillatory Integrals 22 2.5. Symbol Calculus 26 2.6. Sobolev Spaces and Operator Bound 28 2.7. Positivity Inequalities of G˚ arding Type 30 2.8. Parametrices 32 2.9. Action of Change of Variables 34 2.10. Tangential Operators 35 2.11. Semi-Classical Operators 41 2.12. Standard Pseudo-Differential Operators 42 2.13. Notes 44 Appendix 44 2.A. Technical Proofs for Pseudo-Differential Calculus 44 2.A.1. Symbol Asymptotic Series: Proof of Lemma 2.4 44 2.A.2. Action on the Schwartz Space: Proof of Proposition 2.10 46 2.A.3. Proofs of Results on Oscillatory Integrals 47 2.A.4. Proofs of the Results on Symbol Calculus 52 2.A.5. Proof of Theorem 2.26: Sobolev Bound 56 2.A.6. Proofs of the G˚ arding Inequalities 58 2.A.7. Parametrix Construction and Properties 60 2.A.8. A Characterization of Ellipticity 61

© 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 I, PNLDE Subseries in Control 97, https://doi.org/10.1007/978-3-030-88674-5 2

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18

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

2.1. Introduction This first chapter serves as a reference for the following ones. Some results such as operator positivity as one can found in G˚ arding type inequalities are stated and proven. Such inequalities take the form, for u in the Schwartz class, S (Rd ), Re(Au, u)L2 (Rd ) ≥ Cu2τ,m/2 , for an operator A of order m ≥ 0 that depends on a large parameter, denoted by τ > 0 here, under some assumption on A, naturally. An example of such an operator would be −Δ + τ 2 for m = 2. Sobolev norms such as .τ, depend also on τ :

D|α| uL2 (Rd ) . uτ,  τ uL2 (Rd ) + |α|=

In a large part of the next chapters the operators will be differential. Yet, it is often convenient to consider them within a larger class of operators, namely pseudo-differential operators, in particular to prove positivity results such as the G˚ arding inequality. To motivate the form of the pseudo-differential operators we shall present below, we first formulate differential operators by means of the Fourier transformation (we refer the reader to Sect. 8.2 for basic aspects of the Fourier transformation). If p(x, ξ, τ ) is a polynomial in (ξ, τ ) of order less than or equal to m, with x, ξ ∈ Rd , and τ ≥ 1, we can write it in the form

p(x, ξ, τ ) = aα (x)ξ α τ k , |α|+k≤m

and we set

p(x, D, τ )u =

|α|+k≤m

aα (x)τ k Dα u.

ˆ(ξ) dξ, for u ∈ S (Rd ), we Observing that Dα u = (2π)−d ∫Rd eix·ξ ξ α u write

τ k ∫ eix·ξ aα (x)ξ α u ˆ(ξ) dξ, p(x, D, τ )u(x) = (2π)−d |α|+k≤m

that is,

Rd

u(ξ) dξ, p(x, D, τ )u(x) = (2π)−d ∫ eix·ξ p(x, ξ, τ )ˆ Rd

or, formally, p(x, D, τ )u(x) = (2π)−d ∫∫

Rd ×Rd

ei(x−y)·ξ p(x, ξ, τ ) u(y) dy dξ.

2.2. CLASSES OF SYMBOLS

19

Note also that (2.1.1)

|p(x, ξ, τ )|  (τ + |ξ|)m ,

and for all α, β ∈ Nd , (2.1.2)

|∂xα ∂ξβ p(x, ξ, τ )|  (τ + |ξ|)m−|β| ,

for |β| ≤ m and ∂xα ∂ξβ p(x, ξ, τ ) ≡ 0 for |β| > m. We wish to generalize such differential operators P (x, D, τ ) that involve a large parameter such as τ to the case of more general functions p(x, ξ, τ ). To maintain some operator “order” properties, it is convenient to preserve the asymptotic behaviors (2.1.1)–(2.1.2). 2.2. Classes of Symbols We introduce the following class of smooth symbols that depend on the parameter τ that is meant to be taken large in the analysis that follows. Definition 2.1 (Symbols with a Large Parameter). Let p, d ∈ N and let a(x, ξ, τ ) ∈ C ∞ (Rp × Rd ), with τ as a parameter in [1, +∞) and m ∈ R, be such that for all multi-indices α ∈ Np , β ∈ Nd there exists Cα,β > 0 such that , x ∈ Rp , ξ ∈ Rd , τ ∈ [1, +∞), |∂xα ∂ξβ a(x, ξ, τ )| ≤ Cα,β λm−|β| τ  1/2 . We write a ∈ Sτm (Rp × Rd ). We call where λτ = |(ξ, τ )| = |ξ|2 + τ 2 symbols of order m the elements of Sτm (Rp × Rd ). We also define Sτ−∞ (Rp × Rd ) = ∩r∈R Sτr (Rp × Rd ) and Sτ∞ (Rp × Rd ) = ∪r∈R Sτr (Rp × Rd ).

(2.2.1)

Note in particular1 that λsτ ∈ Sτs (Rp × Rd ) for all s ∈ R. We shall often simply write Sτm (resp. Sτ−∞ , Sτ∞ ), especially when the values of p and d clear from the context, for instance in the case p = d with d the dimension of the ambient space, which will occur most often in this book. For a ∈ Sτm (Rp × Rd ) we call principal symbol, σ(a), the equivalence class of a in Sτm (Rp × Rd )/Sτm−1 (Rp × Rd ). Note that this notion differs from the usual notion of principal symbol for a (pseudo-)differential operator. The reader may be used to viewing |ξ|2 as the principal symbol of |ξ|2 + τ 2 V (x). To the contrary, here the term τ 2 V (x) is also in Sτ2 and needs to be considered also in the principal part. This is a particular case of the examples given below in (2.2.2)–(2.2.3) the case of differential operators. The best possible constants Cα,β in (2.2.1) yield semi-norms that allow one to endow the space Sτm (Rp × Rd ) with a Fr´echet space structure. 1We shall use λ to measure symbol growth at ∞ instead of |ξ| + τ as the latter is not τ smooth at ξ = 0 and thus does not belong to any symbol space.

20

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

Very basic properties of symbols are given by the following propositions whose proof is left as an exercise. Proposition 2.2. The following maps Sτm → Sτm−|β| a → ∂xα ∂ξβ a, for α and β multi-indices, and 

Sτm × Sτm → Sτm+m



(a, b) → ab, are continuous. Proposition 2.3. Let a ∈ Sτ0 and F ∈ C ∞ (C; C). Then F (a) ∈ Sτ0 . The following lemma allows one to define an asymptotic series of symbols. Lemma 2.4. Let m ∈ R and aj ∈ Sτm−j with j ∈ N. Then there exists a ∈ Sτm such that ∀N ∈ N,

a−

N

j=0

aj ∈ Sτm−N −1 .

We then write a ∼ j aj . The symbol a is unique up to Sτ−∞ . If there exists a closed set U such that supp(aj ) ⊂ U for all j ∈ N, then the symbol a can be chosen to satisfy supp(a) ⊂ U . We refer to Sect. 2.A.1 for a proof. It is then natural to identify a0 with the principal symbol of a. If moreover a(x, ξ, τ ) ∈ Sτm is a polynomial function in (ξ, τ ) of order m, which corresponds to the introductory example of this chapter, we have

(2.2.2) a(x, ξ, τ ) = aα (x)ξ α τ k , |α|+k≤m

and we call principal symbol the representative of σ(a) in Sτm /Sτm−1 that is homogeneous of degree m in (ξ, τ ). This gives

aα (x)ξ α τ k . (2.2.3) σ(a)(x, ξ, τ ) = |α|+k=m

The case of symbols as in (2.2.2) will occur very naturally in the chapters that follow. 2.2.1. Homogeneous and Polyhomogeneous Symbols. The case of polynomial symbols motivates the study of homogeneous symbols. Let a ∈ Sτm be moreover homogeneous of degree m with respect to (ξ, τ ), that is, a(x, tξ, tτ ) = tm a(x, ξ, τ ),

for t > 0 and λτ ≥ r0 ,

2.3. CLASSES OF PSEUDO-DIFFERENTIAL OPERATORS

21

for some r0 > 0. If τ ≥ 1, we write a(x, ξ, τ ) = τ m a(x, ξ/τ, 1), which allows one to differentiate with respect to the parameter τ . We find

ξj ∂ξj a(x, ξ/τ, 1), ∂τ a(x, ξ, τ ) = mτ m−1 a(x, ξ/τ, 1) − τ m−2 1≤j≤d

which gives |∂τ a(x, ξ, τ )|  τ m−1 (1 + |ξ/τ |2 )m/2 + τ m−2 |ξ|(1 + |ξ/τ |2 )(m−1)/2  τ −1 λm τ . Further such computations yield, for r ∈ N and τ ≥ 1, ∂τr a ∈ τ −r Sτm . In particular we have the following result. Lemma 2.5. Let a ∈ Sτm be homogeneous of degree m with respect to (ξ, τ ) and such that, for some C > 0, supp(a) ⊂ {(x, ξ, τ ) ∈ Rd × Rd × R+ ; τ ≥ C|ξ|}. Then for all r ∈ N, ∂τr a ∈ Sτm−r . The proof simply uses that τ  λτ in supp(a). Definition 2.6. Let m ∈ R and let am−j ∈ Sτm−j for each

j ∈ N be homogeneous of degree m − j. If a ∈ Sτm is such that a ∼ j∈N am−j in the sense of Lemma 2.4, one says that a is polyhomogeneous symbol. One m . We have S m ⊂ S m . writes a ∈ Sτ,ph τ τ,ph For m ∈ N, the case of symbols a(x, ξ, τ ) ∈ Sτm with a polynomial in (ξ, τ ) provides a particular class of polyhomogeneous symbols. Observe that in this case, we have ∂τr a ∈ Sτm−r without any assumption on the support of a as in Lemma 2.5. 2.3. Classes of Pseudo-Differential Operators With the symbol classes we have just introduced, we can define pseudodifferential operators. Definition 2.7 (Pseudo-differential operators with a large parameter). If a ∈ Sτm (Rd × Rd ), we set ˆ(ξ) dξ, a(x, D, τ )u(x) = Op(a)u(x) := (2π)−d ∫ eix·ξ a(x, ξ, τ ) u Rd

for u ∈ S (Rd ). We denote by Ψm τ the set of these pseudo-differential oper, σ(A) will be its principal symbol in Sτm /Sτm−1 , that is ators. For A ∈ Ψm τ σ(A) = σ(a). If moreover a(x, ξ, τ ) is a polynomial function in (ξ, τ ) of order m, we say that Op(a) ∈ Dτm .

22

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

We shall keep the notation D m for standard differential operators of order m, that is operators of the form Op(a) with

aα (x)ξ α , x ∈ Rd , ξ ∈ Rd . a(x, ξ) = |α|≤m

(See Sect. 2.12 below.) Pseudo-differential operators with a large parameter like those we have just defined are also referred to as semi-classical operators. We shall avoid the use of this terminology and confine ourselves to calling them pseudodifferential operators with a large parameter. We shall also call standard operators, those that are associated with symbols that do not depend on a parameter such as τ (see Sect. 2.12). In the literature semi-classical operators often refer to operators associated with small parameters h > 0. These operators coincide with those we have just defined if proper scalings and changes of variables are performed. We refer to Sect. 2.11 for this identification. Remark 2.8. Formally, we can write Op(a)u(x) = (2π)−d ∫∫

ei(x−y)·ξ a(x, ξ, τ ) u(y) dy dξ.

Rd ×Rd

Such a double integral may not have a meaning in the classical sense, e.g. Lebesgue integration. Yet it has a very precise definition and meaning in the sense of the so-called oscillatory integrals. Details are given in Sect. 2.4. Remark 2.9. The pseudo-differential operators defined above apply to functions defined in the whole Rd , through the use of the Fourier transformation. Below in Sect. 2.10 we shall introduce tangential pseudo-differential operators that can act on functions defined on a half-space. Pseudo-differential operators can also be defined on an open set of Rd . We refer to [162] for a presentation of such operators. For the definition of pseudo-differential operators on a manifold, we refer to [162, 175]. Above, pseudo-differential operators are defined through their action on Schwartz functions. In fact we have the following continuity result. Proposition 2.10. Let a ∈ Sτm . We have Op(a) : S (Rd ) → S (Rd ) continuously. Note that the continuity property is by no mean uniform w.r.t. the large parameter τ . We refer to Appendix 2.A.2 for a proof. Below we shall extend the action of pseudo-differential operators to temperate distributions and to adapted Sobolev spaces. 2.4. Oscillatory Integrals Oscillatory integrals are useful for the definition the Schwartz kernel of pseudo-differential operators (and many other operators) and also for the understanding of the pseudo-differential calculus. For a review of the notion

2.4. OSCILLATORY INTEGRALS

23

of Schwartz kernel we refer to Sect. 8.3.4. We shall give a precise meaning to integrals of this form: (2π)−d ∫∫

u ∈ S (Rd ), x ∈ Rd ,

ei(x−y)·ξ a(x, ξ, τ )u(y) dydξ,

Rd ×Rd

for a ∈ Sτ+∞ (Rd × Rd ) and view the Schwartz kernel of Op(a), Aτ (x, y) = (2π)−d ∫ ei(x−y)·ξ a(x, ξ, τ )dξ, Rd

as a distribution. In fact these two integrals are perfectly well-defined if |a(x, ξ, τ )|  ξm ,  with m < −d where . = 1 + |.|2 . This holds in particular if a ∈ m d d Sτ (R × R ) and m < −d. Yet, for m ≥ −d, the meaning of the two integrals may not be clear according to classical integration theories. This type of integral is called oscillatory because of the phase term ei(x−y)·ξ . We shall in fact introduce more general phase functions, to be denoted by ϕ here, and give a precise meaning to the following type of integral. ∫ eiϕ(x,ξ) a(x, ξ, τ )dξ,

Rd

in the sense of distributions for a ∈ Sτm (Rp × Rd ), possibly with p = d. For a review of distribution theory we refer to Chap. 8. Theorem 2.11. Let p, d ∈ N and let ϕ : Rp × Rd → C be C ∞ and such that: (1) Im ϕ ≥ 0. (2) ϕ is homogeneous of degree 1 in ξ, for say |ξ| ≥ 1. (3) For all α, β, there exists Cα,β > 0 such that |ξ||β| |∂xα ∂ξβ ϕ(x, ξ)| ≤ Cα,β |ξ|,

x ∈ Rp , ξ ∈ Rd .

(4) There exists C > 0 such that |dx ϕ|2 + |ξ|2 |dξ ϕ|2 ≥ C|ξ|2 ,

x ∈ Rp , ξ ∈ Rd .

Then, the functional Iϕ (a, u, τ ) =

∫∫

Rp ×Rq

eiϕ(x,ξ) a(x, ξ, τ )u(x) dξ dx,

that is well-defined for u ∈ S (Rp ) and a ∈ Sτ−d−ε (Rp × Rq ), ε > 0, can be extended in a unique manner by continuity to all a ∈ Sτm (Rp × Rq ), for all m ∈ R. Moreover as a distribution in S  (Rp ), the map u → Iϕ (a, u, τ ) is of order ≤ k for all k > m + d. A proof is given in Sect. 2.A.3.1. Following this theorem, we keep the notation ∫ eiϕ(x,ξ) a(x, ξ, τ )dξ,

Rd

24

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

to denote the distribution in S  (Rp ), u → Iϕ (a, u, τ ), even if the integral is not defined according to classical integration theories. Observe now that ϕ : Rd × Rd × Rd → R (x, y, ξ) → (x − y) · ξ fulfills the assumption made on the phase function in Theorem 2.11. Hence for a ∈ Sτm (Rd × Rd ), the map w → (2π)−d

∫∫∫

ei(x−y)·ξ a(x, ξ, τ )w(x, y) dξ dy dx,

Rd ×Rd ×Rd

is a distribution in S (R2d ) in the x, y variables of order ≤ k with k > m + d. This allows one to write the Schwartz kernel of the operator Op(a) as Aτ (x, y) = (2π)−d ∫ ei(x−y)·ξ a(x, ξ, τ ) dξ, Rd

and with the kernel theorem (see Theorem 8.45), we have for u, v ∈ S (Rd ), Op(a)u(x), v(y)S  (Rd ),S (Rd ) = Aτ , u ⊗ vS  (R2d ),S (R2d ) = (2π)−d

∫∫∫

ei(x−y)·ξ a(x, ξ, τ ) v(x)u(y) dξ dy dx.

Rd ×Rd ×Rd

Note that the kernel theorem states that  d d Op(a)u(x) = Aτ (x, .), u(.)S  (Rd ),S (Rd ) ∈ S (R ) for u ∈ S (R ). Yet, in the present case we have Op(a)u(x) ∈ S (Rd ) by Proposition 2.10. Note also that Theorem 2.11 gives a precise meaning to the formula Op(a)u(x) = Aτ (x, .), u(.)S  (Rd ),S (Rd ) = (2π)−d ∫∫

ei(x−y)·ξ a(x, ξ, τ ) u(y) dξ dy,

Rd ×Rd

for any given value of x by considering the phase ϕx (y, ξ) = (x − y) · ξ with the variable x as a parameter. Remark 2.12. Let a ∈ Sτm (Rp × Rd ) and χ ∈ Cc∞ (R) be such that  χ(0) = 1. As χ(ελτ )a(x, ξ) converges to a in Sτm (Rp × Rd ), m > m, by Proposition 2.59, we conclude by Theorem 2.11 that we have Iϕ (a, u, τ ) =

∫∫

eiϕ(x,ξ) a(x, ξ, τ )u(x)dξ dx

Rp ×Rd

= lim

∫∫

ε→0 Rp ×Rd

eiϕ(x,ξ) χ(ελτ )a(x, ξ, τ )u(x)dξ dx,

for u ∈ S (Rp ), that is, ∫ eiϕ(x,ξ) a(x, ξ, τ )dξ = lim ∫ eiϕ(x,ξ) χ(ελτ )a(x, ξ, τ )dξ,

Rd

in the sense of distributions.

ε→0 Rd

2.4. OSCILLATORY INTEGRALS

25

We say that the oscillatory integral is regularized in this limiting process. This terminology is also used for the successive applications of the operator L=

∇x ϕ¯ · Dx + |ξ|2 ∇ξ ϕ¯ · Dξ , |dx ϕ|2 + |ξ|2 |dξ ϕ|2

in the proof of Theorem 2.11 given in Sect. 2.A.3.1. Remark 2.13. Regularization allows one to generalize to oscillatory integrals the usual calculus rules for absolutely convergent integrals: integration by parts, homogeneous change of variables, the Fubini theorem, limits and differentiations under the sum sign. The oscillatory integrals we have introduced up to now allow us to justify the formal form given in Remark 2.8 above for the action of pseudodifferential operators. We shall also introduce a second form of oscillatory integrals that will be useful for the derivation of symbol calculus of pseudodifferential operators. To that purpose we need to introduce the following symbol class. Definition 2.14. Let ρ ∈ (−∞, 1], p ∈ N, and m ∈ R. We say that p p a ∈ Am ρ (R ) if for all α ∈ N there exists Cα such that |∂θα a(θ)| ≤ Cα θm−ρ|α| ,

θ ∈ Rp ,

√ with θ = 1 + θ2 . p m p −∞ p m p We set A+∞ ρ (R ) = ∪m Aρ (R ) and Aρ (R ) = ∩m Aρ (R ). We introduce the following semi-norms on Am ρ : m (a) = sup θ−m+ρ|α| |∂θα a(θ)|. Nρ,k |α|≤k θ∈Rp

They provide a Fr´echet space topology. Remark 2.15. As in the case of Sτm (see Proposition 2.59), we can prove p m p m p that Cc∞ (Rp ) ⊂ A−∞ ρ (R ) is dense in Aρ (R ) for the topology of Aρ (R ),  if m > m. p m p  We note that Am ρ (R ) injects in a continuous manner in Aρ (R ) if ρ ≥ ρ . Theorem 2.16. Let ρ ∈ (−∞, 1] and p ∈ N. Let ϕ ∈ C ∞ (Rp \ 0; R) be homogeneous of degree μ with μ > 1 − ρ that satisfies dθ ϕ(θ) = 0, if θ = 0. Then, the integral Iϕ (a) = ∫ eiϕ(θ) a(θ) dθ, p that is well-defined for a ∈ A−∞ ρ (R ), can be uniquely extended in a continm p uous manner to all a ∈ Aρ (R ), m ∈ R:   Iϕ (a) ≤ CN m (a), ρ,k

for a certain k (k depends on μ, ρ, and m).

26

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

A proof is given in Sect. 2.A.3.2. Remark 2.17. Here also, for this second form of oscillatory integrals, one can prove that if a depends on a parameter in a smooth way, then so does Iϕ (a). In fact, as a → Iϕ (a) is linear and continuous, we see that differentiations commute with the integration sign. Moreover, the content of Remark 2.13 also applies to the second form of oscillatory integrals presented in Theorem 2.16. 2.5. Symbol Calculus We shall compose pseudo-differential operators or compute their adjoints in what follows. Such operations yield a calculus at the level of operator symbols that we provide here. The symbol of the adjoint operator can be obtained as follows. Theorem 2.18 (Formal Adjoint). Let a ∈ Sτm . Then Op(a)∗ = Op(a∗ ) for a certain a∗ ∈ Sτm that admits the following asymptotic expansion:

1 ∂xα ∂ξα a ¯(x, ξ, τ ). a∗ (x, ξ, τ ) ∼ |α| α i α! ¯. More precisely, we have The principal symbol of a∗ is then simply a

1 ∂α∂αa ¯(x, ξ, τ ) + RN (x, ξ, τ ), a∗ (x, ξ, τ ) = |α| α! x ξ i |α|≤N with RN (x, ξ, τ ) ∈ Sτm−N −1 given by RN (x, ξ, τ ) =

1

∫

|α|=N +1 0

×

∫∫

Rd ×Rd

(N + 1)(1 − σ)N (2π)d iN +1 α!

e−iy·η ∂xα ∂ηα a ¯(x − σy, ξ − η, τ ) dηdydσ.

The map a → a∗ from Sτm to itself is continuous. The map a → RN from Sτm to Sτm−N −1 is also continuous. We refer to Sect. 2.A.4 for a proof. The integral form of the remainder RN is to be understood in the sense of Theorem 2.16. It follows by duality using Proposition 2.10 and Theorem 2.18 that pseudo-differential operators can be uniquely extended to S  (Rd ). Proposition 2.19. Let a ∈ Sτm . We have Op(a) : S  (Rd ) → S  (Rd ) continuously. We mention here the case of the adjoint of differential operators. Remark 2.20. Let a(x, ξ) (resp. a(x, ξ, τ )) be polynomial in ξ (resp. (ξ, τ )) of degree m and then Op(a)∗ = Op(a∗ ) ∈ D m (resp. Dτm )and the polynomial function a∗ is given by

2.5. SYMBOL CALCULUS

1

|α|≤m

i|α| α!

a∗ (x, ξ) =

27

∂ξα ∂xα a ¯(x, ξ),



resp. a∗ (x, ξ, τ ) =

 1 α α ∂ ∂ a ¯ (x, ξ, τ ) . x ξ |α| |α|≤m i α!

Here, the formula is thus exact at the level of the symbol with a finite summation. With Proposition 2.19, we now know that pseudo-differential operators can act on S  (Rd ). In particular, for a given ξ ∈ Rd , they can act on eξ (x) = eix·ξ . In fact, this allows one to retrieve the symbol of the operator. Proposition 2.21. Let a ∈ Sτm . Then, Op(a)eξ (x) = a(x, ξ, τ )eξ (x). This identity is often written as a(x, D, τ )eix·ξ = a(x, ξ, τ )eix·ξ . We refer to Sect. 2.A.4 for a proof. The composition formula for pseudo-differential operators is given by the following theorem. 

Theorem 2.22 (Composition). Let a ∈ Sτm and b ∈ Sτm . Then Op(a) ◦  Op(b) = Op(c) for a certain c ∈ Sτm+m that admits the following asymptotic expansion: (2.5.1)

c(x, ξ, τ ) = (a ◦ b)(x, ξ, τ ) ∼

α

1 ∂ α a(x, ξ, τ ) ∂xα b(x, ξ, τ ). i|α| α! ξ

More precisely, we have c(x, ξ, τ ) =

1 ∂ξα a(x, ξ, τ )∂xα b(x, ξ, τ ) |α| i α!

|α|≤N

+ RN (x, ξ, τ ),



with RN (x, ξ, τ ) ∈ Sτm+m −N −1 given by RN (x, ξ, τ ) =

1

∫

|α|=N +1 0

×

∫∫

(N + 1)(1 − σ)N (2π)d iN +1 α!

Rd ×Rd

e−iy·η ∂ξα a(x, ξ − η, τ )∂xα b(x − σy, ξ, τ ) dηdydσ. 



The map (a, b) → c from Sτm × Sτm to Sτm+m is continuous. The map   (a, b) → RN from Sτm × Sτm to Sτm+m −N −1 is also continuous. We refer to Sect. 2.A.4 for a proof. The first term in the expansion (2.5.1) of a ◦ b, the principal symbol, is 1 ab(x, ξ, τ ); the second term is i j ∂ξj a(x, ξ, τ ) ∂xj b(x, ξ, τ ). The principal symbol of the commutator [Op(a), Op(b)] is then as follows.

28

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER 

Corollary 2.23 (Commutator). Let a ∈ Sτm and b ∈ Sτm . Then  [Op(a), Op(b)] = Op(a) ◦ Op(b) − Op(b) ◦ Op(a) = Op(c) with c ∈ Sτm+m −1 and principal symbol 1  σ(c) = {a, b} ∈ Sτm+m −1 , i where the Poisson bracket is given by 

 ∂ξj a∂xj b − ∂xj a∂ξj b (x, ξ, τ ). {a, b}(x, ξ, τ ) = 1≤j≤d

Remark 2.24. Let a(x, ξ), b(x, ξ) (resp. a(x, ξ, τ ), b(x, ξ, τ )) be polynomials in ξ (resp. (ξ, τ )) of degrees m and m and then Op(a)Op(b) = Op(c) ∈   D m+m (resp. Dτm+m ) and the polynomial function c is given by

1 ∂ξα a(x, ξ) ∂xα b(x, ξ), c(x, ξ) = |α| |α|≤m i α!  

1 resp. c(x, ξ, τ ) = ∂ξα a(x, ξ, τ ) ∂xα b(x, ξ, τ ) . |α| |α|≤m i α! As for the adjoint formula, the composition formula is exact at the level of the symbol with a finite summation in this case. 2.6. Sobolev Spaces and Operator Bound We now introduce Sobolev spaces and Sobolev norms that are adapted to the scaling parameter τ . The natural norm on L2 (Rd ) is written as u2L2 (Rd ) := ( ∫ |u(x)|2 dx)1/2 , Rd

v (x)dx. and is associated with the inner product (u, v)L2 (Rd ) := ∫Rd u(x)¯ Let s ∈ R; we then set (2.6.1)

uτ,s := Λsτ uL2 (Rd ) ,

with Λsτ := Op(λsτ ) ∈ Ψsτ ,

and Hτs (Rd ) := {u ∈ S  (Rd ); uτ,s < ∞}, 1/2  where, as introduced above, λτ = |(ξ, τ )| = |ξ|2 +τ 2 . The space Hτs (Rd ) is algebraically equal to the standard Sobolev space H s (Rd ). For a fixed value of τ , the norm .τ,s is equivalent to the standard Sobolev norm that we denote by .s . However, these two norms are not uniformly equivalent as τ goes to +∞. In fact we only have uτ,s  us ,

if s ≤ 0,

For s ∈ N, we have uτ,s 

and

|α|+k≤s

us  uτ,s ,

τ k Dα uL2 (Rd ) .

if s ≥ 0.

2.6. SOBOLEV SPACES AND OPERATOR BOUND

29

If Ω is an open set of Rd , this allows one to also define the space Hτs (Ω) for s ∈ N. This space coincides algebraically with the Sobolev space H s (Ω) but is equipped with the norm

τ k Dα uL2 (Ω) , (2.6.2) uHτs (Ω) := |α|+k≤s

and associated inner product. Note that from the density of S (Rd ) in L2 (Rd ), we deduce the counterpart density for Hτs (Rd ). Proposition 2.25. For any s ∈ R, the space of Schwartz functions S (Rd ) is dense in the Sobolev space Hτs (Rd ). Proof. Let f ∈ Hτs (Rd ). Then g = Λsτ f ∈ L2 (Rd ). If we pick a sequence, (gn )n ⊂ S (Rd ) that converges to g in L2 (Rd ). Setting fn = s −s Λ−s τ gn , we have, as Λτ ◦ Λτ = Id, fn − f τ,s = Λsτ (fn − f )L2 (Rd ) = gn − gL2 (Rd ) → 0,    The spaces Hτs (Rd ) and Hτ−s (Rd ) are in duality, i.e. Hτ−s (Rd ) = Hτs (Rd ) in the sense of distributional duality with L2 (Rd ) = Hτ0 (Rd ) as a pivot space. In particular we have which concludes the proof.

f, g¯H s (Rd ),Hτ−s (Rd ) = (f, g)L2 (Rd ) ,

for f, g ∈ S (Rd )

τ

and |(f, g)L2 (Rd ) | ≤ f τ,s gτ,−s ,

f, g ∈ S (Rd ).

We have the following continuity result. Theorem 2.26. If a(x, ξ, τ ) ∈ Sτm and s ∈ R, we then have Op(a) : → Hτs−m (Rd ) continuously, uniformly in τ ∈ [1, +∞).

Hτs (Rd )

We refer to Sect. 2.A.5 for a proof. Note that for a ∈ Sτm , m ∈ R, we have   (Op(a)u, u)L2 (Rd )  ≤ Cu2 (2.6.3)

u ∈ S (Rd ),  −m/2 m/2  = Λτ Op(a)u, Λτ u L2 (Rd ) , where τ,m/2 ,

as we may write (Op(a)u, u)L2 (Rd ) −m/2

Λτ

m/2

Op(a) ∈ Ψτ

by Theorem 2.22.

Remark 2.27. We also observe that we have 

uτ,m ≤ Cτ m−m uτ,m ,

m ≥ m,

u ∈ S (Rd ),

for some constant C depending on m and m . Hence, we can have uτ,m  uτ,m , by choosing τ sufficiently large. This property will be exploited on numerous occasions in what follows.

30

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

2.7. Positivity Inequalities of G˚ arding Type The following G˚ arding inequalities are important results that we shall use on many occasions in what follows. Proofs are given in Sect. 2.A.6. Theorem 2.28 (Local G˚ arding Inequality). Let V be an open set of Rd . m d d Let a(x, ξ, τ ) ∈ Sτ (R × R ), with principal part am . If there exist C0 > 0 and R > 0 such that (2.7.1) x ∈ V, ξ ∈ Rd , τ ∈ [1, +∞), |(ξ, τ )| ≥ R, Re am (x, ξ, τ ) ≥ C0 λm τ , then, for any 0 < C1 < C0 there exists τ∗ ≥ 1 such that Re(Op(a)u, u)L2 (Rd ) ≥ C1 u2τ,m/2 ,

u ∈ S (Rd ), supp(u) ⊂ V , τ ≥ τ∗ .

The positivity and the ellipticity of the principal symbol of a thus imply positivity for the operator Op(a). The value of τ∗ depends on C0 , C1 and a finite number of constants Cα,β associated with the symbol a(x, ξ, τ ) (see Definition 2.1). We shall often use this result in the case V is bounded, that is V is compact. Yet, the cases where V is unbounded, e.g., V = Rd , are contained in the statement. We also shall need a microlocal version of the G˚ arding inequality, meaning that the ellipticity of the symbol a(x, ξ, τ ) only holds in a conic2 region in phase space. Theorem 2.29 (Microlocal G˚ arding Inequality). Let K be a compact d set of R and let W be a conic open set of Rd × Rd × R+ contained in K × Rd × R+ . Let also χ ∈ Sτ0 be homogeneous of degree 0 (for λτ ≥ 1) and be such that supp(χ) ⊂ W . Let a(x, ξ, τ ) ∈ Sτm , with principal part am homogeneous of degree m. If there exist C0 > 0 and R > 0 such that (2.7.2) (x, ξ, τ ) ∈ W, τ ∈ [1, +∞), |(ξ, τ )| ≥ R, Re am (x, ξ, τ ) ≥ C0 λm τ , then for any 0 < C1 < C0 , N ∈ N, there exist CN and τ∗ ≥ 1 such that Re(Op(a)Op(χ)u, Op(χ)u)L2 (Rd ) ≥ C1 Op(χ)u2τ,m/2 − CN u2τ,−N , for u ∈ S (Rd ) and τ ≥ τ∗ . Corollary 2.30. Let K be a compact set of Rd and let W be a conic open set of Rd × Rd × R+ contained in K × Rd × R+ . Let also χ ∈ Sτ0 be homogeneous of degree 0 (for λτ ≥ 1) and be such that supp(χ) ⊂ W . Let a(x, ξ, τ ) ∈ Sτm , with principal part am homogeneous of degree m. If there exist C0 > 0 and R > 0 such that |am (x, ξ, τ )| ≤ C0 λm τ ,

(x, ξ, τ ) ∈ W, τ ∈ [1, +∞),

2The sense we give to “conic” is made precise in Sect. 1.7.

|(ξ, τ )| ≥ R,

2.7. POSITIVITY INEQUALITIES OF G˚ ARDING TYPE

31

then, for any C1 > C0 , m ∈ R, and N ∈ N, there exist CN and τ∗ ≥ 1 such that Op(a)Op(χ)uτ,m ≤ C1 Op(χ)uτ,m+m + CN uτ,−N , for u ∈ S (Rd ) and τ ≥ τ∗ . 2(m+m )

Proof. The operator Q = C12 Λτ



− Op(a)∗ Λ2m τ Op(a) has





2 q0 = C12 λτ2(m+m ) − λ2m τ |am |

for principal symbol. For (x, ξ, τ ) ∈ W and |(ξ, τ )| ≥ R, we have 

q0 (x, ξ, τ ) ≥ (C12 − C02 )λτ2(m+m ) . The microlocal G˚ arding inequality of Theorem 2.29 gives C12 Op(χ)u2m+m − Op(a)Op(χ)u2m = (QOp(χ)u, Op(χ)u)L2 (Rd ) ≥ −CN u2τ,−N , for τ chosen sufficiently large, which gives the result.



The G˚ arding inequalities stated above are written for a scalar operator. They have the following system counterparts. For a = (aij )1≤i,j≤n with m aij ∈ Sτ ij , and U = t (U1 , . . . , Un ) ∈ (S (Rd ))n , we set

Op(a)U (x) = V (x) ∈ Rn , with Vi (x) = Op(aij )Ui (x), 1≤j≤n

and U 2τ,k =

1≤j≤n

Uj 2τ,k .

Theorem 2.31 (G˚ arding Inequality for Systems). Let V be an open set of Rd . Let aij (x, ξ, τ ) ∈ Sτm , 1 ≤ i, j ≤ n, with principal part aij m . We set ij t am = (am )1≤i,j≤n and Re am = (am + a ¯m )/2. If there exist C0 > 0 and R > 0 such that, for z ∈ Cn , (2.7.3)



Re am (x, ξ, τ )z, z

 Cn

2 d ≥ C 0 λm τ zCn , x ∈ V, ξ ∈ R , τ ∈ [1, +∞), |(ξ, τ )| ≥ R,

then for any 0 < C1 < C0 , there exists τ∗ ≥ 1 such that Re(Op(a)U, U )(L2 (Rd ))n ≥ C1 U 2τ,m/2 , for U ∈ (S (Rd ))n , supp(U ) ⊂ V , and τ ≥ τ∗ . Theorem 2.32 (Microlocal G˚ arding Inequality for Systems). Let K be a compact set of Rd and let W be a conic open set of Rd × Rd × R+ contained in K × Rd × R+ . Let also χ ∈ Sτ0 be homogeneous of degree 0 (for λτ ≥ 1) and be such that supp(χ) ⊂ W . Let aij (x, ξ, τ ) ∈ Sτm , 1 ≤ i, j ≤ n, with homogeneous principal part aij m. ij ∗ We set am = (am )1≤i,j≤n and Re am = (am + am )/2. If there exist C0 > 0 and R > 0 such that, for z ∈ Cn ,

32



2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

Re am (x, ξ, τ )z, z

 Cn

2 ≥ C 0 λm τ zCn , (x, ξ, τ ) ∈ W, τ ∈ [1, +∞), |(ξ, τ )| ≥ R,

then for any 0 < C1 < C0 , N ∈ N, there exist CN and τ∗ ≥ 1 such that Re(Op(a)Op(χ)U, Op(χ)U )(L2 (Rd ))n ≥ C1 Op(χ)U 2τ,m/2 − CN U 2τ,−N , for U ∈ (S (Rd ))n and τ ≥ τ∗ . 2.8. Parametrices Elliptic operators can be inverted up to some regularizing operator. Proposition 2.33. Let m ∈ R and let p ∈ Sτm be elliptic, that is, for some C > 0 and R > 0, |p(x, ξ, τ )| ≥ Cλm τ ,

x ∈ Rn ,

ξ ∈ Rn , τ ∈ [τ0 , +∞),

λτ ≥ R.

 ∈ S −N such that For any N ∈ N∗ there exist qN ∈ Sτ−m and rN , rN τ

qN ◦ p = 1 + rN ,

 p ◦ qN = 1 + rN .

Moreover, qN is unique in Sτ−m /Sτ−m−N . There also exist q ∈ Sτ−m and  ∈ S −∞ such that r ∞ , r∞ τ q ◦ p = 1 + r∞ ,

 p ◦ q = 1 + r∞ ,

with q unique in Sτ−m /Sτ−∞ . This proposition is a particular case of the following (micro-)local version. Proposition 2.34. Let m ∈ R and let W be a conic open set of Rd × × R+ . Let then p ∈ Sτm be elliptic in W , that is, for some C > 0 and R > 0,

Rd

|p(x, ξ, τ )| ≥ Cλm τ , Let then χ ∈ Sτ0  Sτ−m and rN , rN

(x, ξ, τ ) ∈ W,

λτ ≥ R.

 ∈ have support in W . For any N ∈ N∗ there exist qN , qN ∈ Sτ−N such that

qN ◦ p = χ + rN ,

  p ◦ qN = χ + rN .

 − q ◦ For any pair of such left and right parametrices we have χ ◦ qN N −m−N  0 . Moreover, we can choose q˜N , q˜N ∈ Sτ such that supp(˜ qN ) ∪ χ ∈ Sτ  ) ⊂ supp(χ) both with χ as principal symbol and such that supp(˜ qN       (x, ξ, τ ), (x, ξ, τ ) = p−1 q˜N qN (x, ξ, τ ) = p−1 q˜N (x, ξ, τ ) and qN

for λτ ≥ R.  ∈ S −∞ such that There also exist q, q  ∈ Sτ−m and r∞ , r∞ τ q ◦ p = χ + r∞ ,

 p ◦ q  = χ + r∞ .

For any pair of such left and right parametrices we have χ ◦ q  − q ◦ χ ∈ Sτ−∞ . q ) ∪ supp(˜ q  ) ⊂ supp(χ) Moreover, we can choose q˜, q˜ ∈ Sτ0 such that supp(˜ both with χ as principal symbol and such that     q(x, ξ, τ ) = p−1 q˜ (x, ξ, τ ) and q  (x, ξ, τ ) = p−1 q˜ (x, ξ, τ ),

2.8. PARAMETRICES

33

for λτ ≥ R. We refer to Appendix 2.A.7 for a proof. Construction of (micro-)local parametrices is of importance as in many cases ellipticity is only known to hold locally or microlocally. Typically the symbol χ will be chosen as a cut-off symbol that is equal to 1 in a conic open subset of W . Parametrices can be handy to prove useful estimates like the following one. Proposition 2.35. Let m ∈ N and P ∈ Dτm be elliptic in Ω and let O be an open subset of Ω such that O  Ω. Then, there exists C > 0 such that   u ∈ C ∞ (Ω). uHτm (O) ≤ C P uL2 (Ω) + uL2 (Ω) , Remark 2.36. One could expect such an elliptic estimate to include boundary conditions on ∂O on the r.h.s. of the inequality. This is not the case here as such boundary “observations” of the function u are simply replaced by an L2 -observation on Ω that is larger than O. Estimations with boundary terms require lots of care. A treatment is done in Chap. 3 of Volume 2, in the case P is of second order. Proof. Let χ, χ1 , χ2 ∈ Cc∞ (Ω) be such that χ = 1 in a neighborhood of O and χ1 = 1 in a neighborhood of supp(χ) and χ2 = 1 in a neighborhood supp(χ1 ) . We set f = P u ∈ L2 (Ω). Then, as P is differential, we may write χ1 P χ2 u = χ1 f. Dτm (Rd )

is elliptic in a neighborhood of supp(χ), there exists Q ∈ As χ1 P ∈ −m d d Ψτ (R ) such that Qχ1 P = χ + R with R ∈ Ψ−∞ τ (R ) by Proposition 2.34. We then have χu = χχ2 u = Qχ1 f − Rχ2 u. With the Sobolev continuity result of Theorem 2.26, we obtain χuHτm (Rd )  χ1 f L2 (Rd ) + χ2 uL2 (Rd )  f L2 (Ω) + χ2 uL2 (Ω) . By the definition of the Hτm (O)-norm in (2.6.2) and because χ = 1 in a neighborhood of O, we have uHτm (O) ≤ χuHτm (Rd ) , which yields the result.  A microlocal version can be obtained by means of Proposition 2.34. We write it here for a pseudo-differential operator. Proposition 2.37. Let a ∈ Sτm and let W be a conic open set of Rd × × R+ such that a is elliptic in W . Let then χ ∈ Sτ0 have support in W . There exists C1 > 0 such that, for any N ∈ N, Rd

Op(χ)uτ,s+m ≤ C1 Op(a)uτ,s + CN uτ,−N , for u ∈ S (Rd ), for some CN > 0, and for τ > 0 chosen sufficiently large.

34

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

Proof. By Proposition 2.34, for q1 = χa−1 ∈ Sτ−m , one has q1 ◦ a = χ + r1 with r1 ∈ Sτ−1 . Then Op(χ)u = Op(q1 )Op(a)u − Op(r1 )u. As q1 ∈ Sτ−m , there exists C0 > 0 such that Op(q1 )vm+s ≤ C0 vs for any v ∈ S (Rd ). Consider now C1 > C0 and N ∈ N. By Proposition 2.34 there exists qN ∈ Sτ−m such that qn ◦ a = χ + rN with rn ∈ Sτ−m−s−N . In fact, that proposition states that we can choose qN with q1 for principal symbol. Then,  >0 qN = q1 + qˆ with qˆ ∈ Sτ−m−1 and for some CN  Op(qN )vm+s ≤ C0 vs + CN vs−1 ≤ C1 vs ,

if τ is chosen sufficiently large by Remark 2.27. Setting v = Op(a)u, one has Op(qN )v = Op(χ)u + rN u yielding Op(χ)um+s ≤ Op(qN )vm+s + Op(rN )um+s ≤ C1 Op(a)us + CN u−N ,



for some CN > 0.

Remark 2.38. If χ is homogeneous of degree 0 in (τ, ξ), one can also write the microlocal elliptic estimate in Proposition 2.37 in the form Op(χ)uτ,s+m ≤ C1 Op(a)Op(χ)uτ,s + CN uτ,−N . ˜ ⊂W In fact, pick χ ˜ ∈ Sτ0 , also homogeneous of degree 0, such that supp(χ) and χ ˜ ≡ 1 on a neighborhood of supp(χ). Then, with Proposition 2.37, one has  Op(χ)Op(χ)u ˜ τ,s+m ≤ C1 Op(a)Op(χ)uτ,s + CN Op(χ)uτ,−N ,  > 0. Then, as r = (1 − χ) ˜ ◦ χ ∈ Sτ−∞ by for some C1 > 0 and some CN pseudo-differential calculus and the support properties of χ and χ, ˜ if one write Op(χ)Op(χ) ˜ = Op(χ) − Op(r), one finds

˜ Op(χ)uτ,s+m ≤ Op(χ)Op(χ)u τ,s+m + Op(r)uτ,s+m ≤ C1 Op(a)Op(χ)uτ,s + CN Op(χ)uτ,−N + Op(r)uτ,s+m  ≤ C1 Op(a)Op(χ)uτ,s + CN uτ,−N ,  > 0. for some CN

2.9. Action of Change of Variables Let X and Y be two open subsets of Rd and κ : X → Y be a smooth diffeomorphism. Let P ∈ Dτm be defined on X and let p(x, ξ, τ ) be its principal symbol. The operator Q on Y , given by   Qu ◦ κ = P (u ◦ κ), u ∈ C ∞ (Y ), is in Dτm and its principal symbol q is given by q(κ(x), ξ, τ ) = p(x, t κ (x)(ξ), τ ), where κ (x) is the differential of κ at x.

2.10. TANGENTIAL OPERATORS

35

A review of the action of change of variables is provided in Sect. 9.1.2. One then sees that q(x, ξ, τ ) follows the natural rules for functions defined on the cotangent bundle T ∗ X under change of variables. See Sect. 9.1.2. This property extends more generally to the principal symbols of pseudodifferential operators. For this more general case we refer to Theorem 18.1.17 in [175]. For operators with a large parameter we refer to [210] where it is done in details. 2.10. Tangential Operators We shall often be interested in boundary problems. Locally, we shall use coordinates so that the geometry is that of the half-space Rd+ = {x ∈ Rd , xd > 0},

x = (x , xd ) with x ∈ Rd−1 , xd ∈ R.

Pseudo-differential operators as defined above are based on the Fourier transformation in all space variables. If one considers a half-space as Rd+ , functions need to be extended to the whole Rd for the Fourier transformation to be used. The definition of the operator then depends on this extension process, and regularity issues occur across xd = 0. Here, we thus introduce pseudo-differential operators that act in the directions that are tangential to the boundary, x in the local setting. Note however that differential operators a(x, D) ∈ D m (resp. a(x, D, τ ) ∈ m Dτ ) are naturally defined in an open set of Rd as they do not rely on the Fourier transformation. We denote the Fourier variable associated with x by ξ  ∈ Rd−1 , and with X = Rd or X = Rd+ , we introduce the following symbol classes of tangential symbols. Definition 2.39 (Tangential Symbols with a Large Parameter). Let a(x, ξ  , τ ) ∈ C ∞ (X × Rd−1 ), with τ as a parameter in [1, +∞) and m ∈ R, be such that for all multi-indices α ∈ Nd , β ∈ Nd−1 , there exists Cα,β > 0 such that (2.10.1) |∂xα ∂ξβ a(x, ξ  , τ )| ≤ Cα,β λT,τ , x ∈ X, ξ  ∈ Rd−1 , τ ∈ [1, +∞),  1/2 where λT,τ = |(ξ  , τ )| = |ξ  |2 + τ 2 . We write a ∈ STm,τ (X × Rd−1 ) or simply STm,τ . Note that λsT,τ ∈ STs ,τ , for all s ∈ R. We also define ST−∞ ,τ = ∩r∈R STr ,τ . m−|β|

For a ∈ STm,τ we call principal symbol, σ(a), the equivalence class of a in STm,τ /STm−1 ,τ . Compare the definition of tangential symbols to that of symbols in the whole phase space given in Definition 2.1: here symbols do not depend on the Fourier variable associated with the normal direction xd . Still, regularity with respect to xd is required.

36

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

Remark 2.40. Observe that if a ∈ STm,τ (X × Rd−1 ), X = Rd , or Rd+ , then a|xd =0 ∈ Sτm (Rd−1 × Rd−1 ). The counterpart of Lemma 2.4 for symbol asymptotic series is given by the following result. with j ∈ N. Then there exists Lemma 2.41. Let m ∈ R and aj ∈ STm−j ,τ a ∈ STm,τ such that ∀N ∈ N, We then write a ∼

a−

N

j=0

j

−1 aj ∈ STm−N . ,τ

aj . The symbol a is unique up to ST−∞ ,τ .

It is then natural to identify a0 with the principal symbol of a. With the symbol classes defined above, we can define tangential pseudodifferential operators. Definition 2.42 (Tangential Pseudo-Differential Operators with a Large Parameter). If a(x, ξ  , τ ) ∈ STm,τ (X × Rd ), we set 



ˆ(ξ  , xd ) dξ  a(x, D , τ )u(x) = OpT (a)u(x) := (2π)1−d ∫ eix ·ξ a(x, ξ  , τ ) u = (2π)

1−d

∫∫

Rd−1 ×Rd−1

e

Rd−1 i(x −y  )·ξ 

a(x, ξ  , τ ) u(y  , xd ) dy  dξ  ,

for x ∈ X and u ∈ S (Rd ) if X = Rd or u ∈ S (Rd+ ) if X = Rd+ . We denote by Ψm T,τ the set of these pseudo-differential operators. For m A ∈ ΨT,τ , σ(A) = σ(a) will be its principal symbol in STm,τ /STm−1 ,τ . If moreover a(x, ξ  , τ ) is a polynomial function in (ξ  , τ ) of order m, we say that OpT (a) ∈ DTm,τ . m Note that we have DTm ⊂ D m , DTm,τ ⊂ Dτm , and yet Ψm T,τ ⊂ Ψτ . In fact m m  observe that ST,τ ∈ Sτ as a differentiation with respect to ξ yields a gain of λT,τ in (2.10.1), while a gain in λτ is needed in the definition of Sτm in (2.2.1). The symbol of the adjoint operator can be obtained as follows.

Theorem 2.43 (Formal Adjoint). Let a ∈ STm,τ . Then OpT (a)∗ = OpT (a∗ ) for a certain a∗ ∈ STm,τ that admits the following asymptotic expansion:

1 ¯(x, ξ  , τ ). ∂ α ∂ α a a∗ (x, ξ  , τ ) ∼ |α| α! x ξ i α The principal symbol of a∗ is then simply a ¯. More precisely, we have

1 ∂xα ∂ξα a ¯(x, ξ  , τ ) + RN (x, ξ  , τ ), a∗ (x, ξ  , τ ) = |α| |α|≤N i α!

2.10. TANGENTIAL OPERATORS

37

−1 with RN (x, ξ, τ ) ∈ STm−N given by ,τ

RN (x, ξ  , τ ) =

1

∫

|α|=N +1 0

×

(N + 1)(1 − σ)N (2π)d−1 iN +1 α! 

∫∫

Rd−1 ×Rd−1



e−iy ·η ∂xα ∂ηα a ¯(x − σy  , xd , ξ  − η  , τ ) dη  dy  dσ.

The proof can be adapted from that of Theorem 2.18 given in Sect. 2.A.4. For the composition of tangential operators we have the following result. 

Theorem 2.44 (Composition). Let a ∈ STm,τ and b ∈ STm,τ . Then OpT (a)◦  that admits the following asympOpT (b) = OpT (c) for a certain c ∈ STm+m ,τ totic expansion: c(x, ξ  , τ ) = (a ◦ b)(x, ξ  , τ ) ∼

1

α

i|α| α!

∂ξα a(x, ξ  , τ ) ∂xα b(x, ξ  , τ ).

More precisely, we have c(x, ξ  , τ ) =

|α|≤N

1 ∂ α a(x, ξ  , τ )∂xα b(x, ξ  , τ ) |α| i α! ξ

+ RN (x, ξ  , τ ),



−N −1 given by with RN (x, ξ  , τ ) ∈ STm+m ,τ

RN (x, ξ  , τ ) = ×

∫∫

1

∫

|α|=N +1 0

Rd−1 ×Rd−1



(N + 1)(1 − σ)N (2π)d−1 iN +1 α!



e−iy ·η ∂ξα a(x, ξ  − η  , τ )∂xα b(x − σy  , xd , ξ  , τ ) dη  dy  dσ.

The first term in the expansion, the principal symbol, is ab(x, ξ  , τ ); the

  second term is 1i d−1 j=1 ∂ξj a(x, ξ , τ ) ∂xj b(x, ξ , τ ). The principal symbol of the commutator [Op(a), Op(b)] is then as follows. 

Corollary 2.45 (Commutator). Let a ∈ STm,τ and b ∈ STm,τ . Then [OpT (a), OpT (b)] = OpT (a) ◦ OpT (b) − OpT (b) ◦ OpT (a) = OpT (c) with c ∈  −1 STm+m and principal symbol ,τ  −1 1 , σ(c) = {a, b} ∈ STm+m ,τ i

where the Poisson bracket only concerns the tangential variables here and is given by {a, b}(x, ξ  , τ ) =

d−1

 j=1

 ∂ξj a∂xj b − ∂xj a∂ξj b (x, ξ  , τ ).

38

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

Remark 2.46. (1) Composition between Op(a) and OpT (b) with a ∈ Sτm and b ∈  STm,τ does not generally yield a pseudo-differential operator unless particular assumptions are made. Typically, if a is supported in  and a region where |(ξ  , τ )|  |ξd |, then Op(a) ◦ OpT (b) ∈ Ψm+m τ  m+m [175, Theorem 18.1.35]. Op(b) ◦ OpT (a) ∈ Ψτ  (2) In the case Op(a) ∈ Dτm and b ∈ STm,τ the composition of the two operators can be easily understood. More generally, if we consider a of the form m

(2.10.2) aj (x, ξ  , τ )ξdj , aj ∈ STm−j a(x, ξ, τ ) = ,τ , j=0

and if we define (2.10.3)

Op(a) :=

m

j=0

OpT (aj )Ddj

we have OpT (b) ◦ Op(a) = Op(c) with c(x, ξ, τ ) =

m

j=0

cj (x, ξ  , τ )ξdj ,



−j cj = b ◦ aj ∈ STm+m . ,τ 

Since [Dd , OpT (b)] = Op(Dd b) ∈ Ψm T,τ , we see by induction that Op(a) ◦ OpT (b) = Op(c ) with c (x, ξ, τ ) =

m

j=0

m+m −1−j

with cj = cj mod ST,τ lar

[Op(a), OpT (b)] =

cj (x, ξ  , τ )ξdj , 

−1−j = aj b mod STm+m . In particu,τ m

j=0

OpT (dj )Ddj ,



−1−j with dj ∈ STm+m . ,τ If now b is also of the form given in (2.10.2), that is, b =

m j m−j  j=0 bj (x, ξ , τ )ξd , with bj ∈ ST,τ , we find we have Op(b)◦Op(a) = Op(c), where

c(x, ξ, τ ) =

m+m

 j=0

cj (x, ξ  , τ )ξdj ,



−j cj = STm+m , ,τ

and [Op(b), Op(a)] = Op(d), where d(x, ξ, τ ) =

m+m

 j=0

dj (x, ξ  , τ )ξdj ,



−j−1 dj = STm+m . ,τ

Further developments on the symbols and operators described in (2.10.2)–(2.10.3) are provided in Chap. 6 of Volume 2.

2.10. TANGENTIAL OPERATORS

39

For u and v defined in Rd+ = {xd > 0}, we introduce the notation (u, v)+ = ∫ u(x)v(x)dx, Rd+

u+ = uL2 (Rd ) . +

For s ∈ R we set ΛsT,τ = OpT (λsT,τ ) ∈ ΨsT,τ , which can be used as a regularity scale, as in the following theorem. Theorem 2.47. If a(x, ξ  , τ ) ∈ STm,τ and s ∈ R, then for some C > 0 we have ΛsT,τ OpT (a)u+ ≤ CΛm+s T,τ u+ , for u ∈ S (Rd+ ). Note that for a ∈ STm,τ , m ∈ R, we have   2 (OpT (a)u, u)+  ≤ CΛm/2 u , T,τ +

u ∈ S (Rd+ ).

Observe also that for s ∈ R, we have ΛsT,τ u2+ = u2L2 (R+ ;Hτs (Rd−1 )) = ∫ u(., xd )2τ,s dxd . R+

Similarly to Remark 2.27, we shall often use the following observation. Remark 2.48. We have 



m ≥ m,

m−m Λm Λm T,τ u+ ≤ Cτ T,τ u+ ,

u ∈ S (Rd+ ),

for some constant C depending on m and m . Hence, we can have 

m Λm T,τ u+  ΛT,τ u+ ,

by choosing τ sufficiently large. We shall often need tangential versions of the G˚ arding inequalities are given by the following two theorems. Theorem 2.49 (Tangential G˚ arding Inequality). Let V be an open set  m d of R+ . Let a(x, ξ , τ ) ∈ ST,τ , with principal part am . If there exist C > 0 and R > 0 such that Re am (x, ξ  , τ ) ≥ Cλm T,τ ,

x ∈ V, ξ ∈ Rd , τ ∈ [1, +∞),

|(ξ  , τ )| ≥ R,

then for any 0 < C  < C there exists τ∗ ≥ 1 such that 2

Re(OpT (a)u, u)+ ≥ C  ΛT,τ u+ , m/2

u ∈ S (Rd+ ), supp(u) ⊂ V , τ ≥ τ∗ .

Theorem 2.50 (Microlocal Tangential G˚ arding Inequality). Let K be a 3 d compact set of R+ and let U be a conic open set of Rd+ ×Rd−1 ×R+ contained 3Here, by conic we mean conic in the (ξ  , τ ) variables, that is (x, ξ  , τ ) ∈ U implies

(x, tξ  , tτ ) ∈ U for all t > 0.

40

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

in K × Rd−1 × R+ . Let also χ ∈ ST0,τ be homogeneous (for |ξ  | + τ ≥ 1) and be such that supp(χ) ⊂ U . Let a(x, ξ  , τ ) ∈ STm,τ , with principal part am homogeneous of degree m. If there exist C > 0 and R > 0 such that Re am (x, ξ  , τ ) ≥ Cλm T,τ ,

(x, ξ  , τ ) ∈ U , τ ∈ [1, +∞),

|(ξ  , τ )| ≥ R,

then for any 0 < C  < C, N ∈ N, there exist CN and τ∗ ≥ 1 such that 2

2

Re(OpT (a)OpT (χ)u, OpT (χ)u)+ ≥ C  ΛT,τ OpT (χ)u+ − CN Λ−N T,τ u+ , m/2

for u ∈ S (Rd+ ) and τ ≥ τ∗ . The proofs of these two theorems can be adapted in a straightforward manner from those of Theorems 2.28 and 2.29, viewing xd as a parameter, and replacing symbol and operator classes by their tangential counterparts. The same comment applies to the following corollary that is the counterpart of Corollary 2.30. Corollary 2.51. Let K be a compact set of Rd+ , 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 (for |ξ  | + τ ≥ 1) and be such that supp(χ) ⊂ U . Let a(x, ξ  , τ ) ∈ STm,τ , with principal part am homogeneous of degree m. If there exist C0 > 0 and R > 0 such that |am (x, ξ  , τ )| ≤ C0 λm T,τ ,

(x, ξ  , τ ) ∈ U , τ ∈ [1, +∞),

|(ξ  , τ )| ≥ R,

then, for any C1 > C0 , m ∈ R, and N ∈ N, there exist CN and τ∗ ≥ 1 such that 



m+m Op(χ)u+ + CN Λ−N Λm T,τ Op(a)Op(χ)u+ ≤ C1 ΛT,τ T,τ u+ ,

for u ∈ S (Rd+ ) and τ ≥ τ∗ . The following proposition will be useful. Proposition 2.52. Let r, s ∈ R and m ∈ N. If a ∈ Sτr (Rd × Rd ). Then, there exists C > 0 such that m Λm T,τ Op(a)uτ,s ≤ CΛT,τ uτ,r+s ,

u ∈ S (Rd ).

Proof. Let α ∈ Nd−1 . Note that Dxα being a differential operator, |α| |α| and it may be considered either as in ΨT,τ (Rd ) or Ψτ (Rd ). With the latter understanding, commutations with Op(a) make sense within pseudodifferential calculus. By induction, we observe that

Aα,β Dxβ , Dxα Op(a) = |β|≤|α|

where Aα,β ∈ Ψrτ (Rd ). With the Sobolev regularity theorem (Theorem 2.26) we have, for |α| ≤ m and τ ≥ 1,

m−|β| β τ Dx uτ,s+r , τ m−|α| Dxα Op(a)uτ,s  |β|≤|α|

2.11. SEMI-CLASSICAL OPERATORS

leading to

|α|≤m

τ m−|α| Dxα Op(a)uτ,s 

|α|≤m

41

τ m−|α| Dxα uτ,s+r .

For any  ∈ R, by means of the Fourier transformation, we note that

m−|α| α τ Dx vτ, , v ∈ S (Rd ). Λm T,τ vτ,  |α|≤m



This yields the result. 2.11. Semi-Classical Operators

Here we make the connection with a form of semi-classical operators that is more usual in the literature. Consider the case a(x, ξ, τ ) ∼ j∈N aj (x, ξ, τ ) where aj ∈ Sτm−j (Rd × Rd ), and moreover, aj is homogeneous of degree m − j in (ξ, τ ) for τ ≥ 1. Then we have, in the sense of oscillatory integrals, Op(a)u(x) = (2π)−d ∫∫

ei(x−y)·ξ a(x, ξ, τ ) u(y) dy dξ

Rd ×Rd

= (2π)−d τ d ∫∫

eiτ (x−y)·ξ a(x, τ ξ, τ ) u(y) dy dξ

Rd ×Rd

after the change of variables ξ → ξ/τ . Observe that aj (x, τ ξ, τ ) = τ m−j aj (x, ξ, 1). We thus have

−j a(x, τ ξ, τ ) ∼ τ m τ aj (x, ξ, 1). j∈N

Setting h = τ −1 , bj (x, ξ) = aj (x, ξ, 1), and

j (2.11.1) h bj (x, ξ), b(x, ξ) ∼ j∈N

we thus obtain (2.11.2)

hm Op(a)u(x) = (2πh)−d ∫∫

ei(x−y)·ξ/h b(x, ξ) u(y) dy dξ

Rd ×Rd

= (2π)−d ∫∫

ei(x−y)·ξ b(x, hξ) u(y) dy dξ

Rd ×Rd

=: Oph (b)u(x). By Remark 2.55 bj (x, ξ) ∈ S m−j and the asymptotic series (2.11.1) can be justified similarly to what it is done in Lemma 2.4. As an example the operator A = −Δ + τ 2 V (x) + q(x) has for principal symbol a(x, ξ, τ ) = |ξ|2 + τ 2 V (x). The semi-classical version with the small parameter h of this operator is then B = h2 A = −h2 Δ + V (x) + h2 q(x). In this quantification its principal symbol is b(x, ξ) = |ξ|2 + V (x), as the symbol of hD is ξ (see the second line in (2.11.2)). In the present book, following a fairly large part of the literature on Carleman estimates, we chose to use a large parameter τ instead of the

42

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

semi-classical small parameter h. Yet both approaches are equivalent and the above lines allow one to go from one formalism to the other one. 2.12. Standard Pseudo-Differential Operators At places we shall also need “standard” pseudo-differential operators, that is operators that do not depend on a large parameter. The associated symbols are given as follows. Definition 2.53 (Standard Symbols). Let p, d ∈ N and let a(x, ξ) ∈ C ∞ (Rp × Rd ), with m ∈ R, be such that for all multi-indices αNp , β ∈ Nd there exists Cα,β > 0 such that |∂xα ∂ξβ a(x, ξ)| ≤ Cα,β ξm−|β| , x ∈ Rp , ξ ∈ Rd ,  1/2 . We write a ∈ S m (Rp × Rd ). The best possible where ξ = 1 + |ξ|2 constants Cα,β in (2.12.1) yield semi-norms that allow one to endow the space S m (Rp × Rd ) with a Fr´echet space structure. We also define S −∞ (Rp × Rd ) = ∩r∈R S r (Rp × Rd ) and S +∞ (Rp × Rd ) = ∪r∈R S r (Rp × Rd ). (2.12.1)

As for the symbols that depend on a large parameter, we shall often simply write S m (resp. S −∞ , S ∞ ), especially when the values of p and d are clear from the context, for instance in the case p = d with d the dimension of the ambient space. For a ∈ S m we call principal symbol, σ(a), the equivalence class of a in S m /S m−1 . With these symbol classes we can define “standard” pseudo-differential operators. Definition 2.54 (Standard a ∈ S m (Rd × Rd ), we set

Pseudo-Differential

Operators). If

ˆ(ξ) dξ a(x, D)u(x) = Op(a)u(x) := (2π)−d ∫ eix·ξ a(x, ξ) u Rd

= (2π)

−d

∫∫

ei(x−y)·ξ a(x, ξ) u(y) dy dξ,

Rd ×Rd

for u ∈ S (Rd ). Oscillatory integrals are to be understood in the sense given in Sect. 2.4; in particular the proof of Theorem 2.11 adapts to standard symbols (see Remark 2.60). We denote by Ψm the set of these pseudo-differential operators. For A ∈ Ψm , σ(A) will be its principal symbol in S m /S m−1 . If moreover a(x, ξ) is a polynomial function in ξ of order m, we say that Op(a) ∈ D m . Remark 2.55. Note that a(x, ξ, 1) ∈ S m if a(x, ξ, τ ) ∈ Sτm . Moreover, the statements of the results on standard pseudo-differential operators that are counterparts of those for pseudo-differential operators with the large parameter τ can be recovered by setting τ to a fixed value, e.g. τ = 1. This includes the oscillatory integrals of Sect. 2.4, the symbol calculus formulae of

2.12. STANDARD PSEUDO-DIFFERENTIAL OPERATORS

43

Sect. 2.5, the Sobolev continuity of Sect. 2.6, and the parametrix construction of Sect. 2.8. For G˚ arding type inequalities (see Sect. 2.7), statements differ as remainder terms associated with lower-order terms appear. In the case of standard pseudo-differential operators they cannot be absorbed by the positive term that is obtained, as is done in the case of pseudo-differential operators with the large parameter τ by letting τ be sufficiently large.   We set Λm = Op ξm . Classical Sobolev norms can then be given by uH s (Rd ) = Λs uL2 (Rd ) ,

u ∈ S (Rd ).

Theorem 2.56. Let V be an open set of Rd . Let a(x, ξ) ∈ S m (Rd × Rd ), with principal part am . If there exist C0 > 0 and R > 0 such that Re am (x, ξ) ≥ C0 ξm ,

x ∈ V, ξ ∈ Rd ,

|ξ| ≥ R,

then, for any 0 < C1 < C0 and s ∈ R, there exists C > 0 such that Re(Op(a)u, u)L2 (Rd ) ≥ C1 u2H m/2 (Rd ) − Cu2H s (Rd ) ,

u ∈ S (Rd ), supp(u) ⊂ V .

We shall also need the following converse result. Theorem 2.57. Let a(x, ξ) ∈ S m (Rd × Rd ), with homogeneous principal part am (x, ξ) for |ξ| ≥ R > 0. Let V be an open set of Rd , and assume that, for some C0 , C0 > 0, we have   Op(a)uL2 (Rd ) ≥ C0 uH m (Rd ) − C0 KuH + uH m−1 (Rd ) , u ∈ S (Rd ), if supp(u) ⊂ V , where K : H m (Rd ) → H is compact with H a Hilbert space. Then, |am (x, ξ)| ≥ C0 |ξ|m ,

x ∈ V, ξ ∈ Rd , |ξ| ≥ R.

A proof is given in Appendix 2.A.8. Tangential symbols and operators are defined similarly. We say that a(x, ξ  ) ∈ STm (Rd × Rd−1 ) if we have |∂xα ∂ξβ a(x, ξ  )| ≤ Cα,β ξ  m−|β| ,

x ∈ Rd , ξ  ∈ Rd−1 ,

1/2  where ξ   = 1+|ξ  |2 . We denote by Ψm T the set of associated operators, that is, 



ˆ(ξ  , xd ) dξ  a(x, D )u(x) = OpT (a)u(x) := (2π)1−d ∫ eix ·ξ a(x, ξ  ) u Rd−1

= (2π)1−d

∫∫

Rd−1 ×Rd−1

e

i(x −y  )·ξ 

a(x, ξ  ) u(y  , xd ) dy  dξ  .

44

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

Remark 2.58. We shall keep the notation DTm for standard tangential differential operators of order m that depend upon the parameter xd , that is operators of the form OpT (a) with

α aα (x)ξ  . a(x, ξ  ) = |α|≤m

m Note that we have DTm ⊂ D m and yet Ψm T ⊂ Ψ . In fact observe that m  as a differentiation with respect to ξ yields a gain of ξ   in T ⊂ S (2.10.1), while a gain in ξ is needed in the definition of Sτm in (2.2.1).

Sm

2.13. Notes Pseudo-differential operators were developed in the late 1950s. Techniques around what we call standard operators here were first developed. Semi-classical techniques, originally introduced for differential operators, were extended to pseudo-differential operators for operators as given in (2.11.2), with a small parameter h > 0. Here, we have made the choice to use a large parameter τ = 1/h mainly because it is a usual practice in the study of Carleman estimates and their application to unique continuation and control theory. Yet, both approaches can be used equivalently. In fact, standard and semi-classical operators can be understood in a unified manner if one uses the framework of the Weyl–H¨ormander calculus (see [173] et [175, chp 18.4–18.6]). Yet, this theory is more involved and we chose not to use it at this stage of this book. We postpone its use to Chap. 14 of Volume 2. Semi-classical operators of the form of Oph (b) as presented in Sect. 2.11 are for instance studied in [119, 244, 291, 332]. For references on standard pseudo-differential operators the reader can consult [26, 162, 175, 304, 317]. Appendix 2.A. Technical Proofs for Pseudo-Differential Calculus 2.A.1. Symbol Asymptotic Series: Proof of Lemma 2.4. Uniqueness up to Sτ−∞ is clear. For the existence we shall need the following result. Proposition 2.59. Let χ ∈ Cc∞ (R). Then χ(ελτ ) converges to χ(0) in for m > 0 as ε → 0. Consequently, if a ∈ Sτm , then χ(ελτ )a(x, ξ, τ ) converges to χ(0)a(x, ξ, τ )   in Sτm with m > m and Sτ−∞ is dense in Sτm for the topology of Sτm , m > m. Sτm

A proof of this proposition is given below. m

ξ, τ ) ∈ Sτ such that a ∼

We now proceed with the existence of a(x, j aj . Note that the convergence of the series j aj (x, ξ, τ ) is not ensured. The idea is to modify the symbols aj such that the sum is locally defined.

2.A. TECHNICAL PROOFS FOR PSEUDO-DIFFERENTIAL CALCULUS

45

Let χ ∈ Cc∞ (R) be such that χ ≡ 1 in a neighborhood of 0. We then introduce   with εj → 0, as j → +∞. Aj (x, ξ, τ ) = 1 − χ(εj λτ ) aj (x, ξ, τ ), d The sequence

(εj )j is to be determined below. For all (ξ, τ ) ∈ R × [1, +∞) the sum j Aj (x, ξ, τ ) is finite. One may thus set a(x, ξ, τ ) = j Aj (x, ξ, τ ). We have a ∈ C ∞ (Rp × Rd ).

−∞ if We observe that aj − Aj ∈ Sτ−∞ and thus j m. We observe that if a ∈ S˜τm , then for all  |α| ≤ r we have ar,α ∈ S˜τm −r and moreover

a → ar,α

(2.A.4)

is continuous,





for the Fr´echet topologies on S˜τm and S˜τm −r . Now, arguing as for (2.A.3) and using (2.A.4), if a ∈ S˜τ−∞ = ∩s∈R S˜τs , with r chosen such that m − r < −d, we have

|Iϕ (a, u, τ )| ≤ qτ (ar,α )μ(∂yα u) |α|≤r

≤

|α|≤r

qτ,r,α (a)μr,α (u),

 where qτ,r,α is a semi-norm on S˜τm and μr,α is a semi-norm on S (Rd ).  As S˜τ−∞ is dense in S˜τm with the topology of S˜τm (adapt the proof of Proposition 2.59), we conclude that the oscillatory integral Iϕ (a, u, τ ) extends in a unique manner by continuity to a ∈ S˜τm . Finally, with the estimate we obtained above, the map u → Iϕ (a, u, τ ) yields a distribution of order ≤ k with k = r > m + d. Since m > m can be chosen arbitrary, we see that the order is ≤ k with k ∈ N such that k > m + d. 

Remark 2.60. The above proof carried out for symbols that depend on τ in an unprescribed way allows one to give a definition of oscillatory integrals, through the same proof in the case of a symbol in Sτm , that actually depend on τ in a precise manner (see Definition 2.1) or in the case of standard symbols, S m , that do not depend on τ (see Definition 2.53). 2.A.3.2. Definitions of Oscillatory Integrals: Proof of Theorem 2.16. We p first sharpen a bit the notation for the semi-norms on Am ρ (R ): m Mρ,α (a) = sup θ−m+ρ|α| |∂θα a(θ)|, θ∈Rp

m m Nρ,k (a) = sup Mρ,α (a), |α|≤k

with α a multi-index. We shall need the following “nonstationary phase” lemma. Lemma 2.61. Let K be a compact of Rp and ϕ ∈ C ∞ (Rp ) be real valued and such that ϕ (θ) ≥ C0 > 0 on K. Then for all functions a ∈ Cc∞ (Rp ) with supp(a) ⊂ K and for all k ∈ N, we have     iλϕ(θ) ∫ e a(θ) dθ ≤ λ−k Ck+1 (ϕ)C(C0 , K) sup ∂θα a(θ), λ ≥ 1, |α|≤k θ∈Rp

where Ck+1 (ϕ) remains bounded if ϕ remains bounded in C k+1 (K). Proof. The proof is quite similar to that of Theorem 2.11. We set L = −i

∇θ ϕ · ∇ θ , |ϕ |2

50

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

and we observe that Leiλϕ = λeiλϕ . We then have ∫ eiλϕ(θ) a(θ) dθ = λ−k ∫ eiλϕ(θ) (t L)k a(θ) dθ. Note that t

p

Lf = i

i=j

∂ θj

(∂θj ϕ)f , |ϕ |2

and that the application of t L, k times, implies differentiations of ϕ up to order k + 1. Estimation of the integrand requires the use of the constant C0 as well as the volume of the compact set K.  Remark 2.62. A more sophisticated version of this lemma, in particular allowing for a complex phase function ϕ, can be found in Theorem 7.7.1 in [176]. Proof of Theorem 2.16. Let χ ∈ Cc∞ (Rp ) be such that χ(θ) = 1 for |θ| < 1 and χ(θ) = 0 if |θ| > 2. We set ψ(θ) = χ(θ) − χ(2θ). We then have 1 ≤ |θ| ≤ 2} =: K. 2 We introduce the following dyadic partition of unity of Rp : (2.A.5)

supp(ψ) ⊂ {θ ∈ Rp ;

1=

∞

χν (θ),

ν=0

with χ0 (θ) = χ(θ), χν (θ) = χ(2−ν θ) − χ(21−ν θ) = ψ(2−ν θ).   We have supp(χν ) ⊂ 2ν−1 ≤ |θ| ≤ 2ν+1 .

m p Lemma 2.63. Let m > 0. The series N ν=0 χν converge to 1 in Aρ (R ).

−N θ). We can then adapt Proof. We observe that N ν=0 χν (θ) = χ(2 the proof of Proposition 2.59.  On the compact set K defined in (2.A.5), we have |ϕ | ≥ C0 > 0. If a ∈ A−∞ ρ , then we have Iϕ (a) = ∫ eiϕ(θ) χ0 (θ)a(θ) dθ +

∞

∫ eiϕ(θ) χν (θ)a(θ) dθ.

ν=1

For the first term we can obtain an estimation as sought in the result of the theorem. We estimate the term of order ν in the sum. We have Iν = ∫ eiϕ(θ) ψ(2−ν θ) a(θ) dθ = 2νp ∫ ei2

νμ ϕ(θ)

ψ(θ) a(2ν θ) dθ,

K

and homogeneity of ϕ. The change of by the change of variables θ → variables allows one to bring the analysis to K that contains supp(ψ). If we 2ν θ

2.A. TECHNICAL PROOFS FOR PSEUDO-DIFFERENTIAL CALCULUS

51

apply Lemma 2.61, we obtain    |Iν | ≤ 2ν(p−μk) Ck+1 (ϕ)C(C0 , K) sup ∂θα ψ(θ)a(2ν θ)  |α|≤k θ∈K

   ≤ 2ν(p−μk) Ck+1 (ϕ)C  (C0 , K) sup 2ν|α|  ∂θα a (2ν θ) |α|≤k θ∈K

m ≤ 2ν(p−μk) Ck+1 (ϕ)C  (C0 , K) sup 2ν|α| Mρ,α (a)2ν θm−ρ|α| . |α|≤k θ∈K

Observing that 2ν |θ| ≥

1 2

implying that we have

2ν |θ| ≤ 2ν θ ≤ C1 2ν |θ|,

C1 > 1,

we find m (a) 2ν(1−ρ)|α|+νm |Iν | ≤ 2ν(p−μk) Ck+1 (ϕ)C  (C0 , K) sup Mρ,α |α|≤k

≤2

ν(p−μk)



m Ck+1 (ϕ)C (C0 , K)2ν(1−ρ)k+νm Nρ,k (a),

as |θ| ≤ 2 in K and as 1 − ρ ≥ 0. We thus obtain   m (a). |Iν | ≤ 2ν m+p+k(1−ρ−μ) C  (C0 , K)Ck+1 (ϕ)Nρ,k The series is then convergent if we choose k such that m + p + k(1 − ρ − μ) < 0, which can be achieved since μ > 1 − ρ. We finally obtain   Iϕ (a) ≤ Ck N m (a), (2.A.6) ρ,k where k depends on ρ, m, and μ. p m p The density of S (Rp ) ⊂ A−∞ ρ (R ) in Aρ (R ) for the topology of p  Am ρ (R ), with m > m (by adapting the proof of Proposition 2.59), allows  p  one to say that Iϕ (a) is well-defined for a ∈ Am ρ (R ). As m and m are p arbitrary here, then Iϕ (a) is well-defined for any a ∈ A∞ ρ (R ). p Finally, let us show that estimation (2.A.6) holds if a ∈ Am ρ (R ). The  p  continuity proven above shows that (working in Am ρ (R ) with m > m) the following limit inversion holds:

iϕ(θ) ∫e χν (θ)a(θ) dθ. Iϕ (a) = ∫ eiϕ(θ) a(θ) dθ = ∫ eiϕ(θ) χν (θ)a(θ) dθ = ν

ν

  m (a), if a ∈ Am (Rp ). The estimations given above then prove Iϕ (a) ≤ Ck Nρ,k ρ 

52

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

2.A.4. Proofs of the Results on Symbol Calculus. First, we provide a complete and detailed proof of Theorem 2.18. Second, we give proof of Theorem 2.22 that uses the line of argumentation of that of Theorem 2.18. Proof of Theorem 2.18. If a ∈ Sτm , we can write, in the sense of oscillatory integrals, Op(a)u(x) = (2π)−d ∫∫ ei(x−y)·ξ a(x, ξ, τ )u(y) dydξ, which we write, for u ∈ S (Rd ), Op(a)u(x) = ∫ Aτ (x, y)u(y) dy = Aτ (x, .), u(.)S  (Rd ),S (Rd ) . Rd

The kernel of the operator is Aτ (x, y) and is a distribution denoted by Aτ (x, y) = (2π)−d ∫ ei(x−y)·ξ a(x, ξ, τ )dξ, Rd

in the sense of Sect. 2.4. If Aτ (x, y) is the distributional kernel of an operator, then the kernel of the adjoint is Bτ (x, y) = Aτ (y, x) if duality is expressed through the sesquilinear product (u, v) = ∫ u¯ v dx,

u, v ∈ S (Rd ).

Let us assume for a moment that a ∈ S (Rd × Rd ). We have Aτ (x, y) = (2π)−d ∫ ei(x−y)·ξ a(x, ξ, τ )dξ = F2−1 (a)(x, x − y, τ ), that is, Aτ (x, x − y) = F2−1 (a)(x, y, τ ), which gives

  a(x, ξ, τ ) = F2 Aτ (x, x − y) (ξ) = ∫ e−iy·ξ Aτ (x, x − y)dy.

As explained above, the kernel of the adjoint operator is Bτ (x, y) = Aτ (y, x), that is, ¯(y, ξ, τ )dξ. Bτ (x, y) = (2π)−d ∫ ei(x−y)·ξ a Then ¯(x − y, η, τ )dη. Bτ (x, x − y) = (2π)−d ∫ eiy·η a We introduce

  a∗ (x, ξ, τ ) = F2 Bτ (x, x − y) = ∫ e−iy·ξ Bτ (x, x − y)dy = (2π)−d ∫∫ eiy·(η−ξ) a ¯(x − y, η, τ ) dηdy.

Then Bτ (x, y) = (2π)−d ∫ ei(x−y)·ξ a∗ (x, ξ, τ )dξ = F2−1 (a∗ )(x, x − y, τ ). At this stage several remarks can be made. Based on Fourier transformations, the map a → a∗ is continuous from S (Rd × Rd ) to S (Rd × Rd ),

2.A. TECHNICAL PROOFS FOR PSEUDO-DIFFERENTIAL CALCULUS

53

and it can be extended from S  (Rd × Rd ) to S  (Rd × Rd ). Similarly, the maps a → Aτ and a → Bτ can also be extended from S  (Rd × Rd ) to S  (Rd × Rd ). With the kernel theorem, we may thus define Op(a)u(x) for a ∈ S  (Rd × Rd ) as Op(a)u(x) = Aτ (x, .), u(.)S  (Rd ×Rd ),S (Rd ) ∈ S  (Rd ). We thus obtain that if a ∈ S  (Rd × Rd ) then a∗ ∈ S  (Rd × Rd ), and (Op(a)u, v) = (u, Op(a∗ )v),

(2.A.7) for u, v ∈ S (Rd ).

We shall now show that if a ∈ Sτm , then a∗ ∈ Sτm and that the map Sτm → Sτm a → a∗ , is continuous. This will prove that (2.A.7) holds with a∗ ∈ Sτm if a ∈ Sτm . We have ¯(x − y, η, τ ) dηdy a∗ (x, ξ, τ ) = (2π)−d ∫∫ eiy·(η−ξ) a

(2.A.8)

= (2π)−d ∫∫ e−iy·η a ¯(x − y, ξ − η, τ ) dηdy = (2π)−d ∫∫ e−iy·η a ¯(x + y, ξ + η, τ ) dηdy. +

2d If a ∈ Sτm (Rd × Rd ), then (a(x + ., ξ + ., τ ) ∈ Am 0 (R ) (with x, ξ, and τ + as parameters), where m = max(m, 0). As (y, η) → −y · η is homogeneous of degree two, Theorem 2.16 gives a∗ (x, ξ, τ ) well-defined and smooth with respect to the two variables x and ξ, and we have   m+ a ¯(x − ., ξ − ., τ ) , |a∗ (x, ξ, τ )|  N0,k

for a some C > 0 and k ∈ N. We choose μ ≥ m+ ≥ 0 (μ yet to be determined). We also have  μ  |a∗ (x, ξ, τ )|  N0,j a ¯(x − ., ξ − ., τ ) , for some C > 0 and some j that depends on μ. Since a ∈ Sτm , we have   ¯(x − y, ξ − η, τ ) ≤ pj (a)η−μ λτ (ξ − η)m , (y, η)−μ sup ∂yα ∂ηβ a |α|+|β|≤j

where λτ (ζ) =

 τ 2 + |ζ|2 and where pj is the semi-norm on Sτm given by

pj (a) = sup x∈Rd ξ∈Rd τ ≥1

sup |α|+|β|≤j

λτ (ξ)|β|−m |∂xα ∂ξβ a(x, ξ, τ )|.

We use the following lemma. Lemma 2.64 (Peetre Inequality). For all ξ, η ∈ Rd , m ∈ R, we have (1 + |ξ − η|)m ≤ (1 + |η|)|m| (1 + |ξ|)m .

54

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

Proof of Lemma 2.64. The case m ≥ 0 can be easily obtained by writing 1 + |ξ − η| ≤ 1 + |ξ| + |η| ≤ (1 + |ξ|)(1 + |η|). If m < 0, then −m > 0, and thus (1 + |ξ − η|)−m ≤ (1 + |η|)−m (1 + |ξ|)−m ,

ξ, η ∈ Rd ,

(1 + |ξ|)−m ≤ (1 + |η|)−m (1 + |ξ + η|)−m ,

ξ, η ∈ Rd ,

that is,

which we write (1 + |ξ|)−m ≤ (1 + |η|)−m (1 + |ξ − η|)−m ,

ξ, η ∈ Rd .

Then we have (1 + |ξ − η|)m ≤ (1 + |η|)−m (1 + |ξ|)m ,

ξ, η ∈ Rd , 

which is precisely the sought result. The Peetre inequality gives  λτ (ξ − η)m = τ m 1 + |ξ/τ − η/τ |2 = τ m ξ/τ − η/τ m ≤ Cm τ m η/τ |m| ξ/τ m ≤ Cm η|m| λm τ (ξ), as here τ ≥ 1. Hence, if we choose μ = |m|, we obtain   ¯(x − y, ξ − η, τ )  pj (a)λm (y, η)−|m| sup ∂yα ∂ηβ a τ (ξ), |α|+|β|≤j

which gives (2.A.9)

|m|

|a∗ (x, ξ, τ )|  N0,j (¯ a(x − ., ξ − ., τ ))  pj (a)λm τ (ξ).

The same argumentation gives (2.A.10)

(ξ), |∂xα ∂ξβ a∗ (x, ξ, τ )|  pj  (∂xα ∂ξβ a)λm−|β| τ

for some j  ∈ N. We thus have a∗ ∈ Sτm and the map a → a∗ is continuous. We now proceed with the proof of the asymptotic series for a∗ (x, ξ, τ ). With the Taylor formula we have

(−y)α α ∂ a ¯(x, ξ, τ ) a ¯(x − y, ξ, τ ) = α! x |α|≤N +

1

∫ (N + 1)(1 − σ)N

|α|=N +1 0

(−y)α α ¯(x − σy, ξ, τ ) dσ. ∂ a α! x

We then consider the term ¯(x, η, τ ) dηdy. a∗α := (2π)−d ∫∫ eiy·(η−ξ) y α ∂xα a

2.A. TECHNICAL PROOFS FOR PSEUDO-DIFFERENTIAL CALCULUS

55

Observe that this is well-defined according to Theorem 2.16 with y α ∂xα a ¯(x − m+ +|α| 2d + y, η, τ ) ∈ A0 (R ) with x and τ as parameters, where m = max(m, 0). Next, observe that y α eiy·(η−ξ) = (−i)|α| ∂ηα eiy·(η−ξ) . Thus, integrations by parts (see Remarks 2.13 and 2.17) give a∗α := (2π)−d i|α| ∫∫ eiy·(η−ξ) ∂xα ∂ηα a ¯(x, η, τ ) dηdy = i|α| ∂xα ∂ξα a ¯(x, ξ, τ ). From (2.A.8), we thus obtain

a∗ (x, ξ  , τ ) = |α|≤N

1 ¯(x, ξ, τ ) ∂α∂αa |α| i α! x ξ

+ RN (x, ξ, τ ),

with RN (x, ξ, τ ) =

1

∫

|α|=N +1 0

(N + 1)(1 − σ)N (2π)d α!

¯(x − σy, ξ, τ ) dσdηdy, × ∫∫ eiy·(η−ξ) (−y)α ∂xα a which can be understood in the sense of Theorem 2.16. As above, integrations by parts give RN (x, ξ, τ ) =

1

∫

|α|=N +1 0

(N + 1)(1 − σ)N (2π)d α!

¯(x − σy, ξ, τ ) dσdηdy, × ∫∫ eiy·(η−ξ) (−i)N +1 ∂xα ∂ηα a which is precisely the remainder term given in the statement of Theorem 2.18. It remains to prove that RN (x, ξ, τ ) ∈ Sτm−N −1 . In fact, for |α| = N + 1, with the same arguments as above used to prove the bound on a∗ (x, ξ, τ ), we show that ¯(x − σy, ξ − η, τ ) dηdy ∫∫ e−iy·η ∂xα ∂ηα a is bounded in Sτm−N −1 uniformly w.r.t. to σ ∈ [0, 1]. Similarly, one obtains that a → RN is continuous from Sτm to Sτm−N −1 . This concludes the proof of Theorem 2.18.  Proof of Proposition 2.21. Let χ ∈ S (Rd ) be such that χ(0) = 1, that is, ∫ χ ˆ = (2π)d . For ξ ∈ Rd we set eξ (x) = eix·ξ . For ε > 0, we set χε (x) = χ(εx) and consider uε = χε eξ ∈ S (Rd ). We have uε → eξ in S  (Rd ) as ε → 0 since χε ϕ → ϕ in S (Rd ) for any ϕ ∈ S (Rd ). By Proposition 2.19 we thus have Op(a)uε → Op(a)eξ in S  (Rd ). We write   Op(a)uε (x) = (2επ)−d ∫ eix·η a(x, η, τ )χ ˆ (η − ξ)/ε dη = (2π)

Rd −d ix·ξ

e

∫ eiεx·η a(x, εη + ξ, τ )χ(η) ˆ dη.

Rd

56

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

Since χ ˆ ∈ S (Rd ), the Lebesgue dominated-convergence theorem applies, and we find that Op(a)uε (x) converges to eix·ξ a(x, ξ, τ ) as ε → 0.  Proof of Theorem 2.22. Arguing as in the proof of Theorem 2.18, we find that Op(a)Op(b) = Op(c) with c(x, ξ, τ ) = (2π)−d ∫∫

Rd ×Rd

e−iz·ζ a(x, ξ − ζ, τ )b(x − z, ξ, τ ) dζdz,

if a, b ∈ S (Rd × Rd ). We prove that the map (a, b) → c ∈ S (Rd × Rd )   yields c ∈ Sτm+m if a ∈ Sτm and b ∈ Sτm . We introduce c˜(x, ξ, y, η, τ ) = (2π)−d ∫∫

Rd ×Rd

e−iz·ζ a(x, η − ζ, τ )b(y − z, ξ, τ ) dζdz.

It is of the same form as (2.A.8) with x, ξ as parameters. As we have, for all multi-indices α, β, 

|∂yα ∂ηβ a(x, η, τ )b(y, ξ, τ )| ≤ Cαβ λτ (η)m−|β| λτ (ξ)m ,  for some Cα,β > 0, where λτ (.) = τ 2 + |.|2 , arguing as in the proof of Theorem 2.18, we obtain the counterpart of (2.A.9)–(2.A.10) 







|∂xα ∂yα ∂ξβ ∂ηβ c˜(x, ξ, y, η, τ )| ≤ Cλτ (ξ)m −|β| λτ (η)m−|β | , for all multi-indices α, α , β, β  , with the constant depending on finitely many  semi-norms on a and b. Thus c(x, ξ, τ ) = c˜(x, ξ, x, ξ, τ ) ∈ Sτm+m . As in the proof of Theorem 2.22, we obtain

1 ˜ N (x, ξ, y, η, τ ), ∂ηα a(x, η, τ )∂yα b(y, ξ, τ ) + R c˜(x, ξ, y, η, τ ) = |α| i α! |α|≤N with ˜ N (x, ξ, y, η, τ ) = R

1

∫

|α|=N +1 0

(N + 1)(1 − σ)N (2π)d iN +1 α!

× ∫∫ e−iz·ζ ∂ηα a(x, η − ζ, τ )∂yα b(y − σz, ξ, τ ) dσdζdz. ˜ N (x, ξ, x, ξ, τ ) ∈ Sτm+m −N −1 . Then, we find RN (x, ξ, τ ) = R



2.A.5. Proof of Theorem 2.26: Sobolev Bound . Observe that it is sufficient to consider the case m = s = 0, that is, if a ∈ Sτ0 , then Op(a) is an endomorphism on L2 (Rd ). In fact, observe that Λrτ = Op(λrτ ) is an   isometry between Hτr (Rd ) and Hτr −r (Rd ), for all r, r ∈ R. We then set s−m ◦ a ◦ λ−s ∈ S 0 ◦ Op(a) ◦ Λ−s B = Λs−m τ τ and have B = Op(b) with b = λτ τ τ by Theorem 2.22. If B is a continuous endomorphism on L2 (Rd ), then the following commutative diagram yields the conclusion: Hτs (Rd ) ⏐ ⏐ Λsτ 

Op(a)

−−−−−−−→

Hτs−m (Rd ) ⏐ ⏐ s−m Λ τ

Op(b)

L2 (Rd )

L2 (Rd ) −−−−−−−−−−−→

2.A. TECHNICAL PROOFS FOR PSEUDO-DIFFERENTIAL CALCULUS

57

We shall need the following classical Lemma. Lemma 2.65 (Schur Lemma). Let Kτ : Rd × Rd → C, with τ ∈ [1, +∞) as a parameter, be continuous, and such that     sup ∫ Kτ (x, y)dy ≤ C0 , sup ∫ Kτ (x, y)dx ≤ C0 , y

x

uniformly in τ . Then the operator with kernel Kτ (x, y) is a continuous endomorphism on L2 (Rd ) with an operator norm less than or equal to C0 . Proof. Let u ∈ Cc∞ (Rd ). We denote by Kτ the operator associated with the kernel Kτ (x, y). By the Cauchy–Schwarz inequality we have     Kτ u(x)2 ≤ ∫ |Kτ (x, y)||u(y)|2 dy ∫ |Kτ (x, y)| dy. It follows that we have  2 ∫ Kτ u(x) dx ≤ C0 ∫∫ |Kτ (x, y)| |u(y)|2 dxdy ≤ C02 ∫ |u(y)|2 dy. We conclude with the density of Cc∞ (Rd ) in L2 (Rd ).



We now prove Theorem 2.26 in the cases m = s = 0. The proof is decomposed into three steps. Step 1: From Sτ0 to Sτ−1 . If a ∈ Sτ0 , let then M be such that sup |a(x, ξ, τ )|2 ≤ M/2. We set  c(x, ξ, τ ) = M − |a(x, ξ, τ )|2 )1/2 and we observe that c(x, ξ, τ ) ∈ S 0 . By Theorem 2.18 we have c∗ (x, ξ, τ ) = c(x, ξ, τ ) + r1 (x, ξ, τ ),

r1 ∈ Sτ−1

and with Theorem 2.22 we obtain c∗ (x, ξ, τ ) ◦ c(x, ξ, τ ) = M − |a(x, ξ, τ )|2 + r1 (x, ξ, τ ) = M − a∗ (x, ξ, τ ) ◦ a(x, ξ, τ ) + r1 (x, ξ, τ ), with r1 , r1 ∈ Sτ−1 . We thus have Op(a)u2L2 (Rd ) ≤ M u2L2 (Rd ) + (Op(r1 )u, u)L2 (Rd ) . It thus remains to prove the L2 -bound of an operator of order −1. Step 2: From Sτ−k to Sτ−2k , with k ≥ 1. We write Op(a)u2L2 (Rd ) = (Op(b)u, u)L2 (Rd ) ,

with b = a∗ ◦ a ∈ Sτ−2k .

The L2 -bound of Op(a) is thus a consequence of that of Op(b). This procedure can be iterated to finally consider a ∈ Sτ−k with k ≥ d + 1.

58

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

Step 3: Case a ∈ Sτ−k with k ≥ d + 1. The kernel Aτ (x, y) of Op(a) is given by Aτ (x, y) = (2π)−d ∫ ei(x−y)·ξ a(x, ξ, τ )dξ, Rd

in the sense of classical integrals. It is continuous in x and y, and we have   Aτ (x, y) ≤ ∫ |a(x, ξ, τ )|dξ ≤ C. Observe that (xj − yj )Aτ (x, y) is the kernel of the commutator [xj , Op(a)]. The composition Theorem 2.22 yields i∂ξj a(x, ξ, τ ) for the (full) symbol of this commutator (it coincides with the principal symbol here—see Corollary 2.23). By induction we find that Op(i|α| ∂ξα a(x, ξ, τ )) has (x−y)α Aτ (x, y) −k−|α|

for kernel. As ∂ξα a ∈ Sτ tion. We thus have

, then (x − y)α Aτ (x, y) is also a bounded func-

(1 + |x − y|)d+1 |Aτ (x, y)| ≤ C, and we obtain the L2 -bound of Op(a) with Lemma 2.65, which concludes the proof.  2.A.6. Proofs of the G˚ arding Inequalities. 2.A.6.1. Proof of the Local G˚ arding Inequality of Theorem 2.28. Observe that V can be replaced by V in inequality (2.7.1). Let then V  be a neighborhood of V such that the (2.7.1) holds for (x, ξ) ∈ V  × Rd with the constant C0 replaced by C0 that satisfies C1 < C0 < C0 and dist(∂V  , V ) ≥ M > 0. This can be done since we have am (x, ξ, τ ) = am (y, ξ, τ ) + O(|x − y|λm τ ),

y ∈ V , x ∈ Rd

by the Taylor formula. We can then pick χ(x) ∈ C ∞ (Rd ) with supp(χ) ⊂ V  such that 0 ≤ χ ≤ 1, χ ≡ 1 in a neighborhood of V , and moreover all derivatives of χ are bounded. We then set a ˜(x, ξ, τ ) = χ(x)am (x, ξ, τ ) + C0 (1 − χ)(x)λm τ that m satisfies a ˜ ∈ Sτ and (2.A.11) Re a ˜(x, ξ, τ ) ≥ C0 λm τ ,

x ∈ Rd , ξ ∈ Rd , τ ∈ [1, +∞),

|(ξ, τ )| ≥ R .

We moreover note that (Op(˜ a)u, u) = (Op(am )u, u) if supp(u) ⊂ V . Without any loss of generality, we may thus consider that the principal symbol am (x, ξ, τ ) satisfies (2.A.11) in the remaining of the proof. We choose L > 0 such that C1 < L < C0 , and we define b smooth in (x, ξ) ∈ Rd × Rd such that 1/2  , for |(ξ, τ )| ≥ R . b(x, ξ, τ ) := Re am (x, ξ, τ ) − Lλm τ 1/2 m/2  Re am (x, ξ, τ )λ−m −L for |(ξ, τ )| ≥ R , we find b ∈ Writing b = λτ τ m/2

by Proposition 2.3, and we set B = Op(b). Pseudo-differential symbol Sτ calculus gives m B ∗ ◦ B = Re Op(am ) − LΛm τ + S = Re Op(a) − LΛτ + T,

2.A. TECHNICAL PROOFS FOR PSEUDO-DIFFERENTIAL CALCULUS

59

with S, T ∈ Ψm−1 , using that a = am mod Sτm−1 , and where Re Op(γ) τ actually means (Op(γ) + Op(γ)∗ )/2. We then have, by (2.6.3), Re(Op(a)u, u)L2 (Rd ) =(Re Op(a)u, u)L2 (Rd ) ≥ L(Λm τ u, u)L2 (Rd ) − (T u, u)L2 (Rd ) ≥ Lu2τ,m/2 − L u2τ,(m−1)/2 ≥ (L − τ −1 L )u2τ,m/2 . We conclude the proof of Theorem 2.28 by taking τ sufficiently large.



2.A.6.2. Proof of the Microlocal G˚ arding Inequality of Theorem 2.29. For a conic set U of Rd × Rd × [0, +∞), we recall that we use the notation SU = {(x, ξ, τ ) ∈ U ; λτ = 1}. Observe that W can be replaced by W in (2.7.2). Since SW is compact, Ssupp(χ) is closed, and Ssupp(χ) ⊂ SW , we have dist(Ssupp(χ) , ∂SW ) ≥ L > 0. ˜∗ be a smooth function on SRd such We set χ∗ = χ|SRd . Let then χ ∗ ∗ ˜∗ ) ⊂ SW . We then define χ(x, ˜ ξ, τ ) ∈ that χ ˜ ≡ 1 on supp(χ ) and supp(χ ∞ d d ˜ ≤ 1, homogeneous C (R × R ) with τ as a parameter in [1, +∞), 0 ≤ χ ˜|SRd = χ ˜∗ , that is, of degree 0 in (ξ, τ ) for λτ ≥ 1, such that χ χ(x, ˜ ξ, τ ) = χ ˜∗ (x, ξ/λτ , τ /λτ ),

x ∈ Rd , ξ ∈ Rd , τ ∈ [1, +∞), λτ ≥ 1.

Then χ(x, ˜ ξ, τ ) ∈ Sτ0 and χ ˜ ≡ 1 in a neighborhood of supp(χ) and supp(χ) ˜ ⊂ ˜(x, ξ, τ ) = χ(x, ˜ ξ, τ )a(x, ξ, τ ) + b(x, ξ, τ ) with W (for λτ ≥ 1). We set a ˜ ξ, τ )λm ˜ ∈ Sτm , and its principal part a ˜m b(x, ξ, τ ) = C0 (1 − χ)(x, τ . We have a satisfies and Re a ˜m (x, ξ, τ ) ≥ C0 λm τ ,

x ∈ Rd , ξ ∈ Rd , τ ≥ τ1 ,

λτ ≥ R0 ,

for C1 < C0 < C0 and for τ1 and R0 chosen sufficiently large. As χ ˜ ≡ 1 on supp(χ) (for λτ ≥ 1) from pseudo-differential calculus, we have Op(a)Op(χ) = Op(χa)Op(χ) ˜ + R with R = Op((1 − χ)a)Op(χ) ˜ ∈ −N ∩N ∈N Ψτ . We then have (Op(a)Op(χ)u, Op(χ)u)L2 (Rd ) = (Op(χa)Op(χ)u, ˜ Op(χ)u)L2 (Rd ) + (Ru, Op(χ)u)L2 (Rd ) = (Op(˜ a)Op(χ)u, Op(χ)u)L2 (Rd ) − (Op(b)Op(χ)u, Op(χ)u)L2 (Rd ) + (Ru, Op(χ)u)L2 (Rd ) = (Op(˜ a)Op(χ)u, Op(χ)u)L2 (Rd ) − (Op(χ)∗ Op(b)Op(χ)u, u)L2 (Rd ) + (Ru, Op(χ)u)L2 (Rd ) . As b vanishes in a neighborhood of the support of χ (for λτ ≥ 1), from pseudo-differential calculus, we find  Ψ−N Op(χ)∗ Op(b)Op(χ) ∈ τ . N ∈N

We then have, by formula (2.6.3),    − (Op(χ)∗ Op(b)Op(χ)u, u)L2 (Rd ) + (Ru, Op(χ)u)L2 (Rd )  ≤ CN u2

τ,−N

60

2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

The result thus follows from the G˚ arding inequality of Theorem 2.28.



2.A.6.3. Proof of the G˚ arding Inequalities for Systems. Here, we prove Theorem 2.31. Arguing as in the proof of Theorem 2.29, Theorem 2.32 can be obtained as of consequence of Theorem 2.31 that is the counterpart of Theorem 2.28 for systems. Arguing as in the proof of Theorem 2.28 above, there exists a neighborhood V  of V such that (2.7.3) holds for V  in place of V with C0 replaced by C0 that satisfies C1 < C0 < C0 . We then pick χ(x) ∈ C ∞ (Rd ) as in the proof  m ij of Theorem 2.28 and set a ˜ij (x, ξ, τ ) = χ(x)aij m (x, ξ, τ ) + C0 (1 − χ)(x)λτ δ , where δ ij stands for the Kronecker symbol. We then have (2.A.12)   2 x ∈ Rd , ξ ∈ Rd , τ ∈ [1, +∞), z ∈ Cn , Re a ˜(x, ξ, τ )z, z Cn ≥ C0 λm τ zCn , for |(ξ, τ )| ≥ R. We moreover note that (Op(˜ a)U, U ) = (Op(am )U, U ) if supp(U ) ⊂ V . Without any loss of generality, we may thus consider that the principal symbol am (x, ξ, τ ) satisfies (2.A.12) in the remaining of the proof. We then choose L > 0 such that C1 < L < C0 , and we define b = (bij ) as the unique positive definite square root of Re am − Lλm τ , that is,  −m 1/2 λτ Re am (x, ξ, τ ) − L , b(x, ξ, τ ) := λm/2 τ for x ∈ Rd , ξ ∈ Rd , τ ∈ [1, +∞), |(ξ, τ )| ≥ R . Using that the eigenvalRe am (x, ξ, τ ) − L satisfy ues μj (x, ξ, τ ), j = 1, . . . , n, of the matrix λ−m τ  μj (x, ξ, τ ) ≥ C0 − L > 0, by functional calculus, using a Dunford–Taylor integral [192, Chapter 5, Theorem 3.35], one finds that b(x, ξ, τ ) is smooth m/2 with respect to (x, ξ) and moreover bij (x, ξ, τ ) ∈ Sτ . The end of the proof is then similar to that of Theorem 2.28.  2.A.7. Parametrix Construction and Properties. In this section we prove Proposition 2.34. On W we define a smooth function ρ such that ρ = p−1 if λτ ≥ R. We then set q1 = χρ, we find q1 ∈ Sτ−m , and with the composition formula of Theorem 2.22 and the last point of Lemma 2.4, we obtain q 1 ◦ p = χ + r1

mod Sτ−∞ ,

with r1 ∈ Sτ−1 , supp(r1 ) ⊂ supp(χ).

We then proceed by induction and assume that for some N ∈ N there qN with exists qN ∈ Sτ−m , with supp(qN ) ⊂ supp(χ) such that qN = ρ˜ q˜N ∈ Sτ0 and qN ◦ p = χ + rN mod Sτ−∞ for some rN ∈ Sτ−N with moreover supp(rN ) ⊂ supp(χ). We then set qN +1 = qN −rN ρ that is well-defined because of the support of rN . We have supp(qN +1 ) ⊂ supp(χ). With the composition formula of Theorem 2.22 and the last point of Lemma 2.4 we obtain (rN ρ) ◦ p = rN − rN +1

mod Sτ−∞ ,

2.A. TECHNICAL PROOFS FOR PSEUDO-DIFFERENTIAL CALCULUS

61

for some rN +1 ∈ Sτ−N −1 that satisfies supp(rN +1 ) ⊂ supp(rN ) ⊂ supp(χ). This yields qN +1 ◦ p = χ + rN +1 mod Sτ−∞ and proves the existence of a left parametrix. The proof a right parametrix can be done similarly.  ∈ S −N we have q ◦p = χ+˜  = χ+˜  , rN and p◦qN rN Now if for some r˜N , r˜N N τ then we find     χ ◦ qN = (qN ◦ p − r˜N ) ◦ qN = qN ◦ χ − r˜N ◦ qN + qN ◦ r˜N ,  − q ◦ χ ∈ S −m−N . implying that χ ◦ qN N τ In the construction above we had AN = qN +1 − qN ∈ Sτ−m−N for N ≥ 1 and supp(AN ) ⊂ supp(χ). We set A0 = q0 ∈ Sτ−m . Then, by Lemma 2.4, q ∼ Aj ∈ Sτ−m can be chosen such that supp(q) ⊂ supp(χ) and

q−

N −1 j=0

Aj = q − qN ∈ Sτ−m−N .

Then, for all N ∈ N, we have q ◦ p = qN ◦ p + (q − qN ) ◦ p = χ mod Sτ−N , that is q ◦ p = χ mod Sτ−∞ . Similarly, we can build q  ∈ Sτ−m such that p ◦ q  = χ mod Sτ−∞ . Now if for some q, q  ∈ Sτ−m we have q ◦ p = χ mod Sτ−∞ and p ◦ q  = χ mod Sτ−∞ , then with the same argument as above we have χ ◦ q  = (q ◦ p) ◦ q  =q◦χ

mod Sτ−∞

mod Sτ−∞ ,

which gives the last point of the proposition.



2.A.8. A Characterization of Ellipticity. Here we prove Theorem 2.57. Let (x0 , ξ0 ) ∈ V × Rd with |ξ0 | ≥ R and let U be an open set such that x0 ∈ U and U  V . Let ϕ, ϕ˜ ∈ Cc∞ (V ) be such that ϕ˜ ≡ 1 in a neighborhood of U and ϕ ≡ 1 in a neighborhood of supp(ϕ). ˜ For u ∈ Cc∞ (U ), we write Op(a)u = ϕOp(a)u + (1 − ϕ)Op(a)(ϕu). ˜ By symbol calculus, adapting Sect. 2.5 to standard operators and symbols, we have (1 − ϕ)Op(a)ϕ˜ ∈ Ψ−∞ and ϕOp(a) = Op(ϕa). Similarly, we have (1 − ϕ)Λm ϕ˜ ∈ Ψ−∞ . We thus obtain   Op(ϕa)uL2 (Rd ) ≥ C0 ϕΛm uL2 (Rd ) − C0 KuH + uH m−1 (Rd ) , for some C0 > 0. With (x0 , ξ0 ) as chosen above, we pick ψ ∈ Cc∞ (U ) such that ψ(x0 ) = 1. We set uλ (x) = λ−m ψ(x)eiλx·ξ0 , for λ ≥ 1. We write λm Op(ϕa)uλ (x) = ψ(x)Op(ϕa)eiλx·ξ0 + [Op(ϕa), ψ]eiλx·ξ0 . We have [Op(ϕa), ψ] ∈ Ψm−1 , and we denote by r0 (x, ξ) ∈ S m−1 its symbol. Proposition 2.21 adapted to standard operators gives   λm Op(ϕa)uλ (x) = ψ(x)a(x, λξ0 ) + r0 (x, λξ0 ) eiλx·ξ0 .

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2. (PSEUDO-)DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

In particular, note that r0 is supported in supp(ϕ) × Rd . As a = am + am−1 with am−1 ∈ S m−1 , we find, using the homogeneity of am ,   Op(ϕa)uλ (x) = ψ(x)am (x, ξ0 ) + λ−m r1 (x, λξ0 ) eiλx·ξ0 , with r1 (x, ξ) ∈ S m−1 supported in supp(ϕ) × Rd . Similarly, we write λm ϕΛm uλ (x) = ψ(x)Λm eiλx·ξ0 + [ϕΛm , ψ]eiλx·ξ0 . We have [ϕΛm , ψ] ∈ Ψm−1 with symbol r2 ∈ S m−1 supported in supp(ϕ) × Rd . Proposition 2.21 gives   ϕΛm uλ = ψ(x)λ−m λξ0 m + λ−m r2 (x, λξ0 ) eiλx·ξ0 . We thus have |Op(ϕa)uλ (x)| = |ψ(x)am (x, ξ0 )| + OL2 (λ−1 ), |ϕΛm uλ (x)| = |ψ(x)λ−m λξ0 m | + OL2 (λ−1 ), where OL2 denotes a function bounded in L2 (Rd ), uniformly in λ. We thus obtain ψam (., ξ0 )L2 (Rd ) = Op(ϕa)uλ L2 (Rd ) + O(λ−1 )

  ≥ C0 λ−m λξ0 m ψL2 (Rd ) − C0 Kuλ H + uλ H m−1 (Rd ) + O(λ−1 ). Observe that uλ converges weakly to 0 in H m (Rd ) as λ → ∞. Thus, as K is a compact operator, up to picking a proper sequence λn → ∞, we obtain Kuλ H → 0. Observe also that uλ H m−1 (Rd ) = O(λ−1 ). We thus obtain (2.A.13)

ψam (., ξ0 )L2 (Rd ) ≥ C0 λ−m λξ0 m ψL2 (Rd ) + o(1).

Letting λ → ∞, we obtain ψam (., ξ0 )L2 (Rd ) ≥ C0 |ξ0 |m ψL2 (Rd ) .   We now choose ψ(x) = χ (x − x0 )/ε /εd/2 , with χ ∈ Cc∞ (Rd ) such that ∫ χ2 = 1. For ε > 0 chosen sufficiently small the function ψ has the properties assumed above. As ψL2 (Rd ) = 1 and |ψ|2 converges to a Dirac measure at x = x0 as ε → 0, we obtain |am (x0 , ξ0 )| ≥ C0 |ξ0 |m . As (x0 , ξ0 ) is arbitrary in V × Rd , with |ξ0 | ≥ R, we have obtained the sought result. 

CHAPTER 3

Carleman Estimate for a Second-Order Elliptic Operator Contents 3.1. 3.2. 3.2.1. 3.2.2. 3.2.3. 3.3. 3.4. 3.4.1. 3.4.2. 3.5. 3.6. 3.6.1. 3.6.2. 3.7. 3.7.1.

Setting 64 Weight Function and Conjugated Operator 64 Conjugated Operator 65 Characteristic Set and Sub-ellipticity Property 65 Invariance Under Change of Variables 68 Local Estimate Away from Boundaries 69 Local Estimates at the Boundary 71 Some Remarks 73 Proofs in Adapted Local Coordinates 74 Patching Estimates 89 Global Estimates with Observation Terms 91 A Global Estimate with an Inner Observation Term 92 A Global Estimate with a Boundary Observation Term 101 Alternative Approach 106 A Modified Carleman Estimate Derivation Away from Boundaries 106 3.7.2. A Modified Carleman Estimate Derivation at a Boundary 107 3.7.3. Alternative Derivation in the Case of Limited Smoothness 110 3.7.4. Valuable Aspects of the Different Approaches 117 3.8. Notes 118 Appendices 120 3.A. Poisson Bracket and Weight Function 120 3.A.1. Smoothness of the Characteristic Set 120 3.A.2. Expression of the Poisson Bracket 121 3.A.3. Construction of a Weight Function 122

© 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 I, PNLDE Subseries in Control 97, https://doi.org/10.1007/978-3-030-88674-5 3

63

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3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

3.A.4. 3.B. 3.B.1. 3.B.2. 3.B.3. 3.B.4. 3.C.

Local Extension of the Domain Where Sub-ellipticity Holds Symbol Positivity Symbol Positivity Away from a Boundary Tangential Symbol Positivity Near a Boundary Proof of Lemma 3.27 Symbol Positivity in the Modified Approach An Explicit Computation

125 125 126 126 126 127 128

3.1. Setting Consider a general second-order elliptic operator P with a principal part of the form (3.1.1)

Di (pij (x)Dj ), with pij (x)ξi ξj ≥ C|ξ|2 , P0 (x, Dx ) = 1≤i,j≤d

1≤i,j≤d

where pij ∈ C ∞ (Rd ; R) with all derivatives bounded and such that pij = pji , 1 ≤ i, j ≤ d. We recall that D = −i∂. The elliptic operator under consideration is then

i b (x)Di + c(x), (3.1.2) P (x, Dx ) = P0 (x, Dx ) + 1≤i≤d

where bi , c ∈ L∞ (Rd ), 1 ≤ i ≤ d. We denote by p the principal symbol of P given by

pij (x)ξi ξj . p(x, ξ) = 1≤i,j≤d

 1 Because of the ellipticity of P , note that |ξ|x = p(x, ξ) 2 defines a norm (for each x ∈ Rd ) equivalent to the usual Euclidean norm, uniformly w.r.t. x locally: for all compact K ⊂ Rd , there exists CK > 0 such that (3.1.3)

1 |ξ| ≤ |ξ|x ≤ CK |ξ|, CK

x ∈ K, ξ ∈ Rd ,

with the associated inner product (ξ, η)x = p˜(x, ξ, η) =

1≤i,j≤d p

ij (x)ξ

i ηj .

The derivation of a Carleman estimate for P will mainly concern the operator P0 . In fact, in the proofs we shall first achieve such an estimate for P0 and a similar estimate for P will be a natural and direct consequence. 3.2. Weight Function and Conjugated Operator Let ϕ(x) be a smooth real valued function. We shall refer to ϕ or to exp(τ ϕ) as the weight function.

3.2. WEIGHT FUNCTION AND CONJUGATED OPERATOR

65

3.2.1. Conjugated Operator. We define the following operator, referred to as the conjugated operator, Pϕ = eτ ϕ P0 e−τ ϕ , to be considered as a differential operator with a large parameter τ , as introduced in Chap. 2. Observe that we have eτ ϕ Dj e−τ ϕ = Dj + iτ ∂j ϕ ∈ Dτ1 . We thus have   

 τϕ e Dj e−τ ϕ pjk eτ ϕ Dk e−τ ϕ Pϕ = 1≤j,k≤d

=

1≤j,k≤d

    Dj + iτ ∂j ϕ pjk Dk + iτ ∂k ϕ ∈ Dτ2 .

We write Pϕ = P2 + iP1 ,

(3.2.1) with (3.2.2) (3.2.3)

P2 = P0 − p(x, τ dϕ) ∈ Dτ2 ,  

  jk P1 = τ Dj p ∂k ϕ + pjk ∂j ϕDk ∈ τ Dτ1 ⊂ Dτ2 . 1≤j,k≤d

Observe that P2 and P1 are both formally self-adjoint, in the sense that (Pj u, v)L2 (Rd ) = (u, Pj v)L2 (Rd ) ,

u, v ∈ S (Rd ),

j = 1, 2.

In fact, we have P2 =

 1 Pϕ + Pϕ∗ , 2

P1 =

 1 Pϕ − Pϕ∗ . 2i

In the framework of the calculus with a large parameter introduced in Chap. 2, their respective principal symbols are real and given by (3.2.4)

p2 (x, ξ, τ ) = p(x, ξ) − p(x, τ dϕ(x)) = |ξ|2x − |τ dϕ(x)|2x ∈ Sτ2 ,

(3.2.5)

p1 (x, ξ, τ ) = 2˜ p(x, ξ, τ dϕ(x)) = 2(ξ, τ dϕ(x))x ∈ τ Sτ1 ⊂ Sτ2 .

Note that the principal symbol of Pϕ in the symbol class Sτ2 is precisely   pϕ (x, ξ, τ ) = p x, ξ + iτ dϕ(x) and p2 = Re pϕ , p1 = Im pϕ (see the definition of the principal symbol of a differential operator in (2.2.3) in Sect. 2.2). 3.2.2. Characteristic Set and Sub-ellipticity Property. We define the characteristic set of pϕ as   Char(pϕ ) = (x, ξ, τ ) ∈ Rd × Rd × [0, +∞); pϕ (x, ξ, τ ) = 0 . For (x, ξ, τ ) ∈ Rd × Rd × [0, +∞), note that we have (x, ξ, τ ) ∈ Char(pϕ )

⇔

p2 (x, ξ, τ ) = p1 (x, ξ, τ ) = 0

⇔

|ξ|x = |τ dϕ|x and (ξ, dϕ)x = 0.

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3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

dϕ(x)

0

Char(pϕ )

Figure 3.1. Form of the characteristic set Char(pϕ ) at the vertical of each point x ∈ Rd , using orthogonality associated with the inner product (., .)x From this observation, the geometry of the characteristic set is illustrated in Fig. 3.1. The symbol pϕ is homogeneous of degree two. This implies that Char(p) is fully described by SChar(p) , with the notation introduced in (1.7.2), that is, Char(p) is the positive cone generated by SChar(p) . Observe that SChar(p) does not intersect {τ = 0}. In fact, SChar(p) has a smooth structure.1 d Lemma 3.1. Let V be an open  set ofd R . Assume  that |dϕ(x)| = 0 for all x ∈ V . Then, the set SChar(p) ∩ V × R × [0, +∞) is a smooth submanifold of V × Rd × (0, +∞) of codimension three.

A proof is given in Appendix 3.A.1. We recall that the Poisson bracket of two functions is d d

∂ ξj f ∂ xj g − ∂xj f ∂ξj g. {f, g} = j=1

j=1

Note that we have {Re f, Im f } = {f¯, f }/(2i). We shall choose a weight function ϕ such that P and ϕ have the following joint property. As it involves only the principal symbol pϕ (x, ξ, τ ), the property statement can be made for either P or P0 . Definition 3.2 (Sub-ellipticity). Let V be a bounded open set in Rd . We say that the weight function ϕ ∈ C ∞ (Rd ; R) and P (resp. P0 ) have the sub-ellipticity property in V if |dϕ| > 0 in V and (3.2.6) ∀(x, ξ) ∈ V × Rd , ∀τ > 0, pϕ (x, ξ, τ ) = 0 1 {pϕ , pϕ }(x, ξ, τ ) = {p2 , p1 }(x, ξ, τ ) > 0. ⇒ 2i 1Elements of differential geometry, including the notion of submanifold, are recalled in Chap. 15 of Volume 2. The reader is also referred to any expository book on differential geometry.

3.2. WEIGHT FUNCTION AND CONJUGATED OPERATOR

67

Remark 3.3. (1) Note that the sub-ellipticity property (3.2.6) can be extended to (x, ξ) ∈ V ×Rd and τ = 0 for (ξ, τ ) = (0, 0) because of the ellipticity of p. In the case of a nonelliptic operator, additional conditions are needed near τ = 0. (2) Observe that the set {(x, ξ, τ ) ∈ Char(p); x ∈ V and |ξ|2 + τ 2 = 1} is compact. From the homogeneity of {p2 , p1 } of order three w.r.t. (ξ, τ ) we deduce the equivalent form of the sub-ellipticity property, for some C > 0, 1 {pϕ , pϕ }(x, ξ, τ ) ≥ Cλ3τ . ∀(x, ξ) ∈ V × Rd , ∀τ > 0, pϕ (x, ξ, τ ) = 0 ⇒ 2i The structure of the Poisson bracket {pϕ , pϕ }/(2i) can be made more explicit by means of Hp , the Hamiltonian vector field associated with p, Hp (x, ξ) = ∇ξ p(x, ξ)∇x − ∇x p(x, ξ)∇ξ . Hamiltonian vector fields and their connection with Poisson bracket and symplectic structures are recalled in Sect. 9.2.2 in the Euclidean setting (and in Sect. 15.7.2 of Volume 2 in the manifold setting). Lemma 3.4. Let ϕ ∈ C ∞ (Rd ). We have   1 {pϕ , pϕ }(x, ξ, τ ) = τ Hp2 ϕ(x, ξ) + Hp2 ϕ(x, τ dϕ) 2i = τ Hp2 ϕ(x, ξ) + τ 3 Hp2 ϕ(x, dϕ). We refer to Appendix 3.A.2 for a proof. Finding a function ϕ so that the sub-ellipticity property holds can be done quite easily as stated in the following lemma whose proof is given in Appendix 3.A.3. Lemma 3.5. Let H ∈ C ∞ (R; R) be such that H  > 0, H  > 0, and H  /H  ≥ C0 > 0. Let V be a bounded open set in Rd and ψ ∈ C ∞ (Rd ; R) be such that |dψ| > 0 in V . Then, for γ > 0 sufficiently large, ϕ = H(γψ) and P have the sub-ellipticity property of Definition 3.2 in V . A typical choice is H = exp, that is, ϕ = exp(γψ) with γ > 0 chosen sufficiently large. Remark 3.6. Observe that in the proof of Lemma 3.5 we actually prove that for γ > 0 sufficiently large, the weight function ϕ = H(γψ) and P have the stronger property: ∀(x, ξ) ∈ V × Rd , ∀τ > 0,

Re pϕ (x, ξ, τ ) = p2 (x, ξ, τ ) = 0 1 {pϕ , pϕ }(x, ξ, τ ) ≥ Cλ3τ . ⇒ 2i If one is only interested in a weight function ϕ that fulfills with P the sub-ellipticity property near a point x0 , one can use the following lemma.

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3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

Lemma 3.7. Let x0 ∈ V and ψ ∈ C ∞ (Rd ; R) be such that ψ(x0 ) = 0 and dψ(x0 ) = 0. Set Vε = {x ∈ V ; |x − x0 | ≤ ε} and Gγ (s) = s + γs2 /2. If γ > 0 is chosen sufficiently large and ε > 0 is chosen sufficiently small, then ϕ = Gγ ◦ ψ and P have the sub-ellipticity property of Definition 3.2 in Vε . A proof is given in Appendix 3.A.3. An important result is the following lemma. It plays a central rˆ ole in the derivation of a Carleman estimate away from a boundary. Lemma 3.8. Let ϕ and P have the sub-ellipticity property of Definition 3.2 in V . Let μ > 0 and ρ = μ(p22 + p21 ) + τ {p2 , p1 }. Then, for all (x, ξ) ∈ V × Rd , and τ ≥ 1, we have ρ(x, ξ, τ ) ≥ Cλ4τ , with C > 0, for μ chosen sufficiently large. We refer to Appendix 3.B.1 for a proof. The following lemma will also be useful. Lemma 3.9. Let V be a bounded open set of Rd such that P and ϕ have the sub-ellipticity property in V . Then, there exists V  an open neighborhood of V such that this property holds also in V  . A proof is given in Appendix 3.A.4. 3.2.3. Invariance Under Change of Variables. In the proofs of Carleman estimates that we give below, we use change of variables. It is legitimate to wonder if the notions introduced above transform naturally and are preserved under such change of variables. Let X and Y be two open subsets of Rd and κ : X → Y be a smooth diffeomorphism. If the operator P0 introduced above is defined on X, then the corresponding operator on Y is given by u ∈ C ∞ (Y ).

(Q0 u) ◦ κ = P0 (u ◦ κ),

The principal symbol q of Q0 is (see Sect. 9.1.2) q(κ(x), ξ) = p(x, t κ (x)ξ). If the weight function ϕ is defined on X, then it naturally transforms into φ(y) = ϕ(x) for y = κ(x), that is φ ◦ κ = ϕ. The principal symbols of the conjugated operators eτ ϕ P0 e−τ ϕ and eτ φ Q0 e−τ φ , defined on X and Y , respectively, are given by pϕ (x, ξ, τ ) = p(x, ξ + iτ dϕ(x)),

qφ (y, η, τ ) = q(y, η + iτ dφ(y)),

yielding, qφ (κ(x), ξ, τ ) = pϕ (x, t κ (x)ξ, τ ),

x ∈ X, ξ ∈ Rn , τ ≥ 1,

3.3. LOCAL ESTIMATE AWAY FROM BOUNDARIES

69

as φ (κ(x)) ◦ κ (x) = ϕ (x), that is, t κ (x)dφ(κ(x)) = dϕ(x), since, for w(x) a vector field, we have dϕ(x), w(x) = ϕ (x)(w(x)) = φ (κ(x))(κ (x)w(x)) = dφ(κ(x)), κ (x)w(x) = t κ (x)dφ(κ(x)), w(x), where ., . denotes the one-form/vector-field duality bracket. We then see that the principal symbol of the conjugated operator Pϕ is naturally transformed through the change of variables κ. In other words, conjugation and change of variables commute well. More details are given in Chap. 9 (see Sect. 9.3.1). Moreover, we have the following proposition. Proposition 3.10. If ϕ and P0 have the sub-ellipticity condition of Definition 3.2 in V ⊂ X, then φ and Q0 have this property in κ(V ) ⊂ Y . We refer to Sect. 9.3.2 for a proof. This is in fact connected to the geometrical invariance of the Poisson bracket. As a consequence, a change of variables can be performed in the proof of a Carleman estimate without affecting the sub-ellipticity property of the weight function with respect to the differential operator P0 (resp. P ). This is an important feature toward the generalization of Carleman estimates to the case of manifolds. In Chap. 5 in Volume 2 we shall derive such estimates for the Laplace–Beltrami operator on a Riemannian manifold. There, the sub-ellipticity property will be stated as a geometrical property, implying right away that it is independent of any local chosen coordinates. This is based on the definition of the Poisson bracket on a manifold, which is recalled in Sect. 15.7.2 in Volume 2. 3.3. Local Estimate Away from Boundaries Away from any boundary, a Carleman estimate for the operator P is based on the G˚ arding inequality and takes the following form. Theorem 3.11. Let V be a bounded open set in Rd , and let ϕ and P have the sub-ellipticity property of Definition 3.2 in V ; then, there exist τ∗ > 0 and C > 0 such that

eτ ϕ Dα u2L2 (Rd ) (3.3.1) τ 3 eτ ϕ u2L2 (Rd ) + τ eτ ϕ Du2L2 (Rd ) + τ −1 |α|=2

≤ Ceτ ϕ P u2L2 (Rd ) , for u ∈ Cc∞ (V ) and τ ≥ τ∗ . Remark 3.12. We observe that the previous estimate is local in the sense that it applies to functions with a prescribed compact support. With a density argument the result of Theorem 3.11 can be extended to functions u ∈ H02 (V ). However, here, we do not treat the case of functions in

70

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

H01 (V ) ∩ H 2 (V ). For such a result one needs local Carleman estimates at the boundary of the open set V as proven in Sect. 3.4. In Sects. 3.5 and 3.6, we shall see how such estimates, at the boundary and away from the boundary, can be patched together to form a global estimate in the case of a bounded open set along with boundary conditions. We shall see that such global estimates require an observation term on the r.h.s. of the Carleman estimate. This observation term is not present in (3.3.1). Remark 3.13. The Carleman estimate of Theorem 3.11 is written for the operator P . If one now considers the operator P˜ given by

˜ bi (x)Di + c˜(x), P˜ = P + 1≤i≤d

where ˜bi , c˜ ∈

L∞ (Rd ),

1 ≤ i ≤ d, we observe that 2

eτ ϕ P u2L2 (Rd )  eτ ϕ P˜ uL2 (Rd ) + eτ ϕ u2L2 (Rd ) + eτ ϕ Du2L2 (Rd ) , and by taking τ ≥ τ∗ with τ∗ sufficiently large, we obtain

τ 3 eτ ϕ u2L2 (Rd ) + τ eτ ϕ Du2L2 (Rd ) + τ −1 eτ ϕ Dα u2L2 (Rd ) |α|=2

≤ Ce

τϕ

2 P˜ uL2 (Rd ) .

Hence, obtaining the Carleman estimate for P yields the same estimate (possibly with different constants C and τ∗ ) for P˜ . Note that the powers of the parameter τ in the different terms on the l.h.s. of (3.3.1) are important in that matter. Remark 3.14. The sub-ellipticity property of Definition 3.2 appears as a sufficient condition to derive the Carleman estimate of Theorem 3.11. We shall see in Theorems 4.5 and 4.7 in Chap. 4 that this condition is in fact necessary to achieve an estimate of the form of (3.3.1). Proof of Theorem 3.11. Observe that, by Remark 3.13 and the form of the estimate we wish to prove, we may use P0 in place of P , with P0 given in (3.1.1) without any loss of generality. We set v = eτ ϕ u. Then, P0 u = f is equivalent to Pϕ v = g = eτ ϕ f or rather P2 v + iP1 v = g. We then obtain (3.3.2)

g2L2 (Rd ) = P1 v2L2 (Rd ) + P2 v2L2 (Rd ) + 2 Re(P2 v, iP1 v)L2 (Rd ) .

As (Pj w1 , w2 )L2 (Rd ) = (w1 , Pj w2 )L2 (Rd ) for w1 , w2 ∈ Cc∞ (V ), we have   2 Re(P2 v, iP1 v)L2 (Rd ) = i [P2 , P1 ]v, v L2 (Rd ) . Let μ > 0 to be fixed below and τ ≥ μ. We write   Pj v2L2 (Rd ) ≥ μτ −1 Pj v2L2 (Rd ) = μτ −1 Pj2 v, v L2 (Rd ) and thus obtain    τ −1 μ(P12 + P22 ) + iτ [P2 , P1 ] v, v

L2 (Rd )

≤ g2L2 (Rd ) .

3.4. LOCAL ESTIMATES AT THE BOUNDARY

71

The principal symbol of the fourth-order operator μ(P12 + P22 ) + iτ [P2 , P1 ] ∈ Dτ4 is ρ = μ(p21 + p22 ) + τ {p2 , p1 } ∈ Sτ4 . Lemma 3.8 gives ρ(x, ξ, τ )  λ4τ for μ > 0 chosen sufficiently large and kept fixed, since the sub-ellipticity property of Definition 3.2 holds. With this positivity result and the G˚ arding inequality of Theorem 2.28, we then obtain τ −1 v2τ,2  g2L2 (Rd ) ,

(3.3.3)

for τ chosen sufficiently large, which reads

Dα v2L2 (Rd )  eτ ϕ f 2L2 (Rd ) . τ 3 v2L2 (Rd ) + τ Dv2L2 (Rd ) + τ −1 |α|=2

Moving back to the function u, we then write eτ ϕ Dj u = (Dj + iτ ∂j ϕ)v,

eτ ϕ Dj Dk u = (Dj + iτ ∂j ϕ)(Dk + iτ ∂k ϕ)v,

yielding (3.3.4)

τ eτ ϕ Du2L2 (Rd )  τ 3 v2L2 (Rd ) + τ Dv2L2 (Rd ) ,

eτ ϕ Dα u2L2 (Rd )  τ 3 v2L2 (Rd ) + τ Dv2L2 (Rd ) τ −1 |α|=2

+τ −1

|α|=2

Dα v2L2 (Rd ) ,

since the derivatives of ϕ are bounded in V . This concludes the proof.



3.4. Local Estimates at the Boundary Here, we consider Ω a smooth open set of Rd in the sense given in Sect. 1.7 and P as in (3.1.2). To prove a Carleman estimate for functions defined in a bounded domain Ω, we can use the previous section to achieve a local estimate in the neighborhood of any point away from the boundary. Yet, we also need a similar estimate in the neighborhood of any point of the boundary of Ω. This is the topic of the present section. For W an open subset of Rd , we set Ω

Cc∞ (W ) = {u = v|W ∩Ω ; v ∈ Cc∞ (Rd ), supp(v) ⊂ W }. We denote by ν the unitary outward-pointing normal vector to ∂Ω. Then, ∂ν v|∂Ω stands for the associated normal derivative at the boundary of a function v, in the sense of the symmetric matrix (pij ), that is,

(3.4.1) ν i pij ∂j v|∂Ω . ∂ν v|∂Ω = 1≤i,j≤d

72

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

Note that if we set ∂ν v|∂Ω = ν · ∇v, the statements of the results of this section would be identical2. We shall prove the following two lemmata. Lemma 3.15. Let V 0 be a bounded open set in Rd such that the boundary ∂Ω is C ∞ in a neighborhood of V 0 , and let ϕ and P have the sub-ellipticity property of Definition 3.2 in a neighborhood of V 0 ∩ Ω. Let y ∈ ∂Ω ∩ V 0 . There exist an open neighborhood V 1 of y in Rd , τ∗ > 0, and C > 0 such that  (3.4.2) τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) ≤ C eτ ϕ P u2L2 (Ω)  + τ 3 |eτ ϕ u|∂Ω |2L2 (∂Ω) + τ |eτ ϕ u|∂Ω |2H 1 (∂Ω) + τ |eτ ϕ ∂ν u|∂Ω |2L2 (∂Ω) , Ω

for u ∈ Cc∞ (V 1 ∩ V 0 ) and τ ≥ τ∗ . The previous lemma is useful if the boundary traces, u|∂Ω and ∂ν u|∂Ω , are known. If the solution satisfies a particular boundary condition, usually one can prove a better estimate. In the following lemma we give a result for a Dirichlet boundary condition that improves upon Lemma 3.15 because of a choice made on the sign of ∂ν ϕ: with the Dirichlet trace one estimates the Neumann trace. Carleman estimates for more general boundary conditions will be proven in Chap. 8 in Volume 2. Lemma 3.16. Let V 0 be a bounded open set in Rd such that the boundary ∂Ω is C ∞ in a neighborhood of V 0 , and let ϕ and P have the sub-ellipticity property of Definition 3.2 in a neighborhood of V 0 ∩ Ω. Let y ∈ ∂Ω ∩ V 0 such that ∂ν ϕ(y) < 0. There exist an open neighborhood V 1 of y in Rd , τ∗ > 0, and C > 0 such that (3.4.3) τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|∂Ω |2L2 (∂Ω)   ≤ C eτ ϕ P u2L2 (Ω) + τ 3 |eτ ϕ u|∂Ω |2L2 (∂Ω) + τ |eτ ϕ u|∂Ω |2H 1 (∂Ω) , Ω

for u ∈ Cc∞ (V 1 ∩ V 0 ) and τ ≥ τ∗ . Note in particular that this lemma, in the case of functions satisfying homogeneous Dirichlet boundary conditions u|∂Ω = 0, yields the most classical form of local Carleman estimates for second-order elliptic operator at a boundary: (3.4.4) τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|∂Ω |2L2 (∂Ω) ≤ Ceτ ϕ P u2L2 (Ω) , for τ ≥ τ∗ . 2We choose to rather use (3.4.1) as it is consistent with a geometrical point of view

where (pij ) is associated with a Riemannian metric. We refer to Chap. 5 in Volume 2 for those developments. See also Sect. 18.9 in Volume 2 that exposes how the Riemannian point of view can be used in the case of operator such as P in an open set of Rd .

3.4. LOCAL ESTIMATES AT THE BOUNDARY

73

3.4.1. Some Remarks. Remark 3.17. In both Lemmata 3.15 and 3.16, we have the occurrence of an open neighborhood V 1 that reduces the analysis near the point y. In fact, if all the required properties are valid in V 0 ∩ Ω, upon patching estimates together, the introduction of V 1 need not be made in the two statements. For the sake of the exposition we decided to postpone those results to Sect. 3.5. Remark 3.18. Observe that we do not estimate the second-order derivatives of the function u in Lemmata 3.15 and 3.16, whereas it is done in the estimate away from the boundary of Theorem 3.11. In fact, an estimation of eτ ϕ Dα u2L2 (Rd ) with |α| = 2 can also be achieved up to the boundary ∂Ω. Yet, this requires to replace the tangential estimates of order 1 for u|∂Ω and of order 0 for ∂ν u|∂Ω on the r.h.s. of (3.4.2) and (3.4.3) by estimates of order 3/2 and 1/2, respectively, as can be expected from Sobolev trace formulae. For such improved inequalities at the boundary we refer to Chap. 8 in Volume 2. Remark 3.19. Observe that a weight function ϕ as in Lemma 3.16 can be easily obtained. Upon possibly reducing the size of V 0 , there exists ˜ = 0 and V 0 ∩ Ω is locally given by ψ˜ : Rd → R of class C ∞ such that dψ(y) ˜ {ψ(x) > 0}. Upon reducing the size of V 0 a second time, we may further ˜ for ˜ assume that |dψ(x)| > 0 in V 0 . A possible choice is then ϕ = exp(γ ψ) γ chosen sufficiently large by Lemma 3.5. We then have ϕ = Cst on the boundary. Observe that the convexity of the level sets of ϕ with respect to the boundary ∂Ω can be further enforced by for instance setting ψ(x) = ˜ ψ(x) − α|x − y|2 , with α > 0 and ϕ = exp(γψ). See Fig. 3.2. Remark 3.20. Note that estimate (3.4.4) is not valid if we have ∂ν ϕ(y) > 0. For such a question we refer to Theorem 4.25 and Corollary 4.26 in Sect. 4.3.

∂Ω = {ψ˜ = 0} ψ=0 y ψ = Cst > 0 ψ˜ = Cst < 0

Figure 3.2. Level of a weight function ψ with convexity with respect to the boundary ∂Ω given by ψ˜ = 0

74

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

Remark 3.21. In both theorems, the r.h.s. of the Carleman estimates exhibits a term involving the H 1 -norm of the trace eτ ϕ u on ∂Ω. The boundary ∂Ω is a submanifold of Rd of codimension one. One thus needs to properly define such a Sobolev norm on a submanifold. This is done for instance in Sect. 18.2 in Volume 2. Below, we shall use local coordinates in which ∂Ω is given by {xd = 0} and Ω = {xd > 0}. Then, the H 1 -norm on ∂Ω is easily defined. In the present section, we shall not give much details on how the estimation in these local coordinates implies the estimation in the original coordinates, in particular for this H 1 -norm at the boundary. The interested reader is referred to Sect. 5.3 of Volume 2 where the derivation of a Carleman estimate is carried out in the case of a Riemannian manifold. Note that, as is often done, one can be inclined to replace the term |eτ ϕ u|∂Ω |2H 1 (∂Ω) on the r.h.s. of the estimations by 2

|eτ ϕ u|∂Ω |2L2 (∂Ω) + |eτ ϕ ∇T u|∂Ω |L2 (∂Ω) , where ∇T u|∂Ω is some tangential gradient of u|∂Ω , meaning differentiation in the directions associated with a frame of tangential vector fields along ∂Ω. First, this object needs to be properly defined; second, a proper L2 -norm needs to be introduced. Above, we wrote |.|L2 (∂Ω) , but this is rather loose at this stage. This issue is addressed in Sect. 5.3 of Volume 2 where such a formulation is used in the case of a Riemannian manifold. 3.4.2. Proofs in Adapted Local Coordinates. For the proofs of Lemmata 3.15 and 3.16 it is more convenient to work in local coordinates yielding a flat boundary and a simple form for the principal part of the operator P . For a point y where ∂Ω is C ∞ there exists an open neighborhood V ⊂ V 0 ⊂ Rd and local coordinates x = (x , xd ) ∈ Rd , where x = (x1 , . . . , xd−1 ) ∈ Rd−1 and y = 0, in which the open set Ω ∩ V is given by {xd > 0} and the operator P takes the form (3.4.5)

R(x, D ) = Di (bij (x)Dj ), P˜ = Dd2 + R(x, D ) + R1 (x, D),

1≤i,j≤d−1

ij  2 where bij = bji and 1≤i,j≤d−1 b ξi ξj ≥ C|ξ | , and R1 (x, D) is a firstorder differential operator with bounded coefficients. Here D stands for D = Dx = (Dx1 , . . . , Dxd−1 ). This local reduction is classical, and a proof is written in Sect. 9.4. Such coordinates are called normal geodesic coordinates.

In what follows, by abuse of notation, we shall write P instead of P˜ . No confusion shall however arise. We set

R(x, D ) = Di (bij (x)Dj ) (3.4.6) P0 = Dd2 + R(x, D ), 1≤i,j≤d−1

as the principal part of P . Note that we purposely choose this principal part to be formally self-adjoint. Note that P0 differs in general by lower terms

3.4. LOCAL ESTIMATES AT THE BOUNDARY

75

from the expression of the operator defined in (3.1.1) in the chosen local coordinates. This has no consequence for the estimates we wish to prove as explained in Remark 3.13. In the chosen normal geodesic coordinates, observe that we have ∂ν = −∂d at the boundary {xd = 0}. Note that if the weight function ϕ and P have the sub-ellipticity property of Definition 3.2, then, according to Proposition 3.10, this property is preserved in the new coordinates. We keep the notation of the sections above for the principal symbol of the operator

r(x, ξ  ) = bij (x)ξi ξj , p(x, ξ) = ξd2 + r(x, ξ  ), 1≤i,j≤d−1

and the associated bilinear forms p˜(x, ξ, η) = ξd ηd + r˜(x, ξ  , η  ),

r˜(x, ξ  , η  ) =

1≤i,j≤d−1

bij (x)ξi ηj .

We also define Pϕ = eτ ϕ P0 e−τ ϕ = P2 + iP1 ∈ Dτ2 where P1 and P2 are P2 = P0 − p(x, τ dϕ),  P1 = τ ∂d ϕDd + Dd ∂d ϕ +

(3.4.7) (3.4.8)



1≤i,j≤d−1

   Di bij ∂j ϕ + bij ∂i ϕDj .

Note that P1 and P2 are formally self-adjoint (like their counterparts in Sects. 3.2 and 3.3). We have the associated (real) principal symbols as in (3.2.4) and (3.2.5) with the above notation Sτ2  p2 (x, ξ, τ ) = p(x, ξ) − p(x, τ dϕ(x)) = ξd2 + q˜2 (x, ξ  , τ ),

(3.4.9)

(3.4.10) Sτ2 ⊃ τ Sτ1  p1 (x, ξ, τ ) = 2˜ p(x, ξ, τ dϕ(x)) = τ q˜1 (x, ξ), where ST2,τ  q˜2 (x, ξ  , τ ) = r(x, ξ  ) − p(x, τ dϕ(x)), S 1  q˜1 (x, ξ) = 2∂d ϕξd + 2˜ r(x, ξ  , dx ϕ),

OpT (˜ q2 ) ∈ DT2,τ ,

Op(˜ q1 ) ∈ D 1 .

Let U be an open subset of Rd , such that 0 ∈ U we denote U+ = U ∩ Rd+ . We define two spaces of functions smooth up the boundary, ∞

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

and (3.4.11)

∞

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

To distinguish the L2 -inner products and L2 -norms in Rd and in Rd−1 in the statements and the proof below and to lighten notation for u and v defined on {xd = 0}, we set

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3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

(u, v)∂ = ∫ u(x )¯ v (x )dx , Rd−1

|u|∂ = |u|L2 (Rd−1 ) .

For u and v defined in Rd+ = {xd > 0}, we also recall the notation v (x)dx, (u, v)+ = ∫ u(x)¯ Rd+

u+ = uL2 (Rd ) . +

Proving Lemmata 3.15 and 3.16 is then equivalent to proving the following two lemmata in the local coordinates we have introduced. Lemma 3.22. Let ϕ and P have the sub-ellipticity property of Definition 3.2 in V + , with V open neighborhood of 0 in Rd . Then, there exists an open neighborhood U of 0 in Rd such that U ⊂ V , and there exist τ∗ > 0 and C > 0 such that  τ 3 eτ ϕ u2+ + τ eτ ϕ Du2+ ≤ C eτ ϕ P u2+ + τ 3 |eτ ϕ u|xd =0+ |2∂  2 + τ |eτ ϕ D u|xd =0+ |∂ + τ |eτ ϕ Dd u|xd =0+ |2∂ , ∞

for u ∈ C c (U+ ) and τ ≥ τ∗ . Lemma 3.23. Let ϕ and P have the sub-ellipticity property of Definition 3.2 in V + , with V open neighborhood of 0 in Rd . Further assume that ∂d ϕ(0) > 0; then, there exists an open neighborhood U of 0 in Rd such that U ⊂ V and there exist τ∗ > 0 and C > 0 such that τ 3 eτ ϕ u2+ + τ eτ ϕ Du2+ + τ |eτ ϕ Dd u|xd =0+ |2∂   2 ≤ C eτ ϕ P u2+ + τ 3 |eτ ϕ u|xd =0+ |2∂ + τ |eτ ϕ D u|xd =0+ |∂ , ∞

for u ∈ C c (U+ ) and τ ≥ τ∗ . Observe that, by Remark 3.13 and the form of the two local Carleman estimates we wish to prove, we may replace P by P0 , with P0 given in (3.4.6) without any loss of generality. This will be done in the proofs below. Lemmata 3.22 and 3.23 are consequences of the following more general proposition. Proposition 3.24. Let ϕ and P have the sub-ellipticity property of Definition 3.2 in V + , with V open neighborhood of 0 in Rd . There exists an open neighborhood U of 0 in Rd such that U ⊂ V and there exist τ∗ > 0 and C > 0 such that Cτ 3 v2+ + Cτ Dv2+ + τ Re B(v) ≤ Pϕ v2+

(3.4.12) ∞

for v ∈ C c (U+ ) and τ ≥ τ∗ where B(v) = 2(∂d ϕDd v|xd =0+ , Dd v|xd =0+ )∂ + (A1 v|xd =0+ , Dd v|xd =0+ )∂ + (Dd v|xd =0+ , A1 v|xd =0+ )∂ + (A2 v|xd =0+ , v|xd =0+ )∂ .

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77

The operators A1 , A1 , and A2 are pseudo-differential operators in the x direction, with differential principal parts: (1) A1 , A1 ∈ Ψ1T,τ and their principal symbols coincide. We denote it by a1 and it is given by r(x, ξ  , dx ϕ). a1 (x, ξ  ) = 2˜ (2) A2 ∈ Ψ2T,τ and its principal symbol, a2 , is given by   a2 (x, ξ  , τ ) = 2∂d ϕ p(x, τ dϕ) − r(x, ξ  ) . Observe that in this proposition, there is no assumption made on the sign of ∂d ϕ at xd = 0. Proof of Lemmata 3.22 and 3.23. By the form of B(v) given in Proposition 3.24 we have |B(v)|  |Dd v|xd =0+ |2∂ + |Dx v|xd =0+ |2∂ + τ 2 |v|xd =0+ |2∂ . This and Proposition 3.24 imply the result of Lemma 3.22, by considering u = eτ ϕ v and arguing as in (3.3.4) at the end of the proof of Theorem 3.11. With the Young inequality we have |B(v) − 2(dd ϕDd v, Dd v)∂ |  τ 2 |v|xd =0+ |2∂ + |Dx v|xd =0+ |2∂ + |Dx v|xd =0+ |∂ |Dd v|xd =0+ |∂  τ 2 |v|xd =0+ |2∂ + (1 + δ −1 )|Dx v|xd =0+ |2∂ + δ|Dd v|xd =0+ |2∂ . If ∂d ϕ > 0 and if we take δ sufficiently small, by Proposition 3.24, we obtain the result of Lemma 3.23, again by considering u = eτ ϕ v and arguing as in (3.3.4).  Proof of Proposition 3.24. We use in the proof the following integration by parts formula (3.4.13)

(Dd u, w)+ = (u, Dd w)+ + i(u|xd =0+ , w|xd =0+ )∂ ,

for u and w in S (Rd+ ) with U open subset of Rd with U ⊂ V to be determined below. Following the previous formula we deduce (Dd2 u, w)+ = (u, Dd2 w)+ + i(u|xd =0+ , Dd w|xd =0+ )∂ + i(Dd u|xd =0+ , w|xd =0+ )∂ , yielding (3.4.14)

(P2 u, w)+ = (u, P2 w)+ + i(u|xd =0+ , Dd w|xd =0+ )∂ + i(Dd u|xd =0+ , w|xd =0+ )∂ ,

(3.4.15)

  (P1 u, w)+ = (u, P1 w)+ + 2iτ (∂d ϕ)u|xd =0+ , w|xd =0+ ∂ .

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3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

We can now compute Pϕ v2+ as in (3.3.2). We have (3.4.16)

Pϕ v2+ = P2 v + iP1 v2+

  = P2 v2+ + P1 v2+ + i (P1 v, P2 v)+ − (P2 v, P1 v)+ .

By (3.4.14) and (3.4.15), we obtain (3.4.17)

i(P1 v, P2 v)+ = i(P2 P1 v, v)+ + (P1 v|xd =0+ , Dd v|xd =0+ )∂

(3.4.18)

+ (Dd P1 v|xd =0+ , v|xd =0+ )∂ ,   i(P2 v, P1 v)+ = i(P1 P2 v, v)+ + 2τ (∂d ϕ)P2 v|xd =0+ , v|xd =0+ ∂ .

With (3.4.17)–(3.4.18) we find   i (P1 v, P2 v)+ − (P2 v, P1 v)+ (3.4.19)   = Re i[P2 , P1 ]v, v + + Re(P1 v|xd =0+ , Dd v|xd =0+ )∂   + Re (Dd P1 − 2τ (∂d ϕ)P2 )v|xd =0+ , v|xd =0+ ∂ , using that the l.h.s. is real as can be seen in (3.4.16). To characterize the trace terms we need the following lemma. Lemma 3.25. The operators P1 ∈ τ D 1 and Dd P1 − 2τ ∂d ϕP2 ∈ Dτ3 can be cast in the following forms r(x, D , τ dx ϕ) P1 = 2τ ∂d ϕDd + 2˜

mod τ D 0 ,

and r(x, D , τ dx ϕ)Dd Dd P1 − 2τ ∂d ϕP2 = 2˜   − 2τ ∂d ϕ R(x, D ) − p(x, τ dϕ)

  mod τ D 0 Dd + DT1,τ .

Proof. Recalling the form of P1 in (3.4.8) the first result in a consequence of applying some commutators. Similarly, note that we also have r(x, D , τ dx ϕ) P1 = 2τ Dd ∂d ϕ + 2˜

mod τ D 0 ,

yielding, as Dd D 0 = D 0 Dd + D 0 , Dd P1 = 2τ Dd2 ∂d ϕ + 2Dd r˜(x, D , τ dx ϕ)

mod τ (D 0 Dd + D 0 ).

We then find

  r(x, D , τ dx ϕ)Dd − 2τ ∂d ϕ R(x, D ) − p(x, τ dϕ) Dd P1 − 2τ ∂d ϕP2 = 2˜ + 2τ [Dd2 , ∂d ϕ] + 2[Dd , r˜(x, D , τ dx ϕ)]

mod τ (D 0 Dd + D 0 ).

We have [Dd2 , ∂d ϕ] ∈ D 0 Dd + D 0 and [Dd , r˜(x, D , τ dx ϕ)] ∈ τ DT1,τ , which yields the result.  From (3.4.19) with the previous lemma we obtain     ˜ (3.4.20) i (P1 v, P2 v)+ − (P2 v, P1 v)+ = Re i[P2 , P1 ]v, v + + τ Re B(v),

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79

with (3.4.21) ˜ B(v) = 2(∂d ϕDd v|xd =0+ , Dd v|xd =0+ )∂ + 2(˜ r(x, D , dx ϕ)v|xd =0+ , Dd v|xd =0+ )∂   + 2 r˜(x, D , dx ϕ)Dd v|xd =0+ , v|xd =0+ ∂     − 2 ∂d ϕ R(x, D ) − p(x, τ dϕ) v|xd =0+ , v|xd =0+ ∂ + (Op(c0 )v|xd =0+ , Dd v|xd =0+ )∂    + Op(˜ c0 )Dd + Op(c1 ) v|xd =0+ , v|xd =0+ ∂ , with Op(c0 ), Op(˜ c0 ) ∈ D 0 and Op(c1 ) ∈ DT1,τ . Using that  ∗ r˜(x, D , τ dx ϕ) = r˜(x, D , τ dx ϕ) mod τ D 0 , ˜ we see that B(v) is of the same form as that of B(v) in the statement of the proposition. Equality (3.4.16) now becomes   ˜ (3.4.22) Pϕ v2+ = P2 v2+ + P1 v2+ + Re i[P2 , P1 ]v, v + + τ Re B(v). We now compute the commutator i[P2 , P1 ]. Its principal symbol, {p2 , p1 }, is polynomial of degree 3 in (ξ, τ ). By (3.4.10), p1 (x, ξ, τ ) takes the form τ q˜1 (x, ξ) where q˜1 is polynomial of degree 1 in ξ. We thus find, using for instance Remark 2.46-(2) in Chap. 2,   (3.4.23) {p2 , p1 } = τ ˜b0 (x)ξd2 + ˜b1 (x, ξ  , τ )ξd + ˜b2 (x, ξ  , τ ) , where ˜bj are real polynomials of degree j in (ξ  , τ ), j = 0, 1, 2. We now distinguish two cases: ∂d ϕ(0) = 0 or ∂d ϕ(0) = 0. The proof is quite different in each case. We start with the simpler case, ∂d ϕ(0) = 0. Case 1: ∂d ϕ(0) = 0. We choose an open set U such that 0 ∈ U , U ⊂ V , with moreover ∂d ϕ(x) = 0 in U + . Recalling that (3.4.24)

p2 (x, ξ, τ ) = ξd2 + q˜2 (x, ξ  , τ ),

(3.4.25)

p1 (x, ξ, τ ) = τ q˜1 (x, ξ),

OpT (˜ q2 ) ∈ DT2,τ ,

q˜1 (x, ξ) = 2∂d ϕξd + 2˜ r(x, ξ  , dx ϕ),

and Op(˜ q1 ) ∈ D 1 , and as ∂d ϕ = 0 we can then write   τ ξd = (2∂d ϕ)−1 p1 (x, ξ, τ ) − 2τ r˜(x, ξ  , dx ϕ) . This allows us to substitute the occurrences of τ ξd and ξd2 in (3.4.23) by that of p1 and p2 , up to some tangential symbols, yielding (3.4.26) {p2 , p1 } = τ b0 (x)p2 (x, ξ, τ ) + b1 (x, ξ  , τ )p1 (x, ξ, τ ) + τ b2 (x, ξ  , τ ), where bj are real polynomials of order j in (ξ  , τ ), j = 0, 1, 2. This form is useful as the first two symbols will yield terms that will be “absorbed” by P2 v+ and P1 v+ in (3.4.22), upon taking the parameter τ sufficiently large. We thus focus our analysis on the last term b2 in (3.4.26).

80

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

By symbol calculus, formula (3.4.26) can be written3 τ OpT (b2 ) = i[P2 , P1 ] − τ b0 P2 − OpT (b1 )P1 − τ Op(c1 ), q1 ) with Op(˜ q1 ) ∈ D 1 . Then, where Op(c1 ) ∈ Dτ1 , exploiting that P1 = τ Op(˜ Equality (3.4.22) becomes   ˜ (3.4.27) Pϕ v2+ = P2 v2+ + P1 v2+ + Re τ OpT (b2 )v, v + + τ Re B(v)   + Re (τ b0 P2 + OpT (b1 )P1 + τ Op(c1 ))v, v + .   We now wish to bound the term Re τ OpT (b2 )v, v + from below yielding some tangential norm of v. The sub-ellipticity property of Definition 3.2 satisfied by the weight function and the operator P reads here, in view of (3.4.26), p2 (x, ξ, τ ) = p1 (x, ξ, τ ) = 0 ⇒ τ b2 (x, ξ  , τ ) ≥ Cλ3τ . Setting ρ(x, ξ  , τ ) = r˜2 (x, ξ  , dx ϕ) + (∂d ϕ)2 q˜2 (x, ξ  , τ ),

Op(ρ) ∈ DT2,τ ,  −1 and observing that p1 (x, ξ, τ ) = 0 is equivalent to ξd = − ∂d ϕ r˜(x, ξ  , dx ϕ), we find  −1 ρ(x, ξ  , τ ) = 0 and ξd = − ∂d ϕ r˜(x, ξ  , dx ϕ) ⇔ p2 (x, ξ, τ ) = p1 (x, ξ, τ ) = 0. We conclude that (3.4.28)

ρ(x, ξ  , τ ) = 0 ⇒ τ b2 (x, ξ  , τ ) ≥ Cλ3T,τ ,

λT,τ = |(ξ  , τ )|.

As q˜2 (x, ξ  , τ ) = r(x, ξ  ) − τ 2 p(x, dϕ(x)), writing ρ(x, ξ  , τ ) = r˜2 (x, ξ  , dx ϕ) + (∂d ϕ)2 r(x, ξ  ) − τ 2 (∂d ϕ)2 p(x, dϕ(x)), and as here ∂d ϕ = 0, we note that τ  |ξ  | if ρ = 0. Hence by (3.4.28), we obtain (3.4.29)

ρ(x, ξ  , τ ) = 0 ⇒ b2 (x, ξ  , τ ) ≥ Cλ2T,τ ,

λT,τ = |(ξ  , τ )|.

We can now give a lemma that is analogous to Lemma 3.8. Lemma 3.26. Let b2 (x, ξ  , τ ) be as given in (3.4.26). If the weight function ϕ and P have the sub-ellipticity property of Definition 3.2, then there exist μ0 > 0 and C > 0 such that 2   2 μλ−2 T,τ ρ (x, ξ , τ ) + b2 (x, ξ , τ ) ≥ CλT,τ ,

for all (x, ξ  ) ∈ U+ × Rd−1 , τ ≥ 1, and μ ≥ μ0 .

3Observe that tangential operators and operators acting in all directions are composed. Yet, both operators are differential, which makes this composition natural. See Remark 2.46 in Chap. 2.

3.4. LOCAL ESTIMATES AT THE BOUNDARY

81

A proof is given in Appendix 3.B.2. We can use the tangential G˚ arding inequality of Theorem 2.49. We then ∞ deduce that there exists C > 0 such that, for all v ∈ C c (U+ ), μ ≥ μ0 fixed in what follows, and τ sufficiently large     2 2 1 Re μOpT (λ−2 (3.4.30) T,τ ρ )v, v + + Re OpT (b2 )v, v + ≥ CΛT,τ v+ . By symbol calculus we write4 −2 2 μOpT (λ−2 T,τ ρ ) = μOpT (λT,τ ρ)OpT (ρ) + OpT (c1 ),

yielding (3.4.31)

c1 ∈ ST1,τ ,

  2 Re μOpT (λ−2 T,τ ρ )v, v +     = Re OpT (ρ)v, OpT (c0 )v + + Re OpT (c1 )v, v + ,

 where the principal part of the tangential symbol c0 ∈ ST0,τ is μλ−2 T,τ ρ(x, ξ , τ ). By formulae (3.4.24) and (3.4.25), we have

(3.4.32)

q˜2 (x, ξ  , τ ) = p2 (x, ξ, τ ) − ξd2 and ξd =

q˜1 (x, ξ) r˜(x, ξ  , dx ϕ) − . 2∂d ϕ ∂d ϕ

Then, we obtain ρ(x, ξ  , τ ) = r˜2 (x, ξ  , dx ϕ) + (∂d ϕ)2 q˜2 (x, ξ  , τ ) = r˜2 (x, ξ  , dx ϕ) + (∂d ϕ)2 p2 (x, ξ, τ )  q˜ (x, ξ) r˜(x, ξ  , d  ϕ) 2 1 x − (∂d ϕ)2 − 2∂d ϕ ∂d ϕ 1 = (∂d ϕ)2 p2 (x, ξ, τ ) − q˜12 (x, ξ) + q˜1 (x, ξ)˜ r(x, ξ  , dx ϕ) 4 = r0 (x)p2 (x, ξ, τ ) + τ −1 r1 (x, ξ)p1 (x, ξ, τ ), where (3.4.33)

r0 (x) = (∂d ϕ(x))2 ,

 1 1 r1 (x, ξ) = r˜(x, ξ  , dx ϕ) − q˜1 (x, ξ) = r˜(x, ξ  , dx ϕ) − ∂d ϕξd . 4 2 The composition of differential operators (see Remark 2.24) thus gives (3.4.34)

(3.4.35)

OpT (ρ) − r0 P2 − τ −1 Op(r1 )P1 = Op(d1 ) ∈ Dτ1 .

With (3.4.30), (3.4.31), and (3.4.35), we find (3.4.36) 2 Re(OpT (b2 )v, v)+ ≥ CΛ1T,τ v+ − Re(OpT (c1 )v, v)+ − Re(r0 P2 v + τ −1 Op(r1 )P1 v + Op(d1 )v, OpT (c0 )v)+ .

b

ij

4For the operator Op (λ−2 ρ2 ) to be well-defined we extend smoothly the coefficients T T,τ

introduced in (3.4.5) to the whole half-space Rd+ .

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3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

Observe that operators acting in all directions, including the xd -direction, are differential here, which makes the above analysis on the half-space Rd+ sensible. Equality (3.4.27) now becomes (3.4.37) 2 ˜ Pϕ v2+ ≥ Cτ Λ1T,τ v+ + P2 v2+ + P1 v2+ + τ Re B(v)   + Re (τ b0 P2 + OpT (b1 )P1 + τ Op(c1 ))v, v + − τ Re(OpT (c1 )v, v)+ − Re(τ r0 P2 v + Op(r1 )P1 v + τ Op(d1 )v, OpT (c0 )v)+ . With (3.4.33)–(3.4.34) and by (3.4.13), we write Re(˜ r(x, D , dx ϕ)P1 v, OpT (c0 )v)+ − 2 Re(Op(r1 )P1 v, OpT (c0 )v)+ = Re(∂d ϕDd P1 v, OpT (c0 )v)+ + 2τ Re(Op(c1 )v, v)+ = Re(P1 v, Dd ∂d ϕOpT (c0 )v)+ − Im(P1 v|xd =0+ , ∂d ϕOpT (c0 )v|xd =0+ )∂ + 2τ Re(Op(c1 )v, v)+ ,

where Op(c1 ) ∈ Dτ1 . We set 1 ˜ B(v) = − τ −1 Im(P1 v|xd =0+ , ∂d ϕOpT (c0 )v|xd =0+ )∂ + B(v), 2 which is precisely of the form given in the statement of the proposition, ˜ observing that the term we add to B(v) only affects lower-order terms. With the Young inequality, we have    Re(τ b0 P2 v + OpT (b1 )P1 v, v)+   1 − Re τ r0 P2 v + r˜(x, D , dx ϕ)P1 v + τ Op(d1 )v, OpT (c0 )v 2 +     1 + Re(P1 v, Dd ∂d ϕOpT (c0 )v)+ + τ Re Op(c1 + c1 ) − OpT (c1 ) v, v +  2     2  τ −1/2 P2 v2+ + P1 v2+ + τ 1/2 Λ1T,τ v+ + Dd v2+ , where we use that [Dd , ∂d ϕOpT (c0 )] ∈ Ψ0T,τ . Inequality (3.4.37) now becomes (3.4.38)

2

Pϕ v2+ ≥ Cτ Λ1T,τ v+ + P2 v2+ + P1 v2+ + τ Re B(v)   − C  τ −1/2 P2 v2+ + P1 v2+   2 − C  τ 1/2 Λ1T,τ v+ + Dd v2+ .

For τ sufficiently large we thus obtain 1 2 (3.4.39) Pϕ v2+ ≥ Cτ Λ1T,τ v+ + P1 v2+ + τ Re B(v) − C  τ 1/2 Dd v2+ . 2 From (3.4.32) we have (3.4.40)

2

τ Dd v2+  τ −1 P1 v2+ + τ Λ1T,τ v+ .

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83

From (3.4.39) and (3.4.40), taking τ sufficiently large, we conclude the proof of Proposition 3.24 in the case ∂d ϕ(0) = 0. Case 2: ∂d ϕ(0) = 0. We consider an open set U = Uε such that 0 ∈ U , U ⊂ V , with moreover |∂d ϕ(x)| ≤ ε in U + , with ε > 0 to be chosen below. Following (3.4.23) we have τ −1 {p2 , p1 } = ˜b0 (x)ξd2 + ˜b1 (x, ξ  , τ )ξd + ˜b2 (x, ξ  , τ ), p2 (x, ξ, τ ) = ξd2 + q˜2 (x, ξ  , τ ),

OpT (˜ q2 ) ∈ DT2,τ ,

where ˜bj are real polynomials of degree j in (ξ  , τ ), j = 0, 1, 2. Then, we can write (3.4.41)

τ −1 {p2 , p1 } = b0 (x)p2 (x, ξ, τ ) + b1 (x, ξ  , τ )ξd + b2 (x, ξ  , τ ),

where bj are real polynomials of order j in (ξ  , τ ), j = 0, 1, 2. By symbol calculus and by (3.4.41) we have       Re i[P2 , P1 ]v, v + = τ Re b0 (x)P2 v, v + + τ Re OpT (b1 )Dd v, v +     + τ Re OpT (b2 )v, v + + τ Re R1 v, v + , where R1 ∈ Dτ1 . We have (3.4.42)     τ | Re b0 (x)P2 v, v + + Re R1 v, v + |  τ −1/2 P2 v2+ 2

+ τ 1/2 Λ1T,τ v+ + Dd v2+ . Equality (3.4.22) then becomes 1 ˜ Pϕ v2+ ≥ P2 v2+ + P1 v2+ + τ Re B(v) (3.4.43) 2     + τ Re OpT (b1 )Dd v, v + + τ Re OpT (b2 )v, v +   2 − C τ 1/2 Λ1T,τ v+ + Dd v2+ ,  for τ chosen sufficiently large. We now wish to bound the terms τ Re OpT (b1 )   Dd v, v + + τ Re OpT (b2 )v, v + from below yielding some tangential norm of v. The sub-ellipticity property of Definition 3.2 fulfilled by the weight function and the operator P reads here (use Remark 3.3-(2)) p2 (x, ξ, τ ) = p1 (x, ξ, τ ) = 0 ⇒ τ b1 (x, ξ  , τ )ξd + τ b2 (x, ξ  , τ ) ≥ Cλ3τ .  q2 (x, ξ  , τ ). As First, p2 (x, ξ, τ ) = 0 means q˜2 (x, ξ  , τ ) ≤ 0 and ξd = ± −˜  r(x, ξ , dx ϕ), and τ = 0, p1 (x, ξ, τ ) = τ q˜1 (x, ξ), with q˜1 (x, ξ) = 2∂d ϕξd + 2˜ we have ⎧  ⎪ ⎨q˜2 (x, ξ, τ ) ≤ 0, p2 (x, ξ, τ ) = p1 (x, ξ, τ ) = 0 ⇔ q2 (x, ξ  , τ ) + r˜(x, ξ  , dx ϕ) = 0, ±∂d ϕ −˜  ⎪ ⎩ q2 (x, ξ  , τ ). ξd = ± −˜

84

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

 Observe that ξd = ± −˜ q2 (x, ξ  , τ ) implies |ξd | ≤ CλT,τ , which yields λτ  λT,τ . Hence, the sub-ellipticity property can be written equivalently:  q2 (x, ξ  , τ ) + r˜(x, ξ  , dx ϕ) = 0 q˜2 (x, ξ  , τ ) ≤ 0 and ± ∂d ϕ −˜  ⇒ ±τ b1 (x, ξ  , τ ) −˜ q2 (x, ξ  , τ ) + τ b2 (x, ξ  , τ ) ≥ Cλ3T,τ . Note now that q˜2 (x, ξ, τ ) ≤ 0 reads r(x, ξ  ) ≤ p(x, τ dϕ(x)) implying |ξ  | ≤ Cτ . Thus, we have λT,τ  τ . Hence, the sub-ellipticity property can be written equivalently:  q2 (x, ξ  , τ ) + r˜(x, ξ  , dx ϕ) = 0 q˜2 (x, ξ  , τ ) ≤ 0 and ± ∂d ϕ −˜  ⇒ ±b1 (x, ξ  , τ ) −˜ q2 (x, ξ  , τ ) + b2 (x, ξ  , τ ) ≥ Cλ2T,τ . Now at x = 0, as ∂d ϕ(0) = 0, we can choose the sign ± above to be − sgn(b1 ) and the sub-ellipticity property then reads (3.4.44)

q˜2 (0, ξ  , τ ) ≤ 0 and r˜(0, ξ  , dx ϕ(0)) = 0  q2 (0, ξ  , τ ) + C0 λ2T,τ , ⇒ b2 (0, ξ  , τ ) ≥ |b1 (0, ξ  , τ )| −˜

for some C0 > 0. If q˜2 (x, ξ  , τ ) > 0, the sub-ellipticity property yields no positivity information. Depending on the sign of q˜2 (x, ξ  , τ ), we have then to consider two cases. We introduce the cut-off symbol χ(x, ξ  , τ ) ∈ ST0,τ homogeneous of degree 0 for |(ξ  , τ )| ≥ 1 such that 0 ≤ χ ≤ 1, χ(x, ξ  , τ ) = 1 if q˜2 (x, ξ  , τ ) ≤ δ1 λ2T,τ /4, and χ(x, ξ  , τ ) = 0 if q˜2 (x, ξ  , τ ) ≥ δ1 λ2T,τ /2, where δ1 > 0 and will be chosen below. Letting α(ξ  , τ ) ∈ ST0,τ be an elliptic symbol to be fixed below, we have, as b1 is a real symbol,         τ  Re OpT ( χ2 b1 )Dd v, v +  ≤ τ  Re OpT ( χα)Dd v, OpT ( χα−1 b1 )v +  + CDd v2+ + Cτ 2 v2+ τ τ 2 ≤ OpT ( χα)Dd v2+ + OpT ( χα−1 b1 )v+ 2 2 + CDd v2+ + Cτ 2 v2+ . We thus find   τ Re OpT (b1 )Dd v, v +     τ ≥ τ Re OpT ((1 − χ2 )b1 )Dd v, v + − Re OpT ( χ2 α2 )Dd2 v, v + 2   τ 2 −2 2 − Re OpT ( χ α b1 )v, v + + τ Re(Dd v|xd =0+ , OpT (c0 )v|xd =0+ )∂ 2  2

− C Dd v2+ + Λ1T,τ v+ ,

3.4. LOCAL ESTIMATES AT THE BOUNDARY

85

with c0 ∈ ST0,τ . As Dd2 = P2 − OpT (˜ q2 ) mod DT1,τ , by an estimate analogous to (3.4.42), we obtain (3.4.45)   τ Re OpT (b1 )Dd v, v +     τ ≥ τ Re OpT ((1 − χ2 )b1 )Dd v, v + + Re OpT ( χ2 α2 q˜2 )v, v + 2   τ 2 −2 2 − Re OpT ( χ α b1 )v, v + + τ Re(Dd v|xd =0+ , OpT (c0 )v|xd =0+ )∂ 2  2

− C τ −1/2 P2 v2+ + τ 1/2 Λ1T,τ v+ + Dd v2+ . √ q2 would suit our purpose here. In fact Setting α2 equal to |b1 |/ −˜ formally if q˜2 < 0, we find  1 2 1 α q˜2 − α−2 b21 = −|b1 | −˜ q2 . 2 2 One can then anticipate that the two terms     τ τ Re OpT ( χ2 α2 )OpT (˜ q2 )v, v + − Re OpT ( χ2 α−2 b21 )v, v + 2 2 in (3.4.45), together with the term   τ Re OpT (b2 )v, v + in (3.4.43) can yield some positivity by the sub-ellipticity condition written in (3.4.44). √ q2 is not a symbol and is not elliptic. We can Yet, the function |b1 |/ −˜ remedy these drawbacks by a slight modification of this function. With δ > δ1 , a parameter to be fixed below, we set  1/4 b1 (0, ξ  , τ )2 + δλ2T,τ 0 α=  1/8 ∈ ST,τ , 4  2 q˜2 (0, ξ , τ ) + δλT,τ which is an elliptic symbol. From (3.4.44), we have the following lemma whose proof can be found in Appendix 3.B.3. Lemma 3.27. If δ > 0 is chosen sufficiently small, we have q˜2 (0, ξ  , τ ) ≤ 0 and r˜(0, ξ  , dx ϕ(0)) = 0 ⇒ b2 (0, ξ  , τ ) + α(ξ  , τ )2 q˜2 (0, ξ  , τ )/2 − α(ξ  , τ )−2 b1 (0, ξ  , τ )2 /2 ≥ C0 λ2T,τ /2. We now choose the value of δ > 0 according to the previous lemma and keep it fixed in what follows. We set r(x, ξ  , dx ϕ)2 + b2 (x, ξ  , τ ) + α(ξ  , τ )2 q˜2 (x, ξ  , τ )/2 m(x, ξ  , τ ) = μ˜ − α(ξ  , τ )−2 b1 (x, ξ  , τ )2 /2 ∈ ST2,τ .

86

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

Applying Lemma 3.56 on the compact set K defined by K = {(ξ  , τ ) ∈ Rd−1 × R+ ; |ξ  |2 + τ 2 = 1 and q˜2 (0, ξ  , τ ) ≤ 0}, we obtain, for μ > 0 chosen sufficiently large (and to be kept fixed in what follows), taking homogeneity into account, q˜2 (0, ξ  , τ ) ≤ 0 ⇒ m(0, ξ  , τ ) ≥ C1 λ2T,τ , for some C1 > 0. By continuity, if δ1 ∈ (0, δ) is chosen sufficiently small, we have q˜2 (x, ξ  , τ ) ≤ δ1 λ2T,τ

⇒ m(x, ξ  , τ ) ≥ C1 λ2T,τ /2,

for x in a small neighborhood W of 0 and ε > 0 chosen such that U = Uε ⊂ W . By the tangential microlocal G˚ arding inequality of Theorem 2.50, as m(x, ξ  , τ )  λ2T,τ in a conic neighborhood of the support of χ and pseudodifferential calculus, we have, for C  > 0,   2 1/2 2 Re OpT ( χ2 m)v, v + ≥ C  OpT ( χ)Λ1T,τ v+ − CΛT,τ v+ . From (3.4.45) we obtain (3.4.46)     τ Re OpT (b2 )v, v + + τ Re OpT (b1 )Dd v, v +     ≥ τ Re OpT ((1 − χ2 )b2 )v, v + + τ Re OpT ((1 − χ2 )b1 )Dd v, v + 2

+ C  τ OpT ( χ)Λ1T,τ v+ + τ Re(Dd v|xd =0+ , OpT (c0 )v|xd =0+ )∂ − μτ OpT ( χ˜ r)v2+   2 − C τ −1/2 P2 v2+ + τ 1/2 Λ1T,τ v+ + Dd v2+ . As r˜ = (2τ )−1 p1 −∂d ϕξd , writing OpT (χ˜ r) = (2τ )−1 OpT (χ)P1 −Op(∂d ϕχ)Dd 0 mod ΨT,τ and using that |∂d ϕ| ≤ ε in U+ , with Corollary 2.51, we find (3.4.47)

r)v2+  τ −1 P1 v2+ + ε2 τ Dd v2+ + τ v2+ . τ OpT ( χ˜

˜ ≤ 1, χ ˜ = 1 on the support of 1 − χ2 , We let χ ˜ ∈ ST0,τ be such that 0 ≤ χ  2 and supported on q˜2 (x, ξ , τ ) ≥ δ1 λT,τ /8. Using that OpT ((1 − χ2 )b1 ) = OpT (χ(1 ˜ − χ2 )b1 ) = OpT ((1 − χ2 )b1 )OpT (χ) ˜ + R, with R ∈ Ψ0T,τ , we find     (3.4.48) τ  Re OpT ((1 − χ2 )b1 )Dd v, v +  2

˜ d v2+  τ OpT ((1 − χ2 )b1 )v+ + τ OpT (χ)D 2

+ Λ1T,τ v+ + Dd v2+ .   ˜ d v2+  τ Re OpT ((1 − χ2 )2 b21 )v, v + + τ OpT (χ)D 2

+ Λ1T,τ v+ + Dd v2+ .

3.4. LOCAL ESTIMATES AT THE BOUNDARY

87

Integrating by parts, we have, with c0 ∈ ST0,τ ,   τ OpT (χ)D ˜ d v2+ ≤ τ Re OpT (χ ˜2 )Dd2 v, v + + τ Re(Dd v|xd =0+ , OpT (c0 )v|xd =0+ )∂

(3.4.49)

+ Cτ 2 v2+ + CDd v2+ . As P2 = Dd2 + OpT (˜ q2 ) mod DT1,τ , we have     (3.4.50) τ Re OpT (χ ˜2 )Dd2 v, v + ≤ − τ Re OpT (χ ˜2 q˜2 )v, v + 2

+ Cτ −1/2 P2 v2+ + Cτ 1/2 Λ1T,τ v+ . With (3.4.48)–(3.4.50), we have     (3.4.51) τ  Re OpT ((1 − χ2 )b1 )Dd v, v +       ˜2 q˜2 )v, v + ≤ C τ Re OpT ((1 − χ2 )2 b21 )v, v + − τ Re OpT (χ + τ Re(Dd v|xd =0+ , OpT (c0 )v|xd =0+ )∂ + τ −1/2 P2 v2+  2 + Λ1T,τ v+ + Dd v2+ . With (3.4.46), (3.4.47), and (3.4.51), we obtain, with c0 ∈ ST0,τ , (3.4.52)     τ Re OpT (b2 )v, v + + τ Re OpT (b1 )Dd v, v + 2

≥ C  τ OpT ( χ)Λ1T,τ v+ + τ Re(Dd v|xd =0+ , OpT (c0 )v|xd =0+ )∂        − C τ  OpT ((1 − χ2 )b2 )v, v +  + τ  OpT (χ ˜2 q˜2 )v, v +     + τ  OpT ((1 − χ2 )2 b21 )v, v +  + τ −1 P1 v2+ + τ −1/2 P2 v2+  2 + τ 1/2 Λ1T,τ v+ + (ε2 τ + 1)Dd v2+ . We now wish to estimate the terms          2 , and  OpT ((1−χ2 )2 b21 )v, v .  OpT ((1−χ2 )b2 )v, v ,  OpT (χ ˜ q ˜ )v, v 2 + + + Observe that ˜2 (1 − χ2 )b2 , (1 − χ2 )b2 = χ

(1 − χ2 )2 b21 = χ ˜2 (1 − χ2 )2 b21 .   Thus, it is sufficient to estimate a generic term of the form  OpT (χ ˜2 r2 )v, v  ˜ of the where r2 ∈ ST2,τ . As q˜2 ≥ δ1 λ2T,τ /16 on a conic neighborhood W ˜ support of χ, ˜ there exists K > 0 such that K q˜2 ± r2 ≥ 2δ2 λ2T,τ on W arding inequality of for δ2 > 0. We can apply the tangential microlocal G˚ Theorem 2.50, and we obtain, with pseudo-differential calculus,     ˜2 q˜2 )v, v + ≥ | Re OpT (χ ˜2 r2 )v, v + | K Re OpT (χ 2

1/2

2

˜ 1T,τ v+ − CΛT,τ v+ , + δ2 OpT (χ)Λ

88

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

which we write (3.4.53)

    ˜2 r2 )v, v + | ≥ −Kτ Re OpT (χ ˜2 q˜2 )v, v + −τ | Re OpT (χ 2

1/2

2

+ δ2 τ OpT (χ)Λ ˜ 1T,τ v+ − Cτ ΛT,τ v+ . ˜2 q˜2 ) = OpT (χ ˜2 )(Dd2 − P2 ) + R1 , with R1 ∈ Ψ1T,τ , we have, with As −OpT (χ an integration by parts and pseudo-differential calculus,   ˜2 q˜2 )v, v + − Kτ Re OpT (χ   ≥ Kτ Re OpT (χ ˜2 )Dd v, Dd v + + τ Re(Dd v|xd =0+ , OpT (c 0 )v|xd =0+ )∂   2 − C τ −1/2 P2 v2+ + τ 1/2 Λ1T,τ v+ + Dd v2+ , 0 where c 0 ∈ ST,τ . Then we write   ˜2 )Dd v, Dd v + ≥ Kτ OpT (χ)D ˜ d v2+ − CDd v2+ , Kτ Re OpT (χ

yielding   ˜2 q˜2 )v, v + − Kτ Re OpT (χ ≥ τ Re(Dd v|xd =0+ , OpT (c 0 )v|xd =0+ )∂   2 − C τ −1/2 P2 v2+ + τ 1/2 Λ1T,τ v+ + Dd v2+ . This last estimate, (3.4.52), and (3.4.53) give     τ Re OpT (b2 )v, v + + τ Re OpT (b1 )Dd v, v + (3.4.54) 2

(4)

≥ C  τ Λ1T,τ v+ + τ Re(Dd v|xd =0+ , OpT (c0 )v|xd =0+ )∂  − C τ −1 P1 v2+ + τ −1/2 P2 v2+  2 + τ 1/2 Λ1T,τ v+ + (ε2 τ + 1)Dd v2+ , (4)

˜ ≥ 1 > 0 and the where c0 ∈ ST0,τ and where we have used that χ + χ tangential G˚ arding inequality of Theorem 2.49. 2 Now we can estimate Dd v2+ from Λ1T,τ v+ and P2 v2+ . In fact, we have   τ Dd v2+ = τ Re Dd2 v, v + + τ Re(Dd v|xd =0+ , iv|xd =0+ )∂     ≤ τ Re P2 v, v + − τ Re OpT (˜ q2 )v, v + 2

+ τ Re(Dd v|xd =0+ , iv|xd =0+ )∂ + CΛ1T,τ v+  2 ≤ τ Re(Dd v|xd =0+ , iv|xd =0+ )∂ + C τ −1/2 P2 v2+ + τ Λ1T,τ v+ .

3.5. PATCHING ESTIMATES

89

This estimate and (3.4.54) give     τ Re OpT (b2 )v, v + + τ Re OpT (b1 )Dd v, v +   2 (5) ≥ C  τ Λ1T,τ v+ + Dd v2+ + τ Re(Dd v|xd =0+ , OpT (c0 )v|xd =0+ )∂  − C τ −1 P1 v2+ + τ −1/2 P2 v2+  2 + τ 1/2 Λ1T,τ v+ + (ε2 τ + 1)Dd v2+ , (5) (5) ˜ where c0 ∈ ST0,τ . Setting B(v) = B(v)+Re(D d v|xd =0+ , OpT (c0 )v|xd =0+ )∂ , which is precisely of the form given in the statement of the proposition, with inequality (3.4.43), we have now reached the following estimate:   2 Pϕ v2+ ≥ C  τ Λ1T,τ v+ + Dd v2+ + τ Re B(v)

1 + P2 v2+ + P1 v2+ 2 − C τ −1 P1 v2+ + τ −1/2 P2 v2+

 2 + τ 1/2 Λ1T,τ v+ + (ε2 τ + 1)Dd v2+ .

For ε sufficiently small and for τ ≥ τ∗ with τ∗ is sufficiently large, we obtain (3.4.12) in the case ∂d ϕ(0) = 0. This concludes the proof of Proposition 3.24.  3.5. Patching Estimates Carleman estimates are local by nature. The previous sections have illustrated how they can be obtained by focusing around a region of interest. They are then applied to functions with support restricted to that region. In particular, at the boundary in both Lemmata 3.15 and 3.16, we have the occurrence of an open neighborhood V 1 that reduces the analysis near a point of interest. In fact, if all the required properties on the operator and the weight function hold in an open set V 0 of Ω, upon patching estimates given by Lemma 3.15 or Lemma 3.16 together, an estimate on the whole open set can be achieved. The following theorem is the nonlocal counterpart of Lemma 3.15. Theorem 3.28. Let V 0 be a bounded open set in Rd such that the boundary ∂Ω is C ∞ in a neighborhood of V 0 , and let ϕ and P have the subellipticity property of Definition 3.2 in V 0 ∩ Ω. There exist τ∗ > 0 and C > 0 such that  τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) ≤ C eτ ϕ P u2L2 (Ω)  + τ 3 |eτ ϕ u|∂Ω |2L2 (∂Ω) + τ |eτ ϕ u|∂Ω |2H 1 (∂Ω) + τ |eτ ϕ ∂ν u|∂Ω |2L2 (∂Ω) , Ω

for u ∈ Cc∞ (V 0 ) and τ ≥ τ∗ .

90

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

The following theorem is the nonlocal counterpart of Lemma 3.16. Theorem 3.29. Let V 0 be a bounded open set in Rd such that the boundary ∂Ω is C ∞ in a neighborhood of V 0 , and let ϕ and P have the sub-ellipticity property of Definition 3.2 in V 0 ∩ Ω assuming moreover that ∂ν ϕ|V 0 ∩∂Ω < 0 in the case V 0 ∩ ∂Ω = ∅. There exist τ∗ > 0, and C > 0 such that τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|∂Ω |2L2 (∂Ω)   ≤ C eτ ϕ P u2L2 (Ω) + τ 3 |eτ ϕ u|∂Ω |2L2 (∂Ω) + τ |eτ ϕ u|∂Ω |2H 1 (∂Ω) , Ω

for u ∈ Cc∞ (V 0 ) and τ ≥ τ∗ . We provide the proof of Theorem 3.29. The proof of Theorem 3.28 can be written mutatis mutandis. Proof of Theorem 3.29. By Remark 3.13, because of the form of the estimate we wish to prove, we see that we may replace P by its principal part P0 given in (3.1.1) without any loss of generality. This allows us to assume that the coefficients of P are smooth. We treat the case Γ0 = V 0 ∩ ∂Ω = ∅. The other case is simpler. Upon smoothly extending the coefficients and the weight function outside Ω near V 0 ∩ ∂Ω, by Lemma 3.9, there exists V 1 an open neighborhood of V 0 ∩ Ω such that the weight function ϕ and the operator P fulfill the sub-ellipticity property of Definition 3.2 in V 1 . Then, if x ∈ V 0 ∩ Ω, there exists an open subset Vx of Rd with x ∈ Vx for which a local Carleman estimate, in the interior of Ω (Theorem 3.11) or at the boundary of Ω for a Ω point of Γ0 (Lemma 3.16), holds for smooth functions in Cc∞ (Vx ). From the covering of the compact set V 0 ∩ Ω by the open sets Vx , x ∈ V 0 ∩ Ω, we can extract a finite covering (Vi )i∈I , such that for all i ∈ I the Carleman estimate in Vi holds for τ ≥ τi > 0, C = Ci > 0: (3.5.1) τ 3 eτ ϕ v2L2 (Vi ∩Ω) + τ eτ ϕ Dv2L2 (Vi ∩Ω) + τ |eτ ϕ ∂ν v|∂Ω |2L2 (V ∩∂Ω) i   2 2 τϕ 3 τϕ τϕ ≤ C e P0 vL2 (Vi ∩Ω) + τ |e v|∂Ω |L2 (V ∩∂Ω) + τ |e v|∂Ω |2H 1 (V ∩∂Ω) , i

Ω for v ∈ Cc∞ (Vi ). Note vanish if Vi ∩ ∂Ω = ∅.

i

that the boundary terms on the l.h.s. and the r.h.s.

Let ( χi )i∈I be a partition of unity of V 0 ∩ Ω subordinated to the open covering Vi , i ∈ I, that is, χi ∈ C ∞ (Rd ), with supp( χi ) ⊂ Vi ,

0 ≤ χi ≤ 1, i ∈ I,

0 and χ = i∈I χi ≡ 1 in a neighborhood of V ∩ Ω. For a reference on partition of unity see for instance [176]. We also provide a construction in Theorem 15.11 in Volume 2.

3.6. GLOBAL ESTIMATES WITH OBSERVATION TERMS Ω

91 Ω

Let now u ∈ Cc∞ (V 0 ). For all i ∈ I, we set ui = χi u ∈ Cc∞ (Vi ). Then, for each ui we have a local Carleman estimate of the form given by (3.5.1). We now observe that we have P0 ui = P0 ( χi u) = χi P0 u + [P0 , χi ]u, where the commutator is a first-order differential operator. For all i ∈ I, we thus obtain (3.5.2)

τϕ e P0 ui 2L2 (Ω)  eτ ϕ P0 u2L2 (Ω) + eτ ϕ u2L2 (Ω) + eτ ϕ Du2L2 (Ω) . i∈I

We also have 

 3 τϕ (3.5.3) τ |e ui |∂Ω |2L2 (∂Ω) + τ |eτ ϕ ui |∂Ω |2H 1 (∂Ω) i∈I

As u = χu =

 τ 3 |eτ ϕ u|∂Ω |2L2 (∂Ω) + τ |eτ ϕ u|∂Ω |2H 1 (∂Ω) .

i∈I

ui , we find

(3.5.4) τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|∂Ω |2L2 (∂Ω) 

 3 τϕ  τ e ui 2L2 (Ω) + τ eτ ϕ Dui 2L2 (Ω) + τ |eτ ϕ ∂ν ui |∂Ω |2L2 (∂Ω) . i∈I

From the local Carleman estimates that hold for each function ui as in (3.5.1) and from (3.5.2)–(3.5.4), we then obtain τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|∂Ω |2L2 (∂Ω)  eτ ϕ P0 u2L2 (Ω) + τ 3 |eτ ϕ u|∂Ω |2L2 (∂Ω) + τ |eτ ϕ u|∂Ω |2H 1 (∂Ω) + eτ ϕ u2L2 (Ω) + eτ ϕ Du2L2 (Ω) . The result then follows by choosing τ > 0 sufficiently large.



3.6. Global Estimates with Observation Terms In some applications it may be useful to obtain a global estimate, that is, an estimate for functions defined in an open connected set Ω of Rd where the equation P u = f is satisfied with u fulfilling some boundary conditions. We present here how such a global estimate can be achieved from the previous results in the case of Dirichlet boundary conditions. In Sect. 3.5, we saw how local estimates could be patched together. One may thus suggest to simply apply this method to obtain a global estimate in Ω. Yet, in Sect. 3.5, patching is done in a region where sub-ellipticity holds for both the weight function ϕ and the operator P . As we shall see, this is a matter that requires some consideration, and, depending on the kind of estimate we wish to obtain, having the sub-ellipticity condition on the whole open set Ω may be impossible.

92

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

Let us consider the case of homogeneous Dirichlet boundary conditions and assume the existence of a smooth weight function ϕ such that we have (3.6.1)

τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ u2L2 (Ω) ≤ Ceτ ϕ P u2L2 (Ω) ,

τ ≥ τ∗ ,

for u ∈ C ∞ (Ω), such that u|∂Ω = 0. Here, we have set C ∞ (Ω) = {u = v|Ω ; v ∈ C ∞ (Rd )}. From Sect. 3.5, we see that if we had a weight function ϕ such that the sub-ellipticity condition holds in the whole Ω and ∂ν ϕ|∂Ω < 0 everywhere on ∂Ω, an estimate of the form of (3.6.1) would follow from Theorem 3.29. Yet, such a weight function cannot exist. This is clear since, with such assumptions, one finds that dϕ must vanish somewhere in Ω, which is inconsistent with the sub-ellipticity condition. This however does not contradict estimate (3.6.1). To that purpose we use the optimality results of Chap. 4. By Theorems 4.5 and 4.7 we necessarily have |dϕ| ≥ C > 0 on Ω. Hence, ϕ reaches its maximum on ∂Ω, say at x0 ∈ ∂Ω. There, its tangential derivative vanishes and thus |dϕ(x0 )| = |∂ν ϕ|∂Ω (x0 )|, yielding ∂ν ϕ|∂Ω (x0 ) > 0. Applying Corollary 4.26 yields however ∂ν ϕ|∂Ω (x0 ) ≤ 0. We have thus reached a contradiction and (3.6.1) cannot hold. Global estimates thus cannot be of the form of (3.6.1). In fact, as opposed to the estimates we derived in the above sections, global estimates are characterized by observation terms. Below, in Theorem 3.34, for Dirichlet boundary conditions, we shall see that a global estimate can be obtained if an observation term of the form τ 3 eτ ϕ u2L2 (ω) is added on the r.h.s. of the estimate, with ω an open subset of Ω. Alternatively, in Theorem 3.40, we shall see that an observation term can take the form τ eτ ϕ ∂ν u2L2 (Γ) , with Γ an open subset of the boundary ∂Ω. In the first case we speak of an inner observation, and in the second case, of a boundary observation. Below we assume that Ω is connected. For nonconnected open sets, an observation term in each connected component is necessary. Such generalizations are left to the reader. 3.6.1. A Global Estimate with an Inner Observation Term. To patch local estimates together to form a global estimate, we choose a global weight function that can be used to derive each of the local estimates of Sects. 3.3 and 3.4. Let Ω be a connected open set of Rd , with Ω only located on one side of ∂Ω. Let Γ0 be an open set of ∂Ω such that ∂Ω is smooth in a neighborhood of Γ0 . Let also ω0 be an open subset of Ω. Note that Γ0 and ω0 need not be connected. Definition 3.30. A real valued function ϕ ∈ C ∞ (Ω) is said to be a global Carleman weight function on Ω adapted to Γ0 and ω0 if it satisfies ∂ν ϕ|∂Ω (x) < 0,

for x ∈ Γ0 ,

and if the sub-ellipticity property of Definition 3.2 for (P, ϕ) is fulfilled in Ω \ ω0 .

3.6. GLOBAL ESTIMATES WITH OBSERVATION TERMS

93

Proposition 3.31. There exists a function ψ ∈ C ∞ (Rd ; R) such that ∂ν ψ|∂Ω (x) < 0,

for x ∈ Γ0 ,

and

|dψ(x)| = 0,

for x ∈ Ω \ ω0 .

The function ϕ = exp(γψ)|Ω , for γ > 0, is a global Carleman weight function on Ω adapted to Γ0 and ω0 in the sense of Definition 3.30 if the parameter γ is chosen sufficiently large. Remark 3.32. An inspection of the proof shows that we construct in fact the function ψ such that ψ = 0 in a neighborhood Γ1 of Γ0 in ∂Ω and such that ψ > 0 in Ω. This yields ϕ ≡ 1 in Γ1 and ϕ > 1 in Ω, which can be useful sometimes in applications. Note also that the construction made in the proof yields a weight function with a finite number of nondegenerate critical points, all located in ω0 . Remark 3.33. In the case Ω is assumed smooth, the case Γ0 = ∂Ω can be considered in Proposition 3.31. Proof of Proposition 3.31. For all y ∈ Γ0 , there exists an open subset Vy of Rd , neighborhood of y, and local coordinates (through a smooth diffeomorphism) such that Ω is given by {xd > 0}. In these local coordinates we define the smooth function ψy (x) = xd . We extract a finite covering V1 , . . . , Vk of Γ0 by such open subsets, that is Γ0 ⊂ V = ∪kj=1 Vj . We denote by ψj the associated function defined in Vj . We choose an open set V (0) of Rd such that Γ0 ⊂ V (0)  V , and we consider a partition of unity ( χj )1≤j≤k of V (0) subordinated to this covering (see for instance Theorem 15.11 in Volume 2) χj ∈ Cc∞ (Rd ),

supp( χj ) ⊂ Vj ,

0 ≤ χj ≤ 1,

for 1 ≤ j ≤ k and χ(0) =

k

χj ≡ 1 in a neighborhood of V (0) .

j=1

By the definition of the functions ψj , there exists K > 0 such that ∂ν ψj |∂Ω ≤ −K in Vj ∩ ∂Ω. We then have (3.6.2)

∂ν (χj ψj )|∂Ω (x) = ( χj ∂ν ψj )|∂Ω (x) ≤ −Kχj (x),

x ∈ ∂Ω.

We also introduce V (1) an open set of Rd and a cut-off function χ(1) ∈ Cc∞ (Rd ) such that 0 ≤ χ(1) ≤ 1 and Γ0 ⊂ V (1)  V (0) ,

supp( χ(1) ) ⊂ V (0) ,

χ(1) ≡ 1 in a neighborhood of V (1) .

For δ > 0 we define the following function on Rd ψ (0) =

k

j=1

χj ψj + δ(1 − χ(1) ).

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3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

ψ (0) |∂Ω = 0 ∂Ω

ψ (0) = δ ψ (0) < 0

ψ (0) |∂Ω > 0

V (1)

ψ (0) > 0 Γ0 Ω V

(∂ν ψ (0) )|∂Ω < 0

(0)

V

Figure 3.3. Local geometry near Γ0 ⊂ ∂Ω. The colored region is the closed set {χ(1) = 1} With (3.6.2) we find that ∂ν ψ (0) |∂Ω (x) =

k

( χj ∂ν ψj )|∂Ω (x) − δ(∂ν χ(1) )|∂Ω (x) ≤ −K + Cδ,

j=1

x ∈ V (0) ∩ ∂Ω. For δ > 0 chosen sufficiently small we thus have (3.6.3)

∂ν ψ (0) |∂Ω (x) ≤ −C < 0,

x ∈ V (0) ∩ ∂Ω.

We also have ψ (0) > 0 on Ω,

ψ (0) = δ on Rd \ V,

ψ (0) ≤ 0 on V (1) \ Ω,

and ψ (0) |∂Ω = δ(1 − χ(1) )|∂Ω yielding ψ (0) |∂Ω (x) = 0 for x ∈ ∂Ω ∩ {χ(1) = 1}, ψ (0) |∂Ω (x) > 0 for x ∈ ∂Ω ∩ {χ(1) < 1}. The local geometry with the open sets V , V (0) , and V (1) is illustrated in Fig. 3.3. From (3.6.3), we deduce that there exists an open neighborhood W 0 of V 0 ∩ ∂Ω in Rd such that |dψ (0) (x)| ≥ C0 > 0,

x ∈ W 0.

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Ω1 W2 W

∂Ω

1

W0 V (0) Γ0 Ω V

Figure 3.4. The neighborhoods W 0 , W 1 , and W 2 near Γ0 ⊂ ∂Ω and the open set Ω1 We now choose W 1 a second open neighborhood of V 0 ∩ ∂Ω in Rd such that W 1  W 0 and Ω \ W 1 is connected.5 In Ω \ W 1 we have ψ (0) ≥ C1 > 0. Thus, there exists a connected open neighborhood Ω1 of Ω \ W 1 in Rd such that ψ (0) ≥ C1 /2 > 0 on Ω1 . In particular Ω1 does not meet an open neighborhood W 2 of ∂Ω ∩ {χ1 = 1} in Rd as ψ (0) vanishes on ∂Ω ∩ {χ1 = 1}. The neighborhoods W 0 , W 1 , and W 2 are represented in Fig. 3.4. We set Ω = Ω ∪ Ω1 . A function f ∈ C ∞ (Ω ) is called a Morse function if it has no degenerate critical point. Morse functions form a dense subset of C ∞ (Ω ) in the C k -topology [260, Corollary 6.8], k ∈ N. Here, density in the C 1 (0) topology suffices. For ε > 0 we select a Morse function ψε such that (0) ψε − ψ (0) C 1 (Ω ) ≤ ε. We let ζ (1) ∈ Cc∞ (W 0 ) be such that ζ (1) ≡ 1 on a neighborhood W 1 . We then set ψε(1) = ζ (1) ψ (0) + (1 − ζ (1) )ψε(0) = ψ (0) + (1 − ζ (1) )(ψε(0) − ψ (0) ).

5The existence of such an open set W 1 is quite intuitive. Yet, one cannot simply use local coordinates near a point of the boundary. One can rather introduce the distance function to the boundary. Then, if one considers a sufficiently small neighborhood of the boundary, this function is smooth and yields a proper set of coordinates. The natural framework for such an argumentation is to view Ω as a Riemannian manifold endowed with the Euclidean metric. For a proof of this result given in the case of Riemannian manifolds we refer the reader to Proposition 17.24 in Volume 2.

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3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR (1)

The function ψε (3.6.4)

coincides with ψ (0) in W 1 . In particular, we have

∂ν ψε(1) |∂Ω (x) ≤ −C < 0 and ψε(1) |∂Ω (x) = 0 for x ∈ V (0) ∩ ∂Ω. (1)

We choose ε sufficiently small to have ψε ≥ C1 /4 on Ω1 . (1) We now prove that ψε is a Morse function for ε > 0 chosen sufficiently small. In fact, in W 0 where |dψ (0) | ≥ C0 > 0 we have dψε(1) = dψ (0) + (1 − ζ (1) )d(ψε(0) − ψ (0) ) − (ψε(0) − ψ (0) )dζ (1) , (1)

yielding |dψε | ≥ C0 /2 in W 0 for ε > 0 chosen sufficiently small. Next, in (1) (0) Ω \ W 0 where ζ (1) vanishes, we simply have ψε = ψε and we conclude (1) that the critical points of ψε in Ω are nondegenerate and located in Ω \ W 0 ⊂ Ω1 . (1)

We denote the critical points of ψε in Ω1 by y1 , . . . , yn . We then consider distinct points y10 , . . . , yn0 ∈ ω0 . As Ω1 is a connected open set, for j ∈ {1, . . . , n}, we consider a non- self-intersecting C ∞ path γj : [0, 1] → Ω1 such that γj (0) = yj0 , γj (1) = yj , and γj (t) = 0 for all t ∈ [0, 1]. In particular, such path does not enter the set W 2 . We require moreover that different paths do not intersect: setting Γj = {γj (t); t ∈ [0, 1]} we require Γj ∩ Γk = ∅ if j = k. These points and paths are illustrated in Fig. 3.5. For

W1 W2 y1 Ω1

Ω y5 y50

Γ0

y40

y10 ω0 y30

y20 y2

y4

∂Ω

y3

Figure 3.5. Pulling back critical points from Ω1 to ω0

3.6. GLOBAL ESTIMATES WITH OBSERVATION TERMS

97

each j, we pick a neighborhood Wj  Ω1 of Γj and wj a smooth vector field on Ω1 such that Wj ∩ Wk = ∅ if k = j and supp(wj ) ⊂ Wj  Ω1

and wj (x) = γj (t) if x = γ(t), t ∈ [0, 1].

We denote by χj (t, x) the flow associated with wj , viz.,   χj (0, x) = x, ∂t χj (t, x) = wj χj (t, x) , t ∈ [0, 1], x ∈ Ω1 . / Wj . For t = 1 we set Observe that χj (t, x) = x for all t ∈ [0, 1] in x ∈ ρj (x) = χj (1, x), which gives a smooth diffeomorphism of Ω1 onto itself. (1) We now set ψ(x) = ψε ◦ ρ1 ◦ · · · ◦ ρn (x). We have ψ > 0 in Ω since (1) ψε ≥ C > 0 in Ω1 . As the diffeomorphisms ρj , j = 1, . . . , n, leave W 2 invariant, by (3.6.4), we have ∂ν ψ |∂Ω (x) ≤ −C < 0 and ψ|∂Ω (x) = 0 x ∈ W 2 ∩ ∂Ω. The chain rule gives dψ(x) = 0 if and only if ρ1 ◦ · · · ◦ ρn (x) = yj for some j ∈ {1, . . . , n}. Since the vector fields wj have disjoint supports, this is equivalent to having x = yj0 . The critical points of ψ are thus all located in ω0 . This concludes the construction of the function ψ. Defining ϕ = exp(γψ) for γ > 0, we have ∂ν ϕ|∂Ω (x) < 0 for x ∈ Γ0 ⊂ W 2 ∩ ∂Ω,

and

|dϕ(x)| = 0 for x ∈ Ω \ ω0 .

Finally, Lemma 3.5 implies that the sub-ellipticity condition of Definition 3.2 is fulfilled in Ω \ ω0 for γ chosen sufficiently large.  We now state the global Carleman estimate we can obtain in the case of Dirichlet boundary conditions and an inner observation. Theorem 3.34 (Global Carleman Estimate—Inner Observation). Let Ω be a bounded connected open set of Rd located on one side of its boundary, and let P be the operator given in (3.1.2). Let Γ0 be an open set of ∂Ω such that ∂Ω is smooth in a neighborhood of Γ0 . Let also W be an open neighborhood of ∂Ω \ Γ0 in Rd . Finally, let ω, ω0 be two nonempty open subsets of Ω such that ω0  ω. Let ϕ be a global weight function adapted to Γ0 and ω0 in the sense of Definition 3.30. Then, there exist τ∗ > 0 and C ≥ 0 such that τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|Γ0 |2L2 (Γ ) 0  τϕ  2 2 2 3 τϕ 3 τϕ ≤ C e P uL2 (Ω) +τ e uL2 (ω) +τ |e u|Γ0 |L2 (Γ ) +τ |eτ ϕ u|Γ0 |2H 1 (Γ ) , 0

0

for τ ≥ τ∗ and u ∈ C ∞ (Ω) vanishing in W ∩ Ω. The function ϕ in the statement is for instance given by Proposition 3.31. As announced above the solution is “controlled” by the data, that is, P u and u|Γ0 , and an inner observation term on ω.

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3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

Remark 3.35. The previous estimate extends by density to functions u ∈ H 2 (Ω) with the same support requirement. Remark 3.36. Observe that Ω need not be smooth away from a neighborhood of Γ0 . In particular, one can consider the case where the connected open set Ω = Ω1 × Ω2 , with Γ0 ∩ ∂Ω1 × ∂Ω2 = ∅ (W is then a neighborhood of ∂Ω1 × ∂Ω2 ). Note that the derivation of a Carleman estimate in the neighborhood of a corner of ∂Ω requires additional work. Note that if ω ⊂ W in Theorem 3.34, then the observation term eτ ϕ uL2 (ω) vanishes in the estimate. This is however not the case in applications we shall make of this theorem (see Sect. 6.5.2). The case Ω smooth and Γ0 = ∂Ω yields the following corollary that provides the form of global Carleman estimates usually stated. Corollary 3.37. Let Ω be a smooth bounded connected open set of Rd located on one side of its boundary, and let P be the operator given in (3.1.2). Let ω, ω0 be two nonempty open subsets of Ω such that ω0  ω. Let ϕ be a global weight function adapted to Γ0 = ∂Ω and ω0 in the sense of Definition 3.30. Then, there exist τ∗ > 0 and C ≥ 0 such that τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|∂Ω |2L2 (∂Ω)   ≤ C eτ ϕ P u2L2 (Ω) +τ 3 eτ ϕ u2L2 (ω) +τ 3 |eτ ϕ u|∂Ω |2L2 (∂Ω) +τ |eτ ϕ u|∂Ω |2H 1 (∂Ω) , for u ∈ C ∞ (Ω) and τ ≥ τ∗ . As explained at the beginning of this section, this estimate fails to hold without the observation term eτ ϕ uL2 (ω) on the r.h.s.. Proof of Theorem 3.34. By Remark 3.13, because of the form of the estimate we wish to prove, we see that we may replace P by its principal part P0 given in (3.1.1) without any loss of generality. This allows us to assume that the coefficients of P are smooth. Let ω1 be an open set of Rd such that ω0  ω1  ω. Let also W0 be an open neighborhood of ∂Ω \ Γ0 in Rd such that W0  W . The geometry is illustrated in Fig. 3.6. We set V 0 = Rd \W0 ∪ ω0 . Since the weight function satisfies the properties of Definition 3.30, we observe that ϕ and P fulfill the sub-ellipticity property of Definition 3.2 in a neighborhood of V 0 ∩ Ω in Ω and ∂ν ϕ|V 0 ∩∂Ω < 0. Ω

Theorem 3.29 thus applies to functions in Cc∞ (V 0 ). Let χ ∈ Cc∞ (V 0 ) be such that χ ≡ 1 in a neighborhood of Ω \ (W ∪ ω1 ). The region where χ varies is illustrated in Fig. 3.6.

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99

∂Ω

Γ0

Ω

W

ω0 ω ω1

W0

Figure 3.6. Geometry for the global Carleman estimate of Theorem 3.34. The function χ used in the proof only varies in the two darker regions surrounded by dotted lines. Between these two regions, in a neighborhood of Ω \ (W ∪ ω1 ), we have χ≡1 Ω

As χu|Ω ∈ Cc∞ (V 0 ), by Theorem 3.29, we have (3.6.5) τ 3 eτ ϕ χu2L2 (Ω) + τ eτ ϕ D( χu)2L2 (Ω) + τ |eτ ϕ ∂ν ( χu)|∂Ω |2L2 (∂Ω)  eτ ϕ P0 ( χu)2L2 (Ω) + τ 3 |eτ ϕ χu|∂Ω |2L2 (∂Ω) + τ |eτ ϕ χu|∂Ω |2H 1 (∂Ω) . We observe that we have P0 ( χu) = χP0 u + [P0 , χ]u, where the commutator is a first-order differential operator. We thus obtain (3.6.6) eτ ϕ P0 ( χu)L2 (Ω)  eτ ϕ P0 uL2 (Ω) + eτ ϕ uL2 (Ω) + eτ ϕ DuL2 (Ω) . We also have (3.6.7)

|eτ ϕ χu|∂Ω |L2 (∂Ω)  |eτ ϕ u|∂Ω |L2 (∂Ω) = |eτ ϕ u|Γ0 |L2 (Γ ) , 0

and (3.6.8)

|eτ ϕ χu|∂Ω |H 1 (∂Ω)  |eτ ϕ u|∂Ω |H 1 (∂Ω)  |eτ ϕ u|Γ0 |H 1 (Γ ) . 0

100

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

  As supp (1 − χ)u ⊂ ω1 , writing u = χu + (1 − χ)u, we find (3.6.9) τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|Γ0 |2L2 (Γ  τ e

τϕ

χu2L2 (Ω)

+ τ e

τϕ

D(χu)2L2 (Ω)

0)

+ τ |e ∂ν (χu)|∂Ω |2L2 (∂Ω)   2 + τ 3 eτ ϕ (1 − χ)u2L2 (ω1 ) + τ eτ ϕ D (1 − χ)u L2 (ω1 ) , 3

τϕ

 τ 3 eτ ϕ χu2L2 (Ω) + τ eτ ϕ D(χu)2L2 (Ω) + τ |eτ ϕ ∂ν (χu)|∂Ω |2L2 (∂Ω) + τ 3 eτ ϕ u2L2 (ω1 ) + τ eτ ϕ Du2L2 (ω1 ) . From the local Carleman estimate (3.6.5) and from (3.6.6)–(3.6.9), we then obtain τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|Γ0 |2L2 (Γ

0) 2 τϕ  e + τ |e u|Γ0 |L2 (Γ ) + τ |e u|Γ0 |2H 1 (Γ ) 0 0 2 2 τϕ τϕ + e uL2 (Ω) + e DuL2 (Ω) + τ 3 eτ ϕ u2L2 (ω1 ) + τ eτ ϕ Du2L2 (ω1 ) .

τϕ

P0 u2L2 (Ω)

3

τϕ

For τ > 0 sufficiently large we have (3.6.10) τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|Γ0 |2L2 (Γ

0) 2 τϕ + τ |e u|Γ0 |L2 (Γ ) + τ |e u|Γ0 |2H 1 (Γ )  e 0 0 2 2 3 τϕ τϕ + τ e uL2 (ω1 ) + τ e DuL2 (ω1 ) .

τϕ

P0 u2L2 (Ω)

3

τϕ

We now aim to remove the last term on the r.h.s. of the previous estimation. ˜ ≤ 1 and χ ˜ ≡ 1 in a neighborhood of Let χ ˜ ∈ Cc∞ (ω; R), be such that 0 ≤ χ ω1 . We observe that   (P0 u, e2τ ϕ χ (3.6.11) ˜2 u)L2 (Ω)   τ −1 eτ ϕ P0 u2L2 (Ω) + τ eτ ϕ u2L2 (ω) . Here, we extend the definition of p(x, ξ) to complex valued cotangent vectors by setting

pij ξi ξj = |ξ|2x , x ∈ Rd , ξ ∈ Cd . p(x, ξ) = 1≤i,j≤d

We also observe that we have, with an integration by parts, Re(P0 u, e2τ ϕ χ ˜2 u)L2 (Ω)

 ij (p Di u, e2τ ϕ χ = Re ˜2 Dj u)L2 (Ω) 1≤i,j≤d

+ (pij Di u, Dj (e2τ ϕ )χ ˜2 u)L2 (Ω) + 2(pij χD ˜ i u, e2τ ϕ Dj ( χ)u) ˜ L2 (Ω) 2 2 2 τϕ −1 2 τ ϕ ≥ eτ ϕ χ|Du| ˜ ˜ x L2 (Ω) − εe χDu L2 (Ω) − Cε τ e uL2 (ω) ,



3.6. GLOBAL ESTIMATES WITH OBSERVATION TERMS

101

for ε > 0. For ε chosen sufficiently small, invoking (3.1.3), we then obtain ˜2 u)L2 (Ω) ≥ Ceτ ϕ Du2L2 (ω1 ) − C  τ 2 eτ ϕ u2L2 (ω) . Re(P0 u, e2τ ϕ χ The previous estimate and (3.6.11) then yield τ eτ ϕ Du2L2 (ω1 )  eτ ϕ P0 u2L2 (Ω) + τ 3 eτ ϕ u2L2 (ω) . 

With (3.6.10) the proof is complete.

3.6.2. A Global Estimate with a Boundary Observation Term. Let Ω be a connected open set of Rd , with Ω only located on one side of ∂Ω. We consider an open set Γ0 of ∂Ω such that Γ0  ∂Ω and ∂Ω is smooth in a neighborhood of Γ0 . Definition 3.38. A real valued function ϕ ∈ C ∞ (Ω) is said to be a global Carleman weight function on Ω adapted to Γ0 if it satisfies ∂ν ϕ|∂Ω (x) < 0,

for x ∈ Γ0 ,

and if the sub-ellipticity property of Definition 3.2 for (P, ϕ) is fulfilled in Ω. As already observed in the beginning of Sect. 3.6, this property cannot hold in the case Γ0 = ∂Ω. In fact, as we have |dϕ| ≥ C > 0 on Ω, ϕ reaches its maximum on ∂Ω, say at x0 ∈ ∂Ω. There its tangential derivative vanishes and thus |dϕ(x0 )| = |∂ν ϕ|∂Ω (x0 )|. As ∂ν ϕ|∂Ω (x0 ) < 0, this yields a contradiction. Thus, there exists a second open set Γ1 of ∂Ω such that Γ0 ∩ Γ1 = ∅. We shall also assume that ∂Ω is smooth in a neighborhood of Γ1 . We use this second set to construct a weight function ϕ adapted to Γ0 . Proposition 3.39. Let Ω be a connected open set of Rd , with Ω only located on one side of ∂Ω. Let Γ0 and Γ1 be open subsets of ∂Ω such that Γ0 ∩ Γ1 = ∅ and ∂Ω is smooth in a neighborhood of Γ0 ∪ Γ1 . There exists a function ψ ∈ C ∞ (Rd ; R) such that ∂ν ψ|∂Ω (x) < 0,

for x ∈ Γ0 ,

and

|dψ(x)| = 0,

for x ∈ Ω.

The function ϕ = exp(γψ)|Ω , for γ > 0, is a global Carleman weight function on Ω adapted to Γ0 in the sense of Definition 3.38 if the parameter γ is chosen sufficiently large. Note that the statement implies that one has no information on the sign of ∂ν ψ, and thus of ∂ν ϕ, on Γ1 . Rd

Proof. Let y0 ∈ Γ1 , and let V be a small open neighborhood of y0 in such that: (1) V ∩ ∂Ω  Γ1 . (2) There exist local coordinates (x , xd ) such that Ω ∩ V and ∂Ω ∩ V are given by {xd > 0} and {xd = 0}, respectively.

102

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

Note that the second point exploits the smoothness of the boundary near Γ1 . We denote by κ the local diffeomorphism V → Rd associated with the local coordinates. We further assume that κ(y0 ) = 0. The local geometry is illustrated in Fig. 3.7. ω0 Γ0

x

κ

y0 W V Ω

Γ1

0 ˜ W

xd ε

∂Ω

xd = −εχ(x )/2

Figure 3.7. Construction of the locally extended set Ω = Ω∪W Let then ε > 0 be such that, in the local coordinates, B(0, ε) ⊂ V . We then choose χ ∈ C ∞ (Rd−1 ) such that 0 ≤ χ ≤ 1 and supp( χ) ⊂ {|x | ≤ ε/2}.   ˜ ⊂ V with We define the set W = κ−1 W ˜ = {x = (x , xd ) ∈ Bε ; χ(x ) > 0 and − εχ(x )/2 < xd ≤ 0}. W This set is represented both in Rd and in local coordinates in Fig. 3.7. We define the open set Ω , smooth extension of Ω, as Ω = Ω ∪ W . We choose an open set ω0 of Rd such that ω0  Ω \ Ω. With Proposition 3.31 we can then obtain a weight function on Ω adapted to Γ0 and ω0 . If we restrict it to Ω, we obtain the sought weight function ψ. Defining ϕ = exp(γψ) for γ > 0 chosen sufficiently large, we obtain the expected properties, using Lemma 3.5.  We now prove a global Carleman estimate in the case of Dirichlet boundary conditions and a boundary observation in an open set Γobs of ∂Ω. We consider Γ0 and Γ1 as above with Γ1  Γobs . The functions we consider have support at the boundary ∂Ω only in the sets Γ0 and Γobs . The properties assumed for ϕ at Γ0 allow one to control the function near Γ0 and its Neumann trace at Γ0 . The Neumann trace needs to be observed at Γobs . Theorem 3.40 (Global Carleman Estimate—Boundary Observation). Let Ω be a bounded connected open set of Rd located on one side of its

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103

boundary, and let P be the operator given in (3.1.2). Let Γ0 and Γobs be two nonempty open sets of ∂Ω such that Γobs \ Γ0 = ∅, and such that ∂Ω is smooth in a neighborhood of Γ0 ∪ Γobs . Let also W be a neighborhood of ∂Ω \ (Γ0 ∪ Γobs ) in Rd . Let ϕ be a global weight function adapted to Γ0 in the sense of Definition 3.38. Then, there exist τ∗ > 0 and C ≥ 0 such that τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|Γ0 |2L2 (Γ ) 0  τϕ 2 2 τϕ ≤ C e P uL2 (Ω) + τ e ∂ν u|Γobs L2 (Γ 3

+ τ |e

τϕ

u|∂Ω |2L2 (∂Ω)

obs )

 + τ |eτ ϕ u|∂Ω |2H 1 (∂Ω) ,

for τ ≥ τ∗ and u ∈ C ∞ (Ω) vanishing in W ∩ Ω. As announced above, the solution is “controlled” by the data, P u and u|∂Ω , and a Neumann type boundary observation term on Γobs ⊂ ∂Ω. Observe that we may have Γ0 ∩ Γobs = ∅. For the construction of the global weight function ϕ, one can pick an open set Γ1 of ∂Ω such that Γ1  Γobs \ Γ0 and then use the weight function provided by Proposition 3.39. Remark 3.41. We insist on the fact that Ω need not be smooth away from a neighborhood of Γ0 ∪ Γobs . In particular, one can consider the case where the connected open set Ω = Ω1 × Ω2 , with (Γ0 ∪ Γobs ) ∩ ∂Ω1 × ∂Ω2 = ∅ (W is then a neighborhood of ∂Ω1 × ∂Ω2 ). Note that if Γobs ⊂ W in Theorem 3.40, then the observation term eτ ϕ ∂ν u|Γobs L2 (Γ ) vanishes in the estimate. This is however not the case obs in applications we shall make of this theorem (see Sect. 7.5.2). If Ω is smooth, one can consider the case Γ0 ∪ Γobs = ∂Ω and Theorem 3.40 yields the following corollary. Corollary 3.42. Let Ω be a smooth bounded connected open set of Rd located on one side of its boundary, and let P be the operator given in (3.1.2). Let Γ0 , Γobs be two nonempty open subsets of ∂Ω such that Γ0 ∪ Γobs = ∂Ω and Γobs \ Γ0 = ∅. Let ϕ be a global weight function adapted to Γ0 in the sense of Definition 3.38. Then, there exist τ∗ > 0 and C ≥ 0 such that τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|∂Ω |2L2 (∂Ω)  ≤ C eτ ϕ P u2L2 (Ω) + τ eτ ϕ ∂ν u|Γobs 2L2 (Γ 3

+ τ |e

τϕ

u|∂Ω |2L2 (∂Ω)

obs )

 + τ |eτ ϕ u|∂Ω |2H 1 (∂Ω) ,

for u ∈ C ∞ (Ω) and τ ≥ τ∗ . Remark 3.43. The previous estimate extends by density to functions u ∈ H 2 (Ω).

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3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

V0 W Γ0

∂Ω

W0 Ω V obs Γobs

Figure 3.8. Geometry for the global Carleman estimate of Theorem 3.40. The function χ associated with the partition of unity of Ω \ W used in the proof only varies in the darker region surrounded by dotted lines. Here, we illustrate a geometrical setting where Γ0 ∩ Γobs = ∅. Yet, the case Γ0 ∩ Γobs = ∅ is also treated in Theorem 3.40 As explained at the beginning of this section, the estimate of Corollary 3.42 fails to hold without the observation term eτ ϕ ∂ν u|Γobs L2 (Γ ) on obs the r.h.s.. Proof of Theorem 3.40. By Remark 3.13, because of the form of the estimate we wish to prove, we see that we may replace P by its principal part P0 given in (3.1.1) without any loss of generality. This allows us to assume that the coefficients of P are smooth. Let W0 be an open neighborhood of ∂Ω \ (Γ0 ∪ Γobs ) in Rd such that W0  W . The geometry is illustrated in Fig. 3.8. For all x ∈ Ω \ W , there exists an open subset Vx of Rd \ W0 with x ∈ Vx such that either: (1) Vx ∩ ∂Ω = ∅. (2) Vx ∩ ∂Ω ⊂ Γ0 ; there ∂Ω is smooth and ∂ν ϕ < 0. (3) Vx ∩ ∂Ω ⊂ Γobs ; there ∂Ω is smooth. From the covering of the compact set Ω \ W by the open sets Vx we can extract a finite covering (Vi )i∈I . We denote by I 0 the set of indices such

3.6. GLOBAL ESTIMATES WITH OBSERVATION TERMS

105

that either Vx ∩ ∂Ω = ∅ or Vx ∩ ∂Ω ⊂ Γ0 and by I obs the set of indices such that Vx ∩ ∂Ω ⊂ Γobs . We set   Vi and V obs = Vi , V0 = i∈I 0

i∈I obs

which forms an open covering of Ω \ W . We also have V 0 ∩ ∂Ω ⊂ Γ0

and

V obs ∩ ∂Ω ⊂ Γobs .

In particular ∂ν ϕ|V 0 ∩∂Ω < 0.

Let χ0 , χ1 ∈ Cc∞ (Rd ) form a partition of unity of Ω \ W subordinated to the open covering by V 0 and V obs (see for instance Theorem 15.11 in Volume 2), that is supp( χ0 ) ⊂ V 0 ,

supp( χ1 ) ⊂ V obs ,

0 ≤ χ0 ≤ 1,

0 ≤ χ1 ≤ 1,

and χ0 + χ1 ≡ 1 in a neighborhood of Ω \ W . Let now u be as in the theorem statement. With the properties of the weight function ϕ in Ω, the Carleman estimate of Theorem 3.28 can be applied to χ1 u, and the Carleman estimate of Theorem 3.29 can be applied to χ0 u. We have 2

2

2

(3.6.12) τ 3 eτ ϕ χ1 uL2 (Ω) + τ eτ ϕ D( χ1 u)L2 (Ω)  eτ ϕ P0 ( χ1 u)L2 (Ω) 2

2

2

+ τ 3 |eτ ϕ χ1 u|∂Ω |L2 (∂Ω) + τ |eτ ϕ χ1 u|∂Ω |H 1 (∂Ω) + τ |eτ ϕ ∂ν ( χ1 u)|∂Ω |L2 (∂Ω) and 2

2

2

(3.6.13) τ 3 eτ ϕ χ0 uL2 (Ω) + τ eτ ϕ D( χ0 u)L2 (Ω) + τ |eτ ϕ ∂ν ( χ0 u)|Γ0 |L2 (∂Ω)  2 2 2 ≤ C eτ ϕ P0 ( χ0 u)L2 (Ω) + τ 3 |eτ ϕ χ0 u|∂Ω |L2 (∂Ω) + τ |eτ ϕ χ0 u|∂Ω |H 1 (∂Ω) . We observe that we have P0 ( χi u) = χi P0 u + [P0 , χi ]u,

i = 0, 1,

where the commutator is a first-order differential operator. We thus obtain (3.6.14) eτ ϕ P0 ( χi u)L2 (Ω)  eτ ϕ P0 uL2 (Ω) + eτ ϕ uL2 (Ω) + eτ ϕ DuL2 (Ω) . We also have (3.6.15) |eτ ϕ χi u|∂Ω |L2 (∂Ω)  |eτ ϕ u|∂Ω |L2 (∂Ω) ,

|eτ ϕ χi u|∂Ω |H 1 (∂Ω)  |eτ ϕ u|∂Ω |H 1 (∂Ω) .

In addition we find (3.6.16)

|eτ ϕ ∂ν ( χ1 u)|∂Ω |L2 (∂Ω)  |eτ ϕ u|∂Ω |L2 (∂Ω) + |eτ ϕ ∂ν u|Γobs |L2 (Γ

obs )

106

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

and τ |eτ ϕ ∂ν u|Γ0 |2L2 (Γ ) + τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) 0

 τϕ 2 2  τ |e ∂ν ( χi u)|Γ0 |L2 (Γ ) + τ 3 eτ ϕ χi uL2 (Ω)

(3.6.17)

0

i=0,1

2



+ τ eτ ϕ D( χi u)L2 (Ω) . From the local Carleman estimates (3.6.12) and (3.6.13) and inequalities (3.6.14)–(3.6.17) we then obtain τ |eτ ϕ ∂ν u|Γ0 |2L2 (Γ ) + τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) 0

 τ |eτ ϕ u|∂Ω |2L2 (∂Ω) + τ |eτ ϕ u|∂Ω |2H 1 (∂Ω) + τ |eτ ϕ ∂ν u|Γobs |2L2 (Γ 3

+ e

τϕ

P0 u2L2 (Ω)

+ e

τϕ

u2L2 (Ω)

+ e

τϕ

obs )

Du2L2 (Ω) . 

For τ > 0 chosen sufficiently large the result follows. 3.7. Alternative Approach

The proof of Theorem 3.11 is mainly based on the sub-ellipticity property (3.2.6). That proof shows that one gives complementary roles to the square terms in (3.3.2), P1 u2L2 and P2 u2L2 , and to the action of the commutator 2 Re(P2 u, iP1 u)L2 = i([P2 , P1 ]u, u)L2 . As the square terms approach zero, the commutator term comes into effect and yields positivity. We describe here a modification of the proof scheme that is quite commonly used. The form of the operator P1 is modified, and the counterpart of Re(P2 u, iP1 u)L2 then exhibits positivity without relying on the square terms. First, we consider estimates away from boundaries. Second, we consider an estimate in the neighborhood of a boundary point. As is now classical, we shall use the operator P0 , principal part of P , in the derivation of the estimates without any loss of generality (see Remark 3.13). 3.7.1. A Modified Carleman Estimate Derivation Away from Boundaries. We use the notation of the proof of Theorem 3.11 and write g = Pϕ v = (P2 + iP1 )v in the form g − μτ P0 (ϕ)v = P2 v + iP 1 v,

with P 1 = P1 + iμτ θ,

θ(x) = P0 ϕ(x),

for 0 < μ < 2. This yields g − μτ θv2L2 (Rd ) = P2 v2L2 (Rd ) + P 1 v2L2 (Rd ) + 2 Re(P2 v, iP 1 v)L2 (Rd ) = P2 v2L2 (Rd ) + P 1 v2L2 (Rd ) + (i[P2 , P1 ]v, v)L2 (Rd ) − 2 Re(P2 v, μτ θv)L2 (Rd ) , and we obtain (3.7.1)

g − μτ θv2L2 (Rd ) ≥ P2 v2L2 (Rd ) + Re(Qv, v)L2 (Rd ) ,

3.7. ALTERNATIVE APPROACH

107

where q = σ(Q) = {p2 , p1 } − 2μτ θp2 . We then have the following lemma whose proof can be found in Appendix 3.B.4. Lemma 3.44. Let V be a bounded open set in Rd and ψ ∈ C ∞ (Rd ; R) be such that |dψ| > 0 in V . If 0 < μ < 2 and ϕ = exp(γψ), then for γ > 0 sufficiently large, there exists Cγ > 0 such that q(x, ξ, τ ) = {p2 , p1 }(x, ξ, τ ) − 2μτ θ(x)p2 (x, ξ, τ ) ≥ Cγ τ λ2τ , for x ∈ V , ξ ∈ Rd , τ ≥ 1. With this positivity result and the G˚ arding inequality of Theorem 2.28 we then conclude that Re(Qv, v)  τ v2τ,1 , for τ taken sufficiently large. With (3.7.1) we thus obtain τ 3/2 vL2 (Rd ) + τ 1/2 DvL2 (Rd ) + P2 vL2 (Rd )  g − μτ θvL2 (Rd ) . Note that we have g − μτ θvL2 (Rd )  gL2 (Rd ) + μτ θvL2 (Rd ) , and, from the form of P2 in (3.2.2), τ −1/2 P0 vL2 (Rd )  τ −1/2 P2 vL2 (Rd ) + τ 3/2 vL2 (Rd ) + τ −1/2 DvL2 (Rd ) . Thus, for τ sufficiently large, we obtain τ 3/2 vL2 (Rd ) + τ 1/2 DvL2 (Rd ) + τ −1/2

|α|=2

Dα vL2 (Rd )  gL2 (Rd ) .

We then conclude as in the end of the proof of Theorem 3.11 and obtain estimate (3.3.1). Remark 3.45. (1) This method, at the symbol level, is a matter of adding a term of the form −2μτ θp2 to the symbol of the commutator i[P2 , P1 ]. As the sign of θp2 is not prescribed, a precise choice of the value of μ is crucial. (2) This method however does not reveal the (necessary and) sufficient sub-ellipticity property of Definition 3.2 that is at the heart of estimates of the form of (3.3.1) in Theorem 3.11. For the necessity of the sub-ellipticity property, see Theorems 4.5 and 4.7 in Chap. 4. 3.7.2. A Modified Carleman Estimate Derivation at a Boundary. Here, following the previous section, we provide a second proof of a Carleman estimate for the operator P in the neighborhood of a point of the boundary. We consider for instance (part of) the setting of Lemma 3.16. For simplicity, we use normal geodesic coordinates in a neighborhood V 0 of the point y ∈ ∂Ω of interest, as used in Sect. 3.4.6. As is done in that section, we set

R(x, D ) = Di (bij (x)Dj ), P0 = Dd2 + R(x, D ), 1≤i,j≤d−1

108

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

with respective principal symbols p(x, ξ) = ξd2 + r(x, ξ  ),

r(x, ξ  ) =

1≤i,j≤d−1

bij (x)ξi ξj .

Observe that, by Remark 3.13, we may replace P by P0 in the derivation of the Carleman estimate without any loss of generality. We define ˜bij (x) by ˜bij = bij if i, j ∈ {1, . . . , d − 1}, ˜bid = ˜bdi = 0 so that

p(x, ξ) = 1≤i,j≤d ˜bij (x)ξi ξj . We consider the conjugated operator Pϕ = eτ ϕ P0 e−τ ϕ . The explicit computations of (3.4.7)–(3.4.8) give Pϕ = P2 + iP1 , with P2 = P0 − p(x, τ dϕ),  P1 = τ ∂d ϕDd + Dd ∂d ϕ +

1≤i,j≤d−1





ij



ij



Di b ∂j ϕ + b ∂i ϕDj

.

The operators P1 and P2 are self-adjoint if considered away from a boundary. With θ = P0 ϕ and P 1 = P1 + iμτ θ, following Sect. 3.7.1, we write P0 v − μτ θv2+ = P2 v2+ + P 1 v2+ + 2 Re(P2 v, iP 1 v)+   = P2 v2+ + P 1 v2+ + i (P1 v, P2 v)+ − (P2 v, P1 v)+ − 2 Re(P2 v, μτ θv)+ . From the computation of the proof of Proposition 3.24, we obtain, see (3.4.20),     ˜ i (P1 v, P2 v)+ − (P2 v, P1 v)+ = Re i[P2 , P1 ]v, v + + τ Re B(v), ˜ where B(v) is of the same form as that of B(v) in the statement of Proposition 3.24, see (3.4.21). We thus obtain, with Q = i[P2 , P1 ] − 2μτ θP2 ,   ˜ P0 v − μτ θv2+ ≥ Re Qv, v + + τ Re B(v). (3.7.2) We set ϕ(x) = exp(γψ(x)) with ψ smooth on V 0 and such that |dψ| ≥ C > 0 in V 0 . The computations of the proof of Lemma 3.44 in Sect. 3.B.4 give q = σ(Q) of the form q(x, ξ, τ ) = {p2 , p1 }(x, ξ, τ ) − 2μτ p2 (x, ξ, τ )P0 ϕ(x) = γq1 + q2 , with qj = τ˜(x)3 q˜j , j = 1, 2, with τ˜ = τ γϕ and with q˜j as given in Sect. 3.B.4, yielding  2 τ (x)3 p x, dψ(x) + 2μ˜ τ (x)p(x, ξ)p(x, dψ(x)) q1 = (4 − 2μ)˜ 2  + τ˜(x) ∂ψ(x)∂ξ p(x, ξ) ,

3.7. ALTERNATIVE APPROACH

and q2 of the form q2 = τ˜(x)3 f (x) + τ˜(x)

1≤i,j≤d

109

kij (x)ξi ξj ,

for some smooth functions f (x) and kij (x) that are bounded and whose iterated derivatives are themselves bounded. We may thus write Q = γQ1 + Q2 + Rτ,γ , with Rτ,γ ∈ τ Dτ1 (for fixed γ) and  2 τ (x)3 p x, dψ(x) + 2μ Q1 = (4 − 2μ)˜ +4

1≤i,j,k, ≤d

Q2 = τ˜(x)3 f (x) +

1≤i,j≤d

Di (˜bij τ˜)(x)p(x, dψ(x))Dj

Di (˜ τ ˜bij ˜bk ∂j ψ∂k ψ)(x)D ,

1≤i,j≤d

Di τ˜(x)kij (x)Dj .

With integration by parts we find   2 τ 3/2 p(x, dψ)v+ + 2μ ∫ τ˜p(x, dψ)B(Dv, Dv) dx Re Q1 v, v + ≥ (4 − 2μ)˜ Rd+

τ 1/2 v|∂ |˜ τ 1/2 Dv|∂ , + 4 ∫ τ˜|B(∇ψ, Dv)|2 dx − C|˜ Rd+

˜j . This gives, for 0 < μ < 2, where B(w, w) ˜ = i,j ˜bij wi w    3/2 2 2 Re Q1 v, v + ≥ C ˜ τ v+ + ˜ τ 1/2 Dv+ − C  |˜ τ 1/2 v|∂ |˜ τ 1/2 Dv|∂ , for v supported in V , as |dψ| ≥ C > 0 in V 0 . We also have   2 2 | Q2 v, v + |  ˜ τ 3/2 v+ + ˜ τ 1/2 Dv+ + |˜ τ 1/2 v|∂ |˜ τ 1/2 Dv|∂ . By choosing γ sufficiently large (to be kept fixed in what follows), we obtain     Re (γQ1 + Q2 )v, v + ≥ C τ 3 v2+ + τ Dv2+ − C  τ |v|∂ |Dv|∂ . As we have

  | Rτ,γ v, v + |  τ 2 v2+ + Dv2+ ,

we find for τ chosen sufficiently large     Re Qv, v + ≥ C τ 3 v2+ + τ Dv2+ − C  τ |v|∂ |Dv|∂ . With (3.7.2) we thus obtain   ˜ P0 v − μτ θv2+ ≥ C τ 3 v2+ + τ Dv2+ + τ Re B(v) − C  τ |v|∂ |Dv|∂ , which yields, for τ chosen sufficiently large,   ˜ − C  τ |v|∂ |Dv|∂ . P0 v2+ ≥ C τ 3 v2+ + τ Dv2+ + τ Re B(v) ˜ With B(v) as described above, we can thus recover the estimate of Lemma 3.22 and its counterpart in Lemma 3.15, as both do not require any assumption on ∂ν ϕ at the boundary.

110

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

We can also recover the estimate of Lemma 3.23 and its counterpart in Lemma 3.16 if we furthermore assume that ∂ν ϕ < 0 near the point y ∈ ∂Ω. This can be achieved by choosing the function ψ such that ∂ν ψ(x) < 0 for x ∈ ∂Ω ∩ V 0 and ϕ = exp(γψ). Construction of such a function is done in Remark 3.19. Remark 3.46. The derivation we carried out here appears simpler than what we did in Sect. 3.4. We recall, however, that this approach does not reveal the sufficient aspect of sub-ellipticity as already pointed out in Remark 3.45-(2). 3.7.3. Alternative Derivation in the Case of Limited Smoothness. The setting is the same as in Sect. 3.7.2. Here however, the proof scheme is only based on integration by parts and does not rely on the pseudodifferential tools of Chap. 2 to obtain positivity properties. Then, it turns out that we may only assume the coefficients of the principal of the operator to be Lipschitz in Ω. We assume the boundary of the open set Ω to be C 1 , allowing one to use integration by parts, that is, the Green formula. We do not rely on the symbol Chap. 2 here.

calculus of ij Let thus P0 = 1≤i,j≤d Di (p (x)Dj ), with pij ∈ W 1,∞ (Ω) and y ∈ ∂Ω. Let V 0 be a bounded open set in Rd such that y ∈ V 0 . We also set P = (pij )1≤i,j≤d and Div = −i div, which yields P0 = Div PD. We consider a function ψ ∈ C 2 (V 0 ) such that |dψ| ≥ C > 0 in V 0 . We set ϕ(x) = exp(γψ(x)). As dϕ = γϕdψ and d2x ϕ = γϕd2x ψ + γ 2 ϕdψ ⊗ dψ, we have, for γ ≥ 1, |dϕ|  γϕ,

(3.7.3)

|d2x ϕ|  γ 2 ϕ.

We consider the conjugated operator Pϕ = eτ ϕ P0 e−τ ϕ . The explicit computation of (3.2.1) gives Pϕ = P2 + iP1 , with P2 = P0 − p(x, τ dϕ),

P1 = τ (D · P∇ϕ + P∇ϕ · D).

The operators P1 and P2 are self-adjoint if considered away from a boundary. We set θ(x) = P0 ϕ(x) and Θ(x) = −γ 2 ϕ(x)p(x, dψ). We then define (3.7.4)

  P 1 = P1 + iτ (μ + 1)Θ − θ = 2τ P∇ϕ · D + iτ (μ + 1)Θ.

Following Sect. 3.7.1, we would be inclined to use μθ instead of (μ + 1)Θ − θ. However, θ involves second-order derivatives of the function ψ, and the computations below would yield a third-order derivative of ψ through a derivative of θ (see the calculation of the term I12 below). Here in the expression of P 1 in (3.7.4), only the term that exhibits the higher order in the parameter γ appears, namely Θ(x), that involves only one derivative of

3.7. ALTERNATIVE APPROACH

111

ψ. This avoids having to require ψ to be C 3 . The analysis of Sect. 3.7.1 still applies as we have, by (3.7.3), |θ − Θ|  γϕ.

(3.7.5) Observe that we have

|Θ|  γ 2 ϕ,

(3.7.6)

|∇Θ|  γ 3 ϕ.

Ω

For v ∈ Cc∞ (V 0 ), we write (3.7.7)

  Pϕ vL2 (Ω) ≥ Pϕ v − τ (μ + 1)Θ − θ vL2 (Ω) − Cτ γ 2 ϕvL2 (Ω) .

  As Pϕ − τ (μ + 1)Θ − θ = P2 + iP 1 , we then write   2 (3.7.8) Pϕ v − τ (μ + 1)Θ − θ vL2 (Ω) = P2 v2L2 (Ω) + P 1 v2L2 (Ω) + 2 Re(P2 v, iP 1 v)L2 (Ω) . We focus on the computation of Re(P2 v, iP 1 v)L2 (Ω) that we write as a sum of four terms Iij , 1 ≤ i ≤ 2, 1 ≤ j ≤ 2, where Iij is the inner product of the ith term in the expression of P2 v and the jth term in the expression of iP 1 v in (3.7.4). Below, we denote by ν = (ν1 , . . . , νd ) the unitary outward-pointing normal to ∂Ω. k

In what follows · stands for the tensor contraction on the last k indices,  k−p p k namely A · (v1 ⊗ · · · ⊗ vk ) = A · (vp+1 ⊗ · · · ⊗ vk ) · (v1 ⊗ · · · ⊗ vp ), for A a k-covariant tensor and v1 , . . . , vk vectors. For the reader who prefers to see tensor indices appear explicitly, the computation of the term I11 is reproduced in Appendix 3.C. Term I11 . With an integration by parts, we have I11 = 2τ Re(Div PDv, i∇ϕ · PDv)L2 (Ω)   = 2τ Re (PDv, ∇(∇ϕ · PDv))L2 (Ω) − (∂ν v|∂Ω , ∇ϕ · P∇v|∂Ω )L2 (∂Ω) , as ∂ν = ν · P∇. We write 2 Re(PDv) · ∇(∇ϕ · PDv) 3

= 2d2x ϕ(PDv, PDv) + 2 Re ∇P · (Dv ⊗ ∇ϕ ⊗ PDv) 2

+ 2 Re(P · ∇Dv) · (∇ϕ ⊗ PDv) 3

= 2d2x ϕ(PDv, PDv) + 2 Re ∇P · (Dv ⊗ ∇ϕ ⊗ PDv) 2

+ 2 Re ∇Dv · (P∇ϕ ⊗ PDv),

112

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

using the symmetry of P. For the third term in the r.h.s. we write, again using the symmetry of P, 2

2

2 Re ∇Dv · (P∇ϕ ⊗ PDv) = 2 Re ∇Dv · (PDv ⊗ P∇ϕ) = ∇(PDv · Dv) · P∇ϕ 3

− ∇P · (Dv ⊗ Dv ⊗ P∇ϕ). With an integration by parts we obtain ∫Ω ∇(PDv · Dv) · P∇ϕ dx = ∫ (P0 ϕ)(PDv · Dv) dx Ω

+ ∫ (∂ν ϕ)(PDv · Dv)|∂Ω dσ, ∂Ω

yielding (3.7.9) 2 Re(PDv, ∇(∇ϕ · PDv))L2 (Ω) = 2 ∫ d2x ϕ(PDv, PDv) dx + ((P0 ϕ)PDv, Dv)L2 (Ω) Ω

  2 + (∂ν ϕ)PDv|∂Ω , Dv|∂Ω L2 (∂Ω) − (∇P · (Dv ⊗ P∇ϕ), Dv)L2 (Ω) 2

+ 2 Re(∇P · (∇ϕ ⊗ PDv), Dv)L2 (Ω) . L ), with the boundary terms We have thus obtained I11 = τ (J11 + L11 + R11   J11 = −2 Re(∂ν v|∂Ω , ∇ϕ · P∇v|∂Ω )L2 (∂Ω) + (∂ν ϕ)PDv|∂Ω , Dv|∂Ω L2 (∂Ω)

and the interior terms, L11 = (ΘPDv, Dv)L2 (Ω) + 2 ∫ d2x ϕ(PDv, PDv) dx Ω

and

 2  L | = ((θ − Θ)PDv, Dv)L2 (Ω) − (∇P · (Dv ⊗ P∇ϕ), Dv)L2 (Ω) |R11  2  + 2 Re(∇P · (∇ϕ ⊗ PDv), Dv)L2 (Ω)  2

 γϕ1/2 DvL2 (Ω) , by (3.7.3) and (3.7.5), yielding 2

I11 ≥ τ (J11 + L11 ) − Cτ γϕ1/2 DvL2 (Ω) . Term I12 . With an integration by parts, we have J L + R12 ), I12 = −τ (μ + 1) Re(Div PDv, Θv)L2 (Ω) = τ (L12 + R12

with L12 = −(μ + 1)(ΘPDv, Dv)L2 (Ω) ,

3.7. ALTERNATIVE APPROACH

113

and J L R12 = (μ + 1) Re(∂ν v|∂Ω , Θv|∂Ω )L2 (∂Ω) , R12 = (μ + 1) Re(DΘ · PDv, v)L2 (Ω) .

With (3.7.6), we have J |  γ 2 |ϕ1/2 ∂ν v|∂Ω |L2 (∂Ω) |ϕ1/2 v|∂Ω |L2 (∂Ω) , |R12 L |R12 |  γ 3 ϕ1/2 DvL2 (Ω) ϕ1/2 vL2 (Ω) .

Term I21 . With an integration by parts, we have I21 = −2τ Re(p(x, τ dϕ)v, iP∇ϕ · Dv)L2 (Ω) = −τ 3 ∫ p(x, dϕ)P∇ϕ · ∇|v|2 dx Ω

= τ 3 (J21 + L21 ), with

  L21 = ∫ div p(x, dϕ)P∇ϕ |v|2 dx.

J21 = − ∫ p(x, dϕ)∂ν ϕ|v|2|∂Ω dσ,

Ω

∂Ω

Term I22 . We have I22 = τ 3 L22 with L22 = (μ + 1) ∫ p(x, dϕ)Θ|v|2 dx. Ω

Gathering the above computations and estimates, we find 2 Re(P2 v, iP 1 v)L2 (Ω) = L + J + RL + RJ , with J = 2(τ J11 + τ 3 J21 ) and L = 2τ (L11 + L12 ) + 2τ 3 (L21 + L22 ), and (3.7.10)

  2 L L + R12 |  τ γ ϕ1/2 DvL2 (Ω) + γ 2 ϕ1/2 DvL2 (Ω) ϕ1/2 vL2 (Ω) , |RL | = τ |R11 J |RJ | = τ |R12 |  τ γ 2 |ϕ1/2 Dv|∂Ω |L2 (∂Ω) |ϕ1/2 v|∂Ω |L2 (∂Ω) .

Set |ν|P = (ν · Pν)1/2 . For any function w, we have ∂ν w = νP∇w, meaning that (∂ν w)ν/|ν|2P is the orthogonal projection in the sense of P of ∇w onto span(ν). Hence, ∇T w = ∇w − (∂ν w)ν/|ν|2P is orthogonal to ν, also in the sense of P. The following lemma then provides a quite natural way to write the boundary terms. Lemma 3.47. We have J = 2τ ∫ j(x)|∂Ω dσ(x) − 2τ 3 ∫ p(x, dϕ)∂ν ϕ|v|2|∂Ω dσ, ∂Ω

∂Ω

2 ¯ + (∂ν ϕ)P∇T v · ∇T v¯. with j(x) = −|ν|−2 P (∂ν ϕ)|∂ν v| − 2 Re(∂ν v)∇T ϕ · P∇T v

Proof. Collecting the terms in J11 and J21 , we have v + (∂ν ϕ)PDv · Dv. j(x) = −2 Re(∂ν v)∇ϕ · P∇¯

114

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

Writing ∇v = (∂ν v)ν/|ν|2P + ∇T v and ∇ϕ = (∂ν ϕ)ν/|ν|2P + ∇T ϕ, we find 2 j(x) = −2|ν|−2 ¯ P (∂ν ϕ)|∂ν v| − 2 Re(∂ν v)∇T ϕ · P∇T v 2 ¯, + |ν|−2 P (∂ν ϕ)|∂ν v| + (∂ν ϕ)P∇T v · ∇T v



which yields the result.

Remark 3.48. For sufficiently smooth coefficients, if one uses normal geodesic coordinates as in Sect. 3.4, yielding pdd = 1 and pdj = 0 for j = d, we have |ν|P = 1, ∂ν ϕ = −∂d ϕ. We then recover in Lemma 3.47 the principal part of the boundary quadratic form obtained in Proposition 3.24. For the interior terms we have L = 2τ 3 ∫ α|v|2 dx + 2τ ∫ βPDv · Dv dx + Y, Y = 4τ ∫ d2x ϕ(PDv, PDv) dx, Ω

with

Ω

Ω

  α = div p(x, dϕ)P∇ϕ + (μ + 1)p(x, dϕ)Θ,

β = −μΘ  γ 2 ϕ. 2

Lemma 3.49. There exists C > 0 such that Y ≥ −Cτ γϕ1/2 DvL2 (Ω) , and there exists C0 > 0 such that, for γ ≥ 1 chosen sufficiently large, we have α ≥ C 0 γ 4 ϕ3 . Remark 3.50. Note that this lemma acts as a replacement for the sub-ellipticity condition obtained by Lemma 3.5 and the symbol positivity property of Lemma 3.8. However, in connection with Remark 3.45-(2), the present approach does not allow one to exhibit a sufficient condition on the weight function ϕ to achieve such positivity. Compare with the subellipticity condition of Definition 3.2 and Lemma 3.8. Here, ϕ is only chosen of the form ϕ = exp(γψ), with |dψ| > 0, and the parameter γ ≥ 1 is chosen sufficiently large. This implies sub-ellipticity by Lemma 3.5. Proof of Lemma 3.49. First, we write d2x ϕ = γ 2 ϕdψ ⊗ dψ + γϕd2x ψ.

(3.7.11) This gives

d2x ϕ(PDv, PDv) = γ 2 ϕ|dψ(PDv)|2 + γϕd2x ψ(PDv, PDv), yielding 2

Y ≥ 4τ γ ∫ ϕd2 ψ(PDv, PDv) dx  −τ γϕ1/2 DvL2 (Ω) . Ω

Second, with θ = P0 ϕ, we write α = ∇(p(x, dϕ)) · P∇ϕ − p(x, dϕ)θ + (μ + 1)p(x, dϕ)Θ = ∇(p(x, dϕ)) · P∇ϕ + μp(x, dϕ)Θ + R, with |R|  γ 3 ϕ3 by (3.7.3) and (3.7.5).

3.7. ALTERNATIVE APPROACH

115

We have ∇(p(x, dϕ)) · P∇ϕ = d2 ϕ(∇ξ p(x, dϕ), P∇ϕ) + ∇x p(x, dϕ) · P∇ϕ ≥ 2d2 ϕ(P∇ϕ, P∇ϕ) − Cγ 3 ϕ3 . With (3.7.11) we obtain

 2 ∇(p(x, dϕ)) · P∇ϕ ≥ 2γ 4 ϕ3 dψ(P∇ψ) − Cγ 3 ϕ3 ≥ 2γ 4 ϕ3 p(x, dψ)2 − Cγ 3 ϕ3 .

As Θ = −γ 2 ϕ(x)p(x, dψ), we find α ≥ (2 − μ)γ 4 ϕ3 p(x, dψ)2 − Cγ 3 ϕ3 . As we have p(x, dψ)  |dψ|  1 and 2 − μ > 0, we obtain α  γ 4 ϕ3 , for γ ≥ 1 chosen sufficiently large.  Remark 3.51. Observe that the positive term γ 2 ϕ|dψ(PDv)|2 in the ex2  pression of Y in the proof is the counterpart of the term τ γ 4 ϕ(x)3 Hp ψ(x, η) =  2 τ γ 2 ϕ(x) Hp ψ(x, ξ) in (3.A.4) found at the symbol level, in the proof of Lemma 3.5 in Appendix 3.A.3. From (3.7.7)–(3.7.8) we have thus obtained, for γ chosen sufficiently large,   2 2 (3.7.12) Pϕ v2L2 (Ω) ≥ Cτ γ 2 τ 2 γ 2 ϕ3/2 vL2 (Ω) + ϕ1/2 DvL2 (Ω) + J + RL + RJ − Cτ 2 γ 4 ϕv2L2 (Ω) . Considering now estimates (3.7.10) we write   2 |RL |  τ γ ϕ1/2 DvL2 (Ω) + γ 2 ϕ1/2 DvL2 (Ω) ϕ1/2 vL2 (Ω) (3.7.13) 2

2

 (τ γ + γ 2 )ϕ1/2 DvL2 (Ω) + τ 2 γ 4 ϕ1/2 vL2 (Ω) . For the term RJ we simply write 2

2

|RJ |  τ 2 γ 3 |ϕ1/2 v|∂Ω |L2 (∂Ω) + γ|ϕ1/2 Dv|∂Ω |L2 (∂Ω) . First, choosing γ ≥ 1 sufficiently large, to be kept fixed in what follows, we obtain   Pϕ v2L2 (Ω) ≥ Cτ τ 2 v2L2 (Ω) + Dv2L2 (Ω)   + J − C  τ 2 |v|∂Ω |L2 (∂Ω) + |Dv|∂Ω |2L2 (∂Ω)   − C  τ 2 v2L2 (Ω) + Dv2L2 (Ω) . Second, choosing τ > 0 sufficiently large, we obtain   Pϕ v2L2 (Ω) ≥ Cτ τ 2 v2L2 (Ω) + Dv2L2 (Ω) + J   − C  τ 2 |v|∂Ω |L2 (∂Ω) + |Dv|∂Ω |2L2 (∂Ω) .

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Considering the form of the term J that corresponds to the principal part of the quadratic form B(v) of Proposition 3.24 (see Remark 3.48 above), we can conclude as in Sect. 3.7.2 and recover the estimate of Theorem 3.28 that does not require any assumption on ∂ν ϕ at the boundary. We can also recover the estimate of Theorem 3.29 if furthermore assuming that ∂ν ϕ < 0 near the point y ∈ ∂Ω. This can be achieved by choosing the function ψ such that ∂ν ψ(x) < 0 for x ∈ ∂Ω ∩ V 0 and ϕ = exp(γψ). Construction of such a function is done in Remark 3.19 in the case of a smooth boundary. Here, if the boundary ∂Ω is assumed to be at least C 2 ˜ in V 0 , then there exists ψ˜ : V 0 → R of class C 2 (V 0 ) such that dψ(y) = 0 0 ˜ and V ∩ Ω is locally given by ψ(x) > 0. As in Remark 3.19, we then set 2 , with the parameter α ≥ 0 allowing one to adjust the ˜ ψ(x) = ψ(x)−α|x−y| convexity of the level sets of ψ with respect to the boundary. See Fig. 3.2.

Theorem 3.52. Let P0 = 1≤i,j≤d Di (pij (x)Dj ) with pij ∈ W 1,∞ (Ω)

and P = P0 + 1≤i≤d bi (x)Di + c(x), where bi , c ∈ L∞ (Rd ), 1 ≤ i ≤ d. Let V 0 be a bounded open set in Rd such that the boundary ∂Ω is C 1 in V 0 . Let ψ ∈ C 2 (V 0 ) be such that |dψ| ≥ C > 0 in V 0 . Then, there exist τ∗ > 0, γ0 ≥ 1, and C0 > 0 such that  τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) ≤ C0 eτ ϕ P u2L2 (Ω) + τ 3 |eτ ϕ u|∂Ω |2L2 (∂Ω)   + τ |eτ ϕ u|∂Ω |2H 1 (∂Ω) + |eτ ϕ ∂ν u|∂Ω |2L2 (∂Ω) , Ω

for u ∈ Cc∞ (V 0 ), τ ≥ τ∗ , γ ≥ γ0 , and ϕ = exp(γψ). If moreover, ∂ν ψ(x) < 0 for x ∈ ∂Ω ∩ V 0 , there exist τ∗ > 0, γ0 ≥ 1, and C0 > 0 such that τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|∂Ω |2L2 (∂Ω)   ≤ C0 eτ ϕ P u2L2 (Ω) + τ 3 |eτ ϕ u|∂Ω |2L2 (∂Ω) + τ |eτ ϕ u|∂Ω |2H 1 (∂Ω) , Ω

for u ∈ Cc∞ (V 0 ), τ ≥ τ∗ , γ ≥ γ0 , and ϕ = exp(γψ). We finish this section by observing that in the case the weight function ψ is chosen nonnegative, which can be done by simply adding a constant to it as Ω is bounded, the dependency upon the parameter γ can be further tracked. Indeed if ψ ≥ 0 we have ϕ1/2 ≤ ϕ ≤ ϕ3/2 . In place of (3.7.12) and (3.7.13) we have   2 2 Pϕ v2L2 (Ω) ≥ Cτ γ 2 τ 2 γ 2 ϕ3/2 vL2 (Ω) + ϕ1/2 DvL2 (Ω) + J 2

+ RL + RJ − Cτ 2 γ 4 ϕ3/2 vL2 (Ω) , and 2

2

|RL |  (τ γ + γ 2 )ϕ1/2 DvL2 (Ω) + τ 2 γ 4 ϕ3/2 vL2 (Ω) .

3.7. ALTERNATIVE APPROACH

117

Then, for γ0 ≥ 1 and τ∗ > 0 both chosen sufficiently large, we have   2 2 Pϕ v2L2 (Ω) ≥ Cτ γ 2 τ 2 γ 2 ϕ3/2 vL2 (Ω) + ϕ1/2 DvL2 (Ω) + J + RJ , for γ ≥ γ0 and τ ≥ τ∗ . From Lemma 3.47 we have 2

2

|J + RJ |  τ γ|ϕ1/2 ∇v|∂Ω |L2 (∂Ω) + τ 3 γ 3 |ϕ3/2 v|∂Ω |L2 (∂Ω) , yielding the following counterpart to the result of Theorem 3.28.

Theorem 3.53. Let P0 = 1≤i,j≤d Di (pij (x)Dj ) with pij ∈ W 1,∞ (Ω)

and P = P0 + 1≤i≤d bi (x)Di + c(x), where bi , c ∈ L∞ (Rd ), 1 ≤ i ≤ d. Let V 0 be a bounded open set in Rd such that the boundary ∂Ω is C 1 in V 0 . Let ψ ≥ 0 be such that ψ ∈ C 2 (V 0 ) and |dψ| ≥ C > 0 in V 0 . Then, there exists τ∗ > 0, γ0 ≥ 1, and C0 > 0 such that 2

2

τ 3 γ 4 eτ ϕ ϕ3/2 uL2 (Ω) + τ γ 2 eτ ϕ ϕ1/2 DuL2 (Ω)  2 ≤ C0 eτ ϕ P u2L2 (Ω) + τ 3 γ 3 |eτ ϕ ϕ3/2 u|∂Ω |L2 (∂Ω)   2 2 + τ γ |eτ ϕ ϕ1/2 u|∂Ω |H 1 (∂Ω) + |eτ ϕ ϕ1/2 ∂ν u|∂Ω |L2 (∂Ω) , Ω

for u ∈ Cc∞ (V 0 ), τ ≥ τ∗ , γ ≥ γ0 , and ϕ = exp(γψ). If moreover, ∂ν ψ(x) < 0 for x ∈ ∂Ω ∩ V 0 , there exist τ∗ > 0, γ0 ≥ 1, and C0 > 0 such that 2

2

2

τ 3 γ 4 eτ ϕ ϕ3/2 uL2 (Ω) + τ γ 2 eτ ϕ ϕ1/2 DuL2 (Ω) + τ γ 2 |eτ ϕ ϕ1/2 ∂ν u|∂Ω |L2 (∂Ω)  2 ≤ C0 eτ ϕ P u2L2 (V 1 ∩Ω) + τ 3 γ 4 |eτ ϕ ϕ3/2 u|∂Ω |L2 (∂Ω)  2 + τ γ 2 |eτ ϕ ϕ1/2 u|∂Ω |H 1 (∂Ω) , Ω

for u ∈ Cc∞ (V 0 ), τ ≥ τ∗ , γ ≥ γ0 , and ϕ = exp(γψ). 3.7.4. Valuable Aspects of the Different Approaches. Observe that in Sect. 3.7.3, the Carleman estimates are obtained in the open set V 0 directly: we do not need to first reduce the analysis in smaller open sets as in the statements of Lemmata 3.15 and 3.16 and, second, to patch such estimates as is done in Sect. 3.5. The proof strategy of the present section may thus appear better; yet, the following arguments need to be taken into account. In fact, such a global derivation can be quite a hurdle in some situations. Assume that, in some setting, one faces a nonclassical situation near a point y ∈ Ω: ellipticity could degenerate there or the boundary could have a corner at such a point. The derivation of a Carleman estimate away from this point can be made as in the present chapter, and one only needs to place effort in the derivation of a local estimate near y. Once this estimate is obtained, it can most likely be patched with the other estimates. In fact, working locally and patching estimates together may often appear as an efficient strategy, avoiding having to derive an estimate in regions where it

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is a known result. We insist on the fact that Carleman estimates are local by nature. A significant advantage of the alternative derivation presented in Sect. 3.7.3, is that, being only based on explicit computations, mainly integrations by parts, it applies in situation where coefficient regularity is an issue. However, this does not prevent one from proving a local estimate with this method. Hence, whatever derivation method one chooses, localizing near the point of interest appears as a reasonable choice. We thus argue that this alternative approach is very valuable, yet one should use it locally, if possible. We shall see in Sect. 4.2.4 a case where the patching procedure cannot be applied as easily as for the estimates derived in the present chapter. Hence, a global derivation can be a valuable alternative. As mentioned earlier, the alternative derivation of Sect. 3.7.3 uses a property of the weight function that is stronger than sub-ellipticity property of Definition 3.2, whereas this latter property is sufficient for the derivation of a Carleman estimate, as shown in Sects. 3.3 and 3.4, and necessary as shown in Theorems 4.5 and 4.7 in Chap. 4. We shall see in Chap. 8 in Volume 2 that deriving estimates in the case of general boundary conditions requires to not only cut the analysis in different regions in space, but rather, in different phase space regions. One then speaks of microlocal Carleman estimates. Once obtained, these estimates need to be patched together. With this in mind, the reader will understand the emphasize we place on the localization-patching approach. To conclude this discussion, we see that it is essential to be able to adapt the proof strategy if constraints are faced. Both approaches are very valuable. 3.8. Notes In this chapter we have placed lots of emphasis on the notion of subellipticity that is at the heart of Carleman estimates. Here, sufficiency of the sub-ellipticity property is proven. Necessity is proven in Chap. 4. The notion of sub-ellipticity can be found in statements in the 1963 book of L. H¨ormander [172]; see for instance Theorems 8.1.1, 8.3.1, and 8.4.1 therein. See also Theorems 28.2.1 and 28.2.3 in [174] where more modern notations are used. Lemma 3.5 allowing to generate a weight function fulfilling the sub-ellipticity property by means of convexification follows from the analysis in Theorem 8.6.2 in [172] and Proposition 28.3.3 in [174]. In these references, setting ϕ = exp(γψ) with ψ assumed to fulfill the so-called strong pseudo-convexity leads to sub-ellipticity for γ chosen sufficiently large. In the elliptic case, the notion of strong pseudo-convexity greatly simplifies: one only requires dϕ to not vanish; Lemma 3.5 appears thus much simpler than the results cited above. The idea of introducing weighted estimates is classical for proving unique continuation properties. A pioneer work is that of T. Carleman [105] where

3.8. NOTES

119

this approach allowed him to remove the assumption of analyticity in a Holmgren-type result: T. Carleman proved the corresponding result in the case of two independent variables assuming that the characteristics of the equations are non-multiple. A.P. Calder´on generalized unique continuation to more variables yet assuming simple multiplicity [100]. His proof is based on a (pseudo-differential) factorization of the operator into first-order factors, for which a Carleman-type estimate is derived. Here, based on the result of the present chapter we cover unique continuation results for secondorder elliptic operators in Chap. 5. We thus postpone some bibliographical discussion to Sect. 4.4. Carleman estimates in the above references concern operators away from a boundary. Second-order elliptic operators estimated at a boundary are treated in [218, 289] and O. Yu. Imanuvilov and J.-P. Puel [179] in the case of Dirichlet boundary conditions. Similar work in the case of parabolic equations can be found in the works of A. Fursikov and O. Yu. Imanuvilov [156], O. Yu. Imanuvilov, J.-P. Puel and M. Yamamoto [180], and E. Fern´andezCara, S. Guerrero, O. Yu. Imanuvilov, and J.-P. Puel [144]. In Chap. 8 ˇ in Volume 2 boundary conditions of Lopatinski˘ı-Sapiro type are treated for second-order operators, which include Dirichlet, Neumann, Robin boundary conditions. We refer to the notes of that chapter for further references. The construction of the global weight function at the beginning of Sect. 3.6.1 uses Morse functions and the associated approximation theorem [260, Corollary 6.8]. Global weight function has enjoyed a popular development following the work of A. Fursikov and O. Yu. Imanuvilov [156] in the close case of parabolic operators. In Sect. 3.6, the global weight function we choose is used to prove the local estimates. This is standard. In some situations, such global weight function may not exist. This is for instance the case of Carleman estimates for operators with principal part D(c(x)D) where c has jumpd across several interfaces. Local estimates may be proved and can still be patched together. Yet, the resulting global weight function may not be locally equal to the weight function used for the proof of the local estimates. We refer to [214] for more details. In the proof of the first Carleman estimate we write in Theorem 3.11 away from a boundary we use the sub-ellipticity property of Definition 3.2. The proof gives complementary roles to the square terms P1 u2L2 and P2 u2L2 in (3.3.2), and to the action of the commutator i([P2 , P1 ]u, u)L2 . As the square terms approach zero, the commutator term comes into effect and yields positivity. Alternatively, as exposed in Sect. 3.7, a modification of the forms of P2 or P1 allows one to only consider a term equivalent to the commutator term that yields positivity without using the two square terms. Such ideas can be found in the early work of L. H¨ ormander [172] for the treatment of principally normal operator by modifying the operator by lower-order terms. The presentation we give in Sect. 3.7 is adapted from the

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approach of A. Fursikov and O. Yu. Imanuvilov that can be found in [156] for the derivation of a Carleman estimate of a parabolic operator. Some extension of the derivation of Carleman estimates to discretized elliptic and parabolic operators can be found in the joint works with F. Boyer and F. Hubert [85, 86, 88] and by S. Ervedoza and F. de Gournay [136], S. Ervedoza and L. Baudouin [59], and S. Ervedoza, L. Baudouin and A. Osses [60]. In these references, estimates are proven following the alternative method described in Sect. 3.7.3, solely based on (discrete) integration by parts, since no robust semi-classical calculus is available for discrete operators. In particular, no condition of the form of the sub-ellipticity has been exhibited in that context. Appendices 3.A. Poisson Bracket and Weight Function 3.A.1. Smoothness of the Characteristic Set. Here, we prove Lemma 3.1. Let (x0 , ξ 0 , τ 0 ) ∈ SChar(p) . As dϕ(x0 ) = 0, from the form of p2 given in (3.2.4), we see that ξ 0 = 0 since |ξ 0 |2 + (τ 0 )2 = 1. Since SChar(p) is given by the equation p1 (x, ξ, τ ) = 0, p2 (x, ξ, τ ) = 0, and 2 λT,τ (ξ, τ ) = |ξ|2 + τ 2 = 1, the proof amounts to showing that dp1 (x0 , ξ 0 , τ 0 ), dp2 (x0 , ξ 0 , τ 0 ), and dλT,τ (ξ 0 , τ 0 ) are of rank three. By the Euler identity for homogeneous functions, since p1 and p2 are homogeneous of degree two in (ξ, τ ), we have for j = 1, 2, dξ,τ pj (x0 , ξ 0 , τ 0 )(ξ 0 , τ 0 ) = ξ 0 · ∇ξ pj (x0 , ξ 0 , τ 0 ) + τ 0 ∂τ pj (x0 , ξ 0 , τ 0 ) = 2pj (x0 , ξ 0 , τ 0 ) = 0. This means that dξ,τ p1 (x0 , ξ 0 , τ 0 ) and dξ,τ p2 (x0 , ξ 0 , τ 0 ) are orthogonal to (ξ 0 , τ 0 ) itself colinear to∇ξ,τ λT,τ (ξ 0 , τ 0 ). Thus,     rank dp1 , dp2 , dλT,τ (x0 , ξ 0 , τ 0 ) = rank dp1 , dp2 (x0 , ξ 0 , τ 0 ) + 1. We have ∂ξi p2 (x, ξ, τ ) = 2

1≤j≤d

pij (x)ξj ,

∂ξi p1 (x, ξ, τ ) = 2τ

1≤j≤d

pij (x)∂j ϕ(x).

0 0 0 Since i,j is symmetric positive definite, we see that ∇ξ p2 (x , ξ , τ ) = 0 since ξ 0 = 0 and ∇ξ p1 (x0 , ξ 0 , τ 0 ) = 0 since dϕ(x0 ) = 0. Defining G(x) = (gij (x))i,j as the inverse of (pij (x))i,j , we have  

pij (x0 )ξi0 ∂j ϕ(x0 ) ∇ξ p1 · G∇ξ p2 (x0 , ξ 0 , τ 0 ) = 4τ

(pij (x))

1≤i,j≤d

= 2p1 (x0 , ξ 0 , τ 0 ) = 0. Since G(x0 ), like (pij (x0 ))i,j , is symmetric positive Definite, we find that rank(∇ξ p1 , ∇ξ p2 )(x0 , ξ 0 , τ 0 ) = 2, which concludes the proof.



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121

3.A.2. Expression of the Poisson Bracket. Here, we prove Lemma 3.4. 1 {pϕ , pϕ }(x, ξ, τ ) = {p2 , p1 } as p2 = Re pϕ and p1 = Im pϕ . We have 2i From (3.2.4)–(3.2.5), using the quadratic structure of p(x, ξ), we have p2 (x, ξ, τ ) = p(x, ξ) − τ 2 p(x, dϕ(x)) and p(x, ξ, τ dϕ(x))=τ ∇ξ p(x, ξ) · ∇x ϕ(x)=τ {p, ϕ}(x, ξ)=τ Hp ϕ(x), p1 (x, ξ, τ )=2˜ since ϕ only depends on the x variable. We thus have {p2 , p1 } = τ {p, {p, ϕ}} − τ 3 {p(x, dϕ(x)), {p, ϕ}}. As {p, {p, ϕ}} = Hp2 ϕ, it now remains to prove that (3.A.1)

−{p(x, dϕ(x)), {p, ϕ}}(x, ξ) = {p, {p, ϕ}}(x, dϕ(x)).

We give two proofs: a first proof by exploiting the invariance of Poisson brackets by symplectomorphisms and a second proof by simply computing the terms in (3.A.1). Finally, the second formula in the statement of the lemma holds as Hp2 ϕ(x, ξ) is homogeneous of degree two in ξ. First Proof of (3.A.1). The present proof follows the method presented in the work of D. Dos Santos Ferreira [123]. The map θ : (x, ξ) → (x, ξ + dϕ(x)) is a symplectomorphism by Proposition 9.4. Thus, for any two functions f and g, we have θ∗ {f, g} = {θ∗ f, θ∗ g} by Proposition 9.6. As θ(x, 0) = (x, dϕ(x)), we obtain (3.A.2) {p, {p, ϕ}}(x, dϕ(x)) = θ∗ {p, {p, ϕ}}(x, 0) = {θ∗ p, θ∗ {p, ϕ}}(x, 0). One has p(x, ξ, dϕ(x)) θ∗ p(x, ξ) = p(x, ξ + dϕ(x)) = p(x, ξ) + p(x, dϕ(x)) + 2˜ = p(x, ξ) + p(x, dϕ(x)) + {p, ϕ}(x, ξ), as p(x, ξ) is quadratic in ξ, and θ∗ {p, ϕ}(x, ξ) = {p, ϕ}(x, ξ + dϕ(x)) = {p, ϕ}(x, ξ) + {p, ϕ}(x, dϕ(x)) = {p, ϕ}(x, ξ) + 2p(x, dϕ(x)) as {p, ϕ}(x, ξ) is linear in ξ and using the Euler identity for homogeneous functions. As the final expression in (3.A.2) is evaluated at ξ = 0, there is no need to consider the quadratic term in the computation. We thus find, using the antisymmetry of Poisson brackets, {p, {p, ϕ}}(x, dϕ(x)) = −{p(x, dϕ(x)), {p, ϕ}}(x, 0). Observing that in fact {p(x, dϕ(x)), {p, ϕ}} is only a function of x concludes the first proof of (3.A.1).

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Second Proof of (3.A.1). For simplicity, we use the Einstein summation convention for repeated indices. With {p, ϕ}(x, ξ) = ∂ξi p(x, ξ)∂xi ϕ(x), we compute   {p, {p, ϕ}}(x, ξ) = ∂ξj p∂ξi p∂x2i xj ϕ + ∂ξj p∂x2j ξi p∂xi ϕ − ∂xj p∂ξ2i ξj p∂xi ϕ (x, ξ). With ξ = dϕ(x), using the Euler identity for homogeneous functions, we find ∂x2j ξi p(x, dϕ)∂xi ϕ(x) = 2∂xj p(x, dϕ(x)), ∂ξ2i ξj p(x, dϕ)∂xi ϕ(x) = ∂ξj p(x, dϕ(x)). This yields   {p, {p, ϕ}}(x, dϕ(x)) = d2x ϕ(x) ∇ξ p(x, dϕ(x)), ∇ξ p(x, dϕ(x)) + ∇ξ p(x, dϕ(x)) · ∇x p(x, dϕ(x)). We also compute − {p(x, dϕ), {p, ϕ}}(x, ξ)   = ∂xj p(x, dϕ) ∂ξj {p, ϕ}(x, ξ)   = ∂ξ p(x, dϕ(x))∂x2 xj ϕ(x) + ∂xj p(x, dϕ(x)) ∂ξ2i ξj p(x, ξ)∂xi ϕ(x)   = d2x ϕ(x) ∇ξ p(x, dϕ(x)), ∇ξ p(x, dϕ(x)) + ∇x p(x, dϕ(x)) · ∇ξ p(x, dϕ(x)), using again the Euler identity. This concludes the second proof of (3.A.1).  Remark 3.54. Developing the expression of the Poisson bracket, we obtain (3.A.3) 1 {pϕ , pϕ }(x, ξ, τ ) = {p2 , p1 }(x, ξ, τ ) 2i   = 2∇ξ p(x, ξ) · ∇x p˜(x, ξ, τ dϕ(x)) + τ d2x ϕ(x) ∇ξ p(x, ξ), ∇ξ p(x, ξ)) − ∇x p(x, ξ) · ∇ξ p(x, τ dϕ(x)) + ∇x p(x, τ dϕ(x)) · ∇ξ p(x, τ dϕ(x))   + τ d2x ϕ(x) ∇ξ p(x, τ dϕ(x)), ∇ξ p(x, τ dϕ(x)) . The structure put forward by Lemma 3.4 is however hidden in such an expression. 3.A.3. Construction of a Weight Function. We begin with the following lemma.

3.A. POISSON BRACKET AND WEIGHT FUNCTION

123

Lemma 3.55. Let G ∈ C ∞ (R) and let ψ ∈ C ∞ (Rd ). Setting ϕ = G ◦ ψ, one has 1 {pϕ , pϕ }(x, ξ, τ ) 2i   2 = τ (G ◦ ψ)(G ◦ ψ)2 (x) Hp ψ(x, η) + 4τ 2 p(x, dψ(x))2 + (G ◦ ψ)3 (x)

1 {pψ , pψ }(x, η, τ ), 2i

for ξ = (G ◦ ψ)(x)η. Proof. First observe that dϕ = (G ◦ ψ)dψ,

d2x ϕ = (G ◦ ψ)d2x ψ + (G ◦ ψ)dψ ⊗ dψ.

From Lemma 3.4 we have 1 {pϕ , pϕ }(x, ξ, τ ) = Hp2 ϕ(x, ξ) + Hp2 ϕ(x, τ dϕ(x)). 2iτ  2 As Hp ϕ = (G ◦ ψ)Hp ψ, we find Hp2 ϕ = (G ◦ ψ) Hp ψ + (G ◦ ψ)Hp2 ψ, which yields 1 {pϕ , pϕ }(x, ξ, τ ) 2iτ  2  2  = (G ◦ ψ)(x) Hp ψ(x, ξ) + Hp ψ(x, τ dϕ(x))   + (G ◦ ψ)(x) Hp2 ψ(x, ξ) + Hp2 ψ(x, τ dϕ(x))   2 2  = (G ◦ ψ)(x) Hp ψ(x, ξ) + (G ◦ ψ)2 Hp ψ(x, τ dψ(x))   + (G ◦ ψ)(x) Hp2 ψ(x, ξ) + (G ◦ ψ)2 (x)Hp2 ψ(x, τ dψ(x)) , using that Hp ϕ(x, ξ) and Hp2 ϕ(x, ξ) are homogeneous of degree one and two in ξ, respectively. With ξ = (G ◦ ψ)(x)η one finds 1 {pϕ , pϕ }(x, ξ, τ ) 2iτ  2  2  = (G ◦ ψ)(G ◦ ψ)2 (x) Hp ψ(x, η) + Hp ψ(x, τ dψ(x))   + (G ◦ ψ)3 (x) Hp2 ψ(x, η) + Hp2 ψ(x, τ dψ(x)) , which by Lemma 3.4 gives 1 {pϕ , pϕ }(x, ξ, τ ) 2iτ =(G ◦ ψ)(x)(G ◦ ψ)2 (x)



2  2  Hp ψ(x, η) + Hp ψ(x, τ dψ(x))

1 {pψ , pψ }(x, η, τ ). 2iτ With the homogeneity of p(x, ξ) and the Euler identity, we conclude the proof.  + (G ◦ ψ)3 (x)

124

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

With the previous lemma we are in a position to give simple proofs for Lemmata 3.5 and 3.7. Proof of Lemma 3.5. Here ϕ = G ◦ ψ with G(s) = H(γs). We set G1 (s) = H  (γs) and G2 = H  (γs). We have G ◦ ψ = γG1 ◦ ψ and G ◦ ψ = γ 2 G2 ◦ ψ. With Lemma 3.55, this yields 1 {pϕ , pϕ }(x, ξ, τ ) 2i = τ γ 4 (G2 G21 )(ψ(x))



Hp ψ(x, η)

2

+ 4τ 2 p(x, dψ(x))2



1 {pψ , pψ }(x, η, τ ) 2i   ≥ 4τ 3 γ 4 (G2 G21 )(ψ(x))p(x, dψ(x))2 − C(γG1 )3 (ψ(x)) τ 3 + |η|3 ,   ≥ 4γ τ˜3 (x)(G2 /G1 )(ψ(x))p(x, dψ(x))2 − C τ˜3 (x) + |ξ|3 , + (γG1 )3 (ψ(x))

using that {pψ , pψ }(x, η, τ ) is homogeneous of degree three in (η, τ ), where −1  −1  η = G (ψ(x)) ξ = γG1 (ψ(x)) ξ and τ˜ = τ G ◦ ψ = τ γG1 ◦ ψ = τ γH  (γψ). Using that (G2 /G1 )(s) = (H  /H  )(γs) ≥ C0 , we obtain (3.A.4)

  1 {pϕ , pϕ }(x, ξ, τ ) ≥ 4C0 γ τ˜3 (x)p(x, dψ(x))2 − C τ˜3 (x) + |ξ|3 . 2i

As dψ(x) = 0 in V , we have p(x, dψ(x)) ≥ C > 0 for x ∈ V . Consider now (x, ξ, τ ) with x ∈ V such that Re pϕ (x, ξ, τ ) = 0. By the form of Re pϕ = p2 in (3.2.4), then |ξ|  τ |dϕ(x)|  τ |G ◦ ψ||dψ(x)|  τ˜(x)|dψ(x)|  τ˜(x). We thus obtain 1 {pϕ , pϕ }(x, ξ, τ ) ≥ τ˜3 (x)(Cγ − C  ), 2i for C > 0 and C  > 0. For γ > 0 chosen sufficiently large, we obtain 1 {pϕ , pϕ }(x, ξ, τ )  τ˜3  λ3τ 2i 

on Char(Re pϕ ).

Proof of Lemma 3.7. For ε > 0 chosen sufficiently small, we have |dψ(x)| ≥ C0 > 0 if x ∈ Vε . This yields p(x, dψ(x)) ≥ C > 0 for x ∈ Vε . We also have |ψ(x)| ≤ C1 |x − x0 | ≤ C1 ε. We choose ε > 0 and γ > 0 such that C1 γε ≤ 1/2. As one has Gγ (s) = 1 + γs, this implies (3.A.5)

Gγ ◦ ψ(x) = 1 + γψ(x)  1,

x ∈ Vε .

3.B. SYMBOL POSITIVITY

125

 Note also that Gγ (s) = γ > 0. With Lemma 3.55 and setting η = ξ/ Gγ ◦  ψ(x) , we have    2  2 1 {pϕ , pϕ }(x, ξ, τ ) = τ γ Gγ ◦ ψ(x) Hp ψ(x, η) + 4τ 2 p(x, dψ(x))2 2i 3 1  {pψ , pψ }(x, η, τ ) + Gγ ◦ ψ(x) 2i     ≥ Cγτ 3 − C  τ 3 + |η|3 ≥ Cγτ 3 − C  τ 3 + |ξ|3 , using (3.A.5) and that {pψ , pψ }(x, η, τ ) is homogeneous of degree three in (η, τ ), and that |η|  |ξ| again by (3.A.5). If (x, ξ, τ ) is such that x ∈ Vε and Re pϕ (x, ξ, τ ) = 0, then by the form of Re pϕ = p2 in (3.2.4) one has |ξ|  τ |dϕ(x)|  τ |Gγ ◦ ψ||dψ(x)|  τ, which implies   1 {pϕ , pϕ }(x, ξ, τ ) ≥ λ3τ Cγ − C  . 2i Choosing γ sufficiently large concludes the proof.



3.A.4. Local Extension of the Domain Where Sub-ellipticity Holds. Here, we prove Lemma 3.9. Since |dϕ| = 0 in V and V is bounded, we have |dϕ| ≥ C > 0 in V . Moreover, there exists W an open neighborhood of V where |dϕ| ≥ C  > 0. Set Sξ,τ = {(ξ, τ ) ∈ Rd × [0, +∞); |ξ|2 + τ 2 = 1}. As the sub-ellipticity property holds in V , by Lemma 3.8, for ρ = μ(p22 + p21 ) + τ {p2 , p1 } and for some μ > 0, we have ρ(x, ξ, τ ) ≥ C > 0,

(x, ξ, τ ) ∈ V × Sξ,τ .

As V × Sξ,τ is compact, there exists an open set V  ⊂ W such that V ⊂ V  and ρ ≥ C  > 0 on V  × Sξ,τ . By homogeneity we obtain ρ(x, ξ, τ ) ≥ C  λ4τ ,

(x, ξ, τ ) ∈ V  × Rd × [0, +∞).

Thus, if (x, ξ, τ ) ∈ V  × Rd × [0, +∞) and pϕ (x, ξ, τ ) = 0, that is p2 (x, ξ, τ ) = p1 (x, ξ, τ ) = 0, then {p2 , p1 }(x, ξ, τ ) > 0. The sub-ellipticity property thus  holds in V  for (P, ϕ). 3.B. Symbol Positivity We start this section by an elementary lemma. Lemma 3.56. Consider two continuous functions, f and g, defined in a compact set K, and assume that f ≥ 0 and moreover f (y) = 0 ⇒ g(y) > 0

for all y ∈ K.

Setting hμ = μf + g, we have hμ ≥ C > 0 for μ > 0 chosen sufficiently large.

126

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

Proof. Let y ∈ K: (1) Either f (y) = 0 and setting μy = 1, we have hμy (y) > 0 (2) Or f (y) > 0 and thus there exists μy > 0 such that hμy (y) > 0 The inequality hμ > 0 holds also in an open neighborhood Vy of y and for any μ ≥ μy . From the covering of the compact set K by the open sets Vy , we extract a finite covering Vy1 , . . . , Vyn and we set μ = max1≤j≤n μyj . We  then obtain hμ > 0 in K, which yields the result since K is compact. 3.B.1. Symbol Positivity Away from a Boundary. Here we prove Lemma 3.8. As pointed out in Remark 3.3-(1), the ellipticity of p allows one to extend to sub-ellipticity property of Definition 3.2 to the case τ = 0. Next we note that ρ is homogeneous of degree 4 in the variables (ξ, τ ). We thus place ourselves on the compact set K = {(x, ξ, τ ); x ∈ V , |ξ|2 + τ 2 = 1, τ ≥ 0}. We apply the result of Lemma 3.56 to ρ on K with f = p22 + p21 and g =  τ {p2 , p1 }, and we conclude by homogeneity. 3.B.2. Tangential Symbol Positivity Near a Boundary. Here, we prove Lemma 3.26. Set f (x, ξ  , τ ) = |(ξ  , τ )|−2 ρ2 (x, ξ  , τ ) and g(x, ξ  , τ ) = b2 (x, ξ  , τ ) and consider the compact set K = {(x, ξ  , τ ); x ∈ U + , ξ 2 + τ 2 = 1, τ ≥ 0}. Applying Lemma 3.56, since (3.4.29) holds, we conclude by homogeneity  since both f and g are homogeneous of degree 2 in (ξ  , τ ). 3.B.3. Proof of Lemma 3.27. Observe that x → x(x2 +δ)−1/4 +|x|1/2 converges uniformly6 to x → 0 on (−∞, 0] as δ goes to 0. Writing  q2 | α2 q˜2 + |b1 | |˜  q˜2 = (b21 + δλ2T,τ )1/2 2 + |b1 | |˜ q2 | 4 1/4 (˜ q2 + δλT,τ )    q˜2 + |˜ q2 | = (b21 + δλ2T,τ )1/2 (˜ q22 + δλ4T,τ )1/4    |˜ q2 | , + |b1 | − (b21 + δλ2T,τ )1/2 6In fact, for x ≤ 0, write

  x(x2 + δ)−1/4 + |x|1/2 = x (x2 + δ)−1/4 − (x2 )−1/4  2 1/4  x (x ) − (x2 + δ)1/4 , = 2 (x + δ)1/4 (x2 )1/4

from which we obtain, as (x2 + δ)1/2 − |x| ≤ δ 1/2 ,   |x(x2 + δ)−1/4 + |x|1/2 | ≤ (x2 )1/4 − (x2 + δ)1/4   −1  2  = (x2 )1/4 + (x2 + δ)1/4 (x + δ)1/2 − |x| ≤ δ 1/4 , which yields the uniform convergence.

3.B. SYMBOL POSITIVITY

we then see that α2 q˜2 + |b1 | set K given by

127

 |˜ q2 | converges uniformly to 0 on the compact

K = {(ξ  , τ ); λT,τ = 1 and q˜2 (0, ξ  , τ ) ≤ 0} as δ goes to 0. Thus, for any ε > 0, for δ > 0 chosen sufficiently small, we have by homogeneity    2 α q˜2 + |b1 | |˜ (3.B.1) q2 | ≤ ελ2T,τ . Similarly, we observe that x → x2 (x2 + δ)−1/2 − |x| converges uniformly to x → 0 on the whole real line, as δ goes to 0. Writing  α−2 b21 − |b1 | |˜ q2 |  b21 2 4 1/4 (˜ q + δλ ) − |b | |˜ q2 | = 2 1 T ,τ 2 (b1 + δλ2T,τ )1/2   b21 − |b | (˜ q22 + δλ4T,τ )1/4 = 1 (b21 + δλ2T,τ )1/2    2 + |b1 | (˜ q2 + δλ4T,τ )1/4 − |˜ q2 |    −2 2 we see that α b1 − b1 |˜ q2 | (0, ξ  , τ ) converges uniformly to 0 on K = {(ξ  , τ ); λT,τ = 1}, as δ goes to 0. Thus for any ε > 0, for δ > 0 chosen sufficiently small, we have by homogeneity     −2 2  α b1 − |b1 | |˜ (3.B.2) q2 | (0, ξ  , τ ) ≤ ελ2T,τ . For ε > 0 chosen sufficiently small (3.B.1) and (3.B.2) together with (3.4.44) yields the result.  3.B.4. Symbol Positivity in the Modified Approach. This section is concerned with the proof of Lemma 3.44. In the proof of Lemma 3.5 in Appendix 3.A.3 we compute in (3.A.4) 1 {p2 , p1 }(x, ξ, τ ) = {pϕ , pϕ }(x, ξ, τ ) 2i   ≥ 4γ τ˜3 (x)p(x, dψ(x))2 − C τ˜3 (x) + |ξ|3 , here for H(s) = exp(s) and ϕ = G(ψ) = H(γψ) = exp(γψ). In particular, here we have τ˜ = τ γϕ. As dϕ = γϕdψ, we have τ dϕ = τ˜dψ, yielding   θ(x)p2 (x, ξ, τ ) = P0 ϕ(x) p2 (x, ξ, τ )    = γϕ(x) − γp(x, dψ(x)) + P0 ψ(x) p(x, ξ) − τ˜2 (x)p(x, dψ(x)) . We thus have −2μτ θ(x)p2 (x, ξ, τ ) ≥ −2μγ τ˜3 p(x, dψ(x))2 + 2μγ τ˜p(x, dψ(x))p(x, ξ)   − C τ˜(x) τ˜(x)2 + |ξ|2 .

128

3. CARLEMAN ESTIMATE FOR A SECOND-ORDER ELLIPTIC OPERATOR

This implies q(x, ξ, τ ) ≥ (4 − 2μ)γ τ˜3 (x)p(x, dψ(x))2 + 2μγ τ˜(x)p(x, dψ(x))p(x, ξ)   − C τ˜3 (x) + |ξ|3 . As 0 < μ < 2 and |dψ| > 0 on V , from the ellipticity of p(x, ξ), we have   q(x, ξ, τ )  τ˜(x) τ˜(x)2 + |ξ|2 (Cγ − C  ), 

which yields the result. 3.C. An Explicit Computation

Here, we reproduce the computation of the term I11 in Sect. 3.7.3, yet making tensor indices appear explicitly. All sums are for indices in {1, . . . , d}.

We have P0 v = i,j Di pij Dj . With an integration by parts, we obtain I11 = 2τ Re

(Di (pij Dj v), ipk ∂ ϕDk v)L2 (Ω)

i,j,k,

= 2τ Re



(pij Dj v, ∂i (pk ∂ ϕDk v))L2 (Ω)

i,j,k,

 − (νi pij Dj v|∂Ω , pk ∂ ϕDk v|∂Ω )L2 (∂Ω) . We write

ij (p Dj v, ∂i (pk ∂ ϕDk v))L2 (Ω) 2 Re i,j,k,

 ij 2 (p Dj v, ∂ i ϕ pk Dk v)L2 (Ω) + (pij Dj v, (∂i pk )∂ ϕ Dk v)L2 (Ω)

= 2 Re

i,j,k,

 + (p Dj v, pk ∂ ϕ ∂k Di v)L2 (Ω) . ij

For the third term in the r.h.s., using the symmetry of P, we write

 k

ij (p Dj v, pk ∂ ϕ ∂k Di v)L2 (Ω) = 2 Re ∫ p ∂ ϕ∂k (pij Dj vDi v) dx i,j,k,

Ω

i,j,k,

 − ∫ pk (∂ ϕ)(∂k pij )Dj vDi v dx . Ω

With an integration by parts we obtain

 i,j,k,

∫ pk ∂ ϕ∂k (pij Dj vDi v) dx =

Ω

 i,j,k,

− ∫ ∂k (pk ∂ ϕ)pij Dj vDi v dx Ω



+ ∫ νk p ∂ ϕp Dj vDi v dσ , k

∂Ω

ij

3.C. AN EXPLICIT COMPUTATION

129

yielding

ij 2 Re (p Dj v, ∂i (pk ∂ ϕDk v))L2 (Ω) i,j,k,

=



  2 ∫ 2∂ i ϕ pij Dj v pk Dk v − ∂k (pk ∂ ϕ)pij Dj vDi v dx i,j,k,

Ω

+ ∫ νk pk ∂ ϕpij Dj vDi v dσ − ∫ (∂k pij )pk ∂ ϕ Dj vDi v dx ∂Ω

Ω

 + 2 Re(pij Dj v, (∂i pk )∂ ϕ Dk v)L2 (Ω) . This is precisely (3.7.9).

CHAPTER 4

Optimality Aspects of Carleman Estimates Contents 4.1. On the Necessity of the Sub-ellipticity Property 4.1.1. Bracket Nonnegativity 4.1.2. Optimal Strength in the Large Parameter and Bracket Positivity 4.2. Limiting Weights and Limiting Carleman Estimates 4.2.1. Limiting Weights 4.2.2. Convexification 4.2.3. Limiting Carleman Estimates Away from a Boundary 4.2.4. Global Limiting Carleman Estimates 4.3. Carleman Weight Behavior at a Boundary 4.4. Notes Appendix 4.A. Some Technical Results 4.A.1. A Linear Algebra Lemma 4.A.2. Sub-ellipticity for First-Order Operators with Linear Symbols 4.A.3. A Particular Class of Limiting Weights

132 132 138 150 150 151 152 155 159 170 171 171 171 173 176

As in Chap. 3, we consider a general second-order elliptic operator P

i P = P0 + b (x)Di + c(x), 1≤i≤d

with principal part (4.0.1) Di (pij (x)Dj ), P0 = 1≤i,j≤d

with p(x, ξ) =

1≤i,j≤d

pij (x)ξi ξj ≥ C|ξ|2 ,

where pij ∈ C ∞ (Rd ; R) with all derivatives bounded and such that pij = pji , 1 ≤ i, j ≤ d, and bi , c ∈ L∞ (Rd ), 1 ≤ i ≤ d. We recall that D = −i∂. © 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 I, PNLDE Subseries in Control 97, https://doi.org/10.1007/978-3-030-88674-5 4

131

132

4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

4.1. On the Necessity of the Sub-ellipticity Property In Chap. 3 all the proofs of Carleman estimates rely on the sub-ellipticity property of the weight function ϕ and P . We recall that the principal symbol of the conjugated operator eτ ϕ P e−τ ϕ is given by pϕ (x, ξ, τ ) = p2 (x, ξ, τ ) + ip1 (x, ξ, τ ), with p2 (x, ξ, τ ) = p(x, ξ) − p(x, τ dϕ(x)) = |ξ|2x − |τ dϕ(x)|2x , p(x, ξ, τ dϕ(x)) = 2(ξ, τ dϕ(x))x . p1 (x, ξ, τ ) = 2˜ See Sects. 3.1 and 3.2 and the notation therein. The Poisson bracket that appears in the sub-ellipticity condition is {p2 , p1 }; see Definition 3.2. In Sect. 4.1.1, we prove that a weak Carleman estimate implies the nonnegativity of the Poisson bracket {p2 , p1 } on the characteristic set of pϕ . By weak we simply mean an estimation of the form (4.1.1)

τ α eτ ϕ uL2  eτ ϕ P uL2

where α is any real number, possibly negative. In Chap. 3 we derived estimates where α = 3/2. In Sect. 4.1.2, we prove that a necessary condition for (4.1.1) to hold is α ≤ 3/2 in dimension d ≥ 2. We then prove that having τ 3/2 eτ ϕ uL2  eτ ϕ P uL2 , necessarily implies the sub-ellipticity condition: The Poisson bracket {p2 , p1 } is positive on the characteristic set of pϕ . The particular case of the dimension d = 1 is also presented. In that case we find α ≤ 2 and we show this is optimal thus improving upon the results of Chap. 3. 4.1.1. Bracket Nonnegativity. The following result shows that even a very weak estimate implies that the bracket {p2 , p1 } is nonnegative on Char(pϕ ). Theorem 4.1. Let d ≥ 1. Let V be a bounded open set in Rd , ϕ(x) ∈ C ∞ (Rd , R), τ∗ > 0 and C0 > 0 such that, for some α ∈ R, (4.1.2)

τ α eτ ϕ uL2 (Rd ) ≤ C0 eτ ϕ P uL2 (Rd ) ,

for all u ∈ Cc∞ (V ) and τ ≥ τ∗ . Then, (4.1.3) ∀x ∈ V, ∀ξ ∈ Rd , ∀τ > 0, pϕ (x, ξ, τ ) = 0 1 {pϕ , pϕ }(x, ξ, τ ) = {p2 , p1 }(x, ξ, τ ) ≥ 0. ⇒ 2i Observe that α is allowed to be negative here. The result of Theorem 4.1 shows that if {p2 , p1 } < 0 at a point (x0 , ξ 0 , τ 0 ) ∈ Char(pϕ ), with τ 0 > 0, then a Carleman estimate cannot hold in a neighborhood of x0 . This is the idea of the proof: Assuming that such

4.1. ON THE NECESSITY OF THE SUB-ELLIPTICITY PROPERTY

133

a point exists, we construct a quasimode, that is, a function vτ supported near x0 , with τ > 0 as a parameter, and such that Pϕ vτ L2  vτ L2 ,

(4.1.4)

thus ruining any hope of achieving an estimate as in (4.1.2). Before proceeding with the actual proof of Theorem 4.1 we expose the main idea that leads to the result. Main Idea in the Proof of Theorem 4.1. The argument amounts to a factorization of Pϕ into two first-order factors, one elliptic, one nonelliptic. Then, a function vτ satisfying (4.1.4) is constructed by only considering the nonelliptic factor. For this reason, here, we replace the operator Pϕ by the first-order operator L = Dx + iτ q(x) with τ > 0 and q real and we restrict ourselves to the one-dimensional case for simplicity. The principal symbol of L is (x, ξ, τ ) = ξ + iτ q(x). Having (x0 , ξ 0 , τ 0 ) = 0 and ¯ }(x0 , ξ 0 , τ 0 )/2i < 0 reads {, ξ 0 = 0,

q(x0 ) = 0, and q  (x0 ) < 0.

Without any loss of generality we assume that x0 = 0. Introduce Q(x) = ∫0x q(s)ds. We have Q(x) = q  (0)x2 /2 + O(x3 ). Let χ ∈ Cc∞ (R) such that Q(x) ≤ −δ on supp(χ ) for some δ > 0 and χ ≡ 1 in a neighborhood V of 0. One sets vτ (x) = χ(x)eτ Q(x) . On the one hand, one finds Lvτ (x) = (Dx χ)(x)eτ Q(x) and Lvτ (x)L2 (R)  e−δτ . On the other hand one has 

vτ (x)2L2 (R) ≥ ∫ eτ q (0)x

2 −τ C|x|3

dx = τ −1/2



eq (0)y

∫

2 −Cτ −1/2 |y|3

dy

y∈τ 1/2 V

V

The Lebesgue dominated-convergence theorem yields 

2

vτ (x)2L2 (R)  τ −1/2 ∫ eq (0)y dy, y∈R

implying (4.1.4).



We now give a detailed proof of the result. Proof of Theorem 4.1. First, assume that dϕ(x0 ) = 0 and d ≥ 1 arbitrary. Then pϕ (x0 , ξ, τ ) = p(x0 , ξ) for any τ > 0 and ξ ∈ Rd . Having pϕ (x0 , ξ, τ ) = 0 thus implies that ξ = 0 since p is elliptic. As p2 and p1 are both homogeneous of degree two in (ξ, τ dϕ(x)) we find that {p2 , p1 } is homogeneous of degree three in (ξ, τ dϕ(x)). With (ξ, τ dϕ(x)) = (0, 0) here we find {p2 , p1 }(x0 , ξ, τ ) = 0. This can be confirmed by inspecting the expression of {p2 , p1 } computed in (3.A.3). Consequently, (4.1.3) holds. Second, assume d = 1 and dϕ(x0 ) = 0. The principal symbol of P reads p(x, ξ) = a(x)ξ 2 , with a(x) ≥ C > 0 in V. This gives pϕ (x, ξ, τ ) = p(x, ξ+iτ dϕ(x)) = a(x) ξ 2 −(τ ϕ (x))2 +2iτ ϕ (x)ξ . Observe that for x = x0 ,

134

4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

then pϕ (x0 , ξ, τ ) = 0 for all (ξ, τ ) ∈ Rd ×R+ and τ > 0. Consequently, (4.1.3) holds trivially. Third, assume d ≥ 2 and dϕ(x0 ) = 0. Let us assume that for some 0 ξ ∈ Rd and τ 0 > 0 we have (4.1.5)

pϕ (x0 , ξ 0 , τ 0 ) = 0 and

1 {pϕ , pϕ }(x0 , ξ 0 , τ 0 ) < 0. 2i

By homogeneity we may assume that τ 0 = 1. Below, we prove that (4.1.5) is in contradiction with the assumed functional inequality (4.1.2). Locally, Σ = {x ∈ Rd ; ϕ(x) = ϕ(x0 )} is a submanifold. Using normal geodesic coordinates in a neighborhood of x0 as given by Theorem 9.7, where x0 = 0 and Σ is locally given by {xd = 0} with x = (x , xd ), x ∈ Rd−1 and xd ∈ R, the operator P takes the form P (x, D) = Dd2 + R(x, D ) + R1 (x, D),

ij ij where R(x, D ) = d−1 i,j=1 r (x)Di Dj , with (r (x)) smooth, symmetric, and uniformly elliptic, and R1 is a smooth first-order differential operator. We choose the orientation of the xd -axis to be such that ∂d ϕ(x0 ) > 0. Without any loss of generality we assume that ϕ(x , xd = 0) = ϕ(0) = 0. For the sake of concision we set  = (x, ξ, τ ),  = (x, ξ  , τ ). Then, at 0 = (x0 , ξ 0 , 1) we have (4.1.5). The expressions in (4.1.5) are of geometrical nature (see Chap. 9). They are thus independent of the chosen local coordinates. For M ∈ N, M ≥ 2, with a Taylor expansion of the coefficients of R(x, D ), R1 (x, D) and the weight function ϕ at x = 0 we write RM (x, D ) =

d−1

˜ D ), rM,ij (x)Di Dj = R(x, D ) + |x|M R(x,

i,j=1

˜ 1 (x, D), R1M (x, D) = R1 (x, D) + |x|M R ˜ ϕM (x) = ϕ(x) + |x|M +2 ϕ(x), ˜ 1 (x, D) are differential operators with smooth coeffi˜ D ) and R where R(x, cients of order two and one, respectively, and ϕ˜ is a smooth real function. The functions ϕM and the coefficients of RM and R1M are polynomial in x. We denote by rM (x, ξ) the principal symbol of RM (x, D ). We set P M (x, D) = Dd2 + RM (x, D ) + R1M (x, D). The conjugated operator Pϕ (x, D, τ ) = eτ ϕ P (x, D)e−τ ϕ reads (4.1.6)

Pϕ (x, D, τ ) = PϕMM (x, D, τ ) + |x|M T2M (x, D, τ ),

4.1. ON THE NECESSITY OF THE SUB-ELLIPTICITY PROPERTY

135

with PϕMM (x, D, τ ) = eτ ϕ P M (x, D)e−τ ϕ and T2M (x, D, τ ) ∈ Dτ2 . We write PϕMM (x, D, τ ) = Q(x, D, τ ) + Q1 (x, D, τ ) with  M M Q(x, D, τ ) = eτ ϕ Dd2 + RM (x, D ) e−τ ϕ  2 = Dd + iτ ∂d ϕM (x) M

+

d−1

i,j=1

M

rM,ij (x)(Di + iτ ∂i ϕM )(Dj + iτ ∂j ϕM ) ∈ Dτ2 ,

and Q1 (x, D, τ ) = eτ ϕ R1M (x, D)e−τ ϕ ∈ Dτ1 . The principal symbol of PϕMM (x, D, τ ) is that of Q(x, D, τ ) and reads M

M

q() = (ξd + iτ ∂d ϕM (x))2 + rM (x, ξ  + iτ dx ϕM (x)). This symbol is polynomial in all its variables. From (4.1.5) we obtain, since M ≥ 2 and x0 = 0, 1 (4.1.7) {¯ q , q}(0 ) < 0. q(0 ) = 0 and 2i Under (4.1.7), for any N ∈ N, we construct a family of functions (vτ )τ >0 , supported near x0 , such that (4.1.8)

Pϕ vτ L2 (Rd )  τ −N ,

vτ L2 (Rd )  τ −d/4 .

We then have Pϕ vτ L2 (Rd )  vτ L2 (Rd ) in contradiction with (4.1.2). The remainder of the proof is dedicated to the construction of vτ . We start by solving the following Eikonal equation, with also a requirement on the Hessian of the imaginary part of the solution. Lemma 4.2 (Eikonal Equation). There exist a neighborhood V 0 of x0 = 0 and a complex valued analytic function w(x) defined in V 0 such that q(x, dw(x), 1) = 0, w(0) = 0, dw(0) = ξ 0 , and d2 Im w(0) = Im d2 w(0) is positive definite. A proof is given below. As Im w(0) = 0 and d Im w(0) = 0 we have Im w(x) = d2 Im w(0)(x, x) + O|x|3 . Upon reducing the size of V 0 , this implies (4.1.9)

C|x|2 ≤ Im w(x) ≤ C  |x|2 ,

x ∈ V 0,

for some C, C  > 0. We use the function given by Lemma 4.2 as a phase. We compute e−iτ w PϕMM (x, D, τ )eiτ w = q(x, τ dw(x), τ ) + ∇ξ q(x, τ dw(x), τ ) · D + P M (x, D) + τ T0M (x), with T0M (x) an analytic function. By homogeneity we have q(x, τ dw(x), τ ) = τ 2 q(x, dw(x), 1) = 0 and ∇ξ q(x, τ dw(x), τ ) = τ ∇ξ q(x, dw(x), 1). If set L1 (x, D) = ∇ξ q(x, dw(x), 1) · D + T0M ∈ D 1 we find e−iτ w PϕMM (x, D, τ )eiτ w = τ L1 (x, D) + P M (x, D).

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4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

Lemma 4.3 (Transport Equation). There exist a neighborhood V 1 of x0 1 with V 1 ⊂ V 0 , and, for any N ∈ N, a function aN τ (x) defined in V , given by

τ −j aj (x), a0 (0) = 1, aN τ (x) = 0≤j≤N

where the functions aj , j = 0, . . . , N , are analytic in V 1 , bounded, with bounded iterated derivatives in V1 , and such that   −N τ L1 (x, D) + P M (x, D) aN ). τ (x) = O(τ A proof is given below. Building an amplitude function aN τ according to Lemma 4.3, we now set iτ w vτ = χaN , τ e

where χ ∈ Cc∞ (V 1 ) is such that 0 ≤ χ ≤ 1 and χ ≡ 1 in a neighborhood V 2  V 1 of 0 and |a0 |2 ≥ 1/2 in supp(χ). By (4.1.9) we have (4.1.10)

x ∈ V 1 \ V 2.

Im w(x) ≥ C1 > 0,

2 For τ chosen sufficiently large, we have |aN τ | ≥ 1/4 in supp(χ). This gives 1 2 −2τ Im w(x) vτ 2L2 (Rd ) = ∫ χ2 (x)|aN dx ≥ ∫ χ2 (x)e−2τ Im w(x) dx. τ (x)| e 4 Rd Rd

With (4.1.9) we then find 1 1 2 2 vτ 2L2 (Rd ) ≥ ∫ χ2 (x)e−Cτ |x| dx = τ −d/2 ∫ χ2 (τ −1/2 y)e−C|y| dy, 4 Rd 4 Rd with the change of variables y = τ 1/2 x. Then, the Lebesgue dominatedconvergence theorem yields (4.1.11)

vτ 2L2 (Rd )  τ −d/2 .

We now compute PϕMM (x, D, τ )vτ = h1 + h2 with h1 = [PϕMM (x, D, τ ), χ]eiτ w aN τ (x),

h2 = χPϕMM (x, D, τ )eiτ w aN τ (x).

As [PϕMM (x, D, τ ), χ] ∈ Dτ1 and Im w ≥ C1 in supp(dχ) by (4.1.10), we have −τ C1 O(1). [PϕMM (x, D, τ ), χ]eiτ w aN τ (x) = τ e

We thus find h1 L2 (Rd )  e−Cτ , for some C > 0. By Lemma 4.3 we have h2 = χeiτ w O(τ −N ) implying h2 L2 (Rd )  τ −N . We thus conclude (4.1.12)

PϕMM (x, D, τ )vτ 

L2 (Rd )

 τ −N .

With the operator T2M (x, D, τ ) ∈ Dτ2 that appears in (4.1.6) one has T2M (x, D, τ )vτ = bτ eiτ w with |bτ |  τ 2 in V 1 that contains the support of χ. With (4.1.9) we have 2

|x|M |T2M (x, D, τ )vτ |  τ 2 |x|M e−Cτ |x| ≤ CM τ 2−M/2 .

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137

Choosing M sufficiently large, from (4.1.6) and (4.1.12) we find Pϕ (x, D, τ )vτ L2 (Rd )  τ −N . We have thus obtained (4.1.8); the proof of Theorem 4.1 is complete.



Proof of Lemma 4.2. We let α( ) ∈ C be such that α( )2 = rM (x, ξ  + iτ dx ϕM (x)) and Re α( ) ≥ 0. We then write

   q() = ξd − ρ+ ( ) ξd − ρ− ( ) , with ρ± ( ) = −iτ ∂d ϕM (x) ± iα( ).

Observe that Im ρ± = −τ ∂d ϕM ± Re α. As Re α ≥ 0 and ∂d ϕM (x0 ) > 0 we find that ρ− ( ) cannot be real for x in a neighborhood of x0 . Since q(0 ) = 0, this implies that (4.1.13)

ξd0 = ρ+ (0 ) = ρ− (0 ),

0 = (x0 , ξ 0 , 1).

In particular, the roots ρ+ ( ) and ρ− ( ) of q() viewed a polynomial function in ξd with  = (x, ξ  , τ ) as parameters are simple for  in a conic neighborhood of 0 . By a classical result the functions ρ± ( ) are analytic with respect to  (see for instance Proposition 6.28 in Volume 2). Set f () = ξd − ρ+ ( ) and e() = ξd − ρ− ( ). Observe that   1 1 q , q}(0 ) < 0, Im ∇ξ f¯ · ∇x f (0 ) = {f¯, f }(0 ) = |e|−2 {¯ 2i 2i since f (0 ) = 0. Then, by Lemma 4.33, as ∇ξ f (0 ) = 0, there exists a symmetric matrix A = (Aij )1≤i,j≤d with positive definite imaginary part such that ∇x f (0 ) + A∇ξ f (0 ) = 0. Let x → w0 (x ), x ∈ Rd−1 , defined in a neighborhood of x = 0, be analytic and such that w0 (0) = 0,

dx w0 (0) = ξ 0 ,

d2x w0 (0) = A ,

where A = (Aij )1≤i,j≤d−1 . By the Cauchy–Kovalevska¨ıa theorem there exists an analytic function x → w(x) solution to ∂d w(x) − ρ+ (x, dx w(x), 1) = f (x, dw(x), 1) = 0,

w|xd =0 (x ) = w0 (x ),

in a neighborhood of x0 = 0, implying q(x, dw(x), 1) = 0. Note that we have dw(x0 ) = ξ 0 . We also claim that d2 w(x0 ) = A. In fact, for j = 1, . . . , d, from the equation we compute 0 = ∂xj f (x, dw(x), 1) = (∂xj f )(x, dw(x), 1)

2 + ∂ξk f (x, dw(x), 1)∂jk w(x), 1≤k≤d

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4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

yielding for x = x0 0 = ∂xj f (0 ) + =

 1≤k≤d

1≤k≤d 0

2 ∂ξk f (0 )∂jk w(x0 )

 2 w(x ) − Ajk ∂ξk f (0 ). ∂jk

For j = 1, . . . , d − 1, this gives (4.1.14)

2 w(x0 ) = Ajd , ∂jd

2 w(x0 ) = ∂ 2 w 0 (0) = A = A using that ∂jk jk for 1 ≤ j, k ≤ d − 1 and jk jk ∂ξd f = 1. Now if j = d, with (4.1.14) and the symmetry of A we find  ∂d2 w(x0 ) = Add .

Proof of Lemma 4.3. The principal part of L1 (x, D) is ∇ξ q(x, dw (x), 1) · D. At x = x0 it reads ∇ξ q(x0 , ξ 0 , 1) · D as dw(x0 ) = ξ 0 . By (4.1.7) we have 1 q , q}(x0 , ξ 0 , 1) < 0, Im (∇ξ q¯ · ∇x q)(x0 , ξ 0 , 1) = {¯ 2i implying ∇ξ q(x0 , ξ 0 , 1) = 0. Thus, ∇ξ q(x, dw(x), 1) = 0 in an open neighborhood W ⊂ V 0 of x0 . The operator L1 (x, D) is thus a genuine transport operator in W with analytic coefficients. By the Cauchy–Kovalevska¨ıa theorem there exists an analytic function a0 (x) in an open neighborhood W 0 ⊂ W of x0 such that a0 (0) = 1 and L1 (x, D)a0 = 0. Iteratively, again with the Cauchy–Kovalevska¨ıa theorem, we choose aj (x), j = 1, . . . , N , analytic in an open neighborhood W j ⊂ W j−1 of x0 , such that j = 0, . . . , N − 1. L1 (x, D)aj+1 + τ −1 P M (x, D)aj = 0,

−1 a (x) we find In W N , setting aN j τ (x) = 0≤j≤N τ   −N −1 M P (x, D)aN . L1 (x, D) + τ −1 P M (x, D) aN τ =τ If we choose V 1 an open neighborhood of x0 such that V 1  W N the result of the lemma follows.  4.1.2. Optimal Strength in the Large Parameter and Bracket Positivity. Here, we show that estimates as strong as those proven in Chap. 3 imply not only the nonnegativity of the Poisson bracket {p2 , p1 } but also its actual positivity on the characteristic set, that is, the sub-ellipticity condition of Definition 3.2 is fulfilled. We start by finding the optimal strength in the large parameter τ associated with the weighted L2 -norm. We see that the cases d = 1 and d ≥ 2 need to be treated separately. Theorem 4.4. Let V be a bounded open set in Rd , ϕ(x) ∈ C ∞ (Rd , R), τ∗ > 0 and C0 > 0 such that, for some α ∈ R, (4.1.15)

τ α eτ ϕ uL2 (Rd ) ≤ C0 eτ ϕ P uL2 (Rd ) ,

4.1. ON THE NECESSITY OF THE SUB-ELLIPTICITY PROPERTY

139

for all u ∈ Cc∞ (V ) and τ ≥ τ∗ . Then, α ≤ 3/2 if d ≥ 2 and α ≤ 2 if d = 1. Moreover, if dϕ(x0 ) = 0 for some x0 ∈ V , then necessarily α ≤ 1. A proof is given below. In dimension d ≥ 2, from Theorem 3.11 and the previous lemma, we see with that α = 3/2 is the optimal power for the large parameter τ . The next theorem shows that achieving the optimal strength in the Carleman estimate is equivalent to having the sub-ellipticity condition. Theorem 4.5. Let d ≥ 2. Let V be a bounded open set in Rd , ϕ(x) ∈ C ∞ (Rd , R), τ∗ > 0 and C0 > 0 such that τ 3/2 eτ ϕ uL2 (Rd ) ≤ C0 eτ ϕ P uL2 (Rd ) ,

(4.1.16)

for all u ∈ Cc∞ (V ) and τ ≥ τ∗ . Then, ϕ and P have the sub-ellipticity condition in V : |dϕ| > 0 in V and ∀(x, ξ) ∈ V × Rd , ∀τ > 0,

pϕ (x, ξ, τ ) = 0 1 {pϕ , pϕ }(x, ξ, τ ) = {p2 , p1 }(x, ξ, τ ) > 0. ⇒ 2i

A proof is given below. Remark 4.6. Note that this result and Theorem 3.11 allow one to conclude that if we have (4.1.16), then the full Carleman estimate holds in V:

3/2−|β| τ ϕ β (4.1.17) τ e D uL2 (Rd )  eτ ϕ P uL2 (Rd ) , |β|≤2

for all u ∈

Cc∞ (V

) and τ ≥ τ∗ for some τ∗ > 0.

In dimension d = 1, Theorem 4.4 suggests that the power 3/2 of the large parameter τ obtained in Theorem 3.11 can be improved. This is indeed true as expressed by the following result. Theorem 4.7. Let d = 1. Let V = (A, B) and ϕ(x) ∈ C ∞ (R, R). If > 0 in [A, B] then pϕ (x, ξ, τ ) is elliptic and there exist C > 0 and τ∗ > 0 such that

2−|β| τ ϕ β (4.1.18) τ e Dx uL2 (R) ≤ Ceτ ϕ P uL2 (R) , |ϕ |

|β|≤2

for all u ∈ Cc∞ (A, B) and τ ≥ τ∗ . Conversely, assume that there exist τ∗ > 0, C0 > 0, and α > 1 such that (4.1.19)

τ α eτ ϕ uL2 (R) ≤ C0 eτ ϕ P uL2 (R) ,

for all u ∈ Cc∞ (A, B) and τ ≥ τ∗ . Then |ϕ | > 0 in [A, B]. And thus estimate (4.1.18) holds. Note that if pϕ (x, ξ, τ ) is elliptic, then the sub-ellipticity condition holds trivially. A proof of Theorem 4.7 is given below.

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4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

Proof of Theorem 4.4. Let x0 ∈ V . First, assume that dϕ(x0 ) = 0 and d ≥ 2. Let ξ 0 ∈ Rd such that p(x0 , ξ 0 + idϕ(x0 )) = 0, that is, p(x0 , ξ 0 ) = p(x0 , dϕ(x0 )),

p˜(x0 , ξ 0 , dϕ(x0 )) = 0,

meaning that ξ 0 and dϕ(x0 ) are orthogonal and of the same norm with respect to the metric induced by (pij ); see Fig. 3.1. Note that this can always be achieved in the case d ≥ 2. Without any loss of generality we assume that x0 = 0 and ϕ(x0 ) = 0. We set ζ 0 = ξ 0 + idϕ(0) and we introduce w(x) = x · ζ 0 and we note that ϕ(x) − Im w(x) = G(x) + |x|3 O(1), with (4.1.20)

G(x) =

1 2 ∂ ϕ(0)xj xk . 2 j,k xj xk

We pick f ∈ Cc∞ (Rd ), f ≡ 0, and set uτ (x) = eiτ w(x) f (τ 1/2 x). We have   2 3 (4.1.21) eτ ϕ uτ 2L2 (Rd ) = ∫ e2τ G(x)+|x| O(1) f (τ 1/2 x) dx Rd

= τ −d/2 ∫ e2G(y)+τ

−1/2 |y|3 O(1))

  f (y)2 dy

Rd

2

∼ τ −d/2 eG f L2 (Rd ) ,

τ →∞

with the change of variables y = τ 1/2 x and the Lebesgue dominated-convergence theorem.

Recalling that p(x, ξ) = 1≤i,j≤d pij (x)ξi ξj , then p(x, Dx ) is the principal part of P . With the Taylor formula we observe that (4.1.22)

e−iτ w(x) p(x, Dx )eiτ w(x) = p(x, Dx + τ ζ 0 )

= p(x, τ ζ 0 ) + pξ (x, τ ζ 0 ) · Dx 1 (2) + pξ (x, τ ζ 0 )(Dx , Dx ). 2     (2) As Dxβ f τ 1/2 x = τ |β|/2 O(1) we have pξ (x, τ ζ 0 )(Dx , Dx )f τ 1/2 x = τ O(1). Taking into account the lower-order terms in P we find     e−iτ w(x) P uτ = p(x, τ ζ 0 )f τ 1/2 x + pξ (x, τ ζ 0 ) · Dx f τ 1/2 x + τ O(1)      = τ 3/2 τ 1/2 p(x, ζ 0 )f τ 1/2 x + pξ (x, ζ 0 ) · (Dx f ) τ 1/2 x  + τ −1/2 O(1) . Next, we write 1

0 p(x, ζ 0 ) = p(0, ζ 0 ) +x · px (0, ζ 0 ) + ∫ (1 − σ)p(2) x (σx, ζ )(x, x)dσ,    0 =0

4.1. ON THE NECESSITY OF THE SUB-ELLIPTICITY PROPERTY

141

which gives

  τ 1/2 p(x, ζ 0 )f τ 1/2 x

  = τ 1/2 x · px (0, ζ 0 )f τ 1/2 x 1  1/2    1/2  0 1/2 (σx, ζ ) τ x, τ x dσ f τ x + τ −1/2 ∫ (1 − σ)p(2) x 0   = τ 1/2 x · px (0, ζ 0 )f τ 1/2 x + τ −1/2 O(1),

as f has compact support. We have thus obtained      e−iτ w(x) P uτ = τ 3/2 τ 1/2 x · px (0, ζ 0 )f τ 1/2 x + pξ (x, ζ 0 ) · (Dx f ) τ 1/2 x  + τ −1/2 O(1) . As ϕ(x) − Im w(x) = G(x) + |x|3 O(1), we then obtain eτ ϕ P uτ 2L2 (Rd )     3 = τ 3 ∫ e2τ G(x)+|x| O(1) τ 1/2 x · px (0, ζ 0 )f τ 1/2 x Rd

= τ 3−d/2

2   + pξ (x, ζ 0 ) · (Dx f ) τ 1/2 x + τ −1/2 O(1) dx   2 G(y)+τ −1/2 |y|3 O(1)  y · px (0, ζ 0 )f (y) ∫ e

Rd

2 + pξ (τ −1/2 y, ζ 0 ) · (Dx f )(y)τ −1/2 O(1) dy,

with the change of variables y = τ 1/2 x. We claim that (4.1.23)

y · px (0, ζ 0 )f (y) + pξ (0, ζ 0 ) · (Dx f )(y) ≡ 0.

Consequently, with the Lebesgue dominated-convergence theorem we find  (4.1.24) eτ ϕ P uτ 2 2 d ∼ τ 3−d/2 ∫ e2G(y) y · px (0, ζ 0 )f (y) L (R ) τ →∞

Rd

2 + pξ (0, ζ 0 ) · (Dx f )(y) dy.

From the assumed functional inequality (4.1.15) with (4.1.21) and (4.1.24) we find that α ≤ 3/2. We now prove Claim (4.1.23). We have pξ (0, ζ 0 ) · (Dx f )(y) = (X + iY )f (y) where X and Y are two constant real vector fields,

X= pij (0)(∂j ϕ(0))∂i , Y =− pij (0)ξj0 ∂i . 1≤i,j≤d

1≤i,j≤d

Note that rank(X, Y ) = 2 as dϕ(0) and ξ 0 are linearly independent and (pij (0)) is invertible. Let us assume that (4.1.23) does not hold. Setting q(y) = y · px (0, ζ 0 ) we obtain (X + iY + q)f = 0. Action of X − iY gives (X 2 + Y 2 )f = Rf where R is a first-order differential operator. For any given y 0 ∈ Rd , set P 0 to be the affine plane generated by X and Y that contains y 0 . We set f 0 = f|P 0 ; it is solution to (X 2 + Y 2 )f 0 = R0 f 0 where

142

4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

R0 is the restriction of R to P 0 , which makes perfect sense as R only implies differentiations in the directions given by X and Y . Thus f 0 is solution in P 0  R2 of a second-order elliptic equation. As f 0 is compactly supported, the unique continuation result of Theorem 5.2 implies that f 0 ≡ 0 in P 0 . As y 0 is arbitrary this gives f ≡ 0, a contradiction. This proves the claim given in (4.1.23). Second, assume that d = 1 and dϕ(x0 ) = 0. The principal symbol of P reads p(x, ξ) = a(x)ξ 2 , with a(x) ≥ C > 0 in V . This gives   pϕ (x, ξ, τ ) = p(x, ξ + iτ dϕ(x)) = a(x) ξ 2 − (τ ϕ (x))2 + 2iτ ϕ (x)ξ . Observe that for x = x0 , then pϕ (x0 , ξ, τ ) = 0 for all (ξ, τ ) ∈ Rd × R+ and τ > 0. As above, without any loss of generality we assume that x0 = 0 and ϕ(x0 ) = 0. Let ξ 0 ∈ Rd and set ζ 0 = ξ 0 + idϕ(0). Introduce the function w(x), G(x), and uτ (x) as above. In the Taylor expansion (4.1.22) we now have p(x, τ ζ 0 ) = 0, and inspecting the computation made above we find eτ ϕ P uτ 2L2 (R)

(4.1.25)

∼

τ →∞

2

τ 7/2 eG p(0, ζ 0 )f L2 (R) .

From the assumed functional inequality (4.1.15) we find with (4.1.21) and (4.1.25) that α ≤ 2. Third, assume that dϕ(x0 ) = 0 and d ≥ 1 arbitrary. Then pϕ (x0 , ξ, τ ) = for any τ > 0 and ξ ∈ Rd . Having pϕ (x0 , ξ, τ ) = 0 thus implies that ξ = 0 since p is elliptic. As above, without any loss of generality we assume that x0 = 0 and ϕ(x0 ) = 0. We now carry out the above analysis with ξ 0 = 0 and ζ 0 = 0. As compared to what we did above we set w = 0 and uτ is simply given by uτ (x) = f (τ 1/2 x) for some f ∈ Cc∞ (Rd ). We now have ϕ(x) = G(x) + |x|3 O(1) with G defined in (4.1.20). The operator P reads

i

Di (pij (x)Dj ) + b (x)Di + c(x) P = p(x0 , ξ)

1≤i,j≤d

=

1≤i,j≤d

Then, P uτ = τ

1≤i,j≤d

1≤i≤d

˜i b (x)Di + c(x). pij (x)Di Dj + 1≤i≤d

pij (x)(Di Dj f )(τ 1/2 x) + τ 1/2

˜i b (x)(Di f )(τ 1/2 x)

1≤i≤d

+ c(x)f (τ 1/2 x).

As f ≡ 0 is compactly supported we have 1≤i,j≤d pij (x)Di Dj f ≡ 0 by the unique continuation result of Theorem 5.2. Inspecting the computation made above we then find 2 

  (4.1.26) pij (0)Di Dj f  . eτ ϕ P uτ 2L2 (Rd ) ∼ τ 2−d/2 eG τ →∞

1≤i,j≤d

L2 (Rd )

4.1. ON THE NECESSITY OF THE SUB-ELLIPTICITY PROPERTY

143

From the assumed functional inequality (4.1.15), with (4.1.21) and (4.1.26), we find α ≤ 1.  Proof of Theorem 4.5. Let x0 ∈ V . As α = 3/2 here we have dϕ(x0 ) = 0 by Theorem 4.4. As in the proof of that lemma we let ξ 0 ∈ Rd be such that p(x0 , ξ 0 + idϕ(x0 )) = 0 and we choose w = x · ζ 0 and G as in (4.1.20). We then set uτ (x) = eiτ w(x) f (τ 1/2 x) with f ∈ Cc∞ (Rd ), f ≡ 0. From the assumed functional inequality (4.1.16), with (4.1.21) and (4.1.24), letting τ go to +∞ we obtain (4.1.27)

 2 2 eG f L2 (Rd ) ≤ C02 ∫ e2G(y) y · px (0, ζ 0 )f (y) + pξ (0, ζ 0 ) · (Dx f )(y) dy, Rd

for any f ∈ Cc∞ (Rd ). Changing f into e−G f yields f 2L2 (Rd ) ≤ C02 L(x, Dx )f 2L2 (Rd ) ,   with L(x, Dx ) = x·px (0, ζ 0 )+pξ (0, ζ 0 )· Dx −(Dx G) . Because of the form of the function G set in (4.1.20), observe that the full symbol of the operator L takes the form L(x, ξ) = (a, x) + (b, ξ) with a, b ∈ Cd . Lemma 4.34 in ¯ L}. We compute Appendix 4.A.2 gives C0−2 ≤ 1i {L,

2 1 ¯ {L, L} = ∂xj xk ϕ(0) ∂ξj p(0, ζ 0 ) ∂ξk p¯(0, ζ 0 ) 2i j,k

+ Im ∂xj p(0, ζ 0 ) ∂ξj p¯(0, ζ 0 ), j

yielding, (4.1.28)

2 1 ≤ ∂xj xk ϕ(0) ∂ξj p(0, ξ 0 + idϕ(0)) ∂ξk p¯(0, ξ 0 − idϕ(0)) 2C02 j,k

+ Im ∂xj p(0, ξ 0 + idϕ(0)) ∂ξj p¯(0, ξ 0 − idϕ(0)). j

Let now τ > 0, x ∈ V , and ξ ∈ Rd be such that p(x, ξ +τ dϕ(x)) = 0. Setting x0 = x, ξ 0 = ξ/τ we can use (4.1.28). By homogeneity we then have (4.1.29)

2 τ3 ≤τ ∂xj xk ϕ(x) ∂ξj p(x, ξ + iτ dϕ(x)) ∂ξk p¯(x, ξ − iτ dϕ(x)) 2 2C0 j,k

+ Im ∂xj p(x, ξ + τ idϕ(x)) ∂ξj p¯(x, ξ − iτ dϕ(x)). j

Observing that 1 {pϕ , pϕ }(x, ξ, τ ) = {p2 , p1 }(x, ξ, τ ) (4.1.30) 2i

2 =τ ∂xj xk ϕ(x) ∂ξj p(x, ξ + iτ dϕ(x)) ∂ξk p¯(x, ξ − iτ dϕ(x)) j,k

+ Im

j

∂xj p(x, ξ + τ idϕ(x)) ∂ξj p¯(x, ξ − iτ dϕ(x)),

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4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

we obtain the positivity of the Poisson bracket at (x, ξ, τ ) in the case x ∈ V : (4.1.31)

τ3 1 {pϕ , pϕ }(x, ξ, τ ) = {p2 , p1 }(x, ξ, τ ) ≥ . 2i 2C02

We now wish to obtain the result in V . We start by proving that dϕ(y) = 0 for y ∈ V . If y ∈ ∂V we consider a sequence (x(k) )k∈N∗ ⊂ V that converges to y. Keeping τ = 0 fixed we also consider (ξ (k) )k∈N∗ ⊂ Rd such that p(x(k) , ξ (k) +iτ dϕ(xk )) = 0 (here we use again that d ≥ 2). We set ζ (k) = ξ (k) + iτ dϕ(x(k) ). We have in particular p(x(k) , ξ (k) ) = τ 2 p(x(k) , dϕ(x(k) )) and the sequence (ξ (k) )k is hence bounded. It converges, up to a subsequence, to a certain ξ ∈ Rd and thus p(y, ξ + iτ dϕ(y)) = 0. In particular p(y, ξ) = τ 2 p(y, dϕ(y)). By (4.1.31) we have {p2 , p1 }(x(k) , ξ (k) , τ ) ≥ τ 3 /(2C02 ). This implies {p2 , p1 }(y, ξ, τ ) ≥

τ3 . 2C02

Setting ζ = ξ + iτ dϕ(y), because of the form of {p2 , p1 } in (4.1.30), we see that ζ = 0. As p(y, ξ) = τ 2 p(y, dϕ(y)) we obtain that dϕ(y) = 0. Let us now prove the sub-ellipticity property up to the boundary. We use the submanifold structure of the characteristic set stated in Lemma 3.1 yet making more explicit how some variables can be chosen functions of the others. Let (y, η) ∈ ∂V × Rd and τ > 0 be such that p(y, η + iτ dϕ(y)) = 0. We define f1 (x, ξ) = p(x, ξ) − τ 2 p(x, dϕ(x)) = 0,

f2 (x, ξ) = p˜(x, ξ, dϕ(x)),

and Z˜ = {(x, ξ) ∈ Rd × Rd ; f1 (x, ξ) = f2 (x, ξ) = 0},

Z = Z˜ ∩ V × Rd .

As dϕ(y) = 0 and η = 0 are orthogonal in the sense of p˜, we see that the partial differentials dξ f1 and dξ f2 form a rank 2 system at (y, η). Up to rearranging the variables, with the implicit function theorem, this implies that in a neighborhood U1 of (y, η) and in a neighborhood U2 of (y, η1 , . . . , ηd−2 ) we have (x, ξ) ∈Z˜ ∩ U1 ⇔ (x, ξ1 , . . . , ξd−2 ) ∈ U2 and (ξd−1 , ξd ) = g(x, ξ1 , . . . , ξd−2 ), with a smooth function g. Consider then a sequence (x(k) )k∈N∗ ⊂ V that converges to y. For k sufficiently large, k ≥ N0 , we have (x(k) , η1 , . . . , ηd−2 ) ∈ U2 and we set ξ (k) = (η1 , . . . , ηd−2 , g(x(k) , η1 , . . . , ηd−2 )). Then (x(k) , ξ (k) ) is in Z and converge to (y, η). By (4.1.31) we have {p2 , p1 }(x(k) , ξ (k) , τ ) ≥ τ 3 /(2C02 ) for all k ≥ N0 . We thus obtain {p2 , p1 }(y, η, τ ) ≥ τ 3 /(2C02 ) by passing to the limit. 

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145

Proof of Theorem 4.7. The principal symbol of P reads p(x, ξ) = a(x)ξ 2 with 0 < C ≤ a(x) ≤ C  . The principal symbol of the conjugated  2 operator Pϕ is pϕ (x, ξ, τ ) = a(x) ξ + iτ ϕ (x) . If ϕ = 0 in [A, B] then |pϕ (x, ξ, τ )|2  (τ 2 + |ξ|2 )2 , for x in a neighborhood of [A, B]. Then, the G˚ arding inequality of Theorem 2.28 yields Pϕ v2L2 (A,B) = (Pϕ∗ Pϕ v, v)L2 (A,B)  v22 , for all u ∈ Cc∞ (A, B). This inequality yields (4.1.18). Conversely, let us assume that (4.1.19) holds for some α > 1. By Theorem 4.4 we have ϕ (x0 ) = 0 for x0 ∈ (A, B). Let now x0 ∈ ∂V = {A, B}. Assume that ϕ (x0 ) = 0. Choosing ξ0 = 0 we then have pϕ (x0 , ξ0 , τ ) = 0. We may also assume that x0 = 0 and ϕ(x0 ) = 0 without any loss of generality. Choose f ∈ Cc∞ (A, B) and set uτ = f (τ 1/2 x). Arguing as in the third part of the proof of Theorem 4.4 we obtain a contradiction as α > 1. Thus  ϕ cannot vanish on the whole [A, B]. Remark 4.8. In the proof of Theorem 4.5 we see that dϕ does not vanish in V by using a sequence of points in V converging to a point at the boundary, knowing that dϕ does not vanish in V by Theorem 4.4. In fact, if x0 ∈ ∂V , again we choose coordinates so as to have x0 = 0. In the construction of the test function uτ = eiτ w(x) f (τ 1/2 x) we can take f ∈ Cc∞ (V ). Then inequality (4.1.16) applies to uτ even in the case where x0 ∈ ∂V , and the computations remain unchanged and we obtain estimate (4.1.27). This directly proves that dϕ cannot vanish at x0 as otherwise ζ 0 = 0 and estimate (4.1.27) cannot hold. Yet such an approach requires that the open set V satisfies the cone property for f (τ 1/2 .) to remain supported in V . This approach in used in the one-dimensional case for the proof of Theorem 4.7 providing a simpler argument. In Theorem 4.5 we only used the estimation of the weighted L2 -norm of the solution by that of P (x, D)u to conclude that the gradient of ϕ does not vanish and that the sub-ellipticity condition holds. It is natural to wonder if this can hold if we consider the estimation of the weighted L2 -norms of some derivative of the function u. The following proposition completes Theorem 4.4 in the case dϕ vanishes at some point. Proposition 4.9. Let V be a bounded open set in Rd , ϕ(x) ∈ C ∞ (Rd , R), τ∗ > 0 and C0 > 0 such that, for some α ∈ R, and some β ∈ Nd , 0 ≤ |β| ≤ 2, (4.1.32)

τ α−|β| eτ ϕ Dβ uL2 (Rd ) ≤ C0 eτ ϕ P uL2 (Rd ) ,

for all u ∈ Cc∞ (V ) and τ ≥ τ∗ . If dϕ(x0 ) = 0 for some x0 ∈ V , then necessarily (1) if |β| = 0 then α ≤ 1;

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4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

(2) if |β| = 1 then α ≤ 3/2; (3) if |β| = 2 then α ≤ 2. A proof is given below. This result is to be compared to the case where dϕ = 0 in V where one finds α ≤ 3/2 for any value of |β| = 0, 1, 2 if d ≥ 2 and α ≤ 2 if d = 1; see Theorems 4.5 and 4.7 in the case |β| = 0 and Theorem 4.10 below in the cases |β| = 1, 2. The result α ≤ 2 in the case |β| = 2 is clearly optimal by simply considering the case ϕ = Cst. The maximal admissible values obtained for α in the cases |β| = 0, 1 are also optimal as revealed by the example treated in Proposition 4.13 below. In the case where dϕ does not vanish in V we have the following result; we only consider the case d ≥ 2 as the case d = 1 is treated in Theorem 4.7. Theorem 4.10. Let d ≥ 2. Let V be a bounded open set in Rd , ϕ(x) ∈ C ∞ (Rd , R) be such that |dϕ| = 0 in V . Assume that for τ∗ > 0, C0 > 0, β ∈ Nd , 1 ≤ |β| ≤ 2, and α ∈ R, we have (4.1.33)

τ α−|β| eτ ϕ Dβ uL2 (Rd ) ≤ C0 eτ ϕ P uL2 (Rd ) ,

for all u ∈ Cc∞ (V ) and τ ≥ τ∗ . If (ϕ )β (x) = 0 in V then α ≤ 3/2. If moreover α = 3/2 then the sub-ellipticity condition holds in V : ∀(x, ξ) ∈ V × Rd , ∀τ > 0,

pϕ (x, ξ, τ ) = 0 1 {pϕ , pϕ }(x, ξ, τ ) = {p2 , p1 }(x, ξ, τ ) > 0. ⇒ 2i

Similarly to Theorem 4.5 if we have (4.1.33) for α = 3/2 and if (ϕ )β does not vanish in V , then the full Carleman estimate (4.1.17) holds in V . Corollary 4.11. Let d ≥ 2. Let V be a bounded open set in Rd and ϕ(x) ∈ C ∞ (Rd , R) be such that |dϕ| = 0 in V . Assume that for τ∗ > 0, C0 > 0, k ∈ N, 1 ≤ k ≤ 2, we have

eτ ϕ Dβ uL2 (Rd ) ≤ C0 eτ ϕ P uL2 (Rd ) , τ 3/2−k β∈Nd |β|=k

for all u ∈ Cc∞ (V ) and τ ≥ τ∗ . Then, the sub-ellipticity condition holds in V . Proof of Proposition 4.9. The case |β| = 0 is contained in Theorem 4.4. For the two other cases we follow the proof of that lemma. Let x0 ∈ V be such that dϕ(x0 ) = 0. We set uτ (x) = f (τ 1/2 x) for some f ∈ Cc∞ (Rd ). Then (4.1.26) holds. We also have (4.1.34)

2

2

eτ ϕ Dβ uτ L2 (Rd ) ∼ τ |β|−d/2 eG f L2 (Rd ) . τ →∞

with the same argument that leads to (4.1.21) (here we take ζ 0 = 0 and w(x) = 0). From the assumed functional inequality (4.1.32), with (4.1.34) and (4.1.26), we find the sought result. 

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Proof of Theorem 4.10. The proof is very similar to that of Theorem 4.4 and that of Theorem 4.5. Let x0 ∈ V . As d ≥ 2 we can find ξ 0 ∈ Rd such that p(x0 , ξ 0 +idϕ(x0 )) = 0. Without any loss of generality we assume that x0 = 0 and ϕ(x0 ) = 0. We construct uτ similarly of the form uτ (x) = eiτ w(x) f (τ 1/2 x), with ζ0 = ξ0 + idϕ(x0 ) and we observe that ζ0β = 0 by the assumption made on ϕ. As we have     e−iτ w(x) Dβ uτ = (D + τ ζ0 )β f τ 1/2 x = τ |β| ζ0β f τ 1/2 x + τ |β|−1/2 O(1), we obtain (4.1.35)

2

2

eτ ϕ Dβ uτ L2 (Rd ) ∼ τ 2|β|−d/2 |ζ0β |2 eG f L2 (Rd ) , τ →∞

where G is given in (4.1.20). The argument is the same as that leads to (4.1.21). From the assumed functional inequality (4.1.33) with (4.1.35) and (4.1.24) letting τ go to +∞ we obtain α ≤ 3/2 and if α = 3/2 2

|ζ0β |2 eG f L2 (Rd )

2  ≤ C02 ∫ e2G(y) y · px (0, ζ 0 )f (y) + pξ (0, ζ 0 ) · (Dx f )(y) dy, Rd

for any f ∈ Cc∞ (Rd ). The remainder of the proof follows similarly as in that  of Theorem 4.5 using that |ζ0β | ≥ |(ϕ (x0 ))β | ≥ C > 0. Remark 4.12. In Theorem 4.10 the assumption (ϕ )β = 0 is not superfluous. If it is not fulfilled we may have an estimate of the form (4.1.36) τ α−|β| eτ ϕ Dβ uL2 (Rd )  eτ ϕ P uL2 (Rd ) ,

with α = 3/2 and |β| > 0,

and yet the sub-ellipticity condition may not be satisfied. Such an example is given by P = −Δ and ϕ(x) = x2d /2 on Rd . We set x = (x1 , . . . , xd ). Then the principal symbol of the conjugated operator is pϕ = p2 + ip1 ,

p2 (x, ξ, τ ) = |ξ|2 − |τ xd |2 ,

p1 (x, ξ, τ ) = 2τ xd ξd ,

and the Poisson bracket is {p2 , p1 }(x, ξ, τ ) = 4τ (ξd2 + (τ xd )2 ). Hence for ξ = 0 and xd = 0 and τ > 0 we have pϕ (x, ξ, τ ) = 0 and

{p2 , p1 }(x, ξ, τ ) = 0.

The sub-ellipticity condition is not satisfied. Still an estimate of the form of (4.1.36) with |β| = 1 is valid as shown by the following proposition.

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4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

Proposition 4.13. Let V be a bounded open set in Rd , d ≥ 2. Let P = −Δ and ϕ(x) = x2d /2. Then, there exist C > 0 and τ∗ > 0 such that τ 3/2 xd eτ ϕ uL2 (Rd ) + τ eτ ϕ uL2 (Rd ) + τ 1/2 + τ −1/2



β∈Nd |β|=1

eτ ϕ Dxβ uL2 (Rd )

eτ ϕ Dxβ uL2 (Rd ) ≤ Ceτ ϕ P uL2 (Rd ) ,

β∈Nd |β|=2

for u ∈ Cc∞ (V ) and τ ≥ τ∗ . Observe that the powers of the large parameter τ associated with some of the terms show the optimality of the results in Proposition 4.9 in the cases |β| = 0, 1. Proof. In fact we write P = Dx2d + M (D )2 where M (D ) is the Fourier  2  2 multiplier Op ξ1 + · · · + ξd−1 . Note in particular that [Dxd , M (D )] = 0. We thus have P = P + P − , with P ± = Dxd ± iM (D ), and as ∂d ϕ = xd we find Pϕ = eτ ϕ P e−τ ϕ = Pϕ+ Pϕ− , with Pϕ± = Dxd + i(τ xd ± M (D )). We observe that (4.1.37) 2

2

Pϕ± vL2 (Rd ) = Dxd v2L2 (Rd ) + (τ xd ± M (D ))vL2 (Rd ) + 2 Re(Dxd , i(τ xd ± M (D ))v)L2 (Rd ) 2

= Dxd v2L2 (Rd ) + (τ xd ± M (D ))vL2 (Rd )   + i[Dxd , (τ xd ± M (D ))]v, v L2 (Rd ) 2

= Dxd v2L2 (Rd ) + (τ xd ± M (D ))vL2 (Rd ) + τ v2L2 (Rd ) . With this estimate for Pϕ+ and Pϕ− , using that Pϕ = Pϕ+ Pϕ− = Pϕ− Pϕ+ , we obtain (4.1.38)

2

2

Pϕ v2L2 (Rd )  Dxd Pϕ+ vL2 (Rd ) + Dxd Pϕ− vL2 (Rd )   2 2 + τ Pϕ+ vL2 (Rd ) + Pϕ− vL2 (Rd ) .

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149

With (4.1.37) we write (4.1.39) 2

2

Pϕ+ vL2 (Rd ) + Pϕ− vL2 (Rd ) ≥ Dxd v2L2 (Rd ) + τ v2L2 (Rd ) 2

+ (τ xd + M (D ))vL2 (Rd ) 2

+ (τ xd − M (D ))vL2 (Rd ) ≥ Dxd v2L2 (Rd ) + τ v2L2 (Rd ) 2

+ τ 2 xd v2L2 (Rd ) + M (D )vL2 (Rd ) . Observing that Dd Pϕ± = Pϕ± Dd + iτ [Dd , xd ] = Pϕ± Dd + τ , we have (4.1.40)

2

2

Dxd Pϕ+ vL2 (Rd ) + Dxd Pϕ− vL2 (Rd ) + τ 2 v2L2 (Rd ) 2

2

 Pϕ+ Dxd vL2 (Rd ) + Pϕ− Dxd vL2 (Rd ) 2

 Dx2d vL2 (Rd ) + τ Dxd v2L2 (Rd ) 2

+ τ 2 xd Dxd v2L2 (Rd ) + M (D )Dxd vL2 (Rd ) , using (4.1.39). Then, with (4.1.38)–(4.1.40) one finds (4.1.41) Pϕ v2L2 (Rd )  τ 2 v2L2 (Rd ) + τ 3 xd v2L2 (Rd ) + τ Dxd v2L2 (Rd ) 2

+ τ M (D )vL2 (Rd ) 2

2

+ τ 2 xd Dxd v2L2 (Rd ) + Dx2d vL2 (Rd ) + M (D )Dxd vL2 (Rd )

2  τ 2 v2L2 (Rd ) + τ 3 xd v2L2 (Rd ) + τ Dxβ vL2 (Rd ) + τ 2 xd Dxd v2L2 (Rd ) +

β∈Nd |β|=1

β∈Nd |β|=1

2

Dxβ Dxd vL2 (Rd ) .

We have eτ ϕ D uL2 (Rd ) = D vL2 (Rd ) , eτ ϕ Dxd uL2 (Rd )  Dxd vL2 (Rd ) + τ xd vL2 (Rd ) . As eτ ϕ Dd e−τ ϕ = Dd + iτ xd and eτ ϕ Dd2 e−τ ϕ = Dd2 + 2iτ xd Dd − τ 2 x2d + τ , we also have eτ ϕ Dxd D uL2 (Rd )  Dxd D vL2 (Rd ) + τ xd D vL2 (Rd ) , eτ ϕ Dx2d uL2 (Rd )  Dx2d vL2 (Rd ) + τ xd Dxd vL2 (Rd ) + τ vL2 (Rd ) + τ 2 x2d vL2 (Rd ) .

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4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

Using that |xd |  1 if x ∈ V , we then find

τϕ β 2

τϕ β 2 τ e D uL2 (Rd ) + τ −1 e Dx Dxd uL2 (Rd ) β∈Nd |β|=1

β∈Nd |β|=1

 τ 2 v2L2 (Rd ) + τ 3 xd v2L2 (Rd ) + τ + τ 2 xd Dxd v2L2 (Rd ) +

β∈Nd |β|=1

β∈Nd |β|=1

2

Dxβ vL2 (Rd ) 2

Dxβ Dxd vL2 (Rd )

 eτ ϕ P u2L2 (Rd ) . Finally, as P = Dd2 + M (D )2 we have

  eτ ϕ Dxβ uL2 (Rd ) = Dxβ vL2 (Rd ) β  ∈Nd−1 |β  |=2

β  ∈Nd−1 |β  |=2

 M (D )2 vL2 (Rd ) = eτ ϕ M (D )2 uL2 (Rd )  eτ ϕ P uL2 (Rd ) + eτ ϕ Dd2 uL2 (Rd ) , which allows us to conclude the proof.



4.2. Limiting Weights and Limiting Carleman Estimates Above, in Theorem 4.1 we saw that a weak form of a Carleman estimate, say τ α eτ ϕ uL2  eτ ϕ P uL2 , for some α ∈ R, implies 1 {pϕ , pϕ }(x, ξ, τ ) ≥ 0. 2i An interesting case is then the following: 1 {pϕ , pϕ }(x, ξ, τ ) = 0. pϕ (x, ξ, τ ) = 0 ⇒ 2i In this section, we study weight functions that fulfill these properties and the estimates that can be obtained with them. pϕ (x, ξ, τ ) = 0 ⇒

4.2.1. Limiting Weights. Following what is said above, we have the following definition. Definition 4.14 (Limiting Carleman Weight). Let V be a bounded open set in Rd . We say that the weight function ϕ ∈ C ∞ (Rd ; R) is a limiting Carleman weight for P (resp. P0 ) in V if |dϕ| > 0 in V and (4.2.1) ∀(x, ξ) ∈ V × Rd , ∀τ > 0, pϕ (x, ξ, τ ) = 0 1 {pϕ , pϕ }(x, ξ, τ ) = {p2 , p1 }(x, ξ, τ ) = 0. ⇒ 2i

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151

Remark 4.15. Note that if ϕ and P have the sub-ellipticity property of Definition 3.2 it cannot hold for −ϕ and P . An important feature of a limiting weight is that if ϕ is a limiting weight for P , then so is −ϕ. Example 4.16. Consider ϕ = xd and P = −Δ in Rd . Then pϕ (x, ξ, τ ) = (ξd + iτ )2 + |ξ  |2 , that is p2 (x, ξ, τ ) = |ξ|2 − τ 2 and p1 = 2τ ξd . We then have {p2 , p1 }(x, ξ, τ ) = 0. Note that this is valid everywhere and not just in Char(pϕ ). We see below that this property can always been obtained, starting with a limiting weight, by a simple modification of the operator. Consider a differential operator Q(x, D) with q(x, ξ) = α(x)p(x, ξ) as principal symbol, with α a nonvanishing smooth function. The conjugated operator eτ ϕ Q(x, D)e−τ ϕ has qϕ (x, ξ, τ ) = α(x)pϕ (x, ξ, τ ) for principal symbol. Assume that ϕ is a limiting weight for P . Observe that for (x, ξ, τ ) ∈ Char(qϕ ) = Char(pϕ ) one has 1 1 {qϕ , qϕ }(x, ξ, τ ) = |α(x)|2 {pϕ , pϕ }(x, ξ, τ ) = 0, 2i 2i meaning that a limiting weight for P is a limiting weight for Q. Since moreover dϕ = 0 in V we may choose α(x) = 1/p(x, dϕ(x)). The function ϕ is then a limiting weight for any operator Q with α(x)p(x, ξ) for principal symbol within addition q(x, dϕ(x)) = 1 in V . The following proposition states that a much stronger property holds in fact, explaining the observation made in Example 4.16. Proposition 4.17. Let V be a connected open set of Rd and let ϕ ∈ The following statements are equivalent:

C ∞ (Rd ; R).

(1) The function ϕ is a limiting weight for P in V and p(x, dϕ(x)) ≡ Cst in V . (2) The function ϕ is such that dϕ = 0 in V and satisfies 1 {pϕ , pϕ }(x, ξ, τ ) = 0, 2i

x ∈ V , ξ ∈ Rd , τ > 0.

One of the arguments used to prove this proposition is based on the following lemma. Lemma 4.18. Let ψ ∈ C ∞ (Rd ; R) and introduce b(x, ξ, η) the bilinear form associated with the quadratic form ξ → Hp2 ψ(x, ξ), where x is treated   as a parameter, that is b(x, ξ, η) = Hp2 ψ(x, ξ + η) − Hp2 ψ(x, −ξ + η) /4. One has b(x, ξ, dψ(x)) = {p, p(x, dψ(x))}(x, ξ). Proposition 4.17 and Lemma 4.18 are proven in Appendix 4.A.3. 4.2.2. Convexification. Let V be an open set of Rd and let ϕ ∈ be a limiting Carleman weight for P in V . For G ∈ C ∞ (R; R),

C ∞ (Rd ; R)

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4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

such that G = 0 in the range of ϕ(x) for x ∈ V , we set φ = G ◦ ϕ. Then, Lemma 3.55 gives 1 {pφ , pφ }(x, ξ, τ ) 2i   2 = τ (G ◦ ϕ)(G ◦ ϕ)2 (x) Hp ϕ(x, η) + 4τ 2 p(x, dϕ(x))2 1 {pϕ , pϕ }(x, η, τ ), x ∈ V , ξ ∈ Rd , τ > 0, 2i with ξ = (G ◦ ϕ)(x)η and where the last term vanishes if pϕ (x, ξ, τ ) = 0. If one assumes moreover that p(x, dϕ(x)) = C0 = 0, with Proposition 4.17 we then obtain   2 1 {pφ , pφ }(x, ξ, τ ) = τ (G ◦ ϕ)(G ◦ ϕ)2 (x) Hp ϕ(x, η) + 4τ 2 C02 , 2i x ∈ V , ξ ∈ Rd , τ > 0. + (G ◦ ϕ)3 (x)

Recall that having p(x, dϕ(x)) = C0 = 0 can be simply obtained by a modification of the operator P ; see the discussion above Proposition 4.17. Consider now the particular case G(s) = s + γs2 /(2τ ) with γ > 0 and τ > 0. One has G (s) = 1 + γs/τ and G = γ/τ . If γ|s|/τ ≤ 1/2 then G = 0. We thus assume that γ > 0 and τ > 0 are chosen such that 2γ sup |ϕ(x)| ≤ τ, x∈V

G

◦ ϕ(x) = 1 + γϕ(x)/τ = 0. With ξ = (1 + γϕ(x)/τ )η, we implying that then have   2 1 (4.2.2) {pφ , pφ }(x, ξ, τ ) = γ(1 + γϕ(x)/τ )2 Hp ϕ(x, η) + 4τ 2 C02 2i  2 = γ Hp ϕ(x, ξ) + 4γ(τ + γϕ(x))2 C02 , for x ∈ V , ξ ∈ Rd , and τ > 0, 4.2.3. Limiting Carleman Estimates Away from a Boundary. Here we derive an estimate for functions supported inside an open set. The estimate we derive does not require any observation term in the spirit of the estimate derived in Sect. 3.3 under sub-ellipticity. Theorem 4.19 (Estimate With a Limiting Weight). Let V be a bounded open set in Rd , and let ϕ ∈ C ∞ (Rd ; R) be a limiting Carleman weight for P in V ; then, there exist τ∗ > 0 and C ≥ 0 such that τ 2 eτ ϕ u2L2 (Rd ) + eτ ϕ Du2L2 (Rd ) ≤ Ceτ ϕ P u2L2 (Rd ) , for τ ≥ τ∗ and u ∈ Cc∞ (V ). Remark 4.20. Note that first two terms in the estimate correspond to a loss of full derivative. Compare for instance with Theorem 3.11. This is quite consistent with the optimality results of Sect. 4.1.2. This weakness in limiting Carleman estimates as stated in Theorem 4.19 prohibits using the

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153

patching procedure presented in Sect. 3.5. Thus, a global estimate cannot be directly deduced from such local estimates. One alternative is to keep track of the convexification parameter γ > 0 that appears in the proof below. A second alternative is to directly derive a global estimate. This last approach is covered in Sect. 4.2.4 below. Proof. Observe that it is equivalent to obtain the estimate for the operator Q = α(x)P (x, D) in place of P (x, D) if α is smooth and α(x) = 0 in V since eτ ϕ αP uL2 (Rd )  eτ ϕ P uL2 (Rd ) . We choose α(x) = 1/p(x, dϕ). The operator Q reads

˜i Q(x, D) = Q0 + b (x)Di + c˜(x), 1≤i≤d

where ˜b and c˜ have the same properties as those of b and c stated under (4.0.1) and with principal part

Di (αpij (x)Dj ). (4.2.3) Q0 = 1≤i,j≤d

It is thus sufficient to consider the case where ϕ is a limiting weight for P0 and P given in (4.0.1) that satisfies moreover p(x, dϕ) = 1. In such case the identity (4.2.2) holds for φ = G ◦ ϕ with G(s) = s + γs2 /(2τ ) and γ > 0 and τ > 0 chosen such that 2γ sup |ϕ(x)| ≤ τ.

(4.2.4)

x∈V

In particular (4.2.5)

G

◦ ϕ = 1 + γϕ/τ  1, implying dφ = (G ◦ ϕ)dϕ  1,

dk φ  1,

k ∈ N,

uniformly in γ > 0 and τ > 0 under condition (4.2.4). Below, under this condition, differential operators in some Dτm will have their symbols satisfying the appropriate estimates uniformly with respect to γ. We first prove an estimate for the operator P0 . We consider the conjugated operator P0,φ = eτ φ P0 e−τ φ that reads P0,φ = P2 + iP1 by (3.2.1) where P2 = P0 − p(x, τ dφ) ∈ Dτ2 ,  

  jk Dj p ∂k φ + pjk ∂j φDk ∈ τ Dτ1 ⊂ Dτ2 . P1 = τ 1≤j,k≤d

Recall that P2 and P1 are both formally selfadjoint. This yields, (4.2.6) P0,φ v2L2 (Rd ) = P2 v2L2 (Rd ) + P1 v2L2 (Rd ) + (i[P2 , P1 ]v, v)L2 (Rd ) , with an integration by parts using that v vanishes in a neighborhood of ∂V . By pseudo-differential calculus, here in the case of differential operators we have i[P2 , P1 ] = Op({p2 , p1 }) + τ R1 ,

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4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

with R1 ∈ Dτ1 , uniformly in γ > 0 under condition (4.2.4) by (4.2.5). By (4.2.2) we write, by symbol calculus,  ∗ {p2 , p1 }(x, ξ, τ ) = C1 γ(τ + γϕ(x))2 + γ Hp ϕ ◦ Hp ϕ(x, ξ) + γr1 (x, ξ), with C1 > 0 and r1 ∈ S 1 polynomial in ξ and where the adjoint notation is that of Theorem 2.18, here for a classical differential operator. An integration by part gives (i[P2 , P1 ]v, v)L2 (Rd ) = C1 γ(τ + γϕ)v2L2 (Rd ) + γOp(Hp ϕ)v2L2 (Rd ) + ((τ R1 + γOp(r1 ))v, v)L2 (Rd )   ≥ Cγτ 2 v2L2 (Rd ) − C  τ 2 v2L2 (Rd ) + Dv2L2 (Rd ) . Choosing γ sufficiently large we find (4.2.7)

(i[P2 , P1 ]v, v)L2 (Rd ) ≥ Cγτ 2 v2L2 (Rd ) − C  v2H 1 (Rd ) .

We then write, with P = (pij )1≤i,j≤d , (P2 v, v)L2 (Rd ) = (PDv, Dv)L2 (Rd ) − (p(x, τ dφ)v, v)L2 (Rd ) ≥ CDv2L2 (Rd ) − τ 2 C  v2L2 (Rd ) , uniformly in γ > 0 by (4.2.5). This gives DvL2 (Rd )  τ vL2 (Rd ) + τ −1 P2 vL2 (Rd ) .

(4.2.8)

With (4.2.6)–(4.2.8) we obtain, γτ 2 v2L2 (Rd ) + γDv2L2 (Rd )  (i[P2 , P1 ]v, v)L2 (Rd ) + γτ −2 P2 v2L2 (Rd )  P0,φ v2L2 (Rd ) , for τ ≥ 1 and using γ/τ  1. Letting γ > 0 and τ ≥ 1 be sufficiently large, under condition (4.2.4), we may replace P0,φ by Pφ = eτ φ P e−τ φ since Pφ = P0,φ + T1 , with T1 ∈ Dτ1 uniformly with respect to γ, and we obtain γ 1/2 τ vL2 (Rd ) + γ 1/2 DvL2 (Rd )  Pφ vL2 (Rd ) . We now keep γ > 0 fixed. As we have τ vL2 (Rd ) + DvL2 (Rd )  τ eτ φ uL2 (Rd ) + eτ φ DuL2 (Rd ) , for v = eτ φ u by a commutator argument, we obtain τ eτ φ uL2 (Rd ) + eτ φ DuL2 (Rd )  eτ φ P uL2 (Rd ) . Observing that τ ϕ ≤ τ φ = τ ϕ + γϕ2 /2 ≤ τ ϕ + C  , we conclude the proof.



4.2. LIMITING WEIGHTS AND LIMITING CARLEMAN ESTIMATES

155

Remark 4.21. Observe that inequality (4.2.7) does not result from a positivity theorem, e.g. a G˚ arding type inequality or a more sophisticated inequality, but from an algebraic simplification at the symbol level thanks to having {pϕ , pϕ } vanishing identically. This suggest that the same argument can be used for a global derivation without requiring any use of local coordinates to reduce the analysis to a half-space as done in Sect. 3.4.2. This is covered in the next section. 4.2.4. Global Limiting Carleman Estimates. Let Ω be a smooth open set of Rd and let ϕ ∈ C ∞ (Rd ; R). For ε > 0, we define the following subsets of the boundary ∂Ω: ∂Ω± ε = {x ∈ ∂Ω; ∂ν ϕ(x) ≶ −ε}. Theorem 4.22. Let Ω be a smooth open set of Rd and let ϕ ∈ C ∞ (Rd ; R) be a limiting Carleman weight for P in Ω. Let also ε > 0. There exist τ∗ > 0 and C ≥ 0 such that τ 2 eτ ϕ u2L2 (Ω) + eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|∂Ω |2L2 (∂Ω+ ) ε  τϕ 2 2 τϕ ≤ C e P uL2 (Ω) + τ |e ∂ν u|∂Ω |L2 (∂Ω− ) ε  2 2 3 τϕ τϕ + τ |e u|∂Ω |L2 (∂Ω) + τ |e u|∂Ω |H 1 (∂Ω) , for τ ≥ τ∗ and u ∈ C ∞ (Ω). In the case of homogeneous Dirichlet boundary conditions the sets ∂Ω± ε can simply be replaced by ∂Ω± 0. Theorem 4.23 (Homogeneous Dirichlet Boundary Conditions). Let Ω be a smooth open set of Rd and let ϕ ∈ C ∞ (Rd ; R) be a limiting Carleman weight for P in Ω. There exist τ∗ > 0 and C ≥ 0 such that  2 τ 2 eτ ϕ u2L2 (Ω) + eτ ϕ Du2L2 (Ω) + τ |∂ν ϕ|1/2 eτ ϕ ∂ν u|∂Ω L2 (∂Ω+ ) 0    2 2 τϕ 1/2 τ ϕ  ≤ C e P uL2 (Ω) + τ |∂ν ϕ| e ∂ν u|∂Ω L2 (∂Ω− ) , 0

τ ≥ τ∗ and for u ∈

C ∞ (Ω)

such that u|∂Ω = 0.

Proof of Theorems 4.22 and 4.23. As in the proof of Theorem 4.19, one can assume that ϕ is a limiting weight for P0 and P given in (4.0.1) that satisfies moreover p(x, dϕ) = 1. We then set φ = G ◦ ϕ with G(s) = s + γs2 /(2τ ) and γ > 0 and τ > 0 chosen such that 2γ sup |ϕ(x)| ≤ τ.

(4.2.9)

x∈Ω

In particular (4.2.10)

G

◦ ϕ = 1 + γϕ/τ  1, implying dφ = (G ◦ ϕ)dϕ  1,

dk φ  1,

k ∈ N,

uniformly in γ > 0 and τ > 0 under condition (4.2.9).

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4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

We set P = (pij )1≤i,j≤d . If we set Div = −i div, it yields P0 = Div PD. We consider the conjugated operator P0,φ = eτ φ P0 e−τ φ . The explicit computation of (3.2.1) gives P0,φ = P2 + iP1 , with P2 = P0 − p(x, τ dφ),

P1 = τ (D · P∇φ + P∇φ · D).

The operators P1 and P2 are formally selfadjoint if considered away from a boundary. For f, g ∈ C ∞ (Ω), with several integrations by parts, we find (4.2.11) (P2 f, g)L2 (Ω) = (f, P2 g)L2 (Ω) + (f|∂Ω , ∂ν g|∂Ω )L2 (∂Ω) − (∂ν f|∂Ω , g|∂Ω )L2 (∂Ω) , (4.2.12) (P1 f, g)L2 (Ω) = (f, P1 g)L2 (Ω) − 2iτ ((∂ν φ)f|∂Ω , g|∂Ω )L2 (∂Ω) , since ∂ν = ν · P∇. These two computations are the counterparts of (3.4.14)– (3.4.15), obtained in a local derivation in well chosen coordinates. For v ∈ C ∞ (Ω), we write P0,φ v2L2 (Ω) = P2 v2L2 (Ω) + P1 v2L2 (Ω) + i(P1 v, P2 v)L2 (Ω)

(4.2.13)

− i(P2 v, P1 v)L2 (Ω) . By (4.2.11)–(4.2.12) we obtain i(P1 v, P2 v)L2 (Ω) = i(P2 P1 v, v)L2 (Ω) − i(P1 v|∂Ω , ∂ν v|∂Ω )L2 (∂Ω) + i(∂ν (P1 v)|∂Ω , v|∂Ω )L2 (∂Ω) , i(P2 v, P1 v)L2 (Ω) = i(P1 P2 v, v)L2 (Ω) − 2τ ((∂ν φ)P2 v|∂Ω , v|∂Ω )L2 (∂Ω) , which gives (4.2.14)   i (P1 v, P2 v)L2 (Ω) − (P2 v, P1 v)L2 (Ω) = Re(i[P2 , P1 ]v, v)L2 (Ω) + Re B(v), where B(v) = −(iP1 v|∂Ω , ∂ν v|∂Ω )L2 (∂Ω) + (M v|∂Ω , v|∂Ω )L2 (∂Ω) , with M = i∂ν ◦ P1 + 2τ (∂ν φ)P2 . With the properties of φ and ϕ, the argument of the proof of Theorem 4.19 applies. We have  ∗ {p2 , p1 }(x, ξ, τ ) = C1 γ(τ + γϕ(x))2 + γ Hp ϕ ◦ Hp ϕ(x, ξ) + γr1 (x, ξ),

4.2. LIMITING WEIGHTS AND LIMITING CARLEMAN ESTIMATES

157

with C1 > 0 and r1 ∈ S 1 polynomial in ξ. An integration by part gives Re(i[P2 , P1 ]v, v)L2 (Ω) ≥ C1 γ(τ + γϕ)v2L2 (Ω) + γOp(Hp ϕ)v2L2 (Ω) + Re((τ R1 + γOp(r1 ))v, v)L2 (Ω) − Cγ|Dv|∂Ω |L2 (∂Ω) |v|∂Ω |L2 (∂Ω)   ≥ Cγτ 2 v2L2 (Ω) − C  τ 2 v2L2 (Ω) + Dv2L2 (Ω) − C  γ|Dv|∂Ω |L2 (∂Ω) |v|∂Ω |L2 (∂Ω) , yielding (4.2.15)

γτ 2 v2L2 (Ω)  Re(i[P2 , P1 ]v, v)L2 (Ω) + Dv2L2 (Ω) + γ|Dv|∂Ω |L2 (∂Ω) |v|∂Ω |L2 (∂Ω) ,

for γ chosen sufficiently large. We write (P2 v, v)L2 (Ω) = (PDv, Dv)L2 (Ω) − (p(x, τ dφ)v, v)L2 (Ω) − (∂ν v|∂Ω , v|∂Ω )L2 (∂Ω) ≥ CDv2L2 (Ω) − τ 2 C  v2L2 (Ω) − |v|∂Ω |L2 (∂Ω) |∂ν v|∂Ω |L2 (∂Ω) . This gives Dv2L2 (Rd )  τ 2 v2L2 (Rd ) + τ −2 P2 v2L2 (Rd ) + |v|∂Ω |L2 (∂Ω) |∂ν v|∂Ω |L2 (∂Ω) . With (4.2.15) we find (4.2.16)

γτ 2 v2L2 (Ω) + γDv2L2 (Ω)  Re(i[P2 , P1 ]v, v)L2 (Ω) + γτ −2 P2 v2L2 (Rd ) + γ|v|∂Ω |L2 (∂Ω) |Dv|∂Ω |L2 (∂Ω) .

From (4.2.13), (4.2.14), and (4.2.16) we find (4.2.17) γτ 2 v2L2 (Ω) + γDv2L2 (Ω) + Re B(v)  P0,φ v2L2 (Rd ) + γ|v|∂Ω |L2 (∂Ω) |Dv|∂Ω |L2 (∂Ω) , for τ ≥ 1 and using γ/τ  1. We now analyze the term B(v). Set |ν|P = (ν · Pν)1/2 . For any function w, we have ∂ν w = νP∇w, meaning that (∂ν w)ν/|ν|2P is the orthogonal projection in the sense of P of ∇w onto span(ν). Hence, ∇T w = ∇w − (∂ν w)ν/|ν|2P is orthogonal to ν, also in the sense of P. For a smooth function f defined on ∂Ω, if f˜ is a smooth extension of f on a neighborhood of ∂Ω in Ω we may then set T f = (∇T f˜)|∂Ω . Using local charts one finds that T is in fact a differential operator of order one on ∂Ω. By abuse of notation we thus simply write T = ∇T .

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4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

Lemma 4.24. We have iP1 = 2τ (∂ν φ)|ν|−2 P ∂ν

mod τ Dτ1 (∂Ω),

M ∈ τ Dτ2 (∂Ω) + τ Dτ1 (∂Ω)∂ν . A proof is given below. Like standard differential operators, differential operators with a large parameter on a manifold can be defined through local charts. This is done in Section 16.3.4 in Volume 2. We then obtain Re B(v) ≥ −2τ ((∂ν φ)|ν|−2 P ∂ν v|∂Ω , ∂ν v|∂Ω )L2 (∂Ω)  3  − C τ |v|∂Ω |2L2 (∂Ω) + τ |v|∂Ω |2H 1 (∂Ω)   − C τ 2 |v|∂Ω |L2 (∂Ω) + τ |v|∂Ω |H 1 (∂Ω) |∂ν v|∂Ω |L2 (∂Ω) . As γ  τ and τ ≥ 1, with (4.2.17) γτ 2 v2L2 (Ω) + γDv2L2 (Ω) − 2τ ((∂ν φ)|ν|−2 P ∂ν v|∂Ω , ∂ν v|∂Ω )L2 (∂Ω)  P0,φ v2L2 (Rd ) + τ 3 |v|∂Ω |2L2 (∂Ω) + τ |v|∂Ω |2H 1 (∂Ω)   + τ 2 |v|∂Ω |L2 (∂Ω) + τ |v|∂Ω |H 1 (∂Ω) |∂ν v|∂Ω |L2 (∂Ω) . Arguing as in the end of the proof of Theorem 4.19, for τ ≥ 1 and γ chosen sufficiently large, we may replace P0,φ by Pφ and obtain (4.2.18) γτ 2 v2L2 (Ω) + γDv2L2 (Ω) − 2τ ((∂ν φ)|ν|−2 P ∂ν v|∂Ω , ∂ν v|∂Ω )L2 (∂Ω)  Pφ v2L2 (Rd ) + τ 3 |v|∂Ω |2L2 (∂Ω) + τ |v|∂Ω |2H 1 (∂Ω)   + τ 2 |v|∂Ω |L2 (∂Ω) + τ |v|∂Ω |H 1 (∂Ω) |∂ν v|∂Ω |L2 (∂Ω) .  Let ε > 0. In ∂Ω+ ε , we have −∂ν φ = −(G ◦ ϕ)∂ν ϕ  ε. For γ kept fixed, we thus obtain  2 τ 2 v2L2 (Ω) + Dv2L2 (Ω) + ετ ∂ν v|∂Ω L2 (∂Ω+ ) ε  2 2 3    Pφ vL2 (Rd ) + τ ∂ν v|∂Ω L2 (∂Ω− ) + τ |v|∂Ω |2L2 (∂Ω) + τ |v|∂Ω |2H 1 (∂Ω) ε   + τ 2 |v|∂Ω |L2 (∂Ω) + τ |v|∂Ω |H 1 (∂Ω) |∂ν v|∂Ω |L2 (∂Ω) .

With the Young inequality we then find  2 τ 2 v2L2 (Ω) + Dv2L2 (Ω) + τ ∂ν v|∂Ω L2 (∂Ω+ ) ε  2 2    Pφ vL2 (Rd ) + τ ∂ν v|∂Ω L2 (∂Ω− ) + τ 3 |v|∂Ω |2L2 (∂Ω) + τ |v|∂Ω |2H 1 (∂Ω) . ε

Arguing as in the end of the proof of Theorem 4.19, the result of Theorem 4.22 follows.

4.3. CARLEMAN WEIGHT BEHAVIOR AT A BOUNDARY

159

In the case of a homogeneous Dirichlet boundary condition we have v|∂Ω = 0 yielding from (4.2.18), keeping the value of γ > 0 fixed, τ 2 v2L2 (Ω) + Dv2L2 (Ω) − 2τ ((∂ν φ)|ν|−2 P ∂ν v|∂Ω , ∂ν v|∂Ω )L2 (∂Ω)  Pφ v2L2 (Rd ) . Arguing as in the end of the proof of Theorem 4.19, the result of Theorem 4.23 follows.  Proof of Lemma 4.24. We have iP1 = 2τ P∇φ · ∇

mod τ Dτ0 (∂Ω)

= 2τ |ν|−4 P (∂ν φ)(Pν · ν)∂ν + 2τ P∇T φ · ∇T = 2τ |ν|−2 P (∂ν φ)∂ν

mod τ Dτ0 (∂Ω)

mod τ Dτ1 (∂Ω).

We then write

  (4.2.19) M = iP1 ∂ν + 2τ (∂ν φ) P0 − p(x, τ dφ) + i[∂ν , P1 ] 2 2 1 = 2τ |ν|−2 P (∂ν φ)∂ν + 2τ (∂ν φ)P0 mod τ Dτ (∂Ω) + τ Dτ (∂Ω)∂ν   2 2 2 = 2τ (∂ν φ) |ν|−2 mod τ Dτ2 (∂Ω) + τ Dτ1 (∂Ω)∂ν , P ∂ν − P · ∇ 2

where · stands for the tensor contraction on the last two indices. Since we have ∂ν2 v|∂Ω ν ⊗ ν + αν ⊗ ∇T ∂ν v|∂Ω + β∇2T v|∂Ω , ∇2 v|∂Ω = |ν|4P for α, β smooth functions on ∂Ω we obtain  2  2 2 (4.2.20) mod Dτ2 (∂Ω) + Dτ1 (∂Ω)∂ν P · ∇2 = |ν|−4 P P · ν ⊗ ν ∂ν 2 = |ν|−2 P ∂ν

mod Dτ2 (∂Ω) + Dτ1 (∂Ω)∂ν .

Together (4.2.19)–(4.2.20) yield the second result.



4.3. Carleman Weight Behavior at a Boundary

For a second-order operator P with principal part 1≤i,j≤d Di (pij (x)Dj ), in Sects. 3.4 and 3.5 Carleman estimates are derived at a boundary, in the case of Dirichlet conditions; see Lemma 3.16 and Theorem 3.29. There, it is assumed that the weight function ϕ satisfies ∂ν ϕ|∂Ω < 0 and this assumption plays an essential rˆ ole in achieving an estimate of the form (4.3.1) τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|∂Ω |2L2 (∂Ω)  eτ ϕ P u2L2 (Ω) + τ 3 |eτ ϕ u|∂Ω |2L2 (∂Ω) + τ |eτ ϕ u|∂Ω |2H 1 (∂Ω) , for functions supported near a point of the boundary. In fact, in some cases if ∂ν ϕ = 0 vanishes on part of the boundary and if the gradient of ϕ does not vanish, that is, ϕ varies along the boundary, a Carleman estimate can sometimes be derived yet, not leading to an estimation of the Neumann

160

4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

trace as in (4.3.1). Here, we shall prove that if one has an estimate of the form (4.3.2) τ α eτ ϕ u2L2 (Ω)  eτ ϕ P u2L2 (Ω) + |eτ ϕ u|∂Ω |2L2 (∂Ω) + |eτ ϕ u|∂Ω |2H 1 (∂Ω) , for functions supported near a point x0 of the boundary, where α ∈ R, then it implies that ∂ν ϕ(x0 ) ≤ 0. Hence, the assumption ∂ν ϕ|∂Ω ≤ 0 is in fact necessary for the derivation of such a Carleman estimate. As a result the sign condition on ∂ν ϕ at the boundary appears as important as the subellipticity property (see Chap. 3 and Sect. 4.1.1). The proof of this result is in fact similar and goes through the construction of a quasimode. Near the boundary we write the principal part of the operator in the form Dx2d + R(x, Dx ), using normal geodesic coordinates (see Sect. 9.4 in Chap. 9), such that Ω = {xd > 0} locally. We denote by p(x, ξ) the associated principal symbol that is homogeneous of order two. As seen in Chap. 3 we may consider P equal to its principal part without any loss of generality. The conjugated operator then reads Pϕ (x, Dx , τ ) = eτ ϕ P e−τ ϕ = (Dxd + iτ ∂d ϕ)2 + R(x, Dx + iτ dx ϕ). If ∂ν ϕ|∂Ω = −∂d ϕ|xd =0+ > 0 at some point of the boundary we shall construct a quasimode, that is, a function vτ , with τ > 0 as a parameter, supported near this point, satisfying homogeneous Dirichlet boundary conditions, and such that Pϕ vτ L2  vτ L2 , thus ruining any hope of achieving a Carleman estimate without a boundary observation of the solution (compare with Theorem 3.28). Theorem 4.25. Assume ∂ν ϕ|∂Ω (x0 ) > 0 for some x0 ∈ ∂Ω. Then, for all open neighborhood V of x0 in Ω, there exists a smooth function vτ satisfying supp(vτ ) ⊂ V, vτ |∂Ω = 0, and for all N ∈ N there exists CN > 0 such that vτ L2 (Ω) ≥ 1,

Pϕ vτ L2 (Ω) ≤ CN τ −N .

Corollary 4.26. Let x0 ∈ ∂Ω and W be an open neighborhood of x0 in Rd , α > 0, τ∗ > 0, and C > 0 such that τ −α eτ ϕ uL2 (Ω) ≤ Ceτ ϕ P uL2 (Ω) , for τ ≥ τ∗ and u = w|W ∩Ω with w ∈ Cc∞ (Rd ) and supp(w) ⊂ W . Then, ∂ν ϕ|∂Ω (x0 ) ≤ 0.

4.3. CARLEMAN WEIGHT BEHAVIOR AT A BOUNDARY

161

Proof of Theorem 4.25. Without any loss of generality we may choose x0 = 0. The principal symbol of Pϕ is given by pϕ (x, ξ, τ ) = p(x, ξ + iτ dϕ(x) = (ξd + iτ ∂d ϕ(x))2 + R(x, ξ  + iτ dx ϕ(x)). Since ∂d ϕ|∂Ω (x0 ) = −∂ν ϕ|∂Ω (x0 ) < 0, observe that we can choose (τ, ξ  ) = (τ 0 , ξ 0 ), with τ0 = 0, such that the two roots of pϕ (x0 , ξd , τ0 ) viewed as a polynomial function in ξd are distinct and both have a positive imaginary part. Two cases need to be considered. First, if dx ϕ(x0 ) = 0, then R(x0 , ξ 0 )  |ξ 0 |2 is real and it suffices to choose |ξ 0 | = 0 small. Second, if dx ϕ(x0 ) = 0, then R(x0 , iτ dx ϕ(x0 )) < 0 meaning that the two roots fulfill the sought property if ξ 0 = 0. Note that by homogeneity we may choose τ 0 = 1. Note that the two roots have the form (4.3.3)

−iτ ∂d ϕ(x0 ) ± α0 ,

with α0 ∈ C \ {0}.

Developing the coefficients of p and also the function ϕ by means of a Taylor expansion at x = x0 = 0 we may write, for any M ∈ N∗ , (4.3.4)

pϕ (x, ξ, τ ) = p(x, ξ + iτ dϕ(x)) = q(x, ξ, τ ) + |x|M tM 2 (x, ξ, τ ),

2 where q is polynomial in all variables and tM 2 ∈ Sτ . Observe that q(0, ξ, 1) = pϕ (0, ξ, 1). Then, in a neighborhood U0 of 0 (x = 0, ξ 0 ) we may write

(4.3.5)

q(x, ξ, 1) = (ξd − ρ1 (x, ξ  ))(ξd − ρ2 (x, ξ  )),

with ρj (x, ξ  ), j = 1, 2, analytic in both variables by a classical result (this result is for instance recalled in Proposition 6.28 in Volume 2). Moreover, there exists C0 > 0 such that Im ρ1 (x, ξ  ) ≥ C0 , Im ρ2 (x, ξ  ) ≥ C0 , |ρ2 (x, ξ  ) − ρ1 (x, ξ  )| ≥ C0 , (4.3.6) (x, ξ  ) ∈ U0 . At the point (x, ξ  ) = (x0 , ξ 0 ) the roots ρ1 and ρ2 of coincide with that of ξd → pϕ (x0 , ξ 0 , ξd , 1). With (4.3.3) we thus have (4.3.7)

ρj (x0 , ξ 0 ) = −i∂d ϕ(x0 ) + (−1)j α0 ,

with α0 ∈ C \ {0}.

We shall construct two functions vτ1 and vτ2 in the form (4.3.8)

vτj (x , xd ) = χ0 (τ 1/2 x )χ1 (τ α1 xd ) × ∫ eiτ φ Rd−1

j (x,ξ  )

aj (x, ξ  , τ )χ0 (τ α2 (ξ  − ξ 0 ))dξ  ,

with 0 < α2 < 1/2 < α1 < 1, τ > 0, and χ0 ∈ Cc∞ (Rd−1 ), χ1 ∈ Cc∞ (R) and χj (y) = 1 for |y| ≤ 1/2,

χj (y) = 0 for |y| ≥ 1,

j = 0, 1.

The constructions of the phase functions φj and the amplitudes aj are given below.

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4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

Lemma 4.27. There exist a neighborhood U1 of (x0 , ξ 0 ) and two analytic complex phase functions φj (x, ξ  ), j = 1, 2 that satisfy the Eikonal equations ∂d φj (x, ξ  ) = ρj (x, dx φj (x, ξ  )),

φj |xd =0 (x , ξ  ) = x · ξ  ,

j = 1, 2,

and (4.3.9)

φj (x, ξ  ) = x · ξ  + xd ρj (x , 0, ξ  ) + x2d gj (x, ξ  ),

j = 1, 2,

with gj analytic in U1 . Lemma 4.28. For all M, N ∈ N there exist a neighborhood W of (x0 , ξ 0 ) and two amplitudes aj (x, ξ  , τ ), j = 1, 2, such that   j j e−iτ φ Pϕ (x, Dx , τ ) eiτ φ aj = (τ −N + τ 2 |x|M )O(1), j = 1, 2, for (x, ξ  ) ∈ W, and a1 |xd =0 = a2 |xd =0 = 1. and a1 |xd =0 = a2 |xd =0 = 1. Moreover aj and its derivatives are bounded uniformly in τ ≥ 1. We then have the following estimations. Lemma 4.29. Let vτj be as given by (4.3.8) with φj and aj given by Lemmata 4.27 and 4.28 for j = 1, 2. For all N ∈ N there exists τ1 > 0 such that Pϕ (vτj ) = τ −N O(1), j = 1, 2,

τ ≥ τ1 ,

for M chosen sufficiently large. Lemma 4.30. For α1 = 1 − ε and α2 = 1/2 − ε with ε chosen sufficiently small, there exist τ2 > 0, C1 > 0 and s > 0 such that we have vτ1 − vτ2 L2 (Rd ) ≥ C1 τ −s , +

τ ≥ τ2 ,

for M chosen sufficiently large. The proofs of the above four lemmata are given below. Setting vτ = C1−1 τ s (vτ1 −vτ2 ) we observe that vτ vanishes at the boundary since vτ1 |xd =0 = vτ2 |xd =0 and that it fulfills the sought quasimode properties. This concludes the proof of Theorem 4.25.  Proof of Lemma 4.27. As ρj (x, ξ) is analytic and so is the initial condition at xd = 0, the Cauchy–Kovalevska¨ıa theorem then yields the existence of φj in a neighborhood of (x0 , ξ 0 ). The expression of φj in (4.3.9) follows  from a Taylor expansion at xd = 0 and using the equation. Proof of Lemma 4.28. To ease notation we suppress the indices and simply write φ(x, ξ  ) and a(x, ξ  , τ ) for the phase and amplitude functions.

4.3. CARLEMAN WEIGHT BEHAVIOR AT A BOUNDARY

163

As Pϕ = P (x, Dx + iτ dϕ(x)) we find 



e−iτ φ(x,ξ ) Pϕ (x, Dx , τ )eiτ φ(x,ξ )   = P x, Dx + τ dφ(x, ξ  ) + iτ dϕ(x)     = τ 2 p x, dφ(x, ξ  ) + idϕ(x) + τ ∇ξ p x, dφ(x, ξ  ) + idϕ(x) · Dx + P (x, Dx ) + τ T0 (x, ξ  ), where T0 is smooth function. We set   L(x, ξ  , Dx ) = ∇ξ p x, dφ(x, ξ  ) + idϕ(x) · Dx + T0 (x, ξ  ), and we find that it reads L(x, ξ  , Dx ) = θ(x, ξ  )Dxd + S(x, ξ  , Dx ) with   θ(x, ξ  ) = 2 i∂d ϕ(x) + ∂d φ(x, ξ  ) and S(x, ξ, Dx ) ∈ DT1 . Because of the form of φ given in Lemma 4.27 the coefficient θ(x, ξ  ) does not vanish in a sufficiently small neighborhood of (x0 , ξ 0 ) since θ(x0 , ξ 0 ) = ±α0 = 0 with α0 as in (4.3.7). From (4.3.4), we have  p(x, dφ(x, ξ  ) + idϕ(x)) = q(x, dφ(x, ξ  ), 1) + |x|M tM 2 (x, dφ(x, ξ ), 1),

and q(x, dφ(x, ξ  ), 1) vanishes by (4.3.5) and Lemma 4.27. We thus have (4.3.10)





e−iτ φ(x,ξ ) Pϕ (x, Dx , τ )eiτ φ(x,ξ )  = τ L(x, ξ  , Dx ) + P (x, Dx ) + |x|M tM 2 (x, dφ(x, ξ ), 1).

We also write, with a Taylor expansion at x = x0 = 0, for any M ∈ N∗ , P (x, Dx ) = P M (x, Dx ) + |x|M tM P (x, D),  L(x, ξ  , Dx ) = LM (x, ξ  , Dx ) + |x|M tM L (x, ξ , D),

where P M (x, Dx ) ∈ D 2 (Rd+ ) and LM (x, ξ  , Dx ) ∈ D 1 (Rd+ ) have analytic 2 d M  1 d coefficients and tM P (x, D) ∈ D (R+ ) and tL (x, ξ , D) ∈ D (R+ ). Note that a Taylor expansion in the ξ  variable is not needed as φ(x, ξ  ) is analytic by Lemma 4.27. Note that LM (x, ξ  , Dx ) takes the form LM (x, ξ  , Dx ) = θM (x, ξ  )Dxd + S M (x, ξ  , Dx ), where θM and S M are obtained by the Taylor expansion at x = x0 of θ and S. In particular θM does not vanish in a small neighborhood U of (x0 , ξ 0 ). In U we introduce inductively: (1) The analytic function a0 (x, ξ  ) as the solution to the transport equation with complex coefficients, by the Cauchy–Kovalevska¨ıa theorem,   M a0 |xd =0 (x , ξ  ) = 1. θ (x, ξ  )Dxd + S M (x, ξ  , Dx ) a0 (x, ξ  ) = 0, defined in a neighborhood U (0) ⊂ U of (x0 , ξ 0 ).

164

4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

(2) For 1 ≤ k ≤ N , the analytic function ak (x, ξ  ) as the solution to the transport equation, again by the Cauchy–Kovalevska¨ıa theorem,   M θ (x, ξ  )Dxd + S M (x, ξ  , Dx ) ak (x, ξ  ) = −P M (x, Dx )ak−1 (x, ξ  ), ak |xd =0 = 0, defined in an open neighborhood U (k) ⊂ U (k−1) of (x0 , ξ 0 ). (3) We set a(x, ξ  , τ ) =

N

τ −k ak (x, ξ  ).

k=0

Observe that all the functions ak and their derivatives are bounded in an open neighborhood W of (x0 , ξ 0 ) such that W  U (M ) and we have   τ L(x, ξ  , Dx ) + P (x, Dx ) a(x, ξ  , τ ) = τ −N P (x, Dx )aN (x, ξ  ) = τ −N O(1). Since   2 M M  2 τ t2 (x, dφ(x, ξ  ), 1) + tM P (x, D) + τ tL (x, D) ak (x, ξ , τ ) = τ O(1), 

this concludes the proof.

Proof of Lemma 4.29. Here, as vτ1 and vτ2 both have the same structure, to ease notation we suppress the indices and simply write v. Similarly we simply write φ(x, ξ  ) and a(x, ξ  , τ ) for the phase and amplitude functions. The action of Pϕ (x, Dx , τ ) on v yields Pϕ (x, Dx , τ )v = A + B + C, where • The term A is a linear combination of terms of the form (β1 )

c(x)τ k+|β1 |/2 χ0

(τ 1/2 x )

  j  × ∫ χ0 (τ α2 (ξ  − ξ 0 ))Dxβ2 χ1 (τ α1 xd )eiτ φ (x,ξ ) aj (x, ξ  , τ ) dξ  , Rd−1

with c(x) smooth and 1 ≤ |β1 | and |β1 + β2 | + k ≤ 2. • The term B is a linear combination of terms of the form (β1 )

c(x)χ0 (τ 1/2 x )τ k+α1 |β1 | χ1

(τ α1 xd )   j  × ∫ χ0 (τ α2 (ξ  − ξ 0 ))Dxβ2 eiτ φ (x,ξ ) aj (x, ξ  , τ ) dξ  , Rd−1

with c(x) smooth and 1 ≤ |β1 | and |β1 + β2 | + k ≤ 2. • The term C is given by χ0 (τ 1/2 x )χ1 (τ α1 xd )

  j  × ∫ χ0 (τ α2 (ξ  − ξ 0 ))Pϕ (x, Dx , τ ) eiτ φ (x,ξ ) aj (x, ξ  , τ ) dξ  . Rd−1

4.3. CARLEMAN WEIGHT BEHAVIOR AT A BOUNDARY

165

Estimation of A. Observe that the terms involved in the definition of A can be written in the following form c(x)τ k+|β1 |/2 A˜ with j  (β ) A˜ = χ0 1 (τ 1/2 x )χ2 (τ α1 xd ) ∫ eiτ φ (x,ξ ) χ0 (τ α2 (ξ  − ξ 0 ))f (x, ξ  , τ )dξ  ,

Rd−1

where f and all its derivative are O(τ |β2 | ) and supp(χ2 ) ⊂ [−1, 1]. We compute dξ φ as dξ φ(x, ξ  ) = x + xd dξ ρj (x , 0, ξ  ) + x2d O(1), (β1 )

by Lemma 4.27. In the support of χ0 with C  and C  positive, we have

(τ 1/2 x )χ2 (τ α1 xd ), since |β1 | > 0,

|dξ φ(x, ξ  )| ≥ |x | − Cxd ≥ C  τ −1/2 − Cτ −α1  τ −1/2 , for τ large as α1 > 1/2. Similarly |dξ φ(x, ξ  )|  τ −1/2 yielding |dξ φ(x, ξ  )|  τ −1/2 .

(4.3.11)

In a similar fashion with Lemma 4.27 we have (4.3.12)

|dγξ φ(x, ξ  )| = xd O(1)  τ −α1  τ −1/2 ,

γ ∈ Nd−1 , |γ| ≥ 2.

We set L= j

∇ ξ  φ · ∇ξ  |dξ φ|2

j

and note that Leiτ φ = iτ eiτ φ . Iterating integrations by parts we obtain, for n ∈ N, (4.3.13)

(β ) A˜ = χ0 1 (τ 1/2 x )χ2 (τ α1 xd )  1 iτ φj (x,ξ )t n  e L χ0 (τ α2 (ξ  − ξ 0 ))f (x, ξ  , τ ) dξ  . × ∫ n Rd−1 (iτ )

We compute 2

(4.3.14)

t

L=L+

dξ φ(dξ φ, dξ φ) Δξ  φ −2 . 2 |dξ φ| |dξ φ|4

To understand the behavior w.r.t. τ we give a precise description of tLn . Lemma 4.31. For all n ∈ N∗ , we have

t n (4.3.15) L = cβ ∂ξβ where cβ = |dξ φ|−4n bβ,γ ∂ξγ1 φ · · · ∂ξγ3n φ, |β|≤n

γ∈Sβ

with bβ,γ ∈ C and  Sβ = γ = (γ1 , · · · , γ3n ) ∈ N3n(d−1) ,

 such that |γ1 + · · · + γ3n | = 4n − |β| and |γj | ≥ 1 . (β1 )

Moreover in the support of χ0

(τ 1/2 x )χ2 (τ α1 xd ) we have |cβ | ≤ Cτ n/2 .

166

4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

  This lemma allows one to estimate tLn χ0 (τ α2 (ξ  − ξ 0 ))f (x, ξ  , τ ) . As the derivatives of f are bounded, the most contribution is associated with derivatives acting on χ0 (τ α2 (ξ  − ξ 0 )). We thus have   |cβ ∂ξβ χ0 (τ α2 (ξ  − ξ 0 ))f (x, ξ  , τ ) | ≤ Cβ τ n/2+|β|α2 ≤ Cn τ n(1/2+α2 ) . Then we can estimate the integrand of (4.3.13) by Cτ −n(1/2−α2 ) . As α2 < 1/2, for any N ∈ N, by choosing n sufficiently large we obtain, for some CN > 0, |A| ≤ CN τ −N . Proof of Lemma 4.31. We proceed by induction and we consider at first n = 1. In (4.3.14) we can write the coefficients associated with firstorder derivative of the form given in (4.3.15) with |β| = 1 and cβ =

∂ξβ φ |dξ φ|2

=

∂ξβ φ|dξ φ|2 |dξ φ|4

.

The coefficients associated with zero-order terms in (4.3.14) also take the form given in (4.3.15) with |β| = 0 and cβ as a sum of two terms: d2ξ φ(dξ φ, dξ φ) |dξ φ|2 Δξ φ Δξ  φ = and |dξ φ|2 |dξ φ|4 |dξ φ|4 Each numerator is composed of a product of 3 terms involving a derivative of φ of a certain order, and the sum of derivatives amount to 4. The form of tLn given in (4.3.15) thus holds for n = 1. Now we assume that (4.3.15) holds for n − 1 and we compute tL(c∂ξβ ) γ for |β| ≤ n − 1 and c = |φξ |−4n+4 ∂ξγ1 φ · · · ∂ξ3n−3 φ where |γ1 + · · · + γ3n−3 | = 4n − 4 − |β| and |γj | ≥ 1. The two zero-order terms of tL in (4.3.14) give a linear combination of terms of the form    γ |dξ φ|−4 ∂ξ2k ξ φ∂ξμ φ∂ξν φ |dξ φ|−4n+4 ∂ξγ1 φ · · · ∂ξ3n−3 φ , k, , μ, ν ∈ {1, . . . , d − 1}. The power of denominator is 4n, the number of terms of numerator is 3n, and the sum of derivatives amounts to 4n − |β|. The action of the first-order term of tL, that is L = |dξ φ|−2 dξ φ · ∂ξ , gives different types of terms. • The terms obtained when the derivative does not act on coefficients have the form    γ3n−3 −4 −4n+4 γ1   ∂ξ  φ · · · ∂ξ  φ ∂ξ ∂ξβ , |dξ φ| ∂ξk φ∂ξμ φ∂ξν φ |dξ φ| for k, , μ, ν ∈ {1, . . . , d − 1}. This coincides with the form given in (4.3.15) if we remark that the sum of all derivatives of φ in the numerator amounts to 4n − (1 + |β|) and 1 + |β| is the order of ∂ξ ∂ξβ .

4.3. CARLEMAN WEIGHT BEHAVIOR AT A BOUNDARY

167

• The terms with a derivative acting on c yield two types of terms. – First, we have terms with a differentiation of a term in the numerator    γ γ |dξ φ|−4 ∂ξk φ∂ξμ φ∂ξν φ |dξ φ|−4n+4 ∂ξγ1 φ · · · ∂ξ ∂ξj φ · · · ∂ξ3n−3 φ ∂ξβ . The sum of derivatives of φ in the numerator amounts to 3 + |γ1 + · · · + γ3n−3 | + 1 = 4n − |β| which coincides with what can be found in (4.3.15). – Second, we have terms with a differentiation of the denominator     γ |dξ φ|−2 ∂ξk φ ∂ξγ1 φ · · · ∂ξ3n−3 φ |dξ φ|−4n+2 ∂ξ2μ ξν φ∂ξ φ ∂ξβ , as the derivative of |φξ |−4n+4 is, up to some constant, |φξ |−4n+2 ∂ξ2k ξ φ∂ξ φ. The sum of derivatives in the numerator amounts to 3 + |γ1 + · · · + γ3n−3 | + 1 = 4n − |β| which coincides with what can be found in (4.3.15). This concludes the induction argument. With the form of cβ in (4.3.15), by (4.3.11) and (4.3.12), in the support (β ) of χ0 1 (τ 1/2 x ), we have τ −3n/2 = τ n/2 , τ −2n thus concluding the proof of Lemma 4.31. |cβ | 



(β )

Estimation of B. On the support of χ1 1 (τ α1 xd ) we have xd  τ −α1 . Since by Lemma 4.27 we have Im φj (x, ξ  ) = xd Im ρj (x , 0, ξ  ) + x2d O(1) and as we have Im ρj (x0 , ξ 0 ) = Im rj > 0 (see the beginning of the proof of Theorem 4.25), then in a neighborhood of (x0 , ξ 0 ), there exists C > 0 such that Im ρj (x , 0, ξ  ) ≥ C. We can thus estimate (β1 )

|B|  τ k+α1 |β1 |+|beta2 | |χ1

(τ α1 xd )|e−Cτ xd  τ 2 e−Cτ

1−α1

≤ CN τ −N ,

for any N ∈ N, as 1 − α1 > 0. Estimation of C. By Lemma 4.28 we have   j  Pϕ (x, Dx , τ ) eiτ φ (x,ξ ) aj (x, ξ  , τ ) = (τ −M + τ 2 |x|M )O(1). As in the support of χ0 (τ 1/2 x )χ1 (τ α1 xd ) we have |x| ≤ Cτ −1/2 . Then we have   χ0 (τ 1/2 x )χ1 (τ α1 xd ) ∫ χ0 (τ α2 (ξ  − ξ 0 ))Pϕ (x, Dx , τ )  Rd−1    j  × eiτ φ (x,ξ ) aj (x, ξ  , τ ) dξ    τ 2−M/2 ,

168

4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

which gives |C| ≤ CN τ −N , for any N ∈ N, for M chosen sufficiently large.



Proof of Lemma 4.30. We first write the phase functions given in Lemma 4.27 as φj (x, ξ  ) = x · ξ 0 + xd ρj (x , 0, ξ 0 ) + x · (ξ  − ξ 0 ) + xd O(ξ  − ξ 0 ) + x2d O(1), for j = 1, 2, yielding    iτ x ·ξ 0 +xd ρj (x ,0,ξ 0 ) iτ x ·(ξ  −ξ 0 )  iτ φj (x,ξ  ) e =e e 1 + τ 1−α1 −α2 O(1) , in the support of χ1 (τ α1 xd )χ0 (τ α2 (ξ  − ξ 0 )), recalling that 0 < α2 < 1/2 < α1 < 1. We also write aj (x, ξ  , τ ) = aj (x , 0, ξ  , τ ) + xd O(1) = 1 + τ −α1 O(1), by Lemma 4.28. This yields, as 1 − α2 > 0,    0  0 vτj (x , xd ) = χ0 (τ 1/2 x )χ1 (τ α1 xd )eiτ x ·ξ +xd ρj (x ,0,ξ )    0  × ∫ eiτ x ·(ξ −ξ ) 1 + τ 1−α1 −α2 O(1) χ0 (τ α2 (ξ  − ξ 0 ))dξ  . Rd−1

With the change of variables ξ  → τ α2 (ξ  − ξ 0 ) we find   iτ x ·ξ 0 +xd ρj (x ,0,ξ 0 ) j  −α2 (d−1) 1/2  α1 vτ (x , xd ) = τ χ0 (τ x )χ1 (τ xd )e  1−α2 x ·ξ   × ∫ eiτ 1 + τ 1−α1 −α2 O(1) χ0 (ξ  )dξ  . Rd−1

For j = 1, 2 we thus write vτj = vτj,a + vτj,p , where vτj,a (x , xd )

=τ

−α2 (d−1)

× ∫ e

χ0 (τ

1/2 

x )χ1 (τ

iτ 1−α2 x ·ξ 

α1

xd )e



iτ x ·ξ 0 +xd ρj (x ,0,ξ 0 )



χ0 (ξ  )dξ  .

Rd−1 d−1 −α2 (d−1)

χ0 (τ 1/2 x )χ1 (τ α1 xd )χ ˇ0 (τ 1−α2 x )    0  0 × eiτ x ·ξ +xd ρj (x ,0,ξ ) ,

= (2π)

τ

where χ ˇ0 is the inverse Fourier transform of χ0 in the x variable, and   iτ x ·ξ 0 +xd ρj (x ,0,ξ 0 ) j,p  1−α1 −α2 d 1/2  α1 vτ (x , xd ) = τ χ0 (τ x )χ1 (τ xd )e O(1). As Im ρj (x , 0, ξ 0 ) ≥ C in a neighborhood of x0 , we have the following estimate: 2

vτj,p L2 (Rd )  τ 2(1−α1 −α2 d) +

∫

∫ χ20 (τ 1/2 x )e−Cτ xd dx dxd ,

[0,τ −α1 ] Rd−1

4.3. CARLEMAN WEIGHT BEHAVIOR AT A BOUNDARY

169

and with the change of variables x → τ 1/2 x we obtain (4.3.16)

2

vτj,p L2 (Rd )  τ 2(1−α1 −α2 d)−(d−1)/2 χ0 2L2 +

 τ 3/2−d/2−2(α1 +α2 d) ,

∫

[0,τ −α1 ]

e−Cτ xd dxd

j = 1, 2,

for τ ≥ 1, since α1 < 1. We now estimate vτ1,a − vτ2,a L2 (Rd ) . We write +

|vτ1,a − vτ2,a | = (2π)d−1 τ −α2 (d−1) χ0 (τ 1/2 x )χ1 (τ α1 xd )|χ ˇ0 |(τ 1−α2 x ) 

× |eiτ xd ρ1 (x ,0,ξ

0 )



0

− eiτ xd ρ2 (x ,0,ξ ) |,

and, with δ > 0, δ/τ

2

vτ1,a − vτ2,a L2 (Rd ) ≥ ∫ +

∫ |vτ1,a − vτ2,a |2 dx dxd .

0 Rd−1

Taking δ sufficiently small we have |τ xd ρj | ≤ 1 in the support of the inte grand and for τ xd ≤ δ. Then since we have |ez − ez | ≥ C|z − z  | for |z| ≤ 1 and |z  | ≤ 1 for some C > 0 we obtain δ/τ

2

vτ1,a − vτ2,a L2 (Rd )  τ 2−2α2 (d−1) ∫ +

∫ χ20 (τ 1/2 x )χ21 (τ α1 xd )|χ ˇ0 |2 (τ 1−α2 x )

0 Rd−1  0

× x2d |ρ1 (x , 0, ξ ) − ρ2 (x , 0, ξ0 )|2 dx dxd δ/τ

 τ 2−2α2 (d−1) ∫ x2d dxd ∫ χ20 (τ 1/2 x )|χ ˇ0 |2 (τ 1−α2 x )dx , 0

Rd−1

(τ α1 x

as χ1 d ) = 1 for τ large since α1 < 1 and using the position of the roots described in (4.3.6). We thus obtain, computing the integral w.r.t. xd and performing the change of variables x → τ 1−α2 x , (4.3.17) 2

vτ1,a − vτ2,a L2 (Rd )  τ −1−2α2 (d−1)+(α2 −1)(d−1) ∫ χ20 (τ α2 −1/2 x )|χ ˇ0 |2 (x )dx +

Rd−1

τ

−d−α2 (d−1)

,

by the Lebesgue dominated-convergence theorem since α2 < 1/2. Finally with (4.3.16) and (4.3.17) we write vτ1 − vτ2 L2 (Rd ) ≥ vτ1,a − vτ2,a L2 (Rd ) − vτ1,p L2 (Rd ) − vτ2,p L2 (Rd ) +

+

≥ Cτ

−d/2−α2 (d−1)/2

+

 3/4−d/4−(α1 +α2 d)

−C τ

+

.

We see that the result of the lemma can be achieved, by taking τ sufficiently large, if we have −d/2 − α2 (d − 1)/2 > 3/4 − d/4 − (α1 + α2 d). Setting α1 = 1 − ε and α2 = 1/2 − ε, with ε small, we obtain the condition 1 > (d + 3)ε. Thus choosing ε sufficiently small we conclude the proof of Lemma 4.30. 

170

4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

4.4. Notes The result of Theorem 4.1, based on the construction of a quasimode, stating the necessity of the nonnegativity of the Poisson bracket {pϕ , pϕ }/(2i) on the characteristic set of pϕ , goes beyond the mere question of deriving Carleman estimate. In fact for a differential operator Q(x, D) with principal symbol q(x, ξ) the following result holds. Theorem 4.32. Assume that for any f ∈ Cc∞ (Rd ) there exists a solution u ∈ D  (Rd ) to Q(x, D)u = f , then the following holds: q(x, ξ) = 0 ⇒

1 {¯ q , q} = {Re q, Im q} ≥ 0, 2i

for all x ∈ Rd We refer to Theorem 6.1.1 in [172] for a proof. As q(x, ξ) is polynomial and homogeneous in ξ the above property reads in fact 1 {¯ q , q} = {Re q, Im q} = 0, 2i by changing ξ into −ξ. Since the estimate of Theorem 4.1 is only valid for τ > 0 one cannot proceed with the change (ξ, τ ) into (−ξ, −τ ) and one only has the condition given in the statement of the theorem. Extension of these ¯ condition consideration to pseudo-operators is known under the ψ (resp. ψ) in connection with the Treves–Nirenberg conjecture. For details we refer to the works of N. Lerner [225, 227] and N. Dencker [117] and the book of N. Lerner [228]. Section 4.1.2 shows that the powers we obtain in the Carleman estimates derived in Chap. 3 are optimal in dimension d ≥ 2. It also shows that sub-ellipticity is a necessary condition for those estimates to hold. Parts of the proofs in this section follow closely that of Theorem 28.2.1 by L. H¨ormander [174]. Note that the proof applies to more general operators than those considered here. Ellipticity is not used in Sect. 4.1.2. Note that this result is already present in the book [172]. In the case of a constant coefficient operator like the Laplace operator, one can write simpler versions of the proofs (see e.g. [208]). Sub-ellipticity is associated with Carleman estimates that only exhibit the loss of a half-derivative. If one considers further iterations of Poisson brackets, say k interactions, a natural generalization of sub-ellipticity yields an estimate with the loss of k/(k + 1) derivative; we refer to the work of N. Lerner [224]. Note that for the bi-Laplace operator, the considered iterated brackets all vanish, and in fact one can only obtain a Carleman estimate with a loss of a full derivative [215]. Limiting Carleman weight in Rd was introduced in the work of C.E. Kenig et al. [195] for the purpose of studying the Calder´on inverse problem of retrieving the principal part or a potential term in a second-order elliptic operator from some knowledge of the Dirichlet-to-Neumann map. In fact, q(x, ξ) = 0 ⇒

4.A. SOME TECHNICAL RESULTS

171

the constraints for a function to be a limiting Carleman weight as given in Definition 4.14 are quite severe. In R2 , limiting weights for the Laplace operator are the harmonic functions with nonvanishing gradient. In Rd with d ≥ 3, again for the Laplace operator, limiting weights only take the form ϕ(x) = aϕ0 (x − x0 ) + b with a = 0 and ϕ0 being one of the following five functions: log |x|, arg(x, ω1 + iω2 ),   |x + ξ|2 arg eiθ (x + iξ)2 , , log |x − ξ|2

x · ξ,

where ω1 , ω2 ∈ Rd are orthogonal and unitary, θ ∈ R, and ξ ∈ Rd \ {0}. These results are reviewed in the work of D. Dos Santos Ferreira [123]. For Riemannian manifolds, such limiting weights may very well not exist; see the works of T. Liimatainen and M. Salo [233] and P. Angulo-Ardoy et al. [37]. Classes of Riemannian metrics for which such weights exist are described in the works of D. Dos Santos Ferreira et al. [124] and P. Angulo-Ardoy et al. [36]. In Sect. 4.3 a quasimode is constructed near a point x0 of the boundary showing that even a weak Carleman estimate implies the sign condition ∂ν ϕ|∂Ω (x0 ) ≤ 0. We thus emphasize that this condition is to be considered with the same attention as the sub-ellipticity condition. The quasimode construction uses some of the ideas of the construction made in the proof of Theorem 4.1. Signs of the roots of the principal symbol are exploited and the proof we give is partly a refinement of the argument developed in a joint work with N. Lerner [211] where such a quasimode is constructed in the case of a transmission problem.

Appendix 4.A. Some Technical Results 4.A.1. A Linear Algebra Lemma. Here we establish a lemma that gives a necessary and sufficient condition for two complex vectors to be related through a symmetric matrix with a positive definite imaginary part. The proof we give is that of [172, Lemma 6.1.4]. Lemma 4.33. Let z, h ∈ Cn be such that z = 0. There exists a symmetric n×n matrix A = (Aij ) with positive definite imaginary part such that h = Az if and only if Im(h, z)Cn = Im

1≤i≤n

hi zi > 0.

172

4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

Proof. Assume that h = Az with A fulfilling the required property. Then  1 (4.A.1) Im(h, z)Cn = Im(Az, z)Cn = (Az, z)Cn − (z, Az)Cn 2i    1 ¯ z)Cn = (Im A)z, z n > 0. = (Az, z)Cn − (Az, C 2i To prove the converse result, we distinguish two cases, whether Re z and Im z are linearly independent or not. First, assume that Re z and Im z are linearly dependent and consider the linear space W = {Bz; B ∈ Mn (C), t B = B}. We claim that W = Cn . In fact, let ζ ∈ Cn . On the one hand, if t ζz = 0 set B = (t ζz)−1 ζ t ζ; it is a symmetric matrix and Bz = ζ. On the other hand, if t ζz = 0, choose η ∈ Cn such that t ηz = 0 and set B = (t ηz)−1 (ζ +η) t (ζ +η); it is a symmetric matrix and we have Bz = ζ + η ∈ W . Since η ∈ W as previously seen, we find ζ ∈ W . The claim is proven. ˜ = h. Consequently, there exists a symmetric matrix A˜ such that Az n Let ξ ∈ R \ {0}, λ, μ ∈ R be such that Re z = λξ and Im z = μξ. We have (λ, μ) = (0, 0). We set A = A˜ + iνB where B ∈ Mn (R) is symmetric, vanishes on span(ξ), and B = Id on the orthogonal of ξ with respect to the real Euclidean inner product. The parameter ν > 0 will be chosen large and ˜ fixed below. We have Az = h, Im A = Im A˜ + νB, and t ξ Im Aξ = t ξ Im Aξ. With the computation in (4.A.1), one has   (λ2 + μ2 )(Im Aξ, ξ)Rn = (Im A)z, z Cn = Im(h, z)Cn > 0, yielding (Im Aξ, ξ)Rn > 0. If ζ ∈ Cn then ζ = αξ + η for α ∈ C and η ∈ Cn such that t ηξ = 0. We have ((Im A)ζ, ζ)Cn = |α|2 (Im Aξ, ξ)Rn + 2 Re α(Im Aξ, η)Cn + ((Im A)η, η)Cn ˜ η)Cn + ((Im A)η, ˜ η)Cn = |α|2 (Im Aξ, ξ)Rn + 2 Re α(Im Aξ, + ν|η|2Cn ≥ |α|2 (Im Aξ, ξ)Rn + (ν − C)|η|2Cn − C  |α||ξ|Rn |η|Cn , for some C, C  > 0. For ν > 0 chosen sufficiently large then ((Im A)ζ, ζ)Cn > 0 for all ζ = 0 yielding that Im A is positive definite and concluding the proof in the first case. Second, assume that Re z and Im z are linearly independent. We claim that for k ∈ Cn (4.A.2)

(k, z)Cn ∈ R ⇔ k = Bz with B ∈ Mn (R) such that t B = B.

If now h ∈ Cn is such that Im(h, z)Cn > 0, we can find k = h − iμz for μ ∈ R such that Im(k, z)Cn = 0. This amounts to picking μ = Im(h, z)Cn /|z|2 . Then, by (4.A.2) we have h = (B + iμ Id)z with B real and symmetric. This concludes the proof in this second case.

4.A. SOME TECHNICAL RESULTS

173

We now prove the property claimed in (4.A.2). We set W = {Bz; B ∈ Mn (R), t B = B}. Let us consider the two isomorphisms j : Cn → R2n ζ → (Re ζ, Im ζ),

t : Cn → R2n ζ → (− Im ζ, Re ζ, ).

For k,  ∈ Cn observe that

  Im(k, )Cn = (Im k, Re )Rn − (Re k, Im )Rn = j(k), t() R2n .

Hence, having (k, z)Cn ∈ R is equivalent to having j(k) in the hyperplane Q orthogonal to t(z) in R2n . If k ∈ W , then (k, z)Cn = (Bz, z)Cn with B real and symmetric yielding (Bz, z)Cn ∈ R. Hence, j(W ) ⊂ Q. As j(W ) is a linear subspace of R2n , it is the intersection of the hyperplanes that contain it. If we show that there is a single hyperplane that contains j(W ), it implies that j(W ) = Q and the claim is proven. Let  ∈ Cn ,  = 0, be such that j(W ) is contained in the orthogonal of t() meaning j(k), t() R2n = 0, for all k ∈ W . Consider k = Bz with B = ξ t ξ with ξ ∈ Rn . Then,   ¯ (4.A.3) 0 = j(k), t() R2n = Im(k, )Cn = Im(t ξz)(t ξ ) = (t ξ Im z)(t ξ Re ) − (t ξ Re z)(t ξ Im ). We show that if ξ is orthogonal to Re z, then it is orthogonal to Re . This means Re  = λ1 Re z for some λ1 ∈ R. In fact, on the one hand, if ξ is such that t ξ Re z = 0 and t ξ Im z = 0, with (4.A.3) we find t ξ Re  = 0. On the other hand, if t ξ Re z = 0 and t ξ Im z = 0, taking η ∈ Rn such that t η Re z = 0 and t η Im z = 0, we have t (η + ξ) Re z = 0 and t (η + ξ) Im z = 0. This gives t η Re  = 0 and t (ξ + η) Re  = 0 as previously seen, which implies t ξ Re  = 0. Arguing similarly we find Im  = λ2 Im z for some λ2 ∈ R. From (4.A.3) we then obtain (λ1 − λ2 )(t ξ Re z)(t ξ Im z) = 0. As ξ can be chosen arbitrary in Rn we find λ1 = λ2 . This implies that  = 0 is such that  = λz with λ ∈ R, yielding in turn t() = 0 and t(z) colinear in R2n . The hyperplane  orthogonal to t() in R2n is then precisely Q. The claim is proven. 4.A.2. Sub-ellipticity for First-Order Operators with Linear Symbols. In the case of a first-order operator with a linear symbol in x and ξ, i.e., L(x, ξ) = (a, x) + (b, ξ), with a, b ∈ Cd , the sub-ellipticity condition holds if and only of {Re L, Im L} > 0 since this Poisson bracket is in fact a constant function and L is not elliptic. The following lemma, which can be found in [174, Lemma 28.2.2], shows that a Carleman type inequality is equivalent to this sub-ellipticity condition and gives a precise knowledge of the constant in the estimate.

174

4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

Lemma 4.34. Let L(x, ξ) = (a, x) + (b, ξ) where a and b are in Cd , then the inequality (4.A.4)

κu2L2 (Rd ) ≤ L(x, D)u2L2 (Rd ) ,

u ∈ Cc∞ (Rd ),

is equivalent to (4.A.5)

¯ L}/i, 0). κ ≤ max({L,

Proof. We can replace Cc∞ (Rd ) in (4.A.4) by S (Rd ); indeed, for v ∈ S (Rd ) taking uδ (x) = χ(δx)v(x), where χ ∈ Cc∞ (Rd ) such that χ(x) = 1 for x in a neighborhood of 0, letting δ go to 0, we obtain the same estimate for v. Remark that if L(x, ξ) = L1 (x, ξ) + iL2 (x, ξ), where Lj (x, ξ) are real ¯ L}/i = 2{L1 , L2 } and (4.A.5) takes the form valued then {L, (4.A.6)

κ ≤ 2 max({L1 , L2 }, 0).

Note in particular that {L1 (x, ξ), L2 (x, ξ)} = i[L1 (x, D), L2 (x, D)] is a constant function. Now, we claim that it suffices to only consider L in one of the following form: Case 1: L(x, ξ) = α1 x1 + α2 x2 , α1 and α2 ∈ C; Case 2: L(x, ξ) = x1 + iβξ1 , β ∈ R. If the claim is true, in the first case, taking uδ (x) = χ(x/δ), where χ ∈ Cc∞ does not vanish identically, and letting δ go to 0, (4.A.4) implies κ ≤ 0. Conversely (4.A.4) clearly holds if κ ≤ 0. In the second case, we have {L1 , L2 } = −β. The case β = 0 is treated above. 2 If β > 0, taking u(x) = e−x1 /(2β) f (x ), with f ∈ Cc∞ (Rd−1 ) and x = (x2 , . . . , xd ), we have L(x, D)u(x) = 0, and (4.A.4) gives κ ≤ 0. If β < 0, integrating by parts, we remark that (x1 + iβDx1 )u2L2 = x1 u2L2 + βDx1 u2L2 + i([x1 , βDx1 ]u, u)L2 = x1 u2L2 + βDx1 u2L2 − βu2L2 . Then (x1 + iβDx1 )u2L2 − (x1 − iβDx1 )u2L2 = −2βu2L2 and (4.A.4) is equivalent to (κ + 2β)u2L2 ≤ (x1 − iβDx1 )u2L2 . As −β > 0, we have seen above that this estimate is equivalent to κ+2β ≤ 0 which is condition (4.A.6). Now we prove the claim. We start by listing simple transformations that preserve estimate (4.A.4), and yield linear symplectomorphisms in (x, ξ) implying that we remain within the class of operators L given in the statement of the lemma. As symplectomorphisms preserve the Poisson bracket by Proposition 9.6, the value of {L1 , L2 } remains unchanged under such transformation.

4.A. SOME TECHNICAL RESULTS

175

Observe that a linear change in the x variables, which preserves estimate (4.A.4), acts on the principal symbol of a differential operator by means of the associated symplectomorphism (See Sect. 9.1.2 and Proposition 9.3). Note also that a partial Fourier transformation in one variable xj , which preserves estimate (4.A.4) by the Plancherel equality, also induces a linear symplectomorphism on the symbol L(x, ξ). Indeed by such a Fourier transformation the multiplication by xj is transformed into −Dξj and Dxj is transformed in ξj . As a result aj xj +bj Dxj is transformed into −aj Dξj +bj ξj . If we want to have the same notation for the transformed operator as for the original operator we rename the variable ξj by xj . Hence we have transformed aj xj + bj Dxj into bj xj − aj Dxj . In T ∗ Rd we thus performed the following transform: (xj , ξj ) → (yj , ηj ) = (ξj , −xj ). Since dηj ∧ dyj = d(−xj ) ∧ dξj = dξj ∧ dxj , this transformation preserves the symplectic form σ (see (9.2.1) for a definition of σ). We can also replace u in (4.A.4) by eiQ(x) v(x) where Q is a real quadratic form. This amounts to conjugating L(x, D) by eiQ(x) . Then L(x, D) is transformed into L(x, D) + HL Q. As HL Q is linear in x variables, with this transformation we remain within the considered class of operators. Note that HL Q = HL1 Q + iHL2 Q. An elementary computation gives {L1 + HL1 Q, L2 + HL2 Q} = {L1 , L2 } + HL1 HL2 Q − HL2 HL1 Q + {HL1 Q, HL2 Q} = {L1 , L2 }, as HL1 and HL2 , constant coefficient vector fields, commute1 and HLj Q only depend on the x variables. Now we explain how the forms given in Case 1 and Case 2 can be obtained through the three simple transformations we have introduced. We first modify L1 . Let L1 (x, ξ) = (a, x) + (b, ξ) with a and b in Rd . If b = 0, by linear change of variables we can write L1 (x, ξ) = x1 if a = 0 otherwise L1 = 0. If we now assume b = 0, by a linear change of variables in x, the constant vector field (b, Dx ) can be written Dx1 . In such coordinates L1 (x, ξ) takes the form (a, x) + ξ1 , with the value of a changed. Taking

Q(x) = a1 x21 /2 + dj=2 aj xj x1 and u = e−iQ v, L1 (x, D)u = e−iQ Dx1 v. The operator L1 now takes the form L1 = Dx1 . By Fourier transformation in the x1 variable, we can obtain the form L1 (x, ξ) = x1 . Next, we modify L2 without altering the form of L1 . Let L2 (x, ξ) = (α, x)+(β, ξ) where α and β are in Rd . By the above method, only acting on the variables x = (x2 , . . . , xd ), if (α2 , . . . , αd , β2 , . . . , βd ) = (0, . . . , 0) we can achieve the form L2 (x, ξ) = α1 x1 + β1 ξ1 + x2 , otherwise we have L2 (x, ξ) = α1 x1 + β1 ξ1 . In any case L2 (x, ξ) = α1 x1 + β1 ξ1 + γx2 . If β1 = 0, then L 1The general formula is [H , H ] = H L1 L2 {L1 ,L2 } . H{L1 ,L2 } = 0.

Here {L1 , L2 } is constant then

176

4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

has the form given in Case 1. If β1 = 0, let Q = α1 x21 /(2β1 ) + γx1 x2 /β1 , we have L2 (x, Dx1 )(e−iQ v) = e−iQ β1 Dx1 v. This last transformation does not  affect L1 . We have thus reached the form of L given in Case 2. 4.A.3. A Particular Class of Limiting Weights. Here, we prove Proposition 4.17 and Lemma 4.18 that provide limiting weight with a Poisson bracket uniformly vanishing. The proofs we write exploit the invariance of a Poisson bracket by a symplectomorphism and are inspired by the proof of Lemma 3.4. Proof of Lemma 4.18. The map Θ : (x, ξ) → (x, ξ + dψ(x)) is a symplectomorphism by Proposition 9.4. Consequently, Θ∗ Hp2 ψ = Θ∗ {p, Hp ψ} = {Θ∗ p, Θ∗ Hp ψ}, by Proposition 9.6. We have Θ∗ p(x, ξ) = p(x, ξ + dψ(x)) = p(x, ξ) + p(x, dψ(x)) + Hp ψ(x, ξ), and, Θ∗ Hp ψ(x, ξ) = Hp ψ(x, ξ + dψ(x)) = Hp ψ(x, ξ) + Hp ψ(x, dψ(x)) = Hp ψ(x, ξ) + 2p(x, dψ(x)), as Hp ψ(x, ξ) is linear in ξ and using the Euler identity for homogeneous functions. We obtain Hp2 ψ(x, ξ + dψ(x)) = Hp2 ψ(x, ξ) + 2{p, p(x, dψ(x))}(x, ξ) + {Hp ψ, p(x, dψ(x))}(x, ξ). The first term is quadratic in ξ; the second is linear; the third term is independent of ξ. We thus obtain Hp2 ψ(x, −ξ + dψ(x)) = Hp2 ψ(x, ξ) − 2{p, p(x, dψ(x))}(x, ξ) + {Hp ψ, p(x, dψ(x))}(x, ξ), which gives  1 2 Hp ψ(x, ξ + dψ(x)) − Hp2 ψ(x, −ξ + dψ(x)) 4 = {p, p(x, dψ(x))}(x, ξ),

b(x, ξ, dψ(x)) =

which concludes the proof of Lemma 4.18.



Proof of Proposition 4.17. First, assume that (1) holds and let us prove that (2) follows. We set C0 = p(x, dϕ(x)). Then (4.A.7)

pϕ (x, ξ, τ ) = p(x, ξ) − C0 τ 2 + 2iτ p˜(x, ξ, dϕ(x)),

with the notation of Sects. 3.1 and 3.2. We recall that p˜(x, ξ, dϕ(x)) = Hp ϕ(x, ξ)/2. This yields (4.A.8)

1 {pϕ , pϕ }(x, ξ, τ ) = {Re pϕ , Im pϕ }(x, ξ, τ ) = τ {p, Hp ϕ}(x, ξ). 2i

4.A. SOME TECHNICAL RESULTS

177

Let t ∈ R. The map θt : (x, ξ) → (x, ξ + tdϕ(x)) is a symplectomorphism by Proposition 9.4. Consequently, θt∗ {p, Hp ϕ} = {θt∗ p, θt∗ Hp ϕ} by Proposition 9.6. The argument is similar to that used in the first part of the proof of Lemma 4.18. One has p(x, ξ, dϕ(x)) θt∗ p(x, ξ) = p(x, ξ + tdϕ(x)) = p(x, ξ) + C0 t2 + 2t˜ = p(x, ξ) + C0 t2 + tHp ϕ(x, ξ), and θt∗ Hp ϕ(x, ξ) = Hp ϕ(x, ξ + tdϕ(x)) = Hp ϕ(x, ξ) + tHp ϕ(x, dϕ(x)) = Hp ϕ(x, ξ) + 2tp(x, dϕ(x)) = Hp ϕ(x, ξ) + 2tC0 as Hp ϕ(x, ξ) is linear in ξ and using the Euler identity for homogeneous functions. One finds {θt∗ p, θt∗ Hp ϕ} = {p, Hp ϕ}. This implies (4.A.9) {p, Hp ϕ}(x, ξ) = θt∗ {p, Hp ϕ}(x, ξ) = {p, Hp ϕ}(x, ξ + tdϕ(x)), x ∈ V , ξ ∈ Rd , t ∈ R. If x ∈ V and ξ ∈ Rd , we write ξ = η + tdϕ(x) with t ∈ R and η ∈ Rd such that p˜(x, η, dϕ(x)) = 0, that is η and dϕ(x) are orthogonal for the inner product associated with pij (x) i,j . With (4.A.9) we have {p, Hp ϕ}(x, ξ) = {p, Hp ϕ}(x, η). It thus remains to prove that {p, Hp ϕ}(x, η) = 0 for any η ∈ Rd such that p˜(x, η, dϕ(x)) = Hp ϕ(x, η)/2 = 0. Consider such a η ∈ Rd . If η = 0 the result holds. Assume η = 0. If one chooses τ > 0 such that p(x, η) = C0 τ 2 we obtain pϕ (x, η, τ ) = 0 by (4.A.7). Since ϕ is a limiting weight, by (4.A.8) this gives {p, Hp ϕ}(x, η) =

1 {pϕ , pϕ }(x, η, τ ) = 0. 2τ i

We have thus proven that (1) implies (2). Second, assume that (2) holds and let us prove that (1) follows. As Hp2 ϕ(x, ξ) is quadratic in ξ we denote by a(x, ξ, η) the associated bilinear form and by Lemma 4.18 we have (4.A.10)

a(x, ξ, dϕ(x)) = {p, p(x, dϕ(x))}(x, ξ).

From Lemma 3.4 we have 1 {pϕ , pϕ }(x, ξ, τ ) = τ Hp2 ϕ(x, ξ) + τ 3 Hp2 ϕ(x, dϕ). 2i The first term is homogeneous of degree two in ξ and the second term is independent of ξ. Since here the Poisson bracket {pϕ , pϕ } vanishes uniformly for x ∈ V , ξ ∈ Rd , and τ > 0, we find (4.A.11)

Hp2 ϕ(x, ξ) = {p, {p, ϕ}}(x, ξ) = 0,

x ∈ V , ξ ∈ Rd .

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4. OPTIMALITY ASPECTS OF CARLEMAN ESTIMATES

From (4.A.11) we conclude that a(x, ξ, η) = 0 for all ξ, η ∈ Rd if x ∈ V , which by (4.A.10) implies   x ∈ V , ξ ∈ Rd . Hp p(x, dϕ(x)) (x, ξ) = {p, p(x, dϕ(x))}(x, ξ) = 0, As a result the function x → p(x, dϕ(x)) is constant on the integral curves of the Hamiltonian vector field Hp , that is, a curve s → (x(s), ξ(s)) such that    d (4.A.12) x(s), ξ(s) = Hp x(s), ξ(s) , ds that is,  

d xk (s) = ∂ξk p x(s), ξ(s) = 2pkj (x(s))ξj (s), ds 1≤i≤d  

d ξk (s) = −∂xk p x(s), ξ(s) = − ∂xk pij (x(s))ξi (s)ξj (s), ds 1≤i,j≤d for k = 1, . . . , d. With the Cauchy–Lipschitz theorem, for (x0 , ξ 0 ) ∈ V ×Rd \ {0} we can define a unique maximal solution s → (x(s), ξ(s)) to (4.A.12) with initial data x(0) = x0 and ξ(0) = ξ 0 . We denote this solution Φs (x0 , ξ 0 ). Proposition 4.35. Let x0 ∈ V . There exist an open neighborhood W of x0 , C > 0, and δ > 0 such that Φs (x, ξ) is well defined for s ∈ (−δ, δ), x ∈ W and ξ ∈ Rd such that |w| ≤ C. If x0 ∈ V we denote by x(s, ξ 0 ) the first component of Φs (x0 , ξ 0 ). We have the following lemma whose proof is given below. Lemma 4.36. Let x0 ∈ V . There exist s0 > 0 and R0 > 0 such that the image of B(0, R0 ) by the map ξ 0 → x(s0 , ξ 0 ) is open. Let x0 ∈ V . Let s0 and R0 be as given by Lemma 4.36. Observe that the image of B(0, R0 ) by the map ξ 0 → x(s0 , ξ 0 ) yields a neighborhood W 0 of x0 since x0 = x(s0 , 0). Moreover, for all x ∈ W 0 , there exists ξ ∈ Rd such that (x, ξ) = Φs0 (x0 , ξ 0 ) for some ξ 0 ∈ B(0, R0 ). One concludes that p(x, dϕ(x)) = p(x0 , dϕ(x0 )). Hence, p(x, dϕ(x)) is locally constant. Since V is connected, we conclude that p(x, dϕ(x)) is constant on V .  Proof of Lemma 4.36. We set Ψ(s, ξ 0 ) = Φs (x0 , ξ 0 ). The first component of Ψ(s, ξ 0 ) is x(s, ξ 0 ). We denote by ξ(s, ξ 0 ) the second component. One has      d x ∇ξ p  (s, ξ 0 ) = (x, ξ)(s, ξ 0 ) . −∇x p ds ξ Setting X(s, ξ 0 ) = ∇ξ0 x(s, ξ 0 ) and Ξ(s, ξ 0 ) = ∇ξ0 ξ(s, ξ 0 ) one finds         d X X X 0 0 0 0 (s, ξ ), , (s, ξ ) = As,ξ0 (0, ξ ) = Ξ Ξ Idd ds Ξ

4.A. SOME TECHNICAL RESULTS

with

 As,ξ0 =

179

  ∇2ξx p ∇2ξξ p  (x, ξ)(s, ξ 0 ) . 2 2 −∇xx p −∇xξ p

We thus have   d X(0, ξ 0 ) = ∇2ξξ p(x0 , ξ 0 ) = P(x0 ), P(x0 ) = pij (x0 ) 1≤i,j≤d , ds that is of rank d from the ellipticity of p. We then obtain   X(s, ξ 0 ) = s P(x0 ) + O(s) . This gives X(s0 , 0) of rank d for s0 > 0 chosen sufficiently small. The conclusion follows by the inverse mapping theorem. 

Part 2

Applications of Carleman Estimates

CHAPTER 5

Unique Continuation Contents 5.1. 5.2. 5.3. 5.3.1. 5.3.2. 5.4. 5.5.

Introduction 183 Local and Global Unique Continuation 184 Quantification of Unique Continuation 186 Quantified Unique Continuation Away from a Boundary 187 Quantified Unique Continuation Up to a Boundary 193 Unique Continuation Initiated at the Boundary 198 Unique Continuation Without Any Prescribed Boundary Condition 202 5.6. Notes 210 Appendix 213 5.A. A Hardy Inequality 213

5.1. Introduction One says that the unique continuation property holds for an operator Q = Q(x, Dx ) if, for a function u satisfying Qu = 0 in some open set Ω, having u ≡ 0 in a connected open set ω  Ω implies that u vanishes everywhere in Ω. Here, we shall consider a second-order elliptic operator P with principal part given by

Di (pij (x)Dj ), with pij (x)ξi ξj ≥ C|ξ|2 , (5.1.1) P0 = 1≤i,j≤d

1≤i,j≤d

with (pij ) smooth with all derivatives bounded in Ω and such that pij = pji , 1 ≤ i, j ≤ d. The elliptic operator under consideration is then

i b (x)Di + c(x), P = P0 + 1≤i≤d

where

bi , c

∈

L∞ (Rd ),

1 ≤ i ≤ 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 I, PNLDE Subseries in Control 97, https://doi.org/10.1007/978-3-030-88674-5 5

183

184

5. UNIQUE CONTINUATION

First, we present local and global versions of the unique continuation property for such operators. Proofs are based on the local Carleman estimate away from boundaries proven in Chap. 3. Second, we provide a quantification of the unique continuation property by means of an interpolation inequality derived from the same Carleman estimate. This quantification can be pushed all the way to the boundary using Carleman estimates in neighborhoods of points of ∂Ω. We also show that the unique continuation property can also be initiated from the boundary. Here, at the boundary, we have chosen to focus on the case of homogeneous Dirichlet boundary condition, first for simplicity and second because the Carleman estimates we have at hand from Chap. 3 have been obtained in this case. For the quantification of unique continuation in the case of ˇ more general boundary conditions, namely Lopatinski˘ı-Sapiro conditions, the reader is referred to Chapter 9 in Volume 2. Finally, we also treat the important case of the quantification of unique continuation in the case no boundary condition is prescribed. 5.2. Local and Global Unique Continuation Let Ω be a bounded open set in Rd . In a neighborhood V of a point x(0) ∈ Ω, we take a function φ such that dφ = 0 in V . We aim to show the following (local) unique continuation result for the operator P . 2 (Ω) satisfying Proposition 5.1. Let u ∈ Hloc   (5.2.1) |P u(x)| ≤ C |u(x)| + |Du(x)| , a.e. in V,

for some C > 0 and u = 0 in {x ∈ V ; φ(x) ≥ φ(x(0) )}. Then u vanishes in a neighborhood of x(0) . Proof. We observe that the form of the operator and (5.2.1) yield   (5.2.2) |P0 u(x)| ≤ C |u(x)| + |Du(x)| , a.e. in V. We may thus assume that P = P0 without any loss of generality. In particular the coefficients of P are then smooth, with all derivatives bounded in Ω. We pick a function ψ whose gradient does not vanish near V (or possibly in a smaller neighborhood of x(0) ) and that satisfies (∇φ(x(0) ), ∇ψ(x(0) )) > 0 and is such that φ − ψ reaches a strict local minimum at x(0) as one moves along the level set {x ∈ V ; ψ(x) = ψ(x(0) )}. For instance, we may choose ψ(x) = φ(x) − c|x − x(0) |2 . We then set ϕ(x) = eγψ(x) and choose γ > 0 according to Lemma 3.5. In the neighborhood V (or possibly in a smaller neighborhood of x(0) ) the geometrical situation we have just described is illustrated in Fig. 5.1. We call W the region {x ∈ V ; φ(x) ≥ φ(x(0) )} (region beneath {φ(x) = (0) φ(x )} in Fig. 5.1). We choose V  and V  neighborhoods of x(0) such that V   V   V and we pick a function χ ∈ Cc∞ (V  ) such that χ ≡ 1 in V  .

5.2. LOCAL AND GLOBAL UNIQUE CONTINUATION

V

185

S

x(0) ∇φ

∇ϕ

B0

φ(x) = φ(x(0) )

W V V

ϕ(x) = ϕ(x(0) ) − ε ϕ(x) = ϕ(x(0) )

Figure 5.1. Local geometry for the unique continuation problem. The colored region S contains the support of [P, χ]u We set v = χu and then v ∈ H02 (V ). Observe that the Carleman estimate of Theorem 3.11 applies to v by Remark 3.12. We have P v = P ( χu) = χ P u + [P, χ]u, where the commutator is a first-order differential operator. We thus obtain, with (5.2.1), τ 3 eτ ϕ χu2L2 (Ω) + τ eτ ϕ D(χu)2L2 (Ω)  eτ ϕ χu2L2 (Ω) + eτ ϕ χDu2L2 (Ω) + eτ ϕ [P, χ]u2L2 (Ω) , for τ ≥ τ∗ . We set S := V  \ (V  ∪ W ) (see the colored region in Fig. 5.1). We have supp([P, χ]u) ⊂ S and supp([D, χ]u) ⊂ S as they are confined in the region where χ varies and u does not vanish. We thus obtain τ 3 eτ ϕ χu2L2 (Ω) + τ eτ ϕ D(χu)2L2 (Ω)  eτ ϕ χu2L2 (Ω) + eτ ϕ D(χu)2L2 (Ω) + eτ ϕ u2L2 (S) + eτ ϕ Du2L2 (S) . Choosing τ sufficiently large, say τ ≥ τ1 , we may ignore the first two terms on the r.h.s. of the previous estimate. We then write eτ ϕ u2L2 (V  ) + τ eτ ϕ Du2L2 (V  )  τ 3 eτ ϕ χu2L2 (Ω) + τ eτ ϕ D(χu)2L2 (Ω)  eτ ϕ u2L2 (S) + eτ ϕ Du2L2 (S) , as χ ≡ 1 in V  . For all ε ∈ R, we set Vε = {x ∈ V ; ϕ(x) ≤ ϕ(x(0) ) − ε}. There exists ε0 > 0 such that S  Vε0 and we choose a ball B0 with center x(0) such that B0 ⊂ V  \ Vε0 . See Fig. 5.1. We then obtain eτ inf B0 ϕ uH 1 (B0 ) ≤ Ceτ supS ϕ uH 1 (S) ,

τ ≥ τ1 .

Since inf B0 ϕ > supS ϕ, letting τ go to ∞, we obtain u ≡ 0 in B0 . With a connectedness argument we can prove the following theorem.



186

5. UNIQUE CONTINUATION

r2 = R/2 R/4 r1

x(0)

x(1)

F

Figure 5.2. Geometry for the proof of Theorem 5.2 Theorem 5.2. Let Ω be a connected open set in Rd and let ω ⊂ Ω, with ω = ∅. If u ∈ H 2 (Ω) satisfies   |P u(x)| ≤ C |u(x)| + |Du(x)| , a.e. in Ω, for some C > 0 and u(x) = 0 in ω, then u vanishes in Ω. Proof. We set F = supp(u) that is a closed set in Ω. Since F cannot be equal to Ω, let us show that F is open. It will then follow that F = ∅. Assume that fr(F ) = F \ F ◦ is not empty and choose x(1) ∈ fr(F ). We set A := Ω \ F . There exists R > 0 such that B(x(1) , R)  Ω and x(0) ∈ B(x(1) , R/4) such that x(0) ∈ A. Since A is open, there exists 0 < r1 < R/4 such that B(x(0) , r1 ) ⊂ A. For r2 = R/2 we have thus obtained r1 < r2 such that B(x(0) , r1 ) ⊂ A,

B(x(0) , r2 )  Ω,

and x(1) ∈ B(x(0) , r2 ).

The Geometry is illustrated in Fig. 5.2. We set Bt = B(x(0) , (1 − t)r1 + tr2 ) for 0 ≤ t ≤ 1. Proposition 5.1 shows that if u vanishes in Bt , with 0 ≤ t ≤ 1, then there exists ε > 0 such that u vanishes in Bt+ε . Since u vanishes in B0 , we thus find that u vanishes in B1 , and in particular in a neighborhood  of x(0) . Hence, x(0) cannot be in fr(F ); the set F is open. 5.3. Quantification of Unique Continuation The Carleman estimate of Theorem 3.11 was central in the proof of the unique continuation result above. In the case of prescribed boundary conditions, say of homogeneous Dirichlet type, a global Carleman estimate, as in Theorem 3.34 gives a quantification of the unique continuation property.

5.3. QUANTIFICATION OF UNIQUE CONTINUATION

187

However, the construction of a global weight function is needed. Moreover, the global aspect of this estimation may be cumbersome. We provide here a local quantification of the unique continuation property that is derived directly from the local Carleman estimates of Sects. 3.3 and 3.4. First, we shall remain away from any boundary, and we shall see that, naturally, this quantified version the unique continuation property does not depend on the boundary properties of the functions. We shall then only rely on the Carleman estimate of Theorem 3.11. Second, we shall derive a quantified version of the unique continuation property that reaches neighborhoods of points at the boundary. We shall prescribe a boundary condition, here of Dirichlet type for simplicity. We shall then also use the Carleman estimate of Theorem 3.29. In each case, the weight functions that are used for the Carleman estimates are quite basic, only local, and do not involve any geometrical argument as in the construction of a global weight function for instance in Proposition 3.31. 5.3.1. Quantified Unique Continuation Away from a Boundary. The unique continuation property can be locally quantified as described in the estimate of the following lemma. Because of its form, this estimate is often referred to as an interpolation inequality. Lemma 5.3 (Local Interpolation Inequality). Let Ω be an open set in Rd and let x(0) ∈ Ω and r > 0 be such that B(x(0) , 4r) ⊂ Ω. Let C0 > 0. There exist C > 0 and δ ∈ (0, 1) such that  δ (5.3.1) uH 1 (B(x(0) ,3r)) ≤ Cu1−δ H 1 (Ω) f L2 (Ω) + uH 1 (B(x(0) ,r)) , for u ∈ H 2 (Ω) satisfying P u = f + g,

with f ∈ L2 (Ω),

  |g(x)| ≤ C0 |u(x)| + |Du(x)| a.e. in Ω.

Proof. With the same argument as in the beginning of the proof of Proposition 5.1, we may simply assume P = P0 . We apply the local Carle(0) 2 man estimate of Theorem 3.11 with the weight function ϕ(x) = e−γ|x−x | . By Lemma 3.5, ϕ satisfies the sub-ellipticity property of Definition 3.2 in B(x(0) , 4r) \ B(x(0) , r/2), if the parameter γ is chosen sufficiently large. Let now χ ∈ Cc∞ (B(x(0) , 4r)) be such that  1 if 3r/4 < |x − x(0) | < 7r/2, χ(x) = 0 if |x − x(0) | < r/2. We apply the Carleman estimate (3.3.1) to v = χu observing that v is supported in B(x(0) , 4r) \ B(x(0) , r/2). We have (5.3.2) τ 3/2 eτ ϕ vL2 (Ω) + τ 1/2 eτ ϕ DvL2 (Ω)  eτ ϕ P vL2 (Ω) ,

τ ≥ τ∗ ≥ 1.

188

5. UNIQUE CONTINUATION

C3

ϕ C2 C1

χ x(0)

r

2r

3r

4r

|x − x(0) |

B3

B2

B1

Figure 5.3. Local geometry for the proof of a local interpolation inequality of Lemma 5.3, with the support property of the cutoff function χ and the behavior of the weight function ϕ. The cutoff function χ vanishes outside the colored regions; the lighter color indicates a region where χ = 1; the darker color indicates the regions where χ varies

We have P v = χP u + [P, χ]u and [P, χ] is a differential operator of order 1 with bounded coefficients supported in B1 ∪ B3 with B1 = {x; 7r/2 ≤ |x − x(0) | ≤ 4r},

B3 = {x; r/2 ≤ |x − x(0) | ≤ 3r/4}.

These regions are illustrated in Fig. 5.3. We write eτ ϕ P vL2 (Ω) ≤ eτ ϕ χP uL2 (Ω) + eτ ϕ [P, χ]uL2 (B1 ∪B3 ) ≤ eτ ϕ f L2 (B(x(0) ,4r)) + eτ ϕ χgL2 (Ω) + eτ ϕ [P, χ]uL2 (B1 ∪B3 ) . We have, by the assumption made on g, eτ ϕ χgL2 (Ω)  eτ ϕ χuL2 (Ω) + eτ ϕ χDuL2 (Ω)  eτ ϕ vL2 (Ω)) + eτ ϕ DvL2 (Ω) + eτ ϕ [D, χ]uL2 (B1 ∪B3 ) .

5.3. QUANTIFICATION OF UNIQUE CONTINUATION

189

From (5.3.2), for τ ≥ 1 chosen sufficiently large we obtain (5.3.3) τ 3/2 eτ ϕ vL2 (Ω) + τ 1/2 eτ ϕ DvL2 (Ω)  eτ ϕ f L2 (B(x(0) ,4r)) + eτ ϕ [P, χ]uL2 (B1 ∪B3 ) + eτ ϕ [D, χ]uL2 (B1 ∪B3 ) . Since ϕ decreases as |x − x(0) | increases, we find (5.3.4)

RHS (5.3.3)  eτ C3 f L2 (Ω) + eτ C3 uH 1 (B3 ) + eτ C1 uH 1 (B1 )    eτ C3 f L2 (Ω) + uH 1 (B(x(0) ,r)) + eτ C1 uH 1 (Ω) , 2

2

where C1 = e−γ(7r/2) and C3 = e−γ(r/2) . As we have χ ≡ 1 on B2 = B(x(0) , 3r) \ B(x(0) , r) we have eτ C2 uH 1 (B2 ) ≤ LHS (5.3.3),

(5.3.5) 2

where C2 = e−γ(3r) for τ ≥ 1. Remark that C1 < C2 < C3 . Inequalities (5.3.3), (5.3.4), and (5.3.5) give   eτ C2 uH 1 (B2 )  eτ C1 uH 1 (Ω) + eτ C3 f L2 (Ω) + uH 1 (B(x(0) ,r)) . Observe that we may in fact write

  eτ C2 uH 1 (B(x(0) ,3r))  eτ C1 uH 1 (Ω) + eτ C3 f L2 (Ω) + uH 1 (B(x(0) ,r)) ,

yielding

 uH 1 (B(x(0) ,3r))  e−τ (C2 −C1 ) uH 1 (Ω) + eτ (C3 −C2 ) f L2 (Ω)  (5.3.6) + uH 1 (B(x(0) ,r)) .

We can then optimize this last estimate by applying Lemma 5.4 below, which yields the result.  Lemma 5.4. Let A ≥ 0, B1 ≥ 0 and B2 ≥ 0. We assume that A ≤ B1 and that there exist τ∗ > 0, μ > 0 and ν > 0 such that (5.3.7)

A ≤ e−ντ B1 + eμτ B2 ,

for τ ≥ τ∗ .

Then, we have A ≤ KB11−δ B2δ ,    δ where K = max δ −δ (1 − δ)−(1−δ) , μν eντ∗ and δ = ν/(ν + μ) ∈ (0, 1). Proof. The result is obvious in the case B1 = 0, as then A = 0, and also in the case B2 = 0, by letting τ go to +∞ in (5.3.7), which also yields A = 0. We can thus assume B1 = 0 and B  −ντ  2 = 0 in what follows. μτ B1 + e B2 . Setting f (τ ) = e−ντ B1 + eμτ B2 , We have A ≤ inf τ ≥τ∗ e  νB1 1/(μ+ν) leading to the value τ0 > 0 that minimizes f is such that eτ0 = μB 2

inf τ >0 f (τ ) = δ −δ (1 − δ)−(1−δ) B11−δ B2δ with δ = ν/(ν + μ) ∈ (0, 1). We then have two cases. Case τ0 ≥ τ∗ : with (5.3.7) we obtain A ≤ δ −δ (1 − δ)−(1−δ) B11−δ B2δ .

190

5. UNIQUE CONTINUATION

Case τ0 < τ∗ : then we have  νB 1/(μ+ν)  μB δ 1 2 ≤ eτ∗ , that is, B1δ ≤ eντ∗ . eτ 0 = μB2 ν  δ We then find A ≤ B1 = B11−δ B1δ ≤ μν eντ∗ B11−δ B2δ . The two found estimations yield the result.



Remark 5.5. In the proof of Lemma 5.4, instead of optimizing the function f (τ ) = e−ντ B1 + eμτ B2 for τ > 0, we can simply pick the value τ = τ1 so as to have e−ντ1 B1 = eμτ1 B2 , that is, eτ1 = (B1 /B2 )1/(μ+ν) . In μ/(μ+ν) ν/(μ+ν) the case τ1 ≥ τ∗ with (5.3.7) we obtain A ≤ 2B1 B2 . In the case 1/(μ+ν) τ δ ντ δ ∗ ∗ ≤ e , that is, B1 ≤ e B2 . We then find τ1 < τ∗ then, (B1 /B2 ) A ≤ B1 = B11−δ B1δ ≤ eντ∗ B11−δ B2δ . We thus obtain the result of the lemma with K replaced by K  = max(2, eντ∗ ). One thus sees that this simple choice for the value of τ yields qualitatively the same result as that given by a more precise optimization procedure. Note also that the factor 2 in the value of K  is not coincidental since supδ∈(0,1) δ −δ (1 − δ)−(1−δ) = 2. Estimations as given in Lemma 5.3 can be propagated. One often says that the resulting inequality quantifies the propagation of “smallness”. Here, we remain away from any boundary. Theorem 5.6 (Propagation of “smallness” Away from Boundaries—Interpolation Inequality). Let Ω be a connected open set in Rd and let ω and U be two open subsets of Ω with U  Ω. Let C0 > 0. There exist C > 0 and δ ∈ (0, 1) such that  δ (5.3.8) uH 1 (U ) ≤ Cu1−δ H 1 (Ω) f L2 (Ω) + uL2 (ω) , for u ∈ H 2 (Ω) satisfying P u = f + g,

with f ∈ L2 (Ω),

  |g(x)| ≤ C0 |u(x)| + |Du(x)| a.e. in Ω.

Proof. Because of the compactness of U , it suffices to prove (5.3.8) with B(y, r) in place of U where y ∈ U and r > 0 satisfying 0 < r ≤ R = dist(U, ∂Ω)/2. Let x(0) be in ω and r0 > 0 such that B(x(0) , r0 )  ω. As Ω is open and connected there exists a path Γ ⊂ Ω from x(0) = Γ(0) to y = Γ(1). Set r1 = dist(Γ, ∂Ω). We have r1 > 0 by compactness. Setting now r = inf(R, r0 , r1 /4), we define a sequence (x(j) )j , for j ≥ 0, by x(j) = Γ(tj ) where t0 = 0 and  inf Aj if Aj = ∅, tj = Aj = {σ ∈ (tj−1 , 1]; Γ(σ) ∈ B(x(j−1) , r)}. 1 if Aj = ∅, The sequence (x(j) )j is finite by a compactness argument. The construction of the sequence is illustrated in Fig. 5.4.

5.3. QUANTIFICATION OF UNIQUE CONTINUATION

191

x(N ) x(N −1) x(N −2) =y

r1 = dist(Γ, ∂Ω) ≥ 4r

x(j+1) x(j) x(j−1)

3r

ω

∂Ω

Γ

x(2)

x

r x

(1)

(0)

Figure 5.4. Construction of the sequence (x(j) )j , j ∈ J, along the path Γ Let (x(0) , · · · , x(N ) ) be such a sequence with x(N ) = y. Note that we have B(x(j+1) , r) ⊂ B(x(j) , 3r)  Ω for j = 0, · · · , N − 1, because of the choice we made for r above. By Lemma 5.3 there exist C > 0 and δ ∈ (0, 1) such that (5.3.9)

uH 1 (B(x(j+1) ,r)) ≤ uH 1 (B(x(j) ,3r))  δ ≤ Cu1−δ + u f  (j) 2 1 1 L (Ω) H (B(x ,r)) , H (Ω)

for j = 0, . . . , N − 1. We may assume that f L2 (Ω) ≤ uH 1 (Ω) , since otherwise the estimate we wish to prove is obvious. We then have  δ f  + u . f L2 (Ω) + uH 1 (B(x(j+1) ,r))  u1−δ (j) 2 1 1 L (Ω) H (B(x ,r)) H (Ω) By induction on j, we find (5.3.10)

 μ + u , f L2 (Ω) + uH 1 (B(y,r))  u1−μ f  (0) 2 1 1 L (Ω) H (B(x ,r)) H (Ω)

where μ = δ N . From Proposition 5.8 below we obtain uH 1 (B(x(0) ,r))  f L2 (Ω) + uL2 (ω) . This estimate and (5.3.10) give (5.3.8) with U = B(y, r).



192

5. UNIQUE CONTINUATION

Lemma 5.7. Let V be an open subset of Ω and let χ ∈ Cc∞ (V ). There exists C > 0 such that   γχDuL2 (Ω) ≤ C χP0 uL2 (Ω) + γ 2 uL2 (V ) , for γ ≥ 1 and u ∈ H 2 (Ω). Proof. On the one hand, by the Young inequality we have (5.3.11)

|(χP0 u, χu)L2 (Ω) | ≤

1 χP0 u2L2 (Ω) + γ 2 χu2L2 (Ω) 4γ 2

 γ −2 χP0 u2L2 (Ω) + γ 2 u2L2 (V ) . On the other hand, by integration by parts we find Re(χP0 u, χu)L2 (Ω) = Re(P0 u, χ2 u)L2 (Ω)

 ij (p Di u, χ2 Dj u)L2 (Ω) = Re 1≤i,j≤d

 + 2(pij Di u, χ(Dj χ)u)L2 (Ω)

 ij (p χDi u, χDj u)L2 (Ω) = Re 1≤i,j≤d

+ 2(pij χDi u, (Dj χ)u)L2 (Ω)



≥ χ|Du|x 2L2 (Ω) − CεχDu2L2 (Ω) − Cε−1 u2L2 (V ) , for ε > 0 by the Young inequality. Here, we have extended the definition of p(x, ξ) = 1≤i,j≤d pij ξi ξj to complex valued co-tangent vectors by setting

pij ζi ζj , x ∈ Ω, ζ ∈ Cd . |ζ|2x = p(x, ζ) = 1≤i,j≤d

From the ellipticity property, we have |ζ|x  |ζ|. This yields, for ε > 0 chosen sufficiently small, χDu2L2 (Ω)  |(χP0 u, χu)L2 (Ω) | + u2L2 (V ) . Combined with (5.3.11) this yields the result.



Proposition 5.8. Let U , V be two open subsets of Ω such that U  V ⊂ Ω. Let C0 > 0. There exists C > 0 such that   uH 1 (U ) ≤ C f L2 (Ω) + uL2 (V ) , if u ∈ H 2 (Ω) and P u = f + g,

with f ∈ L2 (Ω),

  |g(x)| ≤ C0 |u(x)| + |Du(x)| a.e. in Ω.

Proof. If P = P0 and g = 0 then the proof of such an estimate can be carried out using standard pseudo-differential operators adapting the proof given for Proposition 2.35—See Remark 2.55.

5.3. QUANTIFICATION OF UNIQUE CONTINUATION

193

In the general case we have P = P0 + R0 with

i b (x)Di + c(x), R0 = 1≤i≤d

∈ where satisfying bi , c

L∞ (Rd ),

1 ≤ i ≤ d. We may thus write P0 u = f + h with h

  |h(x)| ≤ C1 |u(x)| + |Du(x)| a.e. in Ω.

We choose χ ∈ Cc∞ (V ) such that χ ≡ 1 in a neighborhood of U . Applying the estimate of Lemma 5.7 we have, for γ ≥ 1, γχDuL2 (Ω)  χf L2 (Ω) + χhL2 (Ω) + γ 2 uL2 (V )  χf L2 (Ω) + χuL2 (Ω) + χDuL2 (Ω) + γ 2 uL2 (V ) . For γ ≥ 1 chosen sufficiently large we obtain χDuL2 (Ω)  χf L2 (Ω) + uL2 (V ) yielding the result as χ ≡ 1 on U .



In fact, one has the following corollary. Corollary 5.9. Let U , V be two open subsets of Ω such that U  V ⊂ Ω. Let C0 > 0. There exists C > 0 such that   uH 2 (U ) ≤ C f L2 (Ω) + uL2 (V ) , if u ∈ H 2 (Ω) and P u = f + g,

with f ∈ L2 (Ω),

  |g(x)| ≤ C0 |u(x)| + |Du(x)| a.e. in Ω.

˜ be an open set such that U  U ˜  V . With ProposiProof. Let U tion 2.35 adapted to standard pseudo-differential operators (see Remark 2.55) we have uH 2 (U )  P uL2 (U˜ ) + uL2 (U˜ )  f L2 (U˜ ) + uH 1 (U˜ ) . We conclude with Proposition 5.8.



5.3.2. Quantified Unique Continuation Up to a Boundary. Here, we use a Carleman estimate at the boundary, as those proven in Sect. 3.4 of Chap. 3, to obtain an interpolation inequality in the neighborhood of a point of the boundary. Then, this inequality can be used to derive an estimate as in Theorem 5.6 above by a propagation of smallness argument; however, here smallness is propagated up to the boundary in the case of homogeneous Dirichlet boundary conditions. We define the following open set (5.3.12)

Ωε = {x ∈ Ω; dist(x, ∂Ω) > ε},

that is not empty for ε > 0 chosen sufficiently small.

194

5. UNIQUE CONTINUATION

Lemma 5.10 (Local Interpolation Inequality at a Boundary). Let y ∈ ∂Ω and let V be an open neighborhood of y in Rd such that ∂Ω is smooth in V . Let C0 > 0. There exist W ⊂ V a neighborhood of y in Rd , ε > 0, δ ∈ (0, 1), and C > 0, such that we have  δ (5.3.13) + u , f  uH 1 (W ∩Ω) ≤ Cu1−δ 2 1 1 L (Ω) H (V ∩Ωε ) H (Ω) for u ∈ H 2 (Ω) satisfying u|V ∩∂Ω = 0 and P u = f + g,

with f ∈ L2 (Ω),

  |g(x)| ≤ C0 |u(x)| + |Du(x)| a.e. in Ω.

The proof follows the same ideas as that of Lemma 5.3 applying the local Carleman estimate at the boundary of Theorem 3.29. Proof. With the same argument as in the beginning of the proof of Proposition 5.1, we may simply assume P = P0 . For simplicity, upon reducing the open set V , we use local coordinates x = (x , xd ) in which the boundary V ∩ ∂Ω is given by {xd = 0} and V ∩ Ω is given by {xd > 0}. We assume moreover that y = 0 in such coordinates. We set x(1) = (0, 2r) where r > 0 is chosen such that B(x(1) , 4r)  V . (1) 2 We set ϕ(x) = e−γ|x−x | ; by Lemma 3.5, the weight function ϕ fulfills the sub-ellipticity property of Definition 3.2 in Ω ∩ B(x(1) , 4r) \ B(x(1) , r/2) if the parameter γ > 0 is chosen sufficiently large (and kept fixed in what follows). We have ∂xd ϕ(x) = −2γ(xd − 2r)ϕ(x), yielding, on the boundary {xd = 0}, ∂ν ϕ(x , 0) = −∂xd ϕ(x) = −4rϕ(x , 0) < 0. Let χ0 ∈ Cc∞ (R) be such that χ0 (s) =



1 if |s| < r0 , 0 if 2r0 < |s|,

where r0 < r/4 and will be chosen below. Let also χ1 ∈ Cc∞ (B(x(1) , 3r)) be such that  1 if 3r/4 < |x − x(1) | < r2 , χ1 (x) = 0 if |x − x(1) | < 5r/8 or r2 < |x − x(1) |, where r2 , r2 are such that 2r < r2 < r2 < 3r and will be chosen below. The geometry associated with the functions we have just introduced is illustrated in Fig. 5.5. We set χ(x) = χ0 (xd )χ1 (x). With V 0 = Ω ∩ B(x(1) , 4r) \ B(x(1) , r/2) as a neighborhood of y = 0, we apply Theorem 3.29 to v = χu. As u|xd =0+ = 0 in V we obtain (5.3.14)

τ 3/2 eτ ϕ vL2 (Ω) + τ 1/2 eτ ϕ DvL2 (Ω)  eτ ϕ P vL2 (Ω) .

5.3. QUANTIFICATION OF UNIQUE CONTINUATION

195

x ∈ Rd−1 V V0

3r/4 5r/8 r/2 y=0

r0

2r0

x(1)

xd

2r r2

r2 3r 4r

∂Ω

Figure 5.5. Geometry near the boundary for the application of the local Carleman estimate of Theorem 3.29. The cutoff function χ1 vanishes outside the colored regions; the lighter color indicates a region where χ1 = 1; the darker color indicates the regions where χ1 varies We have P v = χP u + [P, χ]u, yielding eτ ϕ P vL2 (Ω)  eτ ϕ χgL2 (Ω) + eτ ϕ χf L2 (Ω) + eτ ϕ [P, χ]uL2 (Ω) . By the assumption made on g, we have eτ ϕ χgL2 (Ω)  eτ ϕ χuL2 (Ω) + eτ ϕ χDuL2 (Ω)  eτ ϕ vL2 (Ω) + eτ ϕ DvL2 (Ω) + eτ ϕ [D, χ]uL2 (Ω) . The operators [P, χ] and [D, χ] are differential operators of order less than or equal to 1 with bounded coefficients and supported in A0 ∪ A1 with A0 = {x ∈ Rd+ ; r0 ≤ xd ≤ 2r0 and |x − x(1) | ≤ r2 }, A1 = {x ∈ Rd+ ; 0 < xd ≤ 2r0 and r2 ≤ |x − x(1) | ≤ r2 }. These two regions are illustrated in Fig. 5.6 that provides a blown-up version of Fig. 5.5 near y = 0.

196

5. UNIQUE CONTINUATION

x

2r0

0 W

r0

xd

A0

A1 ∂Ω

2r r1 r2 r2

Figure 5.6. Close-up of the geometry near the boundary for the application of the local Carleman estimate of Theorem 3.29. The darker colored region A0 ∪ A1 indicates where the cutoff function χ varies. The lighter colored region represents the open set W of Rd From (5.3.14), for τ ≥ 1 chosen sufficiently large we obtain (5.3.15) τ 3/2 eτ ϕ vL2 (Ω) + τ 1/2 eτ ϕ DvL2 (Ω)  eτ ϕ χf L2 (Ω) + eτ ϕ [P, χ]uL2 (A0 ∪A1 ) + eτ ϕ [D, χ]uL2 (A0 ∪A1 ) . 2

2

On A0 , we have ϕ ≤ e−γ(2r−2r0 ) . On A1 we have ϕ ≤ e−γr2 . We thus obtain (5.3.16) τ 3/2 eτ ϕ vL2 (Ω) + τ 1/2 eτ ϕ DvL2 (Ω)    eτ C3 f L2 (Ω) + uH 1 (V ∩Ωr ) + eτ C1 uH 1 (Ω) , 0

2 e−γr2 ,

2 e−γ(2r−2r0 ) ,

C3 = recalling the definition of the set Ωr0 where C1 = in (5.3.12). We now introduce W = {x ∈ Rd ; xd ∈ (−r0 , r0 )} ∩ {x ∈ Rd ; |x − x(1) | < r1 } with r1 = r + r2 /2; this is an open neighborhood of y = 0 in Rd as 2r < r1 . Since r1 < r2 and xd < r0 , we have χ0 χ1 ≡ 1, and thus u ≡ v in W ∩ Ω. If we restrict the l.h.s. of (5.3.16) to W ∩ Ω, as on this set we have 2 ϕ ≥ e−γr1 , we obtain

5.3. QUANTIFICATION OF UNIQUE CONTINUATION

(5.3.17)

197

eτ C2 uH 1 (W ∩Ω) ≤ τ 3/2 eτ ϕ vL2 (Ω) + τ eτ ϕ DvL2 (Ω) , 2

where C2 = e−γr1 for τ ≥ 1. Observe that we have C1 < C2 < C3 . Then (5.3.16) and (5.3.17) give (5.3.18)   uH 1 (W ∩Ω)  eτ (C3 −C2 ) f L2 (Ω) + uH 1 (V ∩Ωr ) + e−τ (C2 −C1 ) uH 1 (Ω) . 0

Then, by Lemma 5.4 we obtain the sought local interpolation inequality at the boundary.  Theorem 5.11 (Propagation of “smallness” Up to Boundaries—Interpolation Inequality). Let Ω be a connected open set in Rd and let ω and U be two open subsets of Ω with U bounded. Assume also that in a neighborhood of U the boundary ∂Ω is smooth. Let C0 > 0. There exist C > 0 and δ ∈ (0, 1) such that  δ (5.3.19) uH 1 (U ) ≤ Cu1−δ H 1 (Ω) f L2 (Ω) + uL2 (ω) , for u ∈ H 2 (Ω) satisfying u|∂Ω = 0 in a neighborhood of U and   P u = f + g, with f ∈ L2 (Ω), |g(x)| ≤ C0 |u(x)| + |Du(x)| a.e. in Ω.

Inequality (5.3.19) is thus identical to (5.3.8) in Theorem 5.6 with U allowed here to touch the smooth parts of the boundary of Ω. If Ω is itself smooth and bounded, we may thus pick U = Ω. Note however that in this latter case if g = 0 the obtained estimate is weaker than the usual elliptic estimate uH 1 (Ω)  f L2 (Ω) obtained in the case of homogeneous Dirichlet boundary conditions. Proof. If U  Ω then the result is given by Theorem 5.6. We thus consider the case ∂Ω ∩ U = ∅. We may assume that f L2 (Ω) ≤ uH 1 (Ω) since otherwise inequality (5.3.19) is obvious. In particular, if (5.3.19) holds for a value δ = δ0 > 0 the estimate also holds for all δ ∈ [0, δ0 ] possibly with a larger constant C = Cδ . The same observation can be made for the estimations (5.3.1) and (5.3.13). Recall the definition of Ωε given in (5.3.12). By Theorem 5.6, for ε > 0 there exists δ > 0 such that  δ f  + u . (5.3.20) uH 1 (Ωε )  u1−δ 2 2 1 L (Ω) L (ω) H (Ω) With a compactness argument and by Lemma 5.10, we can find a finite number of open sets Wj of Rd , j ∈ J, such that ∂Ω ∩ U ⊂ ∪j∈J Wj

198

5. UNIQUE CONTINUATION

and such that, for some values δ = δj ∈ (0, 1) and εj > 0, we have  δj 1−δj uH 1 (Wj ∩Ω)  uH 1 (Ω) f L2 (Ω) + uH 1 (Ωε ) (5.3.21) . j

For ε ∈ (0, 1) we set ˜ε = {x ∈ U, dist(x, ∂Ω) < ε}. U ˜ε ⊂ (∪j∈J Wj ). Applying (5.3.21) for There exists ε ∈ (0, 1) such that U each j ∈ J, using now δ  = min δj ∈ (0, 1),

and ε = min εj ∈ (0, 1)

j∈J

j∈J

in place of δj and εj (note that the set Ωε increases as ε decreases) we obtain  δ   + u . (5.3.22) uH 1 (U˜  )  u1−δ f  2 1 1 L (Ω) H (Ω  ) H (Ω) ε

ε

By (5.3.20), we have, for some δ  ∈ (0, 1), (5.3.23)

f L2 (Ω) + uH 1 (Ω

ε )



 u1−δ H 1 (Ω)



f L2 (Ω) + uL2 (ω)

δ ,

as the estimate of f L2 (Ω) is clear by the assumption made above. Then estimates (5.3.22) and (5.3.23) give  δ uH 1 (U˜  )  u1−δ f  (5.3.24) + u , δ = δ  δ  . L2 (Ω) L2 (ω) H 1 (Ω) ε

˜ε , estimate (5.3.20) together Taking now ε ∈ (0, ε ), since we have U ⊂ Ωε ∪ U with (5.3.24) yield the result.  5.4. Unique Continuation Initiated at the Boundary On a bounded smooth open set Ω ⊂ Rd , for an operator P0 as in (5.1.1), the unique H 2 -solution of P0 u = 0 in Ω and u = 0 on ∂Ω, is u = 0 because of the Poincar´e inequality. However, the following problem |P0 u(x)| ≤ C|u(x)| a.e. in Ω, and u = 0 on ∂Ω, may have a nontrivial solution, for example an eigenfunction for the operator P0 . If one adds the condition ∂ν u = 0 on ∂Ω, one can prove that u = 0. This is the subject of this section. In fact, it suffices to have ∂ν u|Γ = 0 and u|Γ = 0 in a small open set Γ of ∂Ω for the following conclusion to hold:    |P0 u(x)| ≤ C |u(x)| + |Du(x)| a.e. in Ω, ⇒ u ≡ 0 in Ω. on Γ, u = 0, ∂ν u = 0 We then say that the unique continuation property is initiated from the boundary. Here, we shall moreover prove a quantified version of this result. Note that we need not assume that Ω is bounded below. We start with the following local interpolation inequality.

5.4. UNIQUE CONTINUATION INITIATED AT THE BOUNDARY

199

Lemma 5.12 (Local Interpolation Inequality with Boundary Observations). Let y ∈ ∂Ω and V be an open neighborhood of y in Rd , such that ∂Ω is smooth in V . Let C0 > 0. There exist W a neighborhood of y in Rd , C > 0, and δ ∈ (0, 1) such that (5.4.1)

 δ f  + |u | + |∂ u | , uH 1 (W ∩Ω) ≤ Cu1−δ 2 1 ν |∂Ω |∂Ω 1 2 L (Ω) H (Ω) H (V ∩∂Ω) L (V ∩∂Ω)

for u ∈ H 2 (Ω) satisfying P u = f + g,

with f ∈ L2 (Ω),

  |g(x)| ≤ C0 |u(x)| + |Du(x)| a.e. in Ω.

Proof. We use a neighborhood U ⊂ V of y in Rd where we can use local coordinates x = (x , xd ) in which the boundary U ∩ ∂Ω is given by {xd = 0} and U ∩ Ω is given by {xd > 0}. We assume moreover that y = 0 in such coordinates. Let r > 0 and x(1) = (0, −r), where r is chosen sufficiently small to (1) 2 have B = B(x(1) , 3r)  U . Let ϕ(x) = e−γ|x−x | . The local geometry is illustrated in Fig. 5.7. By Lemma 3.5, ϕ satisfies the sub-ellipticity property of Definition 3.2 in B ∩ Ω for γ ≥ 0 chosen sufficiently large, to be kept fixed in what follows. Let χ ∈ C0∞ (U ) be such that χ(x) = 1 in B(x(1) , 5r/2) and χ(x) = 0 in U \ B(x(1) , 11r/4). We apply the local Carleman estimate of Theorem 3.28 to v = χu. This estimate is well adapted to cases where no boundary condition is prescribed as in the present case. We obtain (5.4.2) τ 3/2 eτ ϕ vL2 (B∩Ω) + τ 1/2 eτ ϕ Dx vL2 (B∩Ω)  eτ ϕ P vL2 (B∩Ω) +τ 3/2 |eτ ϕ v|xd =0+ |L2 (B∩∂Ω) +τ 1/2 |eτ ϕ Dx v|xd =0+ |L2 (B∩∂Ω) . We have P v = χP u + [P, χ]u, yielding eτ ϕ P vL2 (B∩Ω)  eτ ϕ χgL2 (B∩Ω) + eτ ϕ χf L2 (B∩Ω) + eτ ϕ [P, χ]uL2 (B∩Ω) . By the assumption made on g, we have eτ ϕ χgL2 (B∩Ω)  eτ ϕ χuL2 (B∩Ω) + eτ ϕ χDuL2 (B∩Ω)  eτ ϕ vL2 (B∩Ω) + eτ ϕ DvL2 (B∩Ω) + eτ ϕ [D, χ]uL2 (B∩Ω) . In Ω, the operators [P, χ] and [D, χ] are differential operators of order less than or equal to 1 with bounded coefficients and supported in A0 = {x ∈ Rd ; 5r/2 ≤ |x − x(1) | ≤ 11r/4} ∩ Ω, that is illustrated in Fig. 5.7.

200

5. UNIQUE CONTINUATION

U

x ∈ Rd−1 A0 3r

x(1) y = 0 r

xd

W

2r Ω B ∂Ω

Figure 5.7. Geometry near the boundary for the application of the local Carleman estimate of Theorem 3.28 with no prescribed boundary condition. The cutoff function χ vanishes outside the colored regions; the lighter color indicates a region where χ = 1; the darker color indicates the regions where χ varies From (5.4.2), for τ ≥ 1 chosen sufficiently large we obtain (5.4.3) τ 3/2 eτ ϕ vL2 (B∩Ω) + τ 1/2 eτ ϕ Dx vL2 (B∩Ω)  eτ ϕ f L2 (B∩Ω) + eτ ϕ [P, χ]uL2 (A0 ∩Ω) + eτ ϕ [D, χ]uL2 (A0 ∩Ω) + τ 3/2 |eτ ϕ v|xd =0+ |L2 (B∩∂Ω) + τ 1/2 |eτ ϕ Dx v|xd =0+ |L2 (B∩∂Ω) . 2

On the set A0 we have ϕ ≤ e−γ(5r/2) . We thus find (5.4.4)

eτ ϕ [P, χ]uL2 (A0 ∩Ω) + eτ ϕ [D, χ]uL2 (A0 ∩Ω)  eC1 τ uH 1 (Ω) , 2

where C1 = e−γ(5r/2) . 2 On {xd = 0}, we have ϕ ≤ e−γr then (5.4.5) τ 3/2 |eτ ϕ v|xd =0+ |L2 (B∩∂Ω) + τ 1/2 |eτ ϕ Dx v|xd =0+ |L2 (B∩∂Ω)    eC3 τ |u|xd =0+ |H 1 (B∩∂Ω) + |Dd u|xd =0+ |L2 (B∩∂Ω) , 2

where C3 = 2e−γr . 2 In Ω, we have ϕ ≤ e−γr ; this implies (5.4.6)

eτ ϕ f L2 (Ω) ≤ CeC3 τ f L2 (Ω) .

5.4. UNIQUE CONTINUATION INITIATED AT THE BOUNDARY

201

In B(x(1) , 2r), χ ≡ 1 yielding u = v. We set W = B(x(1) , 2r) ∩ Ω. As in this 2 set ϕ ≥ e−γ(2r) we have (5.4.7)

eC2 τ uH 1 (W ) ≤ τ 3/2 eτ ϕ vL2 (B∩Ω) + τ 1/2 eτ ϕ Dx vL2 (B∩Ω) 2

where C2 = e−γ(2r) for τ ≥ 1. Gathering (5.4.3)–(5.4.7) we obtain uH 1 (W )  e−(C2 −C1 )τ uH 1 (Ω)  + e(C3 −C2 )τ f L2 (Ω) + |u|xd =0+ |H 1 (B∩∂Ω)  + |Dd u|xd =0+ |L2 (B∩∂Ω) . Observe that C1 < C2 < C3 . Applying Lemma 5.4, we obtain the result.



Theorem 5.13 (Propagation of “smallness” Initiated from the Boundary—Interpolation Inequality). Let Ω be a connected open set in Rd and let U be an open subset of Ω with U bounded. Assume also that in a neighborhood of U the boundary ∂Ω is smooth. Let also y ∈ ∂Ω and V a open neighborhood of y in Rd , such that ∂Ω is smooth in V . Let C0 > 0. There exist C > 0 and δ ∈ (0, 1) such that (5.4.8)

 δ f  + |u | + |∂ u | uH 1 (U ) ≤ Cu1−δ ν |∂Ω L2 (V ∩∂Ω) , |∂Ω H 1 (V ∩∂Ω) L2 (Ω) H 1 (Ω)

for u ∈ H 2 (Ω) satisfying u|∂Ω = 0 in a neighborhood of U and   P u = f + g, with f ∈ L2 (Ω), |g(x)| ≤ C0 |u(x)| + |Du(x)| a.e. in Ω. Proof. We may assume that f L2 (Ω) ≤ uH 1 (Ω) since otherwise inequality (5.4.8) is obvious. With Lemma 5.12 there exist W neighborhood of y in Rd and δ1 ∈ (0, 1) such that  1 f L2 (Ω) + uH 1 (W ∩Ω)  u1−δ H 1 (Ω) f L2 (Ω) + |u|∂Ω |H 1 (V ∩∂Ω) δ1 + |∂ν u|∂Ω |L2 (V ∩∂Ω) . Next, by Theorem 5.11 there exists δ2 ∈ (0, 1) such that  δ2 2 , uH 1 (U )  u1−δ H 1 (Ω) f L2 (Ω) + uL2 (W ∩Ω) Together, these two inequalities give the result with δ = δ1 δ2 .



202

5. UNIQUE CONTINUATION

5.5. Unique Continuation Without Any Prescribed Boundary Condition In Sect. 5.3.2, for a quantified unique continuation result that goes from the interior of Ω up to its boundary, we assumed that the functions under consideration satisfy homogeneous Dirichlet boundary conditions. More general boundary conditions are considered in Chapter 9 in Volume 2. Yet, in Theorem 5.2, we obtained a unique continuation result up to the boundary without assuming any boundary condition. Here, we provide a quantification of unique continuation in that later case. The estimate we obtain is naturally weaker than those obtained in the case of prescribed boundary conditions. A central argument is also a Hardy-type inequality. Lemma 5.14 (Unique Continuation Estimate at a Boundary). Let y ∈ ∂Ω and let V be an open neighborhood of y in Rd such that ∂Ω is smooth in V . Let C0 > 0. There exist W ⊂ V a neighborhood of y in Rd , ε > 0, and C > 0, such that we have   uH 2 (Ω) (5.5.1) , uH 1 (W ∩Ω) ≤ CuH 2 (Ω) h f L2 (Ω) + uH 1 (V ∩Ωε ) with

  log 2 + log(2 + r) h(r) = , log(2 + r)

for u ∈ H 2 (Ω) satisfying P u = f + g,

with f ∈ L2 (Ω),

  |g(x)| ≤ C0 |u(x)| + |Du(x)| a.e. in Ω.

Observe that the function h : R+ → R+ is decreasing and limr→∞ h(r) = 0; in particular we use the convention h(+∞) = 0. We also recall that the set Ωε is defined in (5.3.12) for ε > 0. Proof. With the same argument as in the beginning of the proof of Proposition 5.1, we may simply assume P = P0 . For simplicity, upon reducing the open set V , we use local coordinates x = (x , xd ) in which the boundary V ∩ ∂Ω is given by {xd = 0} and V ∩ Ω is given by {xd > 0}. We assume moreover that y = 0 in such coordinates. We choose b > 0 such that V 0 = B d−1 (0, 4b) × (−4b, 4b) ⊂ V . We introduce ρ ∈ C ∞ (R+ ) nondecreasing such that  0 if s ≤ b, ρ(s) = s if 2b ≤ s. We define the functions ψ and ϕ both in C ∞ (Rd ) by ψ(x , xd ) = xd − ρ(|x |),

ϕ(x) = eγψ(x) , γ ≥ 1.

5.5. UNIQUE CONTINUATION WITHOUT BOUNDARY CONDITION

203

By Lemma 3.5, the weight function ϕ fulfills the sub-ellipticity property of Definition 3.2 in V 0 ∩ Rd+ if the parameter γ > 0 is chosen sufficiently large (and kept fixed in what follows). As ∂ν ϕ|∂Ω = −∂xd ϕ|xd =0+ = −γϕ|xd =0+ ≤ −1, Theorem 3.29 applies in V 0 ∩ Rd+ . We define χ0 , χ1 ∈ C ∞ (R) by   1 if s < b, 1 if − 2b < s, χ0 (s) = χ1 (s) = 0 if 2b < s, 0 if s < −3b, and we set on Rd

  χ(x) = χ0 (xd )χ1 ψ(x) .

The local geometry is represented in Fig. 5.8, displaying some level sets of the function ψ (and thus ϕ). We introduce the two regions A0 = {x ∈ Rd ; b ≤ xd ≤ 2b and − 3b ≤ ψ(x)}, A1 = {x ∈ Rd ; xd ≤ 2b and − 3b ≤ ψ(x) ≤ −2b}, and we see that supp(dχ) ⊂ A0 ∪ A1 . These two regions are illustrated in Fig. 5.8. We apply Theorem 3.29 to v = χu supported in V 0 , yielding (5.5.2) τ 3/2 eτ ϕ vL2 (Rd ) + τ 1/2 eτ ϕ DvL2 (Rd ) +

 e P vL2 (Rd ) + τ τϕ

+

3/2

+

|e v|xd =0+ |L2 (Rd−1 ) + τ 1/2 |eτ ϕ D v|xd =0+ |L2 (Rd−1 ) . τϕ

We have P v = χP u + [P, χ]u, yielding eτ ϕ P vL2 (Rd )  eτ ϕ χgL2 (Rd ) + eτ ϕ χf L2 (Rd ) + eτ ϕ [P, χ]uL2 (Rd ) . +

+

+

+

By the assumption made on g, we have eτ ϕ χgL2 (Rd )  eτ ϕ χuL2 (Rd ) + eτ ϕ χDuL2 (Rd ) +

+

+

 eτ ϕ vL2 (Rd ) + eτ ϕ DvL2 (Rd ) + eτ ϕ [D, χ]uL2 (Rd ) . +

+

+

The operators [P, χ] and [D, χ] are differential operators of order less than or equal to one with bounded coefficients and supported in A0 ∪A1 . From (5.5.2), for τ ≥ 1 chosen sufficiently large we obtain (5.5.3) τ 3/2 eτ ϕ vL2 (Rd ) + τ 1/2 eτ ϕ DvL2 (Rd ) +

+

 eτ ϕ χf L2 (Rd ) + eτ ϕ [P, χ]uL2 (A0 ∪A1 ) + eτ ϕ [D, χ]uL2 (A0 ∪A1 ) +

+ τ 3/2 |eτ ϕ v|xd =0+ |L2 (Rd−1 ) + τ 1/2 |eτ ϕ D v|xd =0+ |L2 (Rd−1 ) . For 0 < a < min(1, b/4) we introduce the open set Wa = {x ∈ Rd ; xd ∈ (a, b), |x | < b}

204

5. UNIQUE CONTINUATION

ψ = −3b ψ = −2b

x

ψ=0

4b 3b 2b b b

0

2b

Wa a

xd A0

A1

∂Ω

Figure 5.8. Geometry near the boundary for the application of the local Carleman estimate of Theorem 3.29. The darker colored region A0 ∪A1 indicates where the cutoff function χ varies. The lighter colored region represents the open set Wa that is illustrated in Fig. 5.8. We have χ ≡ 1, and thus u ≡ v, in W2a . We estimate the l.h.s. of (5.5.3) from below as follows, with τ ≥ 1, (5.5.4)

τ 3/2 eτ ϕ vL2 (Rd ) + τ 1/2 eτ ϕ DvL2 (Rd ) +

+

 e vL2 (W2a ) + e DvL2 (W2a )  eτ C2 vH 1 (W2a ) , τϕ

τϕ

where C2 = inf W2a ϕ = e2γa > 1. On A0 , we have ϕ ≤ C3 = e2γb . On A1 we have ϕ ≤ e−2γb ≤ 1. We thus obtain (5.5.5) eτ ϕ χf L2 (Rd ) + eτ ϕ [P, χ]uL2 ((A0 ∪A1 )∩Rd ) + eτ ϕ [D, χ]uL2 ((A0 ∪A1 )∩Rd ) + + +    eτ C3 f L2 (Rd ) + uH 1 (V ∩Ωb ) + eτ uH 1 (Rd ) . +

+

We recall that the set Ωb is defined in (5.3.12) for b > 0; here A0 ⊂ Ωb .

5.5. UNIQUE CONTINUATION WITHOUT BOUNDARY CONDITION

205

As ϕ|xd =0+ ≤ 1, we write, for τ ≥ 1, τ 3/2 |eτ ϕ v|xd =0+ |L2 (Rd−1 ) + τ 1/2 |eτ ϕ D v|xd =0+ |L2 (Rd−1 )

(5.5.6)

 τ 3/2 eτ |v|xd =0+ |H 1 (Rd−1 )  τ 3/2 eτ vH 2 (Rd ) . +

From (5.5.3)–(5.5.6) we thus obtain (5.5.7)   eτ C2 vH 1 (W2a )  τ 3/2 eτ uH 2 (Rd ) + eτ C3 f L2 (Rd ) + uH 1 (V ∩Ωb ) . +

+

We have 1 < C2 < C3 and we write (5.5.7) in the form   vH 1 (W2a )  τ 3/2 e−τ (C2 −1) uH 2 (Rd ) + eτ (C3 −C2 ) f L2 (Rd ) + uH 1 (V ∩Ωb ) , +

+

for some τ ≥ τ∗ . Observing that C2 − 1 =

e2γa

− 1 ≥ 2γa, we simply write

τ 3/2 e−τ (C2 −1)  a−3/2 (aτ )3/2 e−2γaτ  a−3/2 e−γaτ . We also write C3 − C2 ≤ C3 and we obtain

  vH 1 (W2a )  a−3/2 e−γaτ uH 2 (Rd ) + eτ C3 f L2 (Rd ) + uH 1 (V ∩Ωb ) , +

+

for τ ≥ τ∗ . Optimizing with respect to τ by applying Lemma 5.4 or rather Remark 5.5, using that 0 < a < 1, yields   C3   γa vH 1 (W2a )  K a−3/2 uH 2 (Rd ) γa+C3 f L2 (Rd ) + uH 1 (V ∩Ωb ) γa+C3 , +

with K =

max(2, eγaτ∗ ) 

a−3/2



≤

C3 γa+C3

+

max(2, eγτ∗ ).

=a

C3 γa+C3

We write  − ρ  γa a γa γa+C3 ,

ρ = 5C3 /2,

and we obtain

γa   C3  − ρ   γa+C 3 vH 1 (W2a )  auH 2 (Rd ) γa+C3 a γa f L2 (Rd ) + uH 1 (V ∩Ωb ) . +

+

Then, the Young inequality with H¨ older exponents yields  − ρ  vH 1 (W2a )  auH 2 (Rd ) + a γa f L2 (Rd ) + uH 1 (V ∩Ωb ) . (5.5.8) +

+

We now set ˜ a = {x ∈ Rd ; 0 < xd < a, |x | ≤ b} W and W = {x ∈ Rd ; −b < xd < b, |x | ≤ b}. ˜ 3a ⊂ W ∩ Rd+ recalling that a ≤ b/4. Pick s ∈ (0, 1); with the We have W Hardy inequality of Proposition 5.18 we write   s/2 s/2 s/2 uH 1 (W u/xd L2 (W ˜ 3a )  a ˜ 2a ) + Du/xd L2 (W ˜ 2a )  as/2 uH s/2+1 (W ∩Rd ) . +

206

5. UNIQUE CONTINUATION

An interpolation argument [74, 236] and the Young inequality with H¨older exponents then yield, for any η > 0,  s/2 1−s/2 uH 1 (W ˜ 3a )  uH 1 (W ∩Rd ) auH 2 (W ∩Rd ) +

+

 ηuH 1 (W ∩Rd ) + η +

˜ 3a ∪ W2a = W ∩ Since W

Rd+ ,

−1

auH 2 (W ∩Rd ) . +

with (5.5.8) we find

vH 1 (W ∩Rd )  ηuH 1 (W ∩Rd ) + (1 + η −1 )auH 2 (Rd ) + + + ρ   − γa f L2 (Rd ) + uH 1 (V ∩Ωb ) . +a +

Choosing η > 0 sufficiently small we thus obtain  − ρ  vH 1 (W ∩Rd )  auH 2 (Rd ) + a γa f L2 (Rd ) + uH 1 (V ∩Ωb ) , +

+

+

for all 0 < a < min(1, b/4). We can then optimize this last estimate with respect to a by applying Lemma 5.15, which concludes the proof.  Lemma 5.15. Let A ≥ 0, B1 ≥ 0 and B2 ≥ 0. We assume that A ≤ B1 and that there exist C0 , a∗ > 0 such that A ≤ aB1 + a−C0 /a B2 ,

(5.5.9) Then, we have (5.5.10)

A ≤ KB1 h(B1 /B2 ),

for 0 < a ≤ a∗ .

  log 2 + log(2 + r) , with h(r) = log(2 + r)

where K = K(a∗ , C0 ), with the natural convention that the r.h.s. vanishes in the case B2 = 0. Proof. First, we consider the case B2 = 0. We then have A ≤ aB1 for any 0 < a ≤ a∗ , implying that A = 0 and (5.5.10). Second, we consider B2 = 0 and we assume that B1 /B2 ≤ M with M ≥ 0 to be fixed below. As the function h given in (5.5.10) is decreasing, we have 1 ≤ h(M )/h(B1 /B2 ) yielding (5.5.11)

A ≤ h(M )B1 h(B1 /B2 ).

Third, we assume B2 = 0 and R := B1 /B2 ≥ M . In (5.5.9) we choose a > 0 to be given by   log 2 + log(2 + R) . a = 2C0 h(R) = 2C0 log(2 + R) Then, a ≤ 2C0 h(M ). We shall choose M ≥ M0 ≥ 0 with   a∗  a∗ if 2C < supR+ h, h−1 2C 0 0 M0 = 0 otherwise. This ensure a ≤ a∗ .

5.5. UNIQUE CONTINUATION WITHOUT BOUNDARY CONDITION

207

For the second term on the r.h.s. of (5.5.9) we write, a−C0 /a B2 = eF B1 ,

F = −C0 log(a)/a − log(R).

We then estimate F as follows F = −C0 log(a)/a − log(R)

  log(2C0 ) + log log(2 + log(2 + R)) 1 ≤ − log(2 + R) 2 log(2 + log(2 + R)) 1 + log(2 + R) − log(R). 2 The first term on the r.h.s. is negative if one chooses R ≥ M ≥ M1 = ee

1/(2C0 ) −2

− 2.

For R ≥ M ≥ 4 we have F ≤

1 1 log(2 + R) − log(R) ≤ − log(R), 2 4

yielding a−C0 /a B2 ≤ R−1/4 B1 ≤ C1 B1 h(R),

with C1 = sup R−1/4 /h(R) < ∞. [4,+∞[

With the choice made for a we have aB1 = 2C0 h(R)B1 . From (5.5.9) we thus obtain A ≤ (2C0 + C1 )B1 h(R), for R ≥ M = max(M0 , M1 , 4). max(2C0 + C1 , h(M )).

We thus obtain the result with K = 

Theorem 5.16 (Propagation of “smallness” Up to Boundaries). Let Ω be a connected open set in Rd and let ω and U be two open subsets of Ω with U bounded. Assume also that in a neighborhood of U the boundary ∂Ω is smooth. Let C0 > 0. There exists C > 0 such that   uH 2 (Ω) (5.5.12) , uH 1 (U ) ≤ CuH 2 (Ω) h f L2 (Ω) + uL2 (ω) with

  log 2 + log(2 + r) , h(r) = log(2 + r)

for u ∈ H 2 (Ω) satisfying P u = f + g,

with f ∈ L2 (Ω),

  |g(x)| ≤ C0 |u(x)| + |Du(x)| a.e. in Ω.

208

5. UNIQUE CONTINUATION

Proof. Recall the definition of Ωε given in (5.3.12). By Theorem 5.6, for ε > 0 there exists δ ∈ (0, 1) such that  δ (5.5.13) + u f  uH 1 (Ωε )  u1−δ 2 2 1 L (Ω) L (ω) H (Ω)  δ f   u1−δ + u . L2 (Ω) L2 (ω) H 2 (Ω) Observe that the result is clear if f L2 (Ω) = 0 and uL2 (ω) = 0 by Theorem 5.2. We may thus assume that f L2 (Ω) + uL2 (ω) > 0. Observe that we may also assume that f L2 (Ω) ≤ uH 2 (Ω) since otherwise inequality (5.5.12) is obvious. We then have  δ 1−δ f L2 (Ω) + uH 1 (Ωε )  uH 2 (Ω) f L2 (Ω) + uL2 (ω) , which we write  (5.5.14)

C1

δ

uH 2 (Ω) f L2 (Ω) + uL2 (ω)

≤

uH 2 (Ω) f L2 (Ω) + uH 1 (Ωε )

,

for some C1 > 0. With a compactness argument and by Lemma 5.14, we can find a finite number of open sets Wj of Rd , j ∈ J, such that ∂Ω ∩ U ⊂ ∪j∈J Wj and such that, for some εj ∈ (0, 1), we have (5.5.15)

 uH 1 (Wj ∩Ω)  uH 2 (Ω) h

 ,

uH 2 (Ω) f L2 (Ω) + uH 1 (U ∩Ωε

j)

For ε ∈ (0, 1) we set ˜ε = {x ∈ U, dist(x, ∂Ω) < ε}. U ˜ε ⊂ (∪j∈J Wj ). Applying (5.3.21) for There exists ε ∈ (0, 1) such that U each j ∈ J, using now ε = min εj ∈ (0, 1) j∈J

in place of εj we obtain uH 1 (U˜

ε )

  uH 2 (Ω) h



uH 2 (Ω) f L2 (Ω) + uH 1 (Ω

, ε )

since, the map r → h(r) is decreasing (note that the set Ωε increases as ε decreases). With (5.5.14) we thus obtain uH 1 (U˜

ε )

   uH 2 (Ω) h C1

uH 2 (Ω) f L2 (Ω) + uL2 (ω)

δ  .

5.5. UNIQUE CONTINUATION WITHOUT BOUNDARY CONDITION

209

for some C1 > 0 and δ ∈ (0, 1). As h(C1 rδ )/h(r) ∼ 1/δ, as r → ∞, we then write   uH 2 (Ω) . (5.5.16) uH 1 (U˜  )  uH 2 (Ω) h ε f L2 (Ω) + uL2 (ω) We note that (5.5.13) reads uH 1 (Ωε )  r−δ uH 2 (Ω) ,

r=

uH 2 (Ω) f L2 (Ω) + uL2 (ω)

.

Here, we assume that f L2 (Ω) ≤ uH 2 (Ω) (see above) implying r ≥ C2 for some C2 > 0. Since h(r)  r−δ for r ≥ C2 , we thus obtain   uH 2 (Ω) . (5.5.17) uH 1 (Ωε )  uH 2 (Ω) h f L2 (Ω) + uL2 (ω) ˜ε ∪ Ωε for 0 < ε < ε , from (5.5.16) and (5.5.17) we deduce the As U ⊂ U result.  The function h that appears in the statement of the quantified unique continuation in Theorem 5.16 may appear to be quite weak in comparison with what one obtains in Theorem 5.11 in the case of homogeneous Dirichlet boundary conditions. The following proposition shows however that the result of Theorem 5.16 is almost sharp in the two dimensional setting with P = −Δ = −∂12 − ∂22 . Proposition 5.17. Let Ω = B 2 (0, 1), ω = B 2 (0, 1/4), and let k : R+ → R+ . Assume that k is nonincreasing and such that, for some C > 0,   uH 2 (Ω) , uH 1 (Ω) ≤ CuH 2 (Ω) k ΔuL2 (Ω) + uL2 (ω) for all u ∈ H 2 (Ω). Then, there exist C > 0 such that lim inf log(r)k(r) ≥ C. r→∞

Proof. We use polar coordinates where Δ = ∂r2 + r−1 ∂r + r−2 ∂θ2 . For n ∈ N, we set vn (r, θ) = rn sin(nθ)

and un (r, θ) = χ(r)vn (r, θ),

where χ ∈ C ∞ ([0, 1]; R), with 0 ≤ χ ≤ 1 and such that  0 if r ≤ 1/4, χ(r) = 1 if 3/4 ≤ r. We observe that Δvn = 0 and un ≡ 0 in ω. We make the following claim (5.5.18) Δun L2 (Ω)  n1/2 (3/4)n ,

un H 1 (Ω)  n1/2 ,

un H 2 (Ω)  n3/2 .

210

5. UNIQUE CONTINUATION

Then, we find (5.5.19)

 un  1  un  2   H (Ω) H (Ω)   n−1 . k n(4/3)n  k Δun L2 (Ω) un H 2 (Ω)

For n ≥ 1, we set r = n(4/3)n ≥ (4/3)n > 1. We thus have n−1 ≥ log(4/3) log−1 (r). Then, (5.5.19) yields the result. claim made in (5.5.18). We compute Δun (r, θ) =   We nnow prove the χ (r)r + (2n + 1)χ (r)rn−1 sin(nθ). This yields 3/4

Δun 2L2 (Ω)  ∫ (r2n + (2n + 1)2 r2n−2 )rdrdθ 1/4

(3/4)2n+2 (2n + 1)2 (3/4)2n + , 2n + 2 2n which yields the first estimate in (5.5.18). To estimate the H 1 -norm of un from below, we simply write 1 2π 1  2 un 2H 1 (Ω) ≥ ∂r un 2L2 (Ω) ≥ ∫ ∫ nrn−1 sin(nθ) rdrdθ  n2 ∫ r2n−1 dr  n. 

0 3/4

3/4

This is the second estimate in (5.5.18). We finally estimate the H 2 -norm of un . We write 2

2

un 2H 2 (Ω)  un 2L2 (Ω) + ∂r2 un L2 (Ω) + r−2 ∂θ2 un L2 (Ω) . We have 2

2

2

2

∂r2 un L2 (Ω)  ∂r2 vn L2 (Ω) + χ ∂r vn L2 (Ω) + χ vn L2 (Ω) 1   ∫ n4 r2n−3 + n2 r2n−1 + r2n+1 dr  n3 . 0

With similar computations for the two other terms we find un 2H 2 (Ω)  n3 . Finally, we write 1

2

un 2H 2 (Ω) ≥ r−2 ∂θ2 un L2 (Ω)  ∫ n4 r2n−1 dr  n3 , 3/4

and we obtain the third estimate in (5.5.18).



5.6. Notes The unique continuation properties in the case of analytic coefficients holds across non-characteristic hypersurfaces by results of E. Holmgren [170] and F. John [187]. For operators with nonanalytic coefficients, the first result on unique continuation for elliptic second-order differential operators goes back to the work of T. Carleman [105], in dimension 2, by introducing weighted estimates. By unique continuation one means that if a solution vanishes one side of an hypersurface, it is in fact identically 0 in a neighborhood of this hypersurface. In [100], C. Calder´ on proved a general result for

5.6. NOTES

211

operators (of arbitrary order) with simple complex roots. See also [174, Section 28.1]. Results on unique continuation in the framework of the more general strong pseudo-convexity condition were obtained by L. H¨ormander [172, Chapter 8]. See also the more modern treatment in [174, Section 28.2-3]. We also refer to N. Lerner [223, 224] for the principally normal operators. For an exposition of unique continuation, including a proof of the Calder´on theorem and results under strong pseudo-convexity, we also refer to the books of C. Zuily [330] and N. Lerner [229]. General results as those mentioned above usually concern unique continuation for operator with homogeneous principal part, thus excluding Schr¨odinger-like and heat-like operators. For results in the case of nonhomogeneous principal parts we refer to works of S. Mizohata [261], H. Kumanogo [200], V. Isakov [183, 184], B. Dehman [114], H. Khalgui Ounaies [196], J.-C. Saut and B. Scheurer [300]. Here, we consider the case of smooth coefficients. However, for secondorder elliptic operators with real coefficients in the principal part, Lipschitz continuity of the coefficients suffices for unique continuation across a C 1 hypersurface. This can be obtained using the Carleman estimate of Theorem 3.52 of Sect. 3.7.3. An important remark is the following. It was shown by A. Pli´s [278] that H¨ older continuity is not enough to get unique continuation: this author constructed a real homogeneous linear differential equation of second-order and of elliptic type on R3 without the unique continuation property although the coefficients are H¨older-continuous with any exponent less than one. The constructions by K. Miller in [251], and later by N. Mandache [241] and N. Filonov in [150], showed that H¨ older continuity is not sufficient to obtain unique continuation for second-order elliptic operators, even in divergence form in dimension greater than two. In dimension two however, boundedness of the coefficients is sufficient to achieve unique continuation in the case of H 1 -solutions; see Schulz [301]. Even though H¨ older regularity in the coefficients of a second-order elliptic operator can be an obstacle to unique continuation in dimension greater than two, discontinuous coefficients can yet be considered. In fact, with the introduction of proper transmission conditions, avoiding the existence of single or double layer potentials, a well-posed elliptic problem can be considered. Derivations of Carleman estimates for such transmission problems allow one to deduce quantifed version of the unique continuation property arguing as in the present chapter. Such derivations can be found 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 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.

212

5. UNIQUE CONTINUATION

For more nonuniqueness results we refer to the works of De Giorgi [113], A. Pli´s [275–277], S. Alinhac and S. Baouendi [22, 25], S. Alinhac and C. Zuily [28], R. Lascar and C. Zuily [202], X. Saint-Raymond [297, 298], S. Alinhac [20], H. Bahouri [47, 48], B. Dehman [115] and [286, 287]. The book of C. Zuily [330] gathers many counter-examples. Here, we only consider bounded coefficients in lower-order terms. There is a large literature addressing unique continuation for classes of operators with lower-order terms with unbounded coefficients; see [54, 75, 83, 122, 194, 306, 308, 327]. Strong unique continuation means that the flatness of the solution near a point implies that the solution vanishes locally. Early result go back to N. Aronszajn [40, 41]. In the work of D. Jerison and C. E. Kenig [185] this property is proven for −Δ + V with V ∈ Ld/2 . For other results we refer to [14, 23, 27, 42, 106, 107, 138, 139, 197, 198, 207, 283–285, 307, 325]. For counter-examples to strong unique continuation see S. Alinhac [19], T. Wolff [326], and S. Alinhac and S. Baouendi [24]. In the present book we chose to not cover strong unique continuation. Here, we prove unique continuation only for second-order elliptic operators and we focus our interest on obtaining local and nonlocal quantifications of this property. In fact, the Cauchy problem for a partial differential equation is well-posed in the sense of Hadamard if the operator is hyperbolic; see P. Lax [206] and S. Mizohata [262]. See also [174, Theorem 23.3.1]. For an elliptic operator, the dependency upon Cauchy data cannot be Lipschitz. This dependency is actually of H¨ older type. Locally the H¨ older behavior takes the form of an interpolation inequality, which can be found in F. John [188], L. E. Payne [269], V. Isakov [184], and H. Bahouri [49]. Following [289], we show how various local interpolation estimates can be propagated like in Theorem 5.6 for instance. The result of Theorem 5.11 providing a quantification of unique continuation up to a boundary under homogeneous Dirichlet condition can be found in [218, 289]. Note that ˇ Lopatinski˘ı-Sapiro conditions are considered in Chapter 9 in Volume 2. If no boundary condition is assumed, log-type estimates can be achieved as in Lemma 5.14 and Theorem 5.16. Such results originate from K-D. Phung [272] with developments by L. Bourgeois [80] and L. Bourgeois and J. Dard´e [82]. Here, we present a version with a slightly improved estimate. The optimality result of Proposition 5.17 comes from [81]. An alternative method to achieve quantification to unique continuation properties is based on the so-called doubling property that we do not cover here: see for instance N. Garofalo and F.-H. Lin [158]. Whereas we only consider scalar equation here, systems raise many open questions as far as unique continuation. Some answers can be found for systems as given by elasticity models, for the Stokes system or more general systems of partial differential equations; we refer for instance to [13, 16, 34, 35, 116, 140, 193, 234]

5.A. A HARDY INEQUALITY

213

Upon addition of assumption on the solution of an equation, say on its support, H¨ ormander’s pseudo-convexity condition can be partly relaxed yet leading to unique continuation; See for instance S. Alinhac [21], some joint work with N. Lerner [230] with development by L. H¨ormander [174, Section 28.4], some joint work with H. Bahouri [51], and X. Saint-Raymond [299]. For higher-order elliptic operators, if strong pseudo-convexity hold, a quantified unique continuation property can be deduced from the general Carleman estimate given in [174, Section 28.2]. At a boundary this can also be done as exposed in D. Tataru [314] and joint work with M. Bellassoued [70]. If strong pseudo-convexity does not hold, unique continuation properties may sometimes still be obtained. A simple example is that of the Bilaplace operator for which no hypersurface can be strongly pseudoconvex. Yet, Carleman estimates can be derived, even near a boundary [215] and a quantifed unique continuation property can be deduced. For the treatment higher-order elliptic operators with nonsmooth coefficients in the lower-order terms we can refer to Wang [324]. Appendix 5.A. A Hardy Inequality Let s ≥ 0. On R+ we can define H s (R+ ) = {v|R+ ; v ∈ H s (R)}, with the following norm uH s (R+ ) =

inf

v∈H s (R) u=v|R +

vH s (R) .

The following proposition is referred to as a Hardy inequality. Proposition 5.18. Let s ∈ [0, 1/2). If u ∈ H s (R+ ) then t−s u ∈ Moreover, there exists C > 0 such that

L2 (R+ ).

t−s uL2 (R+ ) ≤ CuH s (R+ ) ,

u ∈ H s (R+ ).

For the proof of the proposition we shall need the more classic Hardy inequality of the following lemma. Lemma 5.19. If f ∈ L2 (R+ ) we have T f L2 (R+ ) ≤ 2f L2 (R+ ) , where T f (x) = x−1 ∫0x f (t)dt for x > 0. Proof of Lemma 5.19. The Cauchy-Schwarz inequality gives x x x |T f (x)|2 ≤ x−2 ∫ t−1/2 dt ∫ t1/2 |f (t)|2 dt = 2x−3/2 ∫ t1/2 |f (t)|2 dt 0

0

0

As x ≥ t in the previous integral, with the Fubini theorem, we obtain +∞

+∞

+∞

+∞

0

0

t

0

∫ |T f (x)|2 dx ≤ 2 ∫ t1/2 |f (t)|2 ∫ x−3/2 dxdt = 4 ∫ |f (t)|2 dt,

214

5. UNIQUE CONTINUATION



which gives the result. Proof of Proposition 5.18. Lemma 5.19 yields, for u ∈ H01 (R+ ) t−1 uL2 (R+ )  u L2 (R+ )  uH 1 (R+ ) .

For θ ∈ [0, 1] we denote by L2 (R+ , t−2θ dt) to be the set of L2 -functions for the measure t−2θ dt on R+ . We then have u ∈ L2 (R+ )

uL2 (R+ ,dt) = uL2 (R+ ) ,

uL2 (R+ ,t−2 dt)  uH 1 (R+ ) ,

u ∈ H01 (R+ ).

An interpolation argument [74, 236] then yields u[L2 (R+ ,dt),L2 (R+ ,t−2 dt)]θ  u[L2 (R+ ),H 1 (R+ )]θ , 0

u ∈ [L2 (R+ ), H01 (R+ )]θ ,

for θ ∈ [0, 1]. We have [L2 (R+ , dt), L2 (R+ , t−2 dt)]θ = L2 (R+ , t−2θ dt) for θ ∈ [0, 1]. We also have [L2 (R+ ), H01 (R+ )]θ = H θ (R+ ), for θ ∈ [0, 1/2) by Theorem 11.6 in [236, Chapter 1]. We thus obtain the result. 

CHAPTER 6

Stabilization of the Wave Equation with an Inner Damping Contents 6.1. 6.2. 6.3. 6.4. 6.4.1. 6.4.2. 6.4.3. 6.5. 6.5.1. 6.5.2.

Introduction and Setting 216 Preliminaries on the Damped Wave Equation 216 Stabilization and Resolvent Estimate 220 Remarks and Non-Quantified Stabilization Results 222 Comparison with Exponential Stability 222 Zero Eigenvalue 222 Non-Quantified Stabilization Results 223 Resolvent Estimate for the Damped Wave Generator 224 Estimations Through an Interpolation Inequality 225 Estimations Through the Derivation of a Global Carleman Estimate 227 6.6. Alternative Proof Scheme of the Resolvent Estimate 229 6.7. Notes 232 Appendices 235 6.A. The Generator of the Damped-Wave Semigroup 235 6.B. Well-Posedness of the Damped Wave Equation 239 6.B.1. Proof of Well-Posedness 239 6.B.2. Other Formulations of Weak Solutions 241 6.C. From a Resolvent to a Semigroup Stabilization Estimate 243 6.D. Proofs of Non-Quantified Stabilization Results 247 6.D.1. Proof of Proposition 6.12 247 6.D.2. Proof of Proposition 6.14 247 6.D.3. Proof of Proposition 6.15 248

© 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 I, PNLDE Subseries in Control 97, https://doi.org/10.1007/978-3-030-88674-5 6

215

216

6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

6.1. Introduction and Setting For Ω a smooth bounded connected open set of Rd and for a second-order elliptic operator P0 given by

(6.1.1) P0 = Di (pij (x)Dj ), with pij (x)ξi ξj ≥ C|ξ|2 , 1≤i,j≤d

1≤i,j≤d

we consider the following wave equation  ∂t2 y + P0 y + α(x)∂t y = 0 in (0, +∞) × Ω, (6.1.2) in Ω, y|t=0 = y 0 , ∂t y|t=0 = y 1 with, say, homogeneous Dirichlet boundary conditions. Here α(x) is a nonnegative bounded function. In the case α ≡ 0 the solution exists for t ∈ R with an energy that remains constant with respect to t. We shall prove here that if α > 0 on a nonempty open subset ω ⊂ Ω then the energy decays to zero as t → +∞. The term α∂t thus acts as a damping and we refer to (6.1.2) as to the damped wave equation. Attention will be placed on the decay rate for the energy. In fact, if ω satisfies the so-called geometrical control condition (GCC) the energy decays exponentially. The geometrical control condition states that every geodesic, for the metric associated with the quadratic form g(x) = (p(x)ij )−1 , travelled at speed one, with reflections at the boundary according to geometrical optics, goes through the open set ω in a finite time. Here, no such assumption is made on the open set ω. We shall prove that in such case the decay rate is at least logarithmic. Here, we shall use many properties of the unbounded operator P0 : 2 L (Ω) → L2 (Ω) with domain D(P0 ) = H 2 (Ω) ∩ H01 (Ω) given by P0 u = P0 u for u ∈ D(P0 ). We refer to Chap. 10 for background material on the operator P0 . On the Sobolev space H01 (Ω) we shall use the following norm

(6.1.3) (pij Di u, Dj u)L2 (Ω) . u2H 1 (Ω) = 0

1≤i,j≤d

which is equivalent to the usual H 1 -norm,

u2H 1 (Ω) = u2L2 (Ω) + (6.1.4) Di u2L2 (Ω) , 1≤i≤d

by the Poincar´e inequality. The norm .H 1 given in (6.1.3) will be often 0 very adapted to the problems involving the operator P0 that we consider. 6.2. Preliminaries on the Damped Wave Equation To define solutions to (6.1.2) we need to provide a proper functional framework, in particular to account for boundary conditions, here of homogeneous Dirichlet type. We have the following existence and uniqueness result.

6.2. PRELIMINARIES ON THE DAMPED WAVE EQUATION

217

Theorem 6.1. For (y 0 , y 1 ) ∈ H01 (Ω) × L2 (Ω) there exists a unique       y ∈ C 2 [0, +∞); H −1 (Ω) ∩ C 1 [0, +∞); L2 (Ω) ∩ C 0 [0, +∞); H01 (Ω) such that (6.2.1) ∂t2 y + P0 y + α(x)∂t y = 0

  in D  (0, +∞) × Ω ,

y|t=0 = y 0 , ∂t y|t=0 = y 1 .

Moreover, there exists C > 0 such that (6.2.2)   t ≥ 0. y(t)H 1 (Ω) + ∂t y(t)L2 (Ω) ≤ C y 0 H 1 (Ω) + y 1 L2 (Ω) ,   If now (y 0 , y 1 ) ∈ H 2 (Ω) ∩ H01 (Ω) × H01 (Ω) there exists a unique       y ∈ C 2 [0, +∞); L2 (Ω) ∩C 1 [0, +∞); H01 (Ω) ∩C 0 [0, +∞); H 2 (Ω)∩H01 (Ω) such that (6.2.3)   ∂t2 y+P0 y+α(x)∂t y = 0 in L∞ [0, +∞); L2 (Ω) ,

y|t=0 = y 0 , ∂t y|t=0 = y 1 .

Moreover, there exists C > 0 such that (6.2.4)   y(t)H 2 (Ω) + ∂t y(t)H 1 (Ω) ≤ C y 0 H 2 (Ω) + y 1 H 1 (Ω) , 0

0

t ≥ 0.

Solutions to (6.2.1) are called weak solutions, whereas solutions to (6.2.3) are called strong solutions. Remark 6.2. It should be noted that a strong solution is also a weak solution; thus because of the uniqueness the two types of  of a weak solution  0 , y 1 ) ∈ H 2 (Ω)∩H 1 (Ω) ×H 1 (Ω). Moreover, solutions coincide in the case (y 0 0   if (yn0 , yn1 ) ⊂ H 2 (Ω)∩H01 (Ω) ×H01 (Ω) converges to (y 0 , y 1 ) in H01 (Ω)⊕L2 (Ω) then the estimates of Theorem 6.1 show that the strong solutions yn associ0 1 to the weak solution with (y ated with (yn0 , yn1 ) converge   y 0associated  ,y )   2 −1 1 2 1 in C [0, +∞); H (Ω) ∩ C [0, +∞); L (Ω) ∩ C [0, +∞); H0 (Ω) . We refer to Appendix 6.B.1 for a full proof of Theorem 6.1. The proof uses the semigroup formulation that we introduce below. Remark 6.3. On D(P0 )=H 2 (Ω)∩H01 (Ω) we have uH 2 (Ω) P0 uL2 (Ω) ; see (10.1.8). We may thus state the continuity inequality (6.2.4) for strong solutions in the equivalent form   (6.2.5) P0 y(t)L2 (Ω) + ∂t y(t)H 1 (Ω) ≤ C P0 y 0 L2 (Ω) + y 1 H 1 (Ω) . 0

0

For a weak solution y(t), we introduce the energy function,   1  (6.2.6) ∂t y(t)2L2 (Ω) + y(t)2H 1 (Ω) E(y)(t) = E y(t), ∂t y(t) = 0 2   1 ∂t y(t)2L2 (Ω) + P0 y(t), y(t) H −1 (Ω),H 1 (Ω) , = 0 2

218

6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

with the H01 -norm defined in (6.1.3) and using (10.1.15). This energy will also be used in the stabilization problem below. Uniqueness of a strong solution can be proven by means of this energy function. In fact, for a such a strong solution we compute     d E(y)(t) = Re ∂t2 y(t), ∂t y(t) L2 (Ω) + Re y(t), ∂t y(t) H 1 (Ω) . 0 dt Using (10.1.15), we have      y(t), ∂t y(t) H 1 (Ω) = P0 y(t), ∂t y(t) H −1 (Ω),H 1 (Ω) = P0 y(t), ∂t y(t) L2 (Ω) , 0

0

since y(t) ∈ D(P0 ). This yields d E(y)(t) = −(α∂t y, ∂t y)L2 (Ω) , dt implying the decay of the energy E(y)(t) as α ≥ 0. We thus have (6.2.7)

0 ≤ E(y)(t) ≤ E(y)(0). As the equation is linear the uniqueness of a strong solution to (6.2.3) follows. Uniqueness of a weak solution requires a more sophisticated argument, that we give in the proof of Theorem 6.1 in Appendix 6.B.1, as the above d E(y)(t) cannot be carried out in such a simple manner computation of dt because of the limited regularity of the solution. To prepare for the proof of the results of Theorem 6.1, it is convenient to cast the damped wave equation (6.1.2) into a semigroup formalism. We refer the reader to Chap. 12 for some elements of semigroup theory. This setting is also central in the analysis of the stabilization property of the damped wave-equation. We set     0  y(t) y 0 −1 0 (6.2.8) , , Y (t) = , Y = A= ∂t y(t) P0 α(x) y1 and if y is a strong solution to (6.2.3) then we have (this is only formal in the case of weak solutions) d Y (t) + AY (t) = 0, Y|t=0 = Y 0 . dt We refer to this equation as to the semigroup equation. We define the unbounded operator A given by (6.2.8) on the Hilbert sum H = H01 (Ω) ⊕ L2 (Ω) with dense domain   (6.2.10) D(A) = H 2 (Ω) ∩ H01 (Ω) × H01 (Ω). (6.2.9)

We endow D(A) with the graph norm V 2D(A) = V 2H + AV 2H ,

V ∈ D(A).

The Hilbert sum H is naturally endowed with the inner product (U, V )H = (u0 , v 0 )H01 (Ω) + (u1 , v 1 )L2 (Ω) ,

U = (u0 , u1 ), V = (v 0 , v 1 ).

6.2. PRELIMINARIES ON THE DAMPED WAVE EQUATION

219

The elliptic operator P0 on L2 (Ω) with dense domain D(P0 ) = H 2 (Ω) ∩ is closed (see Sect. 10.1). It follows that (A, D(A)) is also a closed operator. This property makes D(A) equipped with the graph norm a complete space. By Proposition 6.22, on D(A) we have the following norm equivalences

H01 (Ω)

(6.2.11)

V D(A)  AV H  V H 2 (Ω)⊕H 1 (Ω) ,

V ∈ D(A).

Theorem 6.4. The unbounded operator (A, D(A)) generates a C0 -semigroup of contraction S(t) = e−tA on H . Proof. As seen above D(A) is dense in H and (A, D(A)) is a closed operator. For λ < 0, λ IdH −A is invertible with a bounded inverse by Proposition 6.25 and moreover by Lemma 6.26 we have (λ IdH −A)−1 L (H) ≤ |λ|−1 . We then obtain the result by the Hille-Yosida theorem (see Theorem 12.6).  Proposition 6.5. Let Y 0 = t (y 0 , y 1 ) ∈ H and set Y (t) = S(t)Y 0 . (1) If Y (t) = t (y(t), z(t)) then       y ∈ C 2 [0, +∞); H −1 (Ω) ∩ C 1 [0, +∞); L2 (Ω) ∩ C 0 [0, +∞); H01 (Ω) , z(t) = ∂t y(t), and y is the unique solution of (6.2.1). d Y (t) + AY (t) = 0, for t ≥ 0, and (2) If moreover, Y 0 ∈ D(A), then dt       y ∈ C 2 [0, +∞); L2 (Ω) ∩C 1 [0, +∞); H01 (Ω) ∩C 0 [0, +∞); H 2 (Ω)∩H01 (Ω) , and y is the unique solution of (6.2.3). The proof of Proposition 6.5 is actually contained in that of Theorem 6.1 given in Appendix 6.B.1. Using the uniqueness part of Theorem 6.1, if Y 0 = t (y 0 , y 1 ) ∈ H and if y(t) is the unique solution of (6.2.1) we then have conversely   t y(t), ∂t y(t) = S(t)Y 0 . In both the strong and weak cases, Proposition 6.5 shows that the solution of the damped wave equation is simply given by the first component of Y (t) = d Y (t) + AY (t) = 0, S(t)Y 0 . In the case Y 0 ∈ D(A), note that we obtain dt for t ≥ 0, by applying Proposition 12.2. Observe that we have   2 for V = t (v 0 , v 1 ) ∈ D(A), Re AV, V H = α1/2 v 1 L2 (Ω) , as (P0 v 0 , v 1 )L2 (Ω) = (v 0 , v 1 )H01 (Ω) using (10.1.17)–(10.1.18). For t (y(t), z(t)) = Y (t) = S(t)Y 0 , with Y 0 ∈ D(A), we thus find 1 d 2 Y (t)2H = − Re(AY (t), Y (t))H = −α1/2 z(t)L2 (Ω) , (6.2.12) 2 dt which is precisely the counterpart of (6.2.7) as we have in fact    1  (6.2.13) E(y)(t) = E y(t), ∂t y(t) = E y(t), z(t) = Y (t)2H . 2

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6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

With (6.2.7) and (6.2.12)–(6.2.13), we saw above that the energy E(y)(t) decays in the case of a strong solution. For a weak a solution, in course of the proof of Theorem 6.1 in Appendix 6.B.1 we see that this decay occurs. In fact, in the case of a weak solution, that is for Y 0 = t (y 0 , y 1 ) ∈ H equality (6.B.2), that holds for Y (t) = S(t)Y 0 implies that E(y)(t) is continuously differentiable and we have 1 d 2 E(y)(t) = −α1/2 ∂t yL2 (Ω) , (6.2.14) E(y)(t) = Y (t)2H , dt 2 even though the computations that led to (6.2.7) could not be carried out in the case of a weak solution. Now that solutions to the damped wave equation have been properly introduced we can place our interest in the study of the nonincreasing behavior of the energy E(y)(t) introduced in (6.2.6) and as expressed in (6.2.7) and (6.2.14). In particular, in Theorem 6.9 below we shall prove that a logarithmic decay rate with respect to time can be achieved for strong solutions, without any particular assumption on the open set ω where α > 0. For weak solutions, we only can prove that the energy tends to zero without any uniform estimation: we only have strong convergence.1 6.3. Stabilization and Resolvent Estimate The following theorem is central in the proof of the stabilization of the damped wave equation (6.1.2). We refer to Appendix 6.C for a proof. Theorem 6.6. Let S(t) = e−tB be a C0 -semigroup on a Hilbert space H with associated (closed) generator B : H → H with dense domain D(B) ⊂ H. We denote by .L (H) the operator norm for linear and bounded maps on H. We assume that we have: (1) the semigroup is bounded, that is, we have the following uniform bound: (6.3.1)

sup S(t)L (H) < +∞; t≥0

(2) for all σ ∈ R, the unbounded operator iσ IdH −B is invertible with a bounded inverse; (3) moreover, there exist K > 0 and σ0 > 0 such that (6.3.2)

(iσ IdH −B)−1 L (H) ≤ KeK|σ| ,

Then, for all k ∈ N there exists C > 0 such C S(t)B −k L (H) ≤  k , log(2 + t)

σ ∈ R, |σ| ≥ σ0 .

t > 0.

Note that the result for the case k = 0 is simply contained in the assumed boundedness of the semigroup on H. 1Here, the choice of words eventually yields a poor phrasing of the result.

6.3. STABILIZATION AND RESOLVENT ESTIMATE

221

Remark 6.7. We recall that the resolvent set ρ(B) of a closed unbounded operator B is the set of λ ∈ C such that the map λ IdH −B from D(B) into H is bijective. Then, the inverse Rλ (B) = (λ IdH −B)−1 that maps H into itself is bounded by the closed graph theorem. The resolvent set ρ(B) is an open set of C. The spectrum of B, denoted by sp(B), is the complement set of ρ(B) in C. See Sect. 11.3. With S(t) bounded one has sp(B) ⊂ {z ∈ C; Re z ≥ 0} by Corollary 12.8. Note that estimation (6.3.2) has the following consequence for a finer localization of the spectrum of the operator B: for some C > 0, we have   sp(B) ⊂ z ∈ C; Re z ≥ C −1 e−C| Im z| . In fact, we prove that the resolvent estimate (6.3.2) remains valid in a neighborhood of the imaginary axis of the form {z ∈ C; | Re z| ≤ e−C0 | Im z| /C0 }, for some C0 > 0. See the beginning of the proof of Theorem 6.6 in Appendix 6.C. We can prove a resolvent estimate of the form (6.3.2) for the damped wave generator (A, D(A)) introduced in (6.2.8) and (6.2.10). Theorem 6.8. Let ω be a nonempty open subset of Ω and α be such that α >0 on ω. Then, the unbounded operator iσ IdH −A with domain D(A) = H 2 (Ω) ∩ H01 (Ω) × H01 (Ω) is invertible on H = H01 (Ω) ⊕ L2 (Ω) for all σ ∈ R and there exist K > 0 and σ0 > 0 such that (iσ IdH −A)−1 L (H ) ≤ KeK|σ| ,

σ ∈ R, |σ| ≥ σ0 .

The proof of Theorem 6.8 is given in Sect. 6.5 below. The main consequence of this estimate, with the result of Theorem 6.6, is the following stabilization theorem, using Proposition 6.5 that allows one to identify strong and weak solutions of the damped wave equation and solutions of the semigroup equation, and on the other hand the connection between the energy E(.)(t) and the norm in H given in (6.2.13). Theorem 6.9. Let ω be a nonempty open subset of Ω and α be such that α > 0 on ω. Let k ∈ N. Then, there exists C > 0 such that, if the initial condition Y 0 = t (y 0 , y 1 ) is in the domain of the operator Ak , the energy of the solution y(t) to (6.2.1) satisfies (6.3.3)

E(y)(t) ≤ 

C log(2 + t)

2

k 0 2k A Y H .

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6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

Remark 6.10. Note that, if α is flat at all orders at ∂Ω, the domain D(Ak ) can be made explicit. By Proposition 6.24 we have D(Ak ) = (k+1)/2 k/2 D(P0 ) × D(P0 ), where, for s ≥ 0, D(Ps0 ) is given by

D(Ps0 ) = {u = j uj φj ; (μsj uj )j ⊂ 2 (C)} ⊂ L2 (Ω), as defined in Sect. 10.1.3. The family of functions (φj )j forms a Hilbert basis of eigenfunctions of P0 , associated with the eigenvalues (μj )j ⊂ R+ , as introduced in Sect. 10.1. Before providing the proof of Theorem 6.8 we make several remarks. 6.4. Remarks and Non-Quantified Stabilization Results 6.4.1. Comparison with Exponential Stability. Observe that the 2 term Ak Y 0 H on the r.h.s. of the norm decay estimate (6.3.3) in Theorem 6.9 cannot be replaced by 2

2

2

Y 0 H = y 0 H 1 (Ω) + y 1 L2 (Ω) = 2E(y)(0). 0

In fact, if we had, for t > 0, E(y)(t) ≤ CE(y)(0)/ log(2 + t)2k , we could then conclude that we have exponential stabilization by the following Proposition. Proposition 6.11. Let us assume that there is a function f : R → R such that limt→+∞ f (t) = 0 and E(y)(t) ≤ f (t)E(y)(0), for any initial condition (y 0 , y 1 ) ∈ H . Then, there is exponential decay of the energy. As exponential stability requires the geometrical control condition (GCC) to be fulfilled by the open set {α > 0}, we see that this may very well not occur in the case of the arbitrary open set ω that we consider in the present stabilization problem. Proof of Proposition 6.11. Let T > 0 be such that R = f (T )−1 > 1. Then E(y)(T ) ≤ R−1 E(y)(0). By induction we have E(y)(jT ) ≤ R−j E(y)(0) for j ∈ N. Let t ≥ T . For some n ∈ N∗ , we have nT ≤ t < (n + 1)T . As the energy is a nonincreasing function we have E(y)(t) ≤ E(y)(nT ) ≤ R−n E(y)(0). As nT > t − T ≥ 0 we find E(y)(t) ≤ R1−t/T E(y)(0),

for t ≥ T,

and for t ∈ [0, T ] as well, as t → E(y)(t) is nonincreasing. This gives the result.  6.4.2. Zero Eigenvalue. Note that σ = 0 is not an eigenvalue for A here: iσ IdH −A is invertible for all σ ∈ R. This is connected to the existence of a boundary ∂Ω and the choice we have made for the boundary condition: here we impose homogeneous Dirichlet boundary condition. Consequently A is invertible. In the case of a manifold without boundary, 0 is an eigenvalue of A. This also holds in the case of homogeneous Neumann boundary conditions. Then, one needs to somehow ignore constant functions that are

6.4. REMARKS AND NON-QUANTIFIED STABILIZATION RESULTS

223

the associated eigenfunctions. Note that this is not an issue as the energy itself is insensitive to such functions. One can then consider the space of solutions after projecting onto a subspace stable by the operator with the constant solutions spanning the kernel of this projection. The same issue also occurs in the case of boundary damping. We refer the reader to Chap. 10 in Volume 2 where the projection described above is used to obtain the proper functional framework. 6.4.3. Non-Quantified Stabilization Results. Observe that in the case Y 0 = t (y 0 , y 1 ) ∈ H , the result of Theorem 6.9 does not provide any convergence towards 0 for the energy E(y)(t) as t → +∞, since this corresponds to the case k = 0. We may however prove that energy of any weak solution to the damped wave equation (6.1.2) goes to 0 as t → +∞. Yet, note that we cannot provide any estimation of the convergence speed. Proposition 6.12. Let S(t) = e−tB be a bounded C0 -semigroup on a Hilbert space H with associated (closed) generator B : H → H with dense domain D(B) ⊂ H. Assume that for any Y 0 ∈ D(B) we have S(t)Y 0 H → 0 as t → +∞. Then the same convergence property hold if Y ∈ H. We refer to Appendix 6.D.1 for a proof. Applied to the damped wave semigroup this proposition and Theorem 6.9 yield the following result as mentioned above. Corollary 6.13. The energy of any weak solution to the damped wave equation (6.1.2) goes to 0 as t → +∞. A second result for the stabilization of weak solutions that we provide is the following one that is independent from Theorem 6.9. Proposition 6.14. Let ω be a nonempty open subset of Ω and α be such that α > 0 on ω. Assume that the following unique continuation property holds: if Y 0 ∈ H and S(t)Y 0 ≡ 0 on (0, +∞) × ω then Y 0 = 0. In such case, the damped wave semigroup S(t) converges strongly to zero, that is, the energy of any weak solution of the damped wave equation (6.1.2) goes to 0 as t → +∞. We refer to Appendix 6.D.2 for a proof. We also refer to comments on this result in Sect. 6.7 We provide a third result based on a resolvent property. Compare with the result of Theorem 6.6. Proposition 6.15. Let S(t) = e−tB be a C0 -semigroup on a Hilbert space H with associated (closed) generator B : H → H with dense domain D(B) ⊂ H. We assume that the semigroup is bounded, that is, we have the following uniform bound (6.4.1)

sup S(t)L (H) < +∞, t≥0

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6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

and that moreover sp(B) ⊂ {Re z > 0}. Then, the semigroup strongly converges to zero as t → +∞, that is, for all y ∈ H we have S(t)y → 0 in H. We refer to Appendix 6.D.3 for a proof. 6.5. Resolvent Estimate for the Damped Wave Generator Here, we give the proof of the resolvent estimate of Theorem 6.8. First, for σ ∈ R, the resolvent operator (iσ IdH −A)−1 is well defined and continuous on H as the spectrum of A is contained in {z ∈ C; Re z > 0} by Proposition 6.25. We shall now estimate the operator norm of this resolvent. Let U ∈ D(A) and F ∈ H be such that (6.5.1)

(iσ IdH −A)U = F,

U = t (u0 , u1 ), F = t (f 0 , f 1 ).

Our goal is to find an estimate of the form U H ≤ ceC|σ| F H . The resolvent equation (6.5.1) reads iσu0 + u1 = f 0 , which we write (6.5.2) iσu0 + u1 = f 0 ,

(iσ − α)u1 − P0 u0 = f 1 ,

(−σ 2 − iσα + P0 )u0 = f,

with f = (iσ − α)f 0 − f 1 .

Multiplication of the second equation by u0 and an integration over Ω give (6.5.3)

2

((−σ 2 + P0 )u0 , u0 )L2 (Ω) − iσα1/2 u0 L2 (Ω) = (f, u0 )L2 (Ω) .

The first term is real and the second term is purely imaginary. From (6.5.3) we have 2

σα1/2 u0 L2 (Ω) = − Im(f, u0 )L2 (Ω) . Since there exist ω0 an open subset of ω and δ > 0 such that α ≥ δ > 0 in ω0 , we obtain (6.5.4)

2

2

δσ0 u0 L2 (ω0 ) ≤ |σ| α1/2 u0 L2 (Ω) ≤ f L2 (Ω) u0 L2 (Ω) ,

for |σ| ≥ σ0 . A key estimate is given by the following observation lemma. We provide two different proofs below. Lemma 6.16. There exists C > 0 such that   u0 H 1 (Ω) ≤ CeC|σ| f L2 (Ω) + u0 L2 (ω0 ) . Then, estimate (6.5.4) yields 1 1   u0 H 1 (Ω)  eC|σ| f L2 (Ω) + u0 L2 2 (Ω) f L2 2 (Ω) , and with the Young inequality we obtain u0 H 1 (Ω)  e2C|σ| f L2 (Ω) .

6.5. RESOLVENT ESTIMATE FOR THE DAMPED WAVE GENERATOR

225

Using the form of f given in (6.5.2) we then obtain   u0 H 1 (Ω)  e2C|σ| f 0 L2 (Ω) + f 1 L2 (Ω) . Finally as u1 = f 0 − iσu0 we obtain (6.5.5)

  u0 H 1 (Ω) + u1 L2 (Ω)  e2C|σ| f 0 L2 (Ω) + f 1 L2 (Ω) ,

yielding the resolvent estimate of Theorem 6.8.



Below, we expose two proof strategies to obtain the estimate of Lemma 6.16. They are related and both based on Carleman estimates for an elliptic operator. The first approach we shall present is based on the quantification of unique continuation obtained in Chap. 5, here applied to the operator Q = Ds2 + P0 + αDs . Recall that the results of Chap. 5 are based on the Carleman estimates derived in Chap. 3. The second approach is based on the derivation of a global Carleman estimate for the same operator Q (patching together local estimates as presented in Sects. 3.5 and 3.6), which then directly yields the estimate of Lemma 6.16. Both approaches are valuable and are worth presenting here, in particular in view of other settings where such strategies can or cannot be applied; see the notes in Sect. 6.7. For a better understanding of the two proofs of Lemma 6.16 that we provide below, we placed diagrams of the two proof strategies in Fig. 6.1. 6.5.1. Estimations Through an Interpolation Inequality. In this section, we provide a proof of Lemma 6.16, and thus Theorem 6.17, that is based on the results on the quantification of the unique continuation property that were obtained in Chap. 5, first, in the interior of a domain and, second, up to (part) of its boundary. The geometrical setting we introduce here is a particular case of what is considered in that chapter. Let S0 > 0 and β ∈ (0, S0 /2). Let also Z = (0, S0 ) × Ω and Y = (β, S0 − β) × Ω. We set z = (s, x) with s ∈ (0, S0 ) and x ∈ Ω. We define the following augmented elliptic operator Q := Ds2 + P0 + R in Z, where R denotes any first-order differential operator in Z with bounded coefficients. Theorem 6.17. Let O be an open set in Z. There exist C > 0 and δ ∈ (0, 1) such that for u ∈ H 2 (Z) that satisfies u(s, x)|x∈∂Ω = 0 for s ∈ (0, S0 ) we have  δ Qu (6.5.6) + u . uH 1 (Y ) ≤ Cu1−δ L2 (Z) L2 (O) H 1 (Z) As Q is elliptic in Z and as Y denotes an open set that only meets smooth parts of the boundary of Z, Theorem 6.17 is a direct consequence of Theorem 5.11. (First) Proof of Lemma 6.16. If we have (−σ 2 − iσα + P0 )u0 = f, we set u = esσ u0 . With Q = Ds2 +P0 +αDs here, we observe that Qu = esσ f . We then apply the interpolation inequality of Theorem 6.17: with S0 > 0,

226

6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

local Carleman estimates away from boundaries for Ds2 + P0 ; Theorem 3.11

interpolation inequality Theorems 5.11 and 6.17

global Carleman estimate Theorem 3.34

local Carleman estimates at points of (0, S0 ) × ∂Ω for Ds2 + P0 ; Theorem 3.29

Lemma 6.16

resolvent estimate Theorem 6.8 Theorem 6.6 stabilization result Theorem 6.9

Figure 6.1. The two proof schemes for the stabilization of the damped wave equation and β ∈ (0, S0 /2) and 0 < β1 < β2 < S0 we have C > 0 and δ0 > 0 such that δ    Qu (6.5.7) + u . uH 1 (Y ) ≤ Cu1−δ L2 (Z) 2 H 1 (Z) L

(β1 ,β2 )×ω0

Next, we note that we have QuL2 (Z)  eC|σ| f L2 (Z) , uH 1 (Y ) ≥ u 2  L

(β,S0 −β);H 1 (Ω)

 ≥ e−C|σ| u0  1 , H (Ω)

uH 1 (Z)  eC|σ| u0 H 1 (Ω) , u 2  L

(β1 ,β2 )×ω0

  eC|σ| u0  2 L (ω0 ) ,

yielding with (6.5.7)   u0 H 1 (Ω)  eC|σ| f L2 (Z) + u0 L2 (ω0 ) . This concludes the proof of the estimate of Lemma 6.16.



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227

6.5.2. Estimations Through the Derivation of a Global Carleman Estimate. In this section, we provide a proof of Lemma 6.16, and thus Theorem 6.17, that is based on the derivation of a global Carleman estimate through the patching a local estimates as presented in Sects. 3.5 and 3.6. Let S0 > 0 and Z = (0, S0 ) × Ω. We choose ω1 an open set of Ω such that ω1  ω0 . We first choose a function ψ1 ∈ C ∞ (Ω) according to Proposition 3.31: ∂ν ψ1 |∂Ω (x) < 0,

for x ∈ ∂Ω,

and

|dψ1 (x)| = 0,

for x ∈ Ω \ ω1 .

Adding a sufficiently large constant to the function, we moreover have ψ1 ≥ C > 0. We also set ψ2 (s) = sin(πs/S0 ) and we define ψ(s, x) = ψ1 (x)ψ2 (s). The critical points of ψ are located in the set {S0 /2} × ω1 . With 0 < s3 < S0 /2 we set s3 = S0 − s3 and B3 = {(s, x) ∈ Z; s3 < s < s3 }. We introduce O an open subset of B3 such that {S0 /2}×ω1 ⊂ O ⊂ (s3 , s3 )× ω0 . We then choose s2 > 0 sufficiently small to have sin(πs2 /S0 ) sup ψ1 < sin(πs3 /S0 ) inf ψ1 , Ω

Ω

and set s2 = S0 − s2 and define B2 = {(s, x) ∈ Z; 0 ≤ s < s2 },

B2 = {(s, x) ∈ Z; s2 < s ≤ S0 }.

We thus have sup ψ = sup ψ < inf ψ. B2

B2

B3

Setting ϕ(s, x) = exp(γψ(s, x)), with γ > 0, we have (6.5.8)

sup ϕ = sup ϕ < inf ϕ. B2

B2

B3

The geometry we have just introduced is illustrated in Fig. 6.2. Let 0 < s1 < s2 and s1 = S0 − s1 . We choose χ(s) ∈ Cc∞ (s1 , s1 ) such that χ ≡ 1 on a neighborhood of (s2 , s2 ). (Second) Proof of Lemma 6.16. We have (−σ 2 − iσα + P0 )u0 = f . We set u = esσ u0 and observe that Qu = esσ f , for Q = Ds2 + P0 + αDs . According to Lemma 3.5, we pick γ > 0 sufficiently large for (Q, ϕ) to fulfill the sub-ellipticity property of Definition 3.2 in  Z \ O. We set Γ0 =  (s1 /2, S0 − s1 /2) × ∂Ω and W = [0, s1 ) ∪ (s1 , S0 ] × Ω. These sets are represented in Fig. 6.2. Observe that ϕ is a global Carleman weight function on Z adapted to Γ0 and O in the sense of Definition 3.30.

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6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

s S0 s1 Γ0 s2 s3 S0 /2 s3 s2

W

B2 O

s1 0

B3

B2

W Ω

ω0

Figure 6.2. Geometry for the derivation of the resolvent estimate through a global Carleman estimate We can then invoke the global Carleman estimate of Theorem 3.34 and apply it to the function χu: there exist τ∗ > 0 and C ≥ 0 such that (6.5.9) τ 3 eτ ϕ χu2L2 (Z) + τ eτ ϕ ∇s,x (χu)2L2 (Z)  eτ ϕ Q(χu)2L2 (Z) + τ 3 eτ ϕ u2L2 (O) , for τ ≥ τ∗ , using that u satisfies homogeneous Dirichlet boundary conditions on (0, S0 ) × ∂Ω. Observe that Qχu = χesσ f + [Q, χ]u where the commutator is a firstorder differential operator in the variable s and supported in B2 ∪ B2 . We thus have (6.5.10) eτ ϕ QχuL2 (Z)  eτ ϕ+sσ f L2 (Z) + eτ ϕ uL2 (B2 ∪B  ) + eτ ϕ ∂s uL2 (B2 ∪B  ) 2

 eτ ϕ+sσ f L2 (Z) + (1 + |σ|)eτ ϕ+sσ u0 L2 (B2 ∪B  ) 2

e

C(τ +|σ|)

f L2 (Ω) + e

τ supB2 ϕ C|σ|

e

0

u L2 (Ω) ,

and (6.5.11)

eτ ϕ uL2 (O)  eC(τ +|σ|) u0 L2 (ω0 ) .

We also have (6.5.12)

eτ ϕ χuL2 (Z) ≥ eτ ϕ uL2 (B3 )  eτ inf B3 ϕ−C|σ| u0 L2 (Ω) ,

and (6.5.13) eτ ϕ ∇s,x (χu)L2 (Z)  eτ ϕ ∇x uL2 (B3 )  eτ inf B3 ϕ−C|σ| ∇x u0 L2 (Ω) .

2

6.6. ALTERNATIVE PROOF SCHEME OF THE RESOLVENT ESTIMATE

229

Collecting estimates (6.5.9)–(6.5.13) we obtain, for some L > 0,   eτ inf B3 ϕ u0 H 1 (Ω) ≤ CeC(τ +|σ|) f L2 (Ω) + u0 L2 (ω0 ) + eτ supB2 ϕ+L(1+|σ|) u0 L2 (Ω) . Setting η = inf B3 ϕ − supB2 ϕ which is positive by (6.5.8), if we choose   τ = η −1 L(1 + |σ|) + 1 + τ∗ , we find eτ inf B3 ϕ − eτ supB2 ϕ+L(1+|σ|) ≥ C > 0. This yields the result of Lemma 6.16.  6.6. Alternative Proof Scheme of the Resolvent Estimate We sketch here yet another proof of Lemma 6.16, based on a Carleman estimate for the operator Pσ = P0 − σ 2 − iσα, for |σ| ≥ σ0 > 0. Once the result of Lemma 6.16 is achieved, Sect. 6.5 shows that the resolvent estimate of Theorem 6.8 follows (see also Fig. 6.1). In fact, observe that Carleman estimates of the form given in Chap. 3 can be obtained for this operator Pσ if one imposes the large parameter to satisfy τ  |σ|. Near a boundary point (an estimate away from a boundary can be deduced from that at the boundary) the estimate takes the following form. It is the counterpart of the result of Theorem 3.29 Theorem 6.18. Let V 0 be a bounded open set in Rd such that the boundary ∂Ω is C ∞ in a neighborhood of V 0 and let ϕ and P have the sub-ellipticity property of Definition 3.2 in a neighborhood of V 0 ∩ Ω with moreover ∂ν ϕ|V 0 ∩∂Ω < 0. For σ0 ≥ 1, there exist τ∗ > 0, and C > 0 such that (6.6.1) τ 3 eτ ϕ u2L2 (Ω) + τ eτ ϕ Du2L2 (Ω) + τ |eτ ϕ ∂ν u|∂Ω |2L2 (∂Ω)   ≤ C eτ ϕ Pσ u2L2 (Ω) + τ 3 |eτ ϕ u|∂Ω |2L2 (∂Ω) + τ |eτ ϕ u|∂Ω |2H 1 (∂Ω) , Ω

for u ∈ Cc∞ (V 0 ) and τ ≥ τ∗ |σ|, for |σ| ≥ σ0 . A proof is given below. With this Carleman estimate we obtain results that are the counterparts of those of Sects. 5.3 and 5.4. First we have the following lemma, counterpart of Lemma 5.3. Lemma 6.19 (Local Interpolation Inequality). Let Ω be an open set in Rd et let x(0) ∈ Ω and r > 0 be such that B(x(0) , 4r) ⊂ Ω. Let C0 > 0. There exist C > 0 and δ ∈ (0, 1) such that  δ (6.6.2) uH 1 (B(x(0) ,3r) ≤ CeC|σ| u1−δ H 1 (Ω) f L2 (Ω) + uH 1 (B(x(0) ,r) , for u ∈ H 2 (Ω) satisfying Pσ u = f + g,

with f ∈ L2 (Ω),

  |g(x)| ≤ C0 |u(x)| + |Du(x)| a.e. in Ω.

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6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

Observe that the constant that appears in the inequality is now of the form K = CeC|σ| (compare with (5.3.1)). In fact, the proof can be carried out mutatis mutandis and one obtains as well (5.3.6):   uH 1 (B(x(0) ,3r)) e−τ (C2 −C1 ) uH 1 (Ω) +eτ (C3 −C2 ) f L2 (Ω) +uH 1 (B(x(0) ,r)) . Here, however we have the condition τ ≥ τ∗ |σ|. Applying the optimization procedure of Lemma 5.4 then yields the announced form of the constant K = CeC|σ| in (6.6.2). Similarly, a local interpolation estimate at the boundary can be obtained, yielding a result that is the counterpart of Lemma 5.12. Lemma 6.20 (Local Interpolation Inequality with Boundary Observations). Let y ∈ ∂Ω and V a open neighborhood of y in Rd , such that ∂Ω is smooth in V . Let C0 > 0. There exist W a neighborhood of y in Rd , C > 0, and δ ∈ (0, 1) such that uH 1 (W ∩Ω) ≤ CeC|σ| u1−δ H 1 (Ω)  δ × f L2 (Ω) + |u|∂Ω |H 1 (V ∩∂Ω) + |∂ν u|∂Ω |L2 (V ∩∂Ω) , for u ∈ H 2 (Ω) satisfying Pσ u = f + g,

with f ∈ L2 (Ω),

  |g(x)| ≤ C0 |u(x)| + |Du(x)| a.e. in Ω.

The propagation procedure described in the proofs of Theorems 5.6 and 5.13 then yields the following global interpolation inequality initiated from an open set ω  Ω. Theorem 6.21. Let Ω be a connected open set in Rd and let ω and U be two open subsets of Ω with U bounded. Assume also that in a neighborhood of U the boundary ∂Ω is smooth. Let C0 > 0. There exist C > 0 and δ ∈ (0, 1) such that  δ uH 1 (U ) ≤ CeC|σ| u1−δ H 1 (Ω) f L2 (Ω) + uL2 (ω) , for u ∈ H 2 (Ω) satisfying u|∂Ω = 0 in a neighborhood of U and   Pσ u = f + g, with f ∈ L2 (Ω), |g(x)| ≤ C0 |u(x)| + |Du(x)| a.e. in Ω. Applied to u0 that satisfies Pσ u0 = f in (6.5.2), with U = Ω and g = 0 here, we then obtain the result of Lemma 6.16. Proof of Theorem 6.18. We only need to prove a local result of the form of Lemma 3.16. Then patching such estimates together as in Sect. 3.5 we obtain the sought result. We use the local setting of Sect. 3.4.2. In particular we use normal geodesic coordinates. Observe that we may assume that α = 0 here, as the introduction of the lower-order term −iσα in the operator Pσ only affects constants in the Carleman estimate (since τ /|σ| is to be chosen large). We thus have Pσ = P0 − σ 2 in the present proof.

6.6. ALTERNATIVE PROOF SCHEME OF THE RESOLVENT ESTIMATE

231

We set v = eτ ϕ u and Pσ,ϕ = eτ ϕ Pσ e−τ ϕ and we write Pσ,ϕ = P2 −σ 2 +iP1 with P2 and P1 with P2 and P1 defined in (3.4.7)–(3.4.8). We then see that identity (3.4.22) becomes (6.6.3)

  2 ˆ Pσ,ϕ v2+ = (P2 − σ 2 )v+ + P1 v2+ + Re i[P2 , P1 ]v, v + + τ Re B(v).

with ˆ ˜ B(v) = B(v) + 2τ σ 2 (∂d ϕv|xd =0+ , v|xd =0+ )∂ . Observe that the modification of the boundary term is harmless since it is small if compared to τ 3 |eτ ϕ u|xd =0+ |∂ that is found on the r.h.s. of the Carleman estimate if one chooses τ /|σ| sufficiently large. Following the computations made in the proof of Proposition 3.24, (3.4.27) now reads   2 ˆ Pσ,ϕ v2+ = (P2 − σ 2 )v+ + P1 v2+ + Re τ OpT (b2 )v, v + + τ Re B(v)   + Re (τ b0 P2 + OpT (b1 )P1 + τ Op(c1 ))v, v + . Computations up to (3.4.36) then remain unchanged and (3.4.37) now becomes 2 2 ˆ Pσ,ϕ v2+ ≥ Cτ Λ1T,τ v+ + (P2 − σ 2 )v+ + P1 v2+ + τ Re B(v)   + Re (τ b0 P2 + OpT (b1 )P1 + τ Op(c1 ))v, v + − τ Re(OpT (c1 )v, v)+ − Re(τ r0 P2 v + Op(r1 )P1 v + τ Op(d1 )v, OpT (c0 )v)+ , which we change into 2 2 ˆ Pσ,ϕ v2+ ≥ Cτ Λ1T,τ v+ + (P2 − σ 2 )v+ + P1 v2+ + τ Re B(v)   + Re (τ b0 (P2 − σ 2 ) + OpT (b1 )P1 + τ Op(c1 ))v, v +

− τ Re(OpT (c1 )v, v)+ − Re(τ r0 (P2 − σ 2 )v + Op(r1 )P1 v

+ τ Op(d1 )v, OpT (c0 )v)+ − C  τ σ 2 v2+ .

With B(v) now given by 1 ˆ B(v) = − τ −1 Im(P1 v|xd =0+ , ∂d ϕOpT (c0 )v|xd =0+ )∂ + B(v), 2 and we obtain in place of (3.4.38), for τ /|σ| chosen sufficiently large, 2

2

Pσ,ϕ v2+ ≥ Cτ Λ1T,τ v+ + (P2 − σ 2 )v+ + P1 v2+ + τ Re B(v)   2 − C  τ −1/2 (P2 − σ 2 )v+ + P1 v2+   2 − C  τ 1/2 Λ1T,τ v+ + Dd v2+ . The end of the proof in the similar to that of Proposition 3.24 and Lemma 3.16 in particular to address the boundary term τ Re B(v). 

232

6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

6.7. Notes Various aspects of stabilization can be found in the expositions by V. Komornik [199], J.-M. Coron [109], and M. Tucsnak and G. Weiss [321]. For the wave equation, exponential decay of the energy can be achieved if the so-called geometrical control condition (GCC) is fulfilled. This result was first proven in dimension one or on manifolds without boundary by J. Rauch and M. Taylor [280]. The GCC expresses that all bicharacteristics (or rays of geometrical optics) reach the damping region in a finite time. The generalization of this exponential decay result to domains with boundaries in the case of homogeneous Dirichlet and Neumann conditions was proven in joint work with C. Bardos and J. Rauch [55, 56]. Bicharacteristics are then replaced by so-called generalized bicharacteristics that obey the laws of reflection at a boundary. There are introduced by R. Melrose and J. Sj¨ostrand [246, 247] to describe the propagation of singularities. The proof in [56] is based on this description of the propagation of singularities. Other methods of proof are based on the use of microlocal defect measures as introduced by L. Tartar [312] and P. G´erard [159]; we refer to the works of P. G´erard and Leichtnam [160], N. Burq [93], N. Burq and P. G´erard [95], J. Sj¨ ostrand [305], L. Miller [252] joint work with N. Burq [97], and work by N. Anantharaman [32]. Note that exponential decay of the energy is equivalent to having an observability estimate for the wave equation (without damping), the observation being located in the region where the damping acts; we refer to the work of A. Haraux [165]. For the wave equation, stabilization (and equivalently observability) can also be expressed by means of the so-called Hautus test for the resolvent; we refer to the works of D. Russell and G. Weiss [295] and of L. Miller [254], and Chapter 6 of the book of M. Tucsnak and G. Weiss [321] that contains many references. Under weaker geometrical conditions one can obtain a polynomial decay rate of the damped wave equation. In existing results the GCC does not hold yet only “few” bicharacteristics are missed. We refer to the works of Z. Liu and B. Rao [237], K.-D. Phung [273], N. Burq and M. Hitrik [96], H. Nishiyama [265], N. Anantharaman and M. L´eautaud [33], N. Burq and C. Zuily [98, 99], and M. L´eautaud and N. Lerner [216]. In the present chapter, no geometrical assumption is formulated. Hence, the logarithmic decay rate obtained in Theorem 6.9 for a damping acting in an arbitrary region appears as the lower bound one can reach. In fact, this result can be proven optimal, as is done in [217] through an example on a surface of revolution in R3 . A first proof of the stabilization result of Theorem 6.9 for the wave equation with a factor log(2 + log(2 + t))/ log(2 + t)k in place of 1/ log(2 + t)k in the semigroup decay rate estimate was given in [217]. This result was improved by N. Burq in [94] achieving the optimal decay rate. More recently,

6.7. NOTES

233

C. Batty and T. Duyckaerts have generalized this stabilization result to other types of resolvent estimates in [58]. See also the work of A. Borichev and Y. Tomilov [78] for some improved result in the case of polynomial stabilization, a case that does not occur in the material we cover. Here, we have followed the proof of [58] in the particular case of the wave equation. We also refer to the recent work of J. Rozendaal, D. Seifert, and R. Stahn [293] for some precised correspondance between the resolvent estimate and the decay rate. Note the result of Theorem 6.9 can be generalized to the case of a Riemannian manifold with boundary. We have chosen to only consider the case of a bounded open set in Rd for the sake of exposition. However, with the Carleman estimates obtained on Riemannian manifold in Chap. 5 in Volume 2 and with the analysis of Chaps. 17 and 18 also in Volume 2, the adaptation of the proof can be carried out. Note that Chap. 10 in Volume 2 is also dedicated to the subject of stabilization, yet in the case of boundary damping; there results are given in the manifold case. Apart from the logarithmic stabilization result of Theorem 6.9 we also provide strong stabilization results, meaning that no decay rate is provided. One result, Proposition 6.14, shows that strong stabilization is a consequence of unique continuation for the wave operator, a topic not covered in the present book; we refer the reader to the works of J. Rauch and M. Taylor [279] and N. Lerner [226]. These two results the solution to vanish in an infinite cylinder. This assumption is relaxed in [288] and the subsequent works of L. H¨ormander [177], D. Tataru [313, 315], and joint work with C. Zuily [290]. For recent results on the quantification of unique continuation for the wave equation we refer to the simultaneous works of R. Bosi, Y. Kurylev and M. Lassas [79] and C. Laurent and M. L´eautaud [205]. Note that the strong stabilization result of Proposition 6.15 can be found in the work of C. Batty [57]; we follow his proof. Observe that the characterization of the spectrum of the generator of the damped wave semigroup is not sufficient to obtain quantified stabilization results. By Proposition 6.15, if sp(A) ⊂ {z ∈ C; Re(z) > 0} the energy of all weak solutions goes to zero. However, having sp(A) ⊂ {z ∈ C; Re(z) ≥ C0 } for some C0 > 0 does not imply exponential stability without an estimation of the resolvent. In [217], Theorem 2.(iii) provides an example of a damped equation on a manifold with such a localization of the spectrum and yet no exponential stabilization is achieved. The first method of proof for the resolvent estimate of Lemma 6.16 we present is based on the quantification of unique continuation of Theorem 5.11 itself based on local Carleman estimates. Exposed in Sect. 6.5.1, this method follows what is often found in the literature going back the works [217, 219]. The second method of proof for Lemma 6.16 exposed in

234

6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

Sect. 6.5.2 is newer in the literature and can be useful in particular situations where local interpolation cannot be easily derived. This was done for instance in the case of discretized operators in joint works with F. Boyer and F. Hubert [85, 86]. This was also done in an unbounded domain in joint work with I. Moyano [212], as the propagation argument of the first method relies on compactness. Note that the resolvent estimate could be deduced from local Carleman estimates (away and at the boundary) for the operator P0 − σ 2 , uniform with respect to σ, for Cτ ≥ |σ| ≥ 1, where τ is the large parameter in the Carleman estimate. This approach is followed for instance by M. Bellassoued [67–69], L. Ouksel [268] and R. Buffe [92]. Here, we have considered the case of homogeneous Dirichlet boundary conditions. For Neumann boundary conditions one can refer to the work of X. Fu [154]. In fact, with the analysis performed in Sect. 4.2 in Volume 2 ˇ describing the Lopatinski˘ı-Sapiro boundary conditions that yield a selfadjoint second-order elliptic operator on a bounded open set (or a manifold) one can adapt the proofs in the present chapter to achieve a logarithmic stabilization result as in Theorem 6.9. Note that if 0 is an eigenvalue of the elliptic operator, as in the case of homogeneous Neumann boundary conditions the analysis needs to be carried out in a quotient space; this is done in Chap. 10 in Volume 2 in the case of a boundary damping. For stabilization with mixed type conditions we refer to X. Fu [155] and joint work with P. Cornilleau [108]. For stabilization under Ventcel type conditions we refer to R. Buffe [92]. We refer also to the work of L. Ouksel [267, 268] for transmission problems in the case of a heterogeneous geometry. The case of transmission problems for heterogeneous types of operators we refer for instance to the works of J. Rauch, X. Zhang, and E. Zuazua [281, 329], T. Duyckaerts [130], I. Khamoun-Fathallah [190] where a wave equation is stabilized throught the coupling with a heat equation. This latter setting is a basic model for the stabilization of fluid-structure couling; we refer to the work of M. Badra and T. Takahashi [46] for a recent reference for these developments and for references. Here, our focus is on damping terms that depend linearly on the solution. Nonlinear approaches are also of interest; we refer to some results by I. Lasiecka and D. Tataru [203], by V. Komornik [199], F. Alabau-Boussouira and K. Ammari [8], F. Alabau-Boussouira [7], and F. Alabau-Boussouira, Y. Privat and E. Tr´elat [12] and the references given therein. For equations with time delay terms we refer to F. Alabau-Boussouira, S. Nicaise, and C. Pignotti [11], S. Nicaise and C. Pignotti [264]. For results on the stabilization of systems of hyperbolic equations through coupling terms and a minimal number of damping terms we refer to F. Alabau-Boussouira [6], F. Alabau-Boussouira and M. L´eautaud [10], and R. Guglielmi [164].

6.A. THE GENERATOR OF THE DAMPED-WAVE SEMIGROUP

235

Appendices 6.A. The Generator of the Damped-Wave Semigroup The damped wave equation associated with the operator P0 reads in a system form (see Sect. 6.2) ∂t Y + AY = 0, with

 A=

Y (0) = Y 0 ∈ H = H01 (Ω) ⊕ L2 (Ω),

 0 −1 , P0 α(x)

D(A) = D(P0 ) ⊕ H01 (Ω).

We recall that D(P0 ) = H 2 (Ω) ∩ H01 (Ω). We refer to Sect. 10.1.1 for properties of the unbounded operator (P0 , D(P0 )) on L2 (Ω). As (P0 , D(P0 )) is a closed operator, we see that A is closed too. We provide some additional properties of the operator A. Proposition 6.22. We have V D(A)  AV H  V H 2 (Ω)⊕H 1 (Ω) ,

V ∈ D(A).

Proof. On the one hand, for V = t (v 0 , v 1 ) ∈ D(A), we have AV H  v 1 H 1 (Ω) + P0 v 0 + αv 1 L2 (Ω)  v 1 H 1 (Ω) + P0 v 0 L2 (Ω) 0

0

 v 1 H 1 (Ω) + v 0 H 2 (Ω)  V H 2 (Ω)⊕H 1 (Ω) , 0

v 0 H 2 (Ω)

using that  (10.1.6). We thus have

0

P0 v 0 L2 (Ω)

for

v0

AV H  V H 2 (Ω)⊕H 1 (Ω) ,

∈ D(P0 ) = H 2 (Ω) ⊕ H01 (Ω) by V ∈ D(A).

On the other hand we write AV H  V D(A)  V H + AV H  V H 2 (Ω)⊕H 1 (Ω)  AV H , 0

using the first equivalence obtained. This concludes the proof.



Proposition 6.23. The adjoint operator A∗ of A on H is defined on the domain D(A∗ ) = D(A) and   0 1 ∗ . A = −P0 α(x) In particular if α = 0 then A is skew-adjoint which expresses that the energy E(y)(t) defined in (6.2.6) is preserved in this case as observed in (6.2.7). Proof. The domain of A∗ is given by D(A∗ ) = {V ∈ H ; ∃C > 0, ∀U ∈ D(A), |(V, AU )H | ≤ CU H },

236

6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

and we recall that, by (10.1.15), (V, V  )H = (v 0 , v 0 )H01 (Ω) + (v 1 , v 1 )L2 (Ω) = P0 v 0 , v 0 H −1 (Ω),H01 (Ω) + (v 1 , v 1 )L2 (Ω) , for V = (v 0 , v 1 ) and V  = (v 0 , v 1 ). First, we prove that D(A∗ ) ⊂ D(A). Let thus V = (v 0 , v 1 ) ∈ D(A∗ ) and U = (u0 , u1 ) ∈ D(A). We have (V, AU )H = −(v 0 , u1 )H01 (Ω) + (v 1 , P0 u0 + αu1 )L2 (Ω) . We first assume u0 = 0. This yields, as v 1 ∈ L2 (Ω), |(v 0 , u1 )H01 (Ω) |  u1 L2 (Ω) ,

u1 ∈ H01 (Ω),

which we write, according to (10.1.15), |P0 v 0 , u1 H −1 (Ω),H01 (Ω) |  u1 L2 (Ω) ,

u1 ∈ H01 (Ω),

implying that P0 v 0 ∈ L2 (Ω) by Proposition 10.10. As v 0 ∈ H01 (Ω) we conclude that v 0 ∈ D(P0 ) by the elliptic regularity theory recalled at the beginning of Sect. 10.1. If we now choose u1 = 0 we find |(v 1 , P0 u0 )L2 (Ω) |  u0 H 1 (Ω) . 0

By Proposition 10.7 we then have

v1

∈ H01 (Ω). We thus D(A∗ ). Let thus V

have D(A∗ ) ⊂ D(A).

Second, we prove that D(A) ⊂ ∈ D(A). In fact, for U ∈ D(A), with the regularity of V and using (10.1.18), we may write (6.A.1)

(V, AU )H = −(v 0 , u1 )H01 (Ω) + (v 1 , P0 u0 + αu1 )L2 (Ω)

= −(P0 v 0 , u1 )L2 (Ω) + (v 1 , u0 )H01 (Ω) + (αv 1 , u1 )L2 (Ω) ,

from which we find |(V, AU )H |  U H .

Finally, if V ∈ D(A∗ ) and U ∈ D(A), with (6.A.1) we write (V, AU )H = (v 1 , u0 )H01 (Ω) + (−P0 v 0 + αv 1 , u1 )L2 (Ω) = (A∗ V, U )H ,

with A∗ as given in the statement of the proposition.



The following proposition makes the domain D(Ak ) explicit by means of the domains D(Ps0 ), for s ≥ 0, that are defined in Sect. 10.1.3. Proposition 6.24. Assume that α ∈ C ∞ (Ω) and if α is flat at all orders (k+1)/2 k/2 at ∂Ω. Let k ∈ N. We have D(Ak ) = D(P0 ) ⊕ D(P0 ). Proof. We have D(A0 ) = D(IdH ) = H = H01 (Ω) ⊕ L2 (Ω),   D(A1 ) = H 2 (Ω) ∩ H01 (Ω) ⊕ H01 (Ω), which coincides with the formula for k = 0 and k = 1 in the statement as 1/2 D(P0 ) = H01 (Ω) and D(P0 ) = H 2 (Ω) ∩ H01 (Ω).

6.A. THE GENERATOR OF THE DAMPED-WAVE SEMIGROUP

237

Let us now assume that this formula holds for some k ∈ N, with k ≥ 1. We recall the inductive definition D(Ak+1 ) = {U ∈ D(Ak ); AU ∈ D(Ak )}. Let us assume that U = (u0 , u1 ) ∈ D(Ak+1 ). Then −u1 = f 0 ,

P0 u0 + αu1 = f 1 ,

with (k+1)/2

F = (f 0 , f 1 ) ∈ D(Ak ) = D(P0 (k+1)/2

By Lemma 10.20, as D(P0 α we obtain

(k+1)/2

u1 ∈ D(P0

),

k/2

) ⊕ D(P0 ).

k/2

) ⊂ D(P0 ) = K k (Ω), by the properties of P0 u0 = f 1 − αu1 ∈ D(P0 ). k/2

(k+1)/2

D(P0 ) ⊕ D(P0 ) and k ≥ 1 we have u0 ∈ D(P0 ). Since U ∈ D(Ak ) =

Consequently, u0 = j∈N αj φj and P0 u0 = j∈N μj αj φj where both series k/2

1+k/2

converge in L2 (Ω). Yet as P0 u0 ∈ D(P0 ) we obtain moreover (μj k/2

2 (C),

meaning precisely

u0

∈

1+k/2 D(P0 ).

proof.

αj ) j ∈

This concludes the inductive 

We now consider the spectral properties of the unbounded operator A. Proposition 6.25. The spectrum of A is contained in {z ∈ C; Re(z) > 0}. For the proof of this proposition we shall need to following lemma. Lemma 6.26. 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).

Proof of Lemma 6.26. Let U = (u0 , u1 ) ∈ D(A) = D(P0 ) ⊕ H01 (Ω). We write ((z IdH −A)U, U )H = (zu0 + u1 , u0 )H01 (Ω) + (−P0 u0 + (z − α)u1 , u1 )L2 (Ω)

= (zu0 + u1 , P0 u0 )L2 (Ω) + (−P0 u0 + (z − α)u1 , u1 )L2 (Ω) = (zu0 , P0 u0 )L2 (Ω) + ((z − α)u1 , u1 )L2 (Ω) + 2i Im(u1 , P0 u0 )L2 (Ω) .

We find, computing the real part, (6.A.2) − Re((z IdH −A)U, U )H = − Re(z)(u0 , P0 u0 )L2 (Ω) + ((α − Re(z))u1 , u1 )L2 (Ω) . As α ≥ 0 and Re(z) < 0, with (10.1.18), we find   2 2 | Re((z IdH −A)U, U )H | ≥ | Re(z)| u0 H 1 (Ω) + u1 L2 (Ω) , 0

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6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

which yields the conclusion for B = A. In the case B = A∗ we similarly have ((z IdH −A∗ )U, U )H = (zu0 , P0 u0 )L2 (Ω) + ((z − α)u1 , u1 )L2 (Ω) − 2i Im(u1 , P0 u0 )L2 (Ω) , which allows one to reach the same conclusion.



Proof of Proposition 6.25. Let z ∈ C. We consider the two cases. Case 1: Re z < 0. By Lemma 6.26 z IdH −A is injective. Moreover, as its adjoint z IdH −A∗ is injective and satisfies (z IdH −A∗ )U H  U H for U ∈ D(A∗ ) by Lemma 6.26, using B = A∗ therein, the map z IdH −A is surjective (see e.g. [90, Theorem 2.20]). The estimation of Lemma 6.26, for B = A, then gives the continuity of the operator (z IdH −A)−1 on H . Case 2: Re z = 0. We start by proving the injectivity of z IdH −A. Let thus U = (u0 , u1 ) ∈ D(A) be such that zU − AU = 0. This gives (6.A.3)

zu0 + u1 = 0,

−P0 u0 + (z − α)u1 = 0.

First, if z = 0 we have u1 = 0 and P0 u0 = 0 with u0 ∈ D(P0 ). Thus u0 = 0. Second, if now z = 0, using (6.A.2) we obtain 0 = Re((z IdH −A)U, U )H = −(αu1 , u1 )L2 (Ω) . This implies u1 vanishes on supp(α) as α ≥ 0. Then u0 also vanishes on supp(α) by (6.A.3) and P0 u0 = P0 u0 = zu1 = −z 2 u0 . With the unique continuation result of Theorem 5.2 we obtain that u0 vanishes in Ω 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 = T + IdH with T = (z − 1) IdH −A. By the first part of the proof, T is invertible with a bounded inverse. The operator T in unbounded on H . We denote by T˜ the restriction of T to D(A) equipped with the graph-norm associated with A. The operator T˜ is bounded by Proposition 6.22. It is also invertible. It is thus a bounded Fredholm operator of index ind T˜ = 0 (see Definition 11.6). Similarly, we denote by ι the injection of D(A) into H and A˜ the restriction of A on D(A) viewed as a bounded operator. We have zι − A˜ = T˜ + ι. Since ι is a compact operator by the RellichKondrachov theorem (see [90, Theorem 9.16] or Theorem 18.7 in Volume 2), we obtain that zι − A˜ is also a bounded Fredholm operator of index 0 by Theorem 11.15. Hence, zι − A˜ is surjective since z IdH −A is injective as  proven above. Consequently, z IdH −A is surjective.

6.B. WELL-POSEDNESS OF THE DAMPED WAVE EQUATION

239

6.B. Well-Posedness of the Damped Wave Equation Here, we prove the well-posedness of the damped wave equation (6.1.2) as stated in Theorem 6.1. We also provided other equivalent formulation of weak solutions. 6.B.1. Proof of Well-Posedness. We give the proof of Theorem 6.1 here. We beginning with strong solutions. If Y 0 ∈ D(A), as (A, D(A)) gen0 on H by 6.4, if we erates a C0 -semigroup   Theorem   set Y (t) = S(t)Y we 0 1 have Y (t) ∈ C [0, +∞); D(A) ∩ C [0, +∞); H and Y (t) is the unique solution to the semigroup equation (6.2.9), by Proposition 12.2. Setting t (y(t), z(t)) = Y (t) we obtain that z = ∂t y from the semigroup equation (6.2.9) and the form of A, and thus we have       y ∈ C 2 [0, +∞); L2 (Ω) ∩C 1 [0, +∞); H01 (Ω) ∩C 0 [0, +∞); H 2 (Ω)∩H01 (Ω) . Using again the semigroup equation  form of A we find that  (6.2.9) and the y(t) solves equation (6.2.3) in C 0 [0, +∞); L2 (Ω) . As S(t) is of contraction on H we have S(t)U H ≤ U H if U ∈ H . As Y 0 ∈ D(A), we thus obtain AS(t)Y 0 H = S(t)AY 0 H ≤ AY 0 H , using Proposition 12.2. From (6.2.11), we thus obtain (6.B.1)

Y (t)H 2 (Ω)⊕H 1 (Ω)  Y 0 H 2 (Ω)⊕H 1 (Ω) . 0

0

This yields     y ∈ L∞ [0, +∞); H 2 (Ω) ∩ H01 (Ω) ∩ W 1,∞ [0, +∞); H01 (Ω)   ∩ W 2,∞ [0, +∞); L2 (Ω) .   Hence, Eq. (6.2.3) is satisfied in L∞ [0, +∞); L2 (Ω) and moreover (6.B.1) yields the continuity inequality (6.2.4). We have thus obtained the existence of a strong solution. If now y(t) is  t (y(t), ∂ y(t)) ∈ C 0 [0, +∞); D(A) ∩ such a solution we find that Y (t) = t   C 1 [0, +∞); H and that it solves the semigroup equation (6.2.9). Since such a solution is unique by Proposition 12.2 we obtain a second proof of the uniqueness of strong solutions (a first proof making use of the energy function is given in Sect. 6.2). We now turn our attention towards weak solutions. For Y 0 = t (y 0 , y 1 ) ∈ Y (t) = t (y(t), H , that is, y0 ∈ H01 (Ω) and y 1 ∈ L2 (Ω), we set   z(t)) = 0 0 0 1 S(t)Y H , yielding y(t) ∈ C [0, +∞); H0 (Ω) , z(t) ∈ ∈ C [0, +∞);   C 0 [0, +∞); L2 (Ω) , y(0) = y 0 , and z(0) = y 1 . We choose χ(t) ∈ Cc∞ (0, +∞). By Lemma 12.4 we have ∫0∞ χ(t)Y (t) dt ∈ D(A) and A ∫0∞ χ(t)Y (t) dt = ∫0∞ χ (t)Y (t) dt in H . This reads, according to (6.2.8),

240

6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING ∞

∞

0 ∞

0

− ∫ χ(t)z(t) dt = ∫ χ (t)y(t) dt ∈ H01 (Ω), ∞

∞

P0 ∫ χ(t)y(t) dt + α ∫ χ(t)z(t) dt = ∫ χ (t)z(t) dt ∈ L2 (Ω). 0

With ϕ, ψ ∈

0

Cc∞ (Ω)

we have, as

0

tP 0

= P0 ,

∞

∞

0

0

− ∫ χ(t)z(t), ϕD  (Ω),Cc∞ (Ω) dt = ∫ χ (t)y(t), ϕD  (Ω),Cc∞ (Ω) dt, and ∞

∞

∫ χ(t)y(t), P0 ψD  (Ω),Cc∞ (Ω) dt + ∫ χ(t)z(t), αψD  (Ω),Cc∞ (Ω) dt 0

0

∞

= ∫ χ (t)z(t), ψD  (Ω),Cc∞ (Ω) dt, 0

which, with Q = (0, +∞) × Ω, can be written as − z(t, x), χ(t)ϕ(x)D  (Q),Cc∞ (Q) = y(t, x), χ (t)ϕ(x)D  (Q),Cc∞ (Q) , and y(t, x), P0 χ(t)ψ(x)D  (Q),Cc∞ (Q) + z(t, x), α(x)χ(t)ψ(x)D  (Q),Cc∞ (Q) = z(t, x), χ (t)ψ(x)D  (Q),Cc∞ (Q) .

or rather z(t, x) − ∂t y(t, x), χ(t)ϕ(x)D  (Q),Cc∞ (Q) = 0, P0 y(t, x) + α(x)z(t, x) + ∂t z(t, x), χ(t)ψ(x)D  (Q),Cc∞ (Q) = 0. For the interested reader basic elements of distribution theory are recalled in Chap. 8. We thus conclude that z = ∂t y and ∂t z + P0 y + αz = 0 in D  (Q), 2 by Proposition 8.38, implying that  ∂t y + P02 y + α∂t y =  0 in the sense of dis1 2 [0, +∞); H −1 (Ω) as tribution and moreover y ∈ C (Ω) ∩ C [0, +∞); L   P0 y + α∂t y ∈ C 0 [0, +∞); H −1 (Ω) . The first component of Y (t) thus provides a solution of the damped wave equation with the regularity properties as stated in Theorem 6.1. We now prove the uniqueness of a weak solution. For this, as the equation is linear, it is sufficient to prove that a weak solution with y|t=0 = y 0 = 0   and ∂t y|t=0 = y 1 = 0 is identically zero. Let y ∈ C 1 [0, +∞); L2 (Ω) ∩   C 0 [0, +∞); H01 (Ω) be such a solution. This function is only defined for t ≥ 0. We extend it by zero in t < 0 and denote extension As   this  by w. 1 2 0 1 w|t=0+ = ∂t w|t=0+ = 0 we observe that w ∈ C R; L (Ω) ∩ C R; H0 (Ω) and moreover w satisfies ∂t2 w + P0 w + α(x)∂t w = 0 in D  (R × Ω).

6.B. WELL-POSEDNESS OF THE DAMPED WAVE EQUATION

241

Next, we choose χ ∈ Cc∞ (R) with supp(χ) ⊂ [−1, 1] and such that ∫R χ(t)dt = t

1. We set wε = w ∗ χε (convolution in time) with χε = ε−1 χ(t/ε). We have ∂t2 wε + P0 wε + α(x)∂t wε = 0

in D  (R × Ω).

and supp(wε ) ⊂ [−ε, +∞)×Ω by the support theorem (see Proposition 8.42). We have wε ∈ C ∞ (R; H01 (Ω)) yielding P0 wε ∈ C ∞ (R; L2 (Ω)), which implies that wε ∈ C ∞ (R; D(P0 )) by elliptic regularity theory as recalled in Sect. 10.1.1. We note then that wε is a strong solution of the damped wave equations for t ∈ [−ε, +∞) (the damped wave equation is insensitive to translations in time). As wε and ∂t wε both vanish at t = −ε then wε is identically zero from the uniqueness of a strong solution. Since χε → δ as ε → 0, meaning that wε → w in D  (R × Ω), we finally find that w and thus y also vanish identically. This gives the uniqueness of a weak solution. We now prove the continuity inequality (6.2.2) for weak solutions. For strong solutions, that is Y 0 ∈ D(A), we have (6.2.12) which we write here, for 0 ≤ t1 ≤ t2 , (6.B.2)

t2

2

Y (t2 )2H − Y (t1 )2H = − ∫ α1/2 zL2 (Ω) dt, t1

  with Y (t) = t (y(t), z(t)) = S(t)Y 0 ∈ C 0 [0, +∞); D(A) by Proposition 12.2. As S(t) is bounded on H we see that both sides of this equality are continuous with respect to Y 0 with the topology of H . Thus, with the density of D(A) in H , we conclude that the equality (6.B.2) also holds for Y 0 ∈ H .   In such case t (y(t), z(t)) = Y (t) = S(t)Y 0 ∈ C 0 [0, +∞); H . This implies estimate (6.2.2). This concludes the proof of Theorem 6.1.  6.B.2. Other Formulations of Weak Solutions. In Theorem 6.1, weak solutions are defined as solutions to the damped wave equation in the sense of distribution. Alternative, yet equivalent formulations    can be given. Consider y, z ∈ C 2 [0, +∞); L2 (Ω) ∩C 1 [0, +∞); H01 (Ω) ∩C 0 [0, +∞); H 2 (Ω) ∩ H01 (Ω) and assume that y is a strong solution as given by Theorem 6.1, that is y|t=0 = y 0 and ∂t y|t=0 = y 1 with the requirement Y 0 =   t (y 0 , y 1 ) ∈ H 2 (Ω) ∩ H 1 (Ω) × H 1 (Ω). Then, with integrations by parts 0 0 and the Green formula (see for instance Proposition 18.30 in Volume 2), for T ≥ 0, we compute (6.B.3)

0 = (y, (∂t2 + P0 − α∂t )z)L2 ((0,T )×Ω) + (∂t y(T ), z(T ))L2 (Ω)   − (y(T ), ∂t z(T ))L2 (Ω) − (y 1 , z(0))L2 (Ω) − (y 0 , ∂t z(0))L2 (Ω) + (αy(T ), z(T ))L2 (Ω) − (αy 0 , z(0))L2 (Ω) .

 1 [0, +∞); Observe that this identity can be extended to a function y ∈ C    L2 (Ω) ∩ C 0 [0, +∞); H01 (Ω) as far as regularity is concerned. This leads one to define weak solutions as follows.

242

6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

Definition 6.27 (Weak Solutions). Let (y 0 , y 1 ) ∈ H01 (Ω) × L2 (Ω). A function y ∈ C 0 ([0, +∞); H01 (Ω)) ∩ C 1 ([0, +∞); L2 (Ω)) such that y(0) = y 0 and ∂t y(0) = y 1 is said to be a weak solution to the damped equation (6.1.2) if identity (6.B.3) holds for all T > 0 and all       z ∈ C 2 [0, T ]; L2 (Ω) ∩ C 1 [0, T ]; H 1 (Ω) ∩ C 0 [0, T ]; H 2 (Ω) . Such solutions are commonly referred to solutions by transposition. Proposition 6.28 (Weak Solutions). Let (y 0 , y 1 ) ∈ H01 (Ω) × L2 (Ω). There exists a unique     y ∈ C 0 [0, +∞); H01 (Ω) ∩ C 1 [0, +∞); L2 (Ω) that is a weak solution to the damped wave equation in the sense of Definition 6.27. Moreover, it coincides with the weak solution given by Theorem 6.1. Proof. We first treat uniqueness. We thus consider y 0 = 0 and y 1 = 0. For ψ 1 ∈ Cc∞ (Ω), by Theorem 6.1 we consider the strong solution       ψ ∈ C 2 [0, +∞); L2 (Ω) ∩ C 1 [0, +∞); H01 (Ω) ∩ C 0 [0, +∞); H 2 (Ω) such that ∂t2 ψ + P0 ψ + α∂t ψ = 0 in L∞ ([0, +∞); L2 (Ω)),

ψ|t=0 = 0, ∂t ψ|t=0 = ψ 1 .

For T ≥ 0 we set z(t) = ψ(T − t) and we have ∂t2 z + P0 z − α∂t z = 0 in L∞ ([0, T ]; L2 (Ω)),

z|t=T = 0, ∂t z|t=T = −ψ 1 .

Identity (6.B.3) then reads (y(T ), ψ 1 )L2 (Ω) = 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 set Y (t) = S(t)Y 0 . We also pick a sequence Y 0,n ⊂ (H 2 (Ω) ∩ H01 (Ω)) × H01 (Ω) such that Y 0,n → Y 0 in H = H01 (Ω) ⊕ L2 (Ω) as n → +∞, set Y n (t) = S(t)Y 0,n . By Proposition 6.5, we have Y n (t) = t (y n (t), ∂t y n (t)) and y n (t) is a strong solution of the boundary damped wave equation in the sense of Theorem 6.1. The semigroup S(t) is of contraction on H as stated in Theorem 6.4; this 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 (6.B.4) y(t) − y n (t)H 1 (Ω) + ∂t y(t) − ∂t y n (t)L2 (Ω)  Y 0 − Y n,0 H ,

t ≥ 0.

From the computation that led to (6.B.3) at the beginning of this section we see that this identity holds for the strong solution y n (t). Yet, all terms

6.C. FROM RESOLVENT TO SEMIGROUP STABILIZATION ESTIMATE

243

in (6.B.3) converge according to (6.B.4). We thus obtain that y(t) is a weak solution. In Remark 6.2 we find that y is also the weak solution given by Theorem 6.1.  6.C. From a Resolvent to a Semigroup Stabilization Estimate First, we note that it is sufficient to prove the result for k = 1 as the estimate for S(t)B −1 can be iterated by writing S(t)B −k = (S(t/k)B −1 ) · · · (S(t/k)B −1 ). Second, we note that the spectrum of B is contained in {z ∈ C; Re(z) ≥ 0} by Corollary 12.8 as here S(t)L (H) ≤ M , for t ≥ 0. We then observe that estimate (6.3.2) holds for all σ ∈ R, as iσ IdH −B is invertible for any σ ∈ R implying that σ → (iσ IdH −B)−1 is continuous with values in L (H) in a neighborhood of the real axis. This claim can be further quantified as follows. There exist C0 and K0 such that (6.C.1)

(z IdH −B)−1 L (H) ≤ K0 eK0 | Im z| ,

z ∈ V,

where V is the following neighborhood of iR V = {z ∈ C; | Re z| ≤ e−C0 | Im z| /C0 }.

(6.C.2) In fact if we write

z IdH −B = (i Im z IdH −B)(IdH +T ),

with T = (Re z)(i Im z IdH −B)−1 ,

we see that T L (H) ≤ (K/C0 )e(K−C0 )| Im z| ≤ K/C0 < 1 if z ∈ V and if C0 > K. Then IdH +T is invertible in L (H) and so is z IdH −B and we have KC0 K| Im z| e . (z IdH −B)−1 L (H) ≤ C0 − K The spectrum of B is thus contained in {z ∈ C; Re z ≥ 0} \ V . We start the proof of the theorem by providing a representation formula for B −1 by means of an integral along a closed contour in C. Let R > 0 to be chosen large below. We define γ as the contour given by γ1 = {z ∈ C; |z| = R, Re z ≤ 0}, positively oriented, and γ2 a path from −iR to +iR such that there is no spectrum of B in the interior of the contour γ. This is feasible by the description of the spectrum of B made above. This contour is illustrated in Fig. 6.3. In the interior of γ the resolvent z → (z IdH −B)−1 is holomorphic (see e.g. Theorem III.6.7 in [192]). By the Cauchy formula we thus have z2  1  dz ∫ 1 + 2 (B − z IdH )−1 , B −1 = 2iπ γ R z and we may thus write (6.C.3)

S(t)B −1 =

z2  dz 1  ∫ 1 + 2 S(t)(B − z IdH )−1 . 2iπ γ R z

We now make the following claim.

244

6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

Im z

iR γ1 V Re z γ2 −iR

Figure 6.3. The contour γ in the complex plane Proposition 6.29. We assume that S(t)L (H) ≤ M , and iσ IdH −B is invertible for any σ ∈ R. Let γ2 be a path from −iR to iR, where Re γ2 > 0 except for the points ±iR. Let K be the domain limited by the segment [−iR, iR] and γ2 . We assume that for z ∈ K, z IdH −B is invertible. We have C 1  e−tz  z2     dz  . (6.C.4) S(t)B −1 L (H) ≤ +  ∫ 1 + 2 (z IdH −B)−1 R 2π γ2 R z L (H) Proof of Proposition 6.29. We choose z ∈ γ1 , z = ±iR and t ≥ 0. We set G(s) = ez(s−t) S(s)(z IdH −B)−1 u for some u ∈ H. As (z IdH −B)−1 u ∈ D(B) we have G (s) = ez(s−t) (z IdH −B)S(s)(z IdH −B)−1 u = ez(s−t) S(s)u, by Proposition 12.2. Observing, as Re z < 0 and S(t)L (H) ≤ M , that lims→+∞ G(s)=0 we have S(t)(z IdH −B)−1 u=G(t)= − ∫t+∞ ez(s−t) S(s)uds, which we write +∞

S(t)(z IdH −B)−1 = − ∫ ez(s−t) S(s)ds, t

z ∈ γ1 , z = ±iR.

We write z = Reiθ , with θ ∈ (π/2, 3π/2) and we find (6.C.5)

+∞  −1 S(t)(z IdH −B)−1 L (H)  ∫ eR(s−t) cos(θ) ds = R cos(θ) . t

We can use this inequality to estimate the part of the integration over γ1 in (6.C.3). Observe that estimate (6.C.5) is poor near θ = π/2 and θ = 3π/2; in fact, the term 1 + z 2 /R2 is introduced in (6.C.3) to compensate for this behavior as we shall now see. We have

6.C. FROM RESOLVENT TO SEMIGROUP STABILIZATION ESTIMATE

(6.C.6)

1+

245

z2 = 1 + e2iθ = eiθ (e−iθ + eiθ ) = 2eiθ cos θ, R2

We thus find, with (6.C.5),   3π/2  −1 dz  z2   z2     ∫ 1+ 2  R cos(θ) dθ  R−1 .  ∫ 1 + 2 S(t)(B − z IdH )−1  R z L (H) R γ1 π/2 This estimate gives the first term in (6.C.4). We now consider the contour integral on γ2 . We introduce t

ht (z) = ∫ e(s−t)z S(s)ds, 0

that is holomorphic in z ∈ C. Using G(s) and G (s) computed above, we obtain ht (z) = (S(t) − e−zt )(z IdH −B)−1 , for z in the resolvent set of B. We then find  dz z2  ∫ 1 + 2 S(t)(B − z IdH )−1 R z γ2   dz z 2  . = ∫ 1 + 2 − ht (z) + e−zt (B − z IdH )−1 R z γ2 We obtain the result of the proposition if we now prove the following estimate   dz  z2    (6.C.7)  R−1 .  ∫ 1 + 2 ht (z)  R z L (H) γ2 As the integrand is holomorphic in C \ {0} we have   z2  z2  dz dz = ∫ 1 + 2 ht (z) , ∫ 1 + 2 ht (z) R z R z γ2 γ3 where γ3 = {z ∈ C, |z| = R, Re z ≥ 0} positively oriented, see Fig. 6.4. For z = Reiθ , with θ ∈ (−π/2, π/2) we have t   −1  1 − e−tR cos(θ) ht (z)L (H) ≤ M ∫ e(s−t)R cos(θ) ds = M R cos(θ) 0



≤ M R cos(θ)

−1

.

Using (6.C.6) as above we find   z2  dz     R−1 .  ∫ 1 + 2 ht (z)  R z γ3 L (H) This conclude the proof of Proposition 6.29.



246

6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

Im z iR

γ3 0 Re z γ2 −iR

Figure 6.4. The contour γ3 in the complex plane Im z iR

γ+

γ0 L Re z

−iR

γ−

Figure 6.5. The contours γ− , γ0 and γ+ in the complex plane We now give a precise form of the contour γ2 as the union of the following three contours γ− , γ0 , and γ+ , in this order, where ⎧ ⎪ s ∈ [0, L], ⎨γ− (s) = s − iR s ∈ [−R, R], γ0 (s) = L + is ⎪ ⎩ γ+ (s) = L − s + iR s ∈ [0, L], as illustrated in Fig. 6.5, where L = e−αR /α with α ≥ max(C0 , K0 , 1) and C0 and K0 as in (6.C.1) and (6.C.2). Thus, the resolvent estimate (6.C.1) holds on γ2 .

6.D. PROOFS OF NON-QUANTIFIED STABILIZATION RESULTS

247

We choose R > 1. On γ− , we have z 2 = s2 − R2 − 2isR and thus     1 + z 2 /R2  = s  s − 2i  s  e−αR . R R R By symmetry the same holds on γ+ . We thus obtain (6.C.8)   L z2  e−tz     R−1 ∫ e(K0 −α)R e−ts ds  R−1 . dz   ∫ 1 + 2 (z IdH −B)−1 R z γ± 0 L (H)  2  On γ0 , we have 1 + Rz 2   1, |z|−1 ≤ L−1 = αeαR , and |e−zt | = e−tL = exp(−te−αR /α). We thus have   e−tz  z2    dz  (6.C.9)  ∫ 1 + 2 (z IdH −B)−1  eCR exp(−te−αR /α). R z γ0 L (H) From Proposition 6.29, (6.C.8) and (6.C.9) we obtain S(t)B −1   R−1 + eCR exp(−te−αR /α). Choosing R =

log t β

with β > α yields

S(t)B −1   (log t)−1 + tC/β exp (−t1−α/β /α)  (log t)−1 . 

This concludes the proof. 6.D. Proofs of Non-Quantified Stabilization Results

6.D.1. Proof of Proposition 6.12. As the C0 -semigroup S(t) is bounded, there exists M > 0 such that (see below Definition 12.1) (6.D.1)

sup S(t)L (H) ≤ M. t≥0

Let Y 0 ∈ H. From the density of D(B) in H, for any ε > 0, there exists Yε0 ∈ D(B) such that Y 0 − Yε0 H ≤ ε. By (6.D.1), we have S(t)(Y 0 − Yε0 )H ≤ M Y 0 − Yε0 H ≤ M ε. We then obtain S(t)Y 0 H ≤ M ε + S(t)Yε0 H and the conclusion follows.  6.D.2. Proof of Proposition 6.14. As the damped wave semigroup S(t) is of contraction, for any Y 0 = t (y 0 , y 1 ) ∈ H , the function t → S(t)Y 0 H is nonincreasing. This is also clear by (6.2.14). By Proposition 6.12, it suffices to prove that the energy of a strong solution goes to 0 as t → +∞. We thus consider Y 0 ∈ D(A). We have S(t)Y 0 D(A) = S(t)Y 0 H + AS(t)Y 0 H . As AS(t)Y 0 = S(t)AY 0 , we see that t → S(t)Y 0 D(A) is nonincreasing, using the decay recalled above.

248

6. STABILIZATION OF THE WAVE EQUATION WITH AN INNER DAMPING

The norm S(t)Y 0 D(A) is thus bounded.2 As a result, there exists an in  creasing sequence (tn )n ⊂ [0, +∞) such that tn → +∞ and S(tn )Y 0 n weakly converges to some U in D(A) as tn → +∞. Recall that D(A) is a Hilbert space if equipped with the graph norm .D(A) since (A, D(A)) is a closed operator. From the compactness of the embedding of D(A) into H by the Rellich-Kondrachov theorem  Theorem 9.16] or Theo (see [90, rem 18.7 in Volume 2), we find that S(tn )Y 0 n converges to U in H . In particular, S(tn )Y 0 H → U H . As t → S(t)Y 0 H is nonincreasing we find limt→+∞ S(t)Y 0 H = U H . We now use the semigroup properties. For some t, t ≥ 0 we write S(t + t )Y 0 = S(t)S(t )Y 0 and note that S(t + t )Y 0 H

→

t →+∞

U H

and

S(t)S(t )Y 0 H

→

t →+∞

S(t)U H .

using the continuity of S(t) on H . This yields S(t)U H = U H for all t ≥ 0. From the estimation of the decay of energy, that is, of the norm in H , given in (6.2.12) with S(t)U = t (y(t), z(t)), we find α1/2 z(t)L2 (Ω) = 0 for all t ≥ 0, implying that z(t) = ∂t y(t) vanishes on (0, +∞)×ω. As U ∈ D(A) we have W (t) = S(t)AU ∈ H and we find d d W (t) = AS(t)U = − S(t)U = −t (z(t), z(t)). dt dt We thus have W (t) = S(t)AU vanishing on (0, +∞) × ω thus implying that AU = 0 by the assumed unique continuation property. This reads, with U = t (u0 , u1 ) ∈ D(P0 ) × H01 (Ω), P0 u0 = 0,

u1 = 0,

implying U = 0 from classical elliptic theory (see for instance the beginning of Sect. 10.1).  6.D.3. Proof of Proposition 6.15. We follow the proof of [57] that is close to that of Theorem 6.6 given in Appendix 6.C. The result of Proposition 6.29 applies: for all R > 0 one has 1  C e−tz  z2     + dz  . S(t)B −1 L (H) ≤  ∫ 1 + 2 (z IdH −B)−1 R 2π γ2 R z L (H) The contour γ2 is chosen above the statement of that proposition: it lies in a neighborhood of the segment [−iR, iR] that is in the resolvent set and Re z > 0 on γ2 except on ±iR. This can be done as the resolvent set ρ(B) is an open set of C.

2The boundedness of S(t)Y 0  D(A) is also clear from (6.2.4) and (6.2.11).

6.D. PROOFS OF NON-QUANTIFIED STABILIZATION RESULTS

249

Let ε > 0. We choose R > 0 such that C/R ≤ ε/2. Since e−tz converges to 0 as t → +∞ for almost every z on γ2 , by the Lebesgue dominatedconvergence theorem   e−tz  z2    = 0. lim  ∫ 1 + 2 (z IdH −B)−1 dz  t→+∞ γ2 R z L (H) Thus, S(t)B −1 L (H) ≤ ε for t sufficiently large, that is, lim S(t)B −1 L (H) = 0.

t→+∞

As D(B) is dense into H the result follows from Proposition 6.12.



CHAPTER 7

Controllability of Parabolic Equations Contents 7.1. 7.2. 7.3. 7.4. 7.5. 7.5.1. 7.5.2.

Introduction and Setting Exact Controllability for a Parabolic Equation Null-Controllability for Semigroup Operators Observability for the Semigroup Parabolic Equation A Spectral Inequality Spectral Inequality Through an Interpolation Inequality Spectral Inequality Through the Derivation of a Global Carleman Estimate 7.5.3. Sharpness of the Spectral Inequality 7.6. Partial Observability and Partial Control 7.7. Construction of a Control Function for a Parabolic Equation 7.8. Dual Approach for Observability and Control Cost 7.9. Properties of the Reachable Set and Generalizations 7.10. Boundary Null-Controllability for Parabolic Equations 7.11. Notes

251 254 256 258 260 261 263 265 267 268 270 273 276 280

7.1. Introduction and Setting Let Ω be a smooth bounded connected open set of Rd in the sense recalled in Sect. 1.7. We consider the second-order elliptic operator

Di (pij (x)Dj ), with pij (x)ξi ξj ≥ C|ξ|2 , (7.1.1) P0 = 1≤i,j≤d

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 I, PNLDE Subseries in Control 97, https://doi.org/10.1007/978-3-030-88674-5 7

251

252

7. CONTROLLABILITY OF PARABOLIC EQUATIONS

where pij ∈ C ∞ (Ω; R) with all derivatives bounded and such that pij = 1 ≤ i, j ≤ d. Let ω be an open subset of Ω. For T > 0, the controlled parabolic equation associated with P0 , in the time interval (0, T ), with homogeneous Dirichlet boundary conditions, and for an initial condition y 0 in L2 (Ω), is given by ⎧ ⎪ in (0, T ) × Ω, ⎨∂t y + P0 y = 1ω v (7.1.2) y=0 on (0, T ) × ∂Ω, ⎪ ⎩ 0 in Ω. y(0) = y pji ,

The function v is the control. The goal is to drive the solution y to a prescribe state at time T > 0, yet only acting in the sub-domain ω. We shall make precise what can actually be achieved below. We denote by S(t) the C0 -semigroup on L2 (Ω) generated by the unbounded operator P0 : L2 (Ω) → L2 (Ω) with domain D(P0 ) = H 2 (Ω)∩H01 (Ω) and defined by P0 u = P0 u for u ∈ D(P0 ). We refer to Chap. 12 for some elements of semigroup theory. For the particular properties of the C0 -semigroup S(t), we refer to Sect. 10.2 in Chap. 10. Of the many properties of the unbounded operator (P0 , D(P0 )), we recall that P0 is maximal monotone and self-adjoint and has a compact resolvent, as reviewed in Sect. 10.1 in Chap. 10. The semigroup S(t) can be expressed by, for t ≥ 0,

−μj t e uj φj , uj = (u, φj )L2 (Ω) , S(t)u = j∈N

where the functions (φj )j∈N are eigenfunctions of P0 and form a Hilbert basis of L2 (Ω), and where (μj )j∈N are the associated real and positive eigenvalues. Extensive details are provided in Sect. 10.2. Observe that we have written boundary conditions explicitly in System (7.1.2). However, in the subsequent analysis the homogeneous Dirichlet boundary condition will not appear explicitly and will be rather hidden in the functional framework. Observe that D(P0 ) = H 2 (Ω) ∩ H01 (Ω) and 1/2 D(P0 ) = H01 (Ω) both of which account for this boundary condition. Remark 7.1. The boundary condition appears however in an explicit manner when one applies one of the Carleman estimates of Chap. 3 near the boundary in what follows. For y 0 ∈ L2 (Ω) and v ∈ L2 ((0, T ) × Ω), we can define a so-called weak solution of the parabolic equation: d y + P0 y = 1ω v, dt

(7.1.3)

y(0) = y 0 ,

as a function in C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) that satisfies t

1/2

1/2

t

(y(t), ψ)L2 (Ω) + ∫ (P0 y, P0 ψ)L2 (Ω) = (y 0 , ψ)L2 (Ω) + ∫ 1ω v(σ), ψL2 (Ω) dσ, 0

0

7.1. INTRODUCTION AND SETTING

253

for all ψ ∈ H01 (Ω) and for all t ∈ [0, T ] (see Definition 10.48 in Chap. 10). By Theorem 10.49, there exists a unique such solution y ∈ C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) ∩ H 1 (0, T ; H −1 (Ω)), given by the Duhamel formula t

(7.1.4)

y(t) = S(t)y 0 + ∫ S(t − σ)1ω v(σ)dσ. 0

Moreover, by Corollary 10.34 and Corollary 10.43 we have (7.1.5)

y ∈ C 0 ((0, T ], H01 (Ω)).

We thus see right away that because of the regularizing effect of the parabolic equation (resp.., semigroup) it is vain to hope to reach an arbitrary target in L2 (Ω) at time t = T . In the case ω = Ω, that is, a control acting everywhere, we shall see in Sect. 7.2 that exact controllability of the parabolic operator can be achieved for the target space H01 (Ω): for any initial condition y 0 ∈ L2 (Ω) for any target state y T ∈ H01 (Ω) there exists v ∈ L2 ((0, T ) × Ω) such that the weak solution y to (7.1.3) satisfies y(T ) = y T . In the case ω  Ω the situation is much different. We shall see that the function y(t) ∈ H01 (Ω) is in fact smooth in Ω \ ω if t > 0, again because of the regularizing effect of the parabolic equation (resp.., semigroup). This ruins any hope to achieve exact controllability in any Sobolev-type space. In such case, one focuses onto the notion of null-controllability: given an initial condition y 0 ∈ L2 (Ω), can one find v ∈ L2 ((0, T ) × Ω) such that the weak solution y to (7.1.3) satisfies y(T ) = 0? Most of Chap. 7 is devoted to the proof of this null-controllability result. Before treating the special case of the controlled parabolic equation (7.1.3), we first present in Sect. 7.3 some basic aspects of null-controllability for semigroup operators that are connected to observability estimates for an adjoint system. Then, in Sect. 7.4 we derive the precise observability inequality for the operator ∂t + P0 that is equivalent to the null-controllability of Eq. (7.1.3). The remainder of Chap. 7 is then devoted to the derivation of this observability inequality. The proof is based on a spectral inequality given and proven in Sect. 7.5. Its proof relies on the Carleman estimates of Chap. 3. Several strategies are proposed for the proof of the spectral inequality, first, through a interpolation inequality in Sect. 7.5.1 or, second, directly from a global Carleman estimate in Sect. 7.5.2. With the spectral inequality, one first proves in Sect. 7.6 a partial observability and a partial controllability in a sequence of growing spaces associated with the spectral family (φj )j∈N that diagonalizes the elliptic operator P0 . Finally, two proofs of the null-controllability of (7.1.3) are provided: the first one consists in a construction of an actual control in L2 ((0, T ) × Ω) in Sect. 7.7, and the second one consists in a proof of the observability inequality in Sect. 7.8. The two arguments can be viewed as dual of each other, and both rely on the results of Sect. 7.6.

254

7. CONTROLLABILITY OF PARABOLIC EQUATIONS

7.2. Exact Controllability for a Parabolic Equation We consider here the case of a control acting everywhere in the controlled parabolic equation (7.1.3), that is, the case ω = Ω. As the solution y is such that y(t) ∈ H01 (Ω) for t > 0, we see that it is sufficient to consider the case y 0 ∈ H01 (Ω). Then, we have the following exact controllability result for the considered parabolic equation. Theorem 7.2. In the case ω = Ω, for any T > 0, any y 0 , y T ∈ weak solution y ∈ to

H01 (Ω) there exists v ∈ L2 ((0, T ) × Ω) such that the C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) ∩ H 1 (0, T ; H −1 (Ω)) d y + P0 y = v, dt

y(0) = y 0 ,

satisfies y(T ) = y T . Proof. We define the operator L : L2 ((0, T ) × Ω) → H01 (Ω) by T

L(v) = ∫ S(T − σ)v(σ)dσ. 0

S(T )y 0

H01 (Ω),

∈ we see that exact controllability is equivalent As we have to having L surjective. By Corollary 11.20 in Sect. 11.6 of Chap. 11, it then amounts to proving the following inequality: (7.2.1)

q F H −1 (Ω) ≤ C0 L∗ q F L2 ((0,T )×Ω) ,

q F ∈ H −1 (Ω).

Here, we identify the Hilbert space L2 ((0, T ) × Ω) to its dual space, while the dual of H01 (Ω) is identified to H −1 (Ω) using classically L2 (Ω) as a pivot space, and L∗ maps H −1 (Ω) into L2 ((0, T ) × Ω). We now identify the (bounded) operator L∗ . At first, we choose q F ∈ L2 (Ω), and we write, by Proposition 10.30, T

q F , LvH −1 (Ω),H01 (Ω) = (q F , Lv)L2 (Ω) = ∫ (q F , S(T − σ)v(σ))L2 (Ω) dσ 0

T

= ∫ (S(T − σ)q F , v(σ))L2 (Ω) dσ 0

= (S(T − .)q F , v)L2 ((0,T )×Ω) . The operator S(t) extends to K −2 (Ω) = D(P0 ) . This extension coincides with the C0 -semigroup generated by the unbounded operator P−2 , which itself the extension of P0 to K −2 (Ω), with domain D(P−2 ) = L2 (Ω); see Sects. 10.1.3 and 10.2. By density, we thus see that the adjoint operator L∗ : H −1 (Ω) → L2 ((0, T ) × Ω) is given by L∗ (q F ) = q, where q(t) = S(T − t)q F . We thus have q ∈ C 0 ([0, T ]; H −1 (Ω)) ∩ L2 (0, T ; L2 (Ω)) ∩ H 1 (0, T ; K −2 (Ω)), and q solves the following backward-in-time semigroup parabolic equation: d − q + P−2 q = 0, and q(T ) = q F , dt by Proposition 10.32.

7.2. EXACT CONTROLLABILITY FOR A PARABOLIC EQUATION

255

Using the spectral decomposition of the semigroup S(t) : H −1 (Ω) → given by Proposition 10.27 that formally coincides with that of 2 S(t) : L (Ω) → L2 (Ω) recalled above, if q F ∈ H −1 (Ω), then we may write

−1/2 F q F = j∈N qjF φj with (μj qj ) ∈ 2 yielding

F −μj (T −t) qj e φj . q(t) = H −1 (Ω)

j∈N

We then find, as (φj )j∈N is a Hilbert basis of L2 (Ω), q2L2 ((0,T )×Ω) = ≥

T

j∈N 0

j∈N

∫ |qjF |2 e−2μj (T −t) dt =

1−

e−2μ0 T 2

j∈N

|qjF |2

1 − e−2μj T 2μj

2

F 2 F μ−1 j |qj |  q H −1 (Ω) ,

as 0 < μ0 ≤ μ1 ≤ · · · . This is precisely the observability inequality (7.2.1).  The following proposition shows that if ω  Ω, exact controllability cannot be achieved for the parabolic equation (7.1.3). Proposition 7.3. In the case ω  Ω, for any T > 0, any y 0 ∈ H01 (Ω), and any v ∈ L2 ((0, T ) × Ω), the weak solution y ∈ C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) ∩ H 1 (0, T ; H −1 (Ω)) to d y + P0 y = 1ω v, dt is smooth in (0, T ] × (Ω \ ω).

y(0) = y 0 ,

Remark 7.4. The weak solution y is given by the Duhamel formula (7.1.4). By Theorem 10.29, we see that S(t)y 0 ∈ C ∞ ((0, T ]; K σ (Ω)) for any σ > 0, for y 0 chosen in K r (Ω) with r arbitrary. The choice of the regularity of the initial condition it thus not important here. Proof. The weak solution is given by (7.1.4). As already pointed out in Remark 7.4, the homogeneous term S(t)y 0 is smooth in the whole Ω if t > 0. Consequently, we only need to consider the Duhamel term w(t) = ∫0t S(t − s)1ω v(s)ds, which we write t

w(t, x) = ∫ ∫ kt−s (x, x )1ω (x )v(s, x )dx ds, 0Ω

(x, x )

is the kernel of the semigroup (see Sect. 10.2.4). Let x0 ∈ where kt Ω \ ω. In an open neighborhood V of x0 the function 1ω (x)v(s, x) vanishes. The estimate of Proposition 10.40 shows that the Lebesgue dominatedconvergence theorem applies yielding, for any α ∈ Nd+1 , t

α α w(t, x) = ∫ ∫ ∂t,x kt−s (x, x )1ω (x )v(s, x )dx ds, ∂t,x 0Ω

α w is bounded yielding the result. Thus ∂t,x

x ∈ V. 

256

7. CONTROLLABILITY OF PARABOLIC EQUATIONS

Because of this negative result for exact controllability in the case ω  Ω, we shall consider the weaker notion of null-controllability that is equivalent to the notion of controllability to the trajectories. 7.3. Null-Controllability for Semigroup Operators Here, we first place ourselves in some abstract setting. We let S(t) be a C0 -semigroup on a Hilbert space H with associated generator B : H → H with dense domain D(B). We also let M : U → H be bounded1 with U a second Hilbert space. Here, we identify H with its dual space and we assume that B is maximal monotone and self-adjoint. Then, for f ∈ L2 (0, T ; H) and y 0 ∈ H the mild solution of d y + By = f, t ∈ (0, T ], y(0) = y 0 ∈ H, dt is given by t

y(t) = S(t)y 0 + ∫ S(t − σ)f (σ) dσ, 0

and y(t) ∈

C 0 ([0, T ]; H)

(see Sect. 12.3).

Then, we consider the control system d (7.3.1) y + By = M v, t ∈ (0, T ], dt with v ∈ L2 (0, T ; U ).

y(0) = y 0 ∈ H,

Definition 7.5. For E a closed subspace of H, we say that this system is null-controllable in E at time T > 0 if one can choose v such that the (mild) solution t

(7.3.2)

y(t) = S(t)y 0 + ∫ S(t − σ)M v(σ) dσ, 0

satisfies ΠE y(T ) = 0, where ΠE stands for the orthogonal projection on E in H. A first observation is that null-controllability in E is equivalent to the ability of reaching in E a natural trajectory of the semigroup S(t). Proposition 7.6 (Controllability to Trajectories in E). Assume that System 7.3.1 is null-controllable in E. Then, for any y 1 ∈ H there exists v ∈ L2 (0, T ; U ) such that the solution 7.3.2 satisfies  ΠE y(T ) − S(T )y 1 ) = 0. 1Here M : U → H is chosen bounded for convenience. This assumption can be relaxed. We refer to the treatment of boundary controllability in Sect. 7.10 where only the operator v → ∫0T S(T − σ)M v(σ) dσ is bounded from U into H.

7.3. NULL-CONTROLLABILITY FOR SEMIGROUP OPERATORS

257

Proof. It suffices to apply the assumed null-controllability property for  the initial condition y 0 − y 1 and to use the linearity of the system. We have the following proposition. Proposition 7.7. The null-controllability in E at time T > 0 of (7.3.1) is equivalent to having the following observability estimate: there exists CE > 0 such that (7.3.3)

S(T )ΠE zH ≤ CE M ∗ S(t)ΠE zL2 (0,T ;U ) ,

z ∈ H.

In particular, in such case, there exists a continuous linear map ΦE : H → L2 ((0, T ), U ) such that for v = ΦE (y 0 ) we have ΠE y(T ) = 0. We have ΦE L (H,L2 ((0,T ),U )) ≤ CE , that is, vL2 (0,T ;U ) ≤ CE y 0 H . Proof. If we define the following operators: L : L2 (0, T ; U ) → H

K:H →H

T

y 0 → ΠE S(T )y 0 ,

v → ∫ ΠE S(T − σ)M v(σ) dσ. 0

From (7.3.2), the null-controllability in E of (7.3.1) is equivalent to having the following inclusion for the ranges of the operators Ran(K) ⊂ Ran(L). With Lemma 11.19, this is equivalent to (7.3.4)

K∗ zH ≤ CE L∗ zL2 (0,T ;U ) ,

z ∈ H,

for some CE > 0. We characterize the adjoint operators of K and L. As B is self-adjoint, we have S(t)∗ = S(t) and thus K∗ z = S(T )ΠE z and (L∗ z)(t) = M ∗ S(T − t)ΠE z,

s ∈ [0, T ].

By (7.3.4), the null-controllability in E is equivalent to having S(T )ΠE zH 1/2 T ≤ CE ∫ M ∗ S(T − t)ΠE z2U dt 0

= CE M ∗ S(t)ΠE zL2 (0,T ;U ) ,

z ∈ H,

for some CE > 0. The second part of the statement of the proposition also follows from Lemma 11.19.  Observe that once one has the existence of a control map like ΦE , then other control maps can be designed. Their norm will however differ. One may, for instance, let v be zero on the time interval [0, T /2] and then work as in the proof of Proposition 7.7 in the time interval [T /2, T ]. Minimizing the norm of the control map in L (H, L2 ((0, T ), U )) can be of interest. Note

258

7. CONTROLLABILITY OF PARABOLIC EQUATIONS

that the proof of Proposition 7.7, and Lemma 11.19 used therein, shows that if we have a bounded linear control map Φ : H → L2 ((0, T ), U ) such that the mild solution y(t) given in (7.3.2) satisfies that ΠE y(T ) = 0 for v = Φ(y 0 ), then the constant CE in the observability inequality (7.3.3) can be chosen equal to ΦL (H,L2 ((0,T ),U )) . Hence, if we denote by CE0 the best possible constant CE in (7.3.3), it implies that there exists a control map Φ0E such that Φ0E L (H,L2 ((0,T ),U )) = CE0 . With these observation in mind, one may wish to single a control map out by means of some criterion. Proposition 7.8. Assume that the observability inequality (7.3.3) holds with CE0 > 0 as the best possible constant. Then, for each y 0 ∈ H, among all controls v in L2 (0, T ; U ) that yield ΠE y(T ) = 0, there exists a unique : y 0 → v min control v min whose norm is minimal. Moreover, the map Φmin E 2 min is linear, bounded on H into L (0, T ; U ), and ΦE L (H,L2 ((0,T ),U )) = CE0 . Proof. We introduce W the space of controls that sends the zero initial condition to a solution at time t = T with a vanishing projection on E: T

W = {w ∈ L2 (0, T ; U ); ΠE ∫ S(t − σ)M w(σ) dσ = 0}. 0

L2 (0, T ; U ),

since S(t − σ)L (H) ≤ CeωT for If (wn )n∈N ⊂ W converges in some C > 0 and ω by (12.1.6), we find that that lim wn ∈ W . This linear space is thus closed. For an initial condition y 0 ∈ H, if v ∈ L2 (0, T ; U ) is a control such that ΠE y(T ) = 0, with y(t) solution to (7.3.1), then Wy0 = v + W is the affine space of all such controls. It follows that v min defined as the projection in the Hilbert space L2 (0, T ; U ) of 0 onto the closed affine subspace Wy0 yields the element of Wy0 with minimal L2 -norm; uniqueness of v min is clear. In particular v min is orthogonal to W . If one considers a control map Φ such that ΦL (H,L2 ((0,T ),U )) = CE0 , as given by Proposition 7.7, then v min L2 (0,T ;U ) ≤ Φ(y 0 )L2 (0,T ;U ) ≤ CE0 y 0 H . It thus remains to prove that Φmin : y 0 → v min is a linear map. Let y 1 , y 2 ∈ H and λ ∈ C and set y 0 = λy 1 + y 2 . Set v 1 = Φmin (y 1 ) and v 2 = Φmin (y 2 ). Observe that v = λv 1 + v 2 is a null-control associated to the initial condition y 0 . Thus, Wy0 = v + W . Since both v 1 and v 2 are orthogonal to W , then so is v and thus v is the projection of 0 onto Wy0 ;  that is, v = Φmin (y 0 ). 7.4. Observability for the Semigroup Parabolic Equation We apply the abstract result of Section 7.3 on null-controllability. Here B = P0 , H = L2 (Ω), D(P0 ) = H 2 (Ω) ∩ H01 (Ω), and M = 1ω .

7.4. OBSERVABILITY FOR THE SEMIGROUP PARABOLIC EQUATION

259

From Proposition 7.7, with E = H, we see that null-controllability holds for the control system (7.1.3) if and only if there exists Cobs > 0 such that S(T )z 0 L2 (Ω) ≤ Cobs S(t)z 0 L2 ((0,T )×ω) ,

z 0 ∈ L2 (Ω),

as M ∗ = M = 1ω here, since P0 is maximal monotone and self-adjoint. In particular this is equivalent to having the following inequality: (7.4.1)

z(T, .)L2 (Ω) ≤ Cobs zL2 ((0,T )×ω) ,

where z(t, .) = S(t)z 0 ∈ C 0 ([0, T ]; L2 (Ω))∩L2 (0, T ; H01 (Ω))∩H 1 (0, T ; H −1 (Ω)) is the weak solution to d (7.4.2) z(0) = z 0 , z + P0 z = 0, dt for z 0 ∈ L2 (Ω) In what follows we shall be interested in the null-controllability of the heat equation in closed subspaces of L2 (Ω) that are generated by a Hilbert basis of eigenfunctions φj of P0 , j ∈ N, as recalled above and introduced in Sect. 10.1. If E is such a closed subspace of L2 (Ω), then null-controllability in E holds for the control system (7.1.3); that is ∀y 0 ∈ L2 (Ω), ∃v ∈ L2 ((0, T ) × Ω) such that ΠE y(T ) = 0, if and only if there exists CE > 0 such that the observability inequality: (7.4.3)

z(T, .)L2 (Ω) ≤ CE z(., .)L2 ((0,T )×ω) ,

holds for z(t, x) solution to (7.4.2) with z 0 ∈ E by Proposition 7.7. Remark 7.9. Note that one often considers the equivalent observability inequality: (7.4.4)

q(0, .)L2 (Ω) ≤ Cq(., .)L2 ((0,T )×ω) ,

where q(t, x) is solution in C 0 ([0, T ]; L2 (Ω))∩L2 (0, T ; H01 (Ω))∩H 1 (0, T ; H −1 (Ω)) to the following backward-in-time equation: (7.4.5)

−

d q + P0 q = 0, dt

q(T ) = q F ,

with qF in L2 (Ω) or in E, as having qF = z 0 then simply yields q(t, x) = z(T − t, x). System (7.4.5) is called the adjoint system of system (7.1.3). In fact, system (7.4.5) is often preferred over system (7.4.2) as, by computing the L2 -inner product of q(t, x) with the equation ∂t y+P0 y = 1ω v, we obtain, after integrations by parts, (7.4.6)

(y(T ), qF )L2 (Ω) − (y 0 , q(0))L2 (Ω) = (1ω v, q)L2 ((0,T )×Ω) .

This identity is often used to prove the equivalence between observability and null-controllability, whereas we chose above to invoke a more abstract argument in the proof of Proposition 7.7 using Lemma 11.19.

260

7. CONTROLLABILITY OF PARABOLIC EQUATIONS

We now wish to proceed with the proof of the null-controllability of system (7.1.3). The method we present here relies on a spectral inequality for the elliptic operator P0 in connection with the spectral decomposition of P0 recalled above and given in Sect. 10.1. We prove this spectral inequality in the next section. This inequality then leads to partial observability inequalities. A first approach is then to deduce a partial control result (Sect. 7.6) in a growing sequence of finite-dimensional subspaces of L2 (Ω) associated with the eigenfunctions of P0 . Then, in Sect. 7.7, the control v in (7.1.3) is built as a sequence of active and passive controls. In fact, the passive mode allows one to take advantage of the natural parabolic exponential decay of the L2 -norm of the solution. A second approach, yet very similar, in the sense that a growing sequence of finite-dimensional subspaces of L2 (Ω) is also used, yields directly an observability inequality of the form (7.4.1) for System (7.4.2). We present this second approach in Sect. 7.8, which can be seen as dual to the first approach as one only works on adjoint equations instead of going back to the controlled equation. 7.5. A Spectral Inequality Let (φj )j be a family of eigenfunctions of P0 that forms a Hilbert basis of L2 (Ω), associated with the nondecreasing sequence of eigenvalues 0 < μ0 ≤ μ 1 ≤ · · · ≤ μ n ≤ · · · These elements are recalled in Sect. 10.1. We prove a spectral inequality that, in particular, measures the loss of orthogonality of the eigenfunctions φj , j ∈ N, when they are restricted to an open subset ω ⊂ Ω such that ω = Ω. It also quantifies how linear combinations of these eigenfunctions can be observed from a sub-domain. Theorem 7.10 (Spectral Inequality). There exists K > 0 such that for all μ ≥ 0 we have (7.5.1)

wL2 (Ω) ≤ eK

√

μ

wL2 (ω) ,

w ∈ span(φj ; μj ≤ μ).

Remark 7.11. (1) With the orthogonality of the φj on Ω, the spectral inequality also reads √ 2 

|αj |2 ≤ e2K μ ∫  αj φj (x) dx, μj ≤μ

ω

μj ≤μ

for all sequences (αj )j∈N∗ ⊂ C and all μ ≥ 0. (2) As a function in L2 (Ω) cannot be characterized by its restriction to ω  Ω, we see that this inequality is only valid for finite linear combinations of eigenfunctions.

7.5. A SPECTRAL INEQUALITY

261

local Carleman estimates away from boundaries for Ds2 + P0 ; Theorem 3.11 interpolation inequality initiated from the boundary; Theorems 5.13 and 7.12

local Carleman estimates at points of {0} × Ω for Ds2 + P0 ; Theorem 3.28

global Carleman estimate with a boundary observation; Theorem 3.40

local Carleman estimates at points of (0, S0 ) × ∂Ω for Ds2 + P0 ; Theorem 3.29

spectral inequality; Theorem 7.10

Figure 7.1. The two proof schemes of the spectral inequality of the Dirichlet Laplace operator Similarly to what we did for the resolvent estimate of Sect. 6.5, we expose two proof strategies to obtain this spectral inequality. The first approach we shall present in Sect. 7.5.1 is based on local Carleman estimates and goes through the derivation of an interpolation inequality. The second approach we present in Sect. 7.5.2 is based on the derivation of a global Carleman estimate (patching together local estimates as presented in Sects. 3.5 and 3.6) that directly yields the estimate of Theorem 7.10. Figure 7.1 shows the structure of the two proof strategies. This figure is the counterpart of Fig. 6.1. In Sect. 7.5.3, we show that the inequality (7.5.1) is a sharp estimation √ as far as the power of μ in exp(K μ) is concerned. 7.5.1. Spectral Inequality Through an Interpolation Inequality. Let S0 > 0 and β ∈ (0, S0 /2). Let also Z = (0, S0 ) × Ω and Y = (β, S0 − β) × Ω. We set z = (s, x) with s ∈ (0, S0 ) and x ∈ Ω. We define the following augmented elliptic operator Q := Ds2 + P0 + R in Z, where R denotes any first-order differential operator in Z with bounded coefficients. We give here an interpolation inequality with a boundary observation at {s = 0} × ω. Theorem 7.12. Let ω be an open set in Ω. There exist C > 0 and δ ∈ (0, 1) such that for u ∈ H 2 (Z) that satisfies u(0,S0 )×∂Ω = 0 for s ∈ (0, S0 )

262

7. CONTROLLABILITY OF PARABOLIC EQUATIONS

we have (7.5.2)

 δ + |u | + |∂ u | . uH 1 (Y ) ≤ Cu1−δ Qu 2 1 s |s=0 |s=0 1 2 L (Z) H (Z) H (ω) L (ω)

As Q is elliptic in Z and as Y denotes an open set that meets smooth parts of the boundary of Z, Theorem 7.12 is a direct consequence of Theorem 5.13.

Proof of Theorem 7.10. There exists (αj )j ⊂ C, such that w = μj ≤μ αj φj . We introduce

αj √ u(s, .) = √ sinh(s μj )φj , μj μj ≤μ which is in H 2 (Z), as the sum is finite, and satisfies Qu = 0,

u|s=0 = 0,

u|(0,S0 )×∂Ω = 0,

where Q = Ds2 + P0 . We can apply the interpolation inequality of Theorem 7.12 to the function u and we obtain, for some δ ∈ (0, 1), δ uH 1 (Y )  u1−δ H 1 (Z) |∂s u|s=0 |L2 (ω) .

(7.5.3) As ∂s u|s=0 =

μj ≤μ αj φj

= w, we have

|∂s u|s=0 |L2 (ω) = wL2 (ω) . √ √ √ √ √ Note that | sinh(s μj )/ μj |  eC μ and | cosh(s μj )|  eC μ , this gives √

u2L2 (Z) + ∂s u2L2 (Z)  eC μ |αj |2 . (7.5.4)

μj ≤μ

By (10.1.12), we also obtain Dx u2L2 (Z)  eC We thus have u2H 1 (Z)  eC

(7.5.5)

√ μ

√

μj ≤μ

μ

μj ≤μ

|αj |2 .

|αj |2 .

Observe now that for s ≥ β we have √ √ sinh(s μj )/ μj ≥ β. Then, for some κ > 0 we have (7.5.6) S0 −β

√ 2 2 2 κ |αj |2 ≤ |αj |2 ∫ μ−1 j | sinh(s μj )| ds = wL2 (Y ) ≤ wH 1 (Y ) . μj ≤μ

μj ≤μ

β

Following (7.5.3), (7.5.4), (7.5.5), and (7.5.6), we obtain √

|αj |2  eC μ w2L2 (ω) . μj ≤μ

7.5. A SPECTRAL INEQUALITY

263

√ √ Finally, we may replace C exp(C μ) by simply exp(K μ), as span(φj ; μj ≤  μ) = {0}, for μ ∈ [0, μ0 ). 7.5.2. Spectral Inequality Through the Derivation of a Global Carleman Estimate. Let S0 > 0 and Z = (0, S0 ) × Ω. We choose ω0 an open set of Ω such that ω0  ω. We first choose a function ψ1 ∈ C ∞ (Ω) according to Proposition 3.31: ∂ν ψ1 |∂Ω (x) < 0,

for x ∈ ∂Ω,

and

|dψ1 (x)| = 0,

for x ∈ Ω \ ω0 .

Adding a sufficiently large constant to the function, we moreover have ψ1 ≥ C > 0. We also require ψ1 to possess a finite number of nondegenerate critical points located in ω0 (see Remark 3.32). ˜ x) = ψ1 (x)ψ2 (s). The We also set ψ2 (s) = sin(πs/S0 ), and we define ψ(s, critical points of ψ˜ are finitely many and are located in the set {S0 /2} × ω0 . With 0 < S0 /2 < s3 < s3 < S0 , we set B3 = {(s, x) ∈ Z; s3 < s < s3 }. We then choose 0 < s1 < S0 /2 such that sin(πs1 /S0 ) sup ψ1 < sin(πs3 /S0 ) inf ψ1 Ω

Ω

and set

s1

= S0 − s1 , and we define

B1 = {(s, x) ∈ Z; 0 ≤ s < s1 , x ∈ / ω0 },

B1 = {(s, x) ∈ Z; s1 < s ≤ S0 }.

Let s2 > 0 be such that S0 /2 < s2 < s3 . Arguing as in the proofs of Propositions 3.31 and 3.39, one can “move” the critical points of the function ψ outside Z through the part of the boundary {0} × ω0 . In fact, one can build a smooth function ρ : Z → Z such that ψ = ψ˜ ◦ ρ satisfies |ψ  | = 0 in Z, ψ = ψ˜ in Z \ [0, s2 ) × ω0 . In particular the composition with ρ does not affect the values of ψ˜ in B1 , B1 , and B3 . From the construction above, we have sup ψ ≤ sup ψ < inf ψ. B1

B1

B3

Setting ϕ(s, x) = exp(γψ(s, x)), with γ > 0, we have (7.5.7)

sup ϕ ≤ sup ϕ < inf ϕ. B1

B1

B3

Let W0 , V1 be open neighborhoods of {0} × ∂Ω such that W0  V1  B1 . Let also s0 be such that s1 < s0 < S0 and define W0 = {(s, x) ∈ Z; s0 ; < s ≤ S0 }. We set W = W0 ∪ W0 . The geometry we have just introduced is illustrated in Fig. 7.2. We choose χ ∈ C ∞ (Z) such that χ ≡ 1 in a neighborhood of Z \ (V1 ∪ B1 ),

χ ≡ 0 in W.

Let s0 ∈ (0, S0 ) be such that (0, s0 ) × ∂Ω ⊂ W0 and S0 − s0 > s0 . We set Γ0 = (s0 /2, S0 − s0 /2) × ∂Ω.

264

7. CONTROLLABILITY OF PARABOLIC EQUATIONS

s S0

W0

s0

B1

s1 s3

B3

s3 s2 S0 /2 Γ0

Γ0

s1

0

B1

B1

V1 W0

ω0

Ω

V1 W0

ω

Figure 7.2. Geometry for the derivation of the spectral inequality through a global Carleman estimate (the s axis is stretched near s = 0 for the sake of clarity)

For w ∈ span{φj ; μj ≤ μ}, there exists (αj )j ⊂ C, such that w = μj ≤μ αj φj . Similarly to what is done in Sect. 7.5.1, we introduce

αj √ u(s, .) = √ sinh(s μj )φj , μj μj ≤μ which is in H 2 (Z), as the sum is finite, and satisfies Qu = 0, where Q =

Ds2

u|s=0 = 0,

u|(0,S0 )×∂Ω = 0,

+ P0 , and moreover ∂s u|s=0 = w. We have uH 1 (Z)  eC

(7.5.8)

√

μ

wL2 (Ω) ,

by (7.5.5). Moreover we have (7.5.9) u2L2 (B3 ) = 

∫

(s3 ,s3 )

μj ≤μ

u(s, .)2L2 (Ω) ds =

μj ≤μ

|αj |2

∫

(s3 ,s3 )

1

2 2 μ−1 j sinh (μj s) ds

|αj |2 = w2L2 (Ω) ,

as sinh(t) ≥ t. According to Lemma 3.5, we pick γ > 0 sufficiently large for (Q, ϕ) to fulfill the sub-ellipticity property of Definition 3.2 in Z. With Γ0 as introduced above, observe that ϕ is a global Carleman weight function on Z adapted to Γ0 in the sense of Definition 3.38. With W as introduced above,

7.5. A SPECTRAL INEQUALITY

265

we can invoke the global Carleman estimate with boundary observation of Theorem 3.40: there exist τ∗ > 0 and C ≥ 0 such that τ 3 eτ ϕ χu2L2 (Z)  eτ ϕ Qχu2L2 (Z) + eτ ϕ ∂s u|s=0 2L2 (ω) ,

(7.5.10)

for τ ≥ τ∗ . Observe that Qχu = [Q, χ]u where the commutator is a firstorder differential operator supported in V1 ∪ B1 . We thus have eτ ϕ QχuL2 (Z)  eτ ϕ uL2 (V1 ∪B  ) + eτ ϕ Ds,x uL2 (V1 ∪B  )

(7.5.11)

e e

1

τ supB

 1 ∪B1

ϕ

1

uH 1 (Z)

√ τ supB  ϕ+C μ 1

wL2 (Ω) ,

by (7.5.8) and we have eτ ϕ ∂s u|s=0 L2 (ω) ≤ eCτ wL2 (ω) .

(7.5.12)

By (7.5.9) and as χ ≡ 1 on B3 , we write eτ ϕ χuL2 (Z)  eτ inf B3 ϕ uL2 (B3 )  eτ inf B3 ϕ wL2 (Ω) . √ By (7.5.7) and (7.5.10)–(7.5.13), we thus obtain for τ = σ μ, with σ > 0 chosen sufficiently large, (7.5.13)

wL2 (Ω) ≤ CeCτ wL2 (ω) ≤ C  eC

 √μ

wL2 (ω) ,

which concludes the proof of the spectral inequality of Theorem 7.10, as we √ √ may replace C  exp(C  μ) by simply exp(K μ), as span(φj ; μj ≤ μ) = {0},  for μ ∈ [0, μ0 ). 7.5.3. Sharpness of the Spectral Inequality. Proposition 7.13. Let ω be a nonempty open set in Ω with ω = Ω. There exist C > 0 and m0 > 0 such that for all μ ≥ m0 there exists a sequence (αj )μj ≤μ , such that 2 √ 

  |αj |2 ≥ CeC μ  αj φj (x) . μj ≤μ

μj ≤μ

L2 (ω)

We denote by kt (x, x ) the heat kernel that we can write kt (x, x ) =

Proof. −tμ j φj (x)φj (x ) for t > 0, with (φj )j∈N ⊂ L2 (Ω) a Hilbert basis j∈N e formed by eigenfunctions of the operator P0 , associated with the eigenvalues (μj )j∈N , sorted as a nondecreasing sequence. The action of the parabolic semigroup S(t) reads S(t)f (x) = ∫ kt (x, x )f (x ) dx (see Sect. 10.2.4). We then write  −tμ   −tμ   e j φj (x)φj (x ) ≤ |kt (x, x )| +  e j φj (x)φj (x ). μj ≤μ

μj >μ

We have, for k ∈ N chosen sufficiently large, as shown in (10.2.7), (7.5.14)

φj L∞  μkj .

266

7. CONTROLLABILITY OF PARABOLIC EQUATIONS

Let T > 0. By Theorem 10.37 in Sect. 10.2.4, there exists δ > 0 such that kt (x, x )  t−d/2 e−δ

|x−y|2 t

x, x ∈ Ω, t ∈ (0, T ].

,

Let x0 be such that dist(x0 , ω) > 0. We then have kt (x, x0 )  e−C0 /t , with C0 > 0, uniformly for x in ω and t ∈ (0, T ]. From (7.5.14), we thus obtain   −tμ

−tμj 2k  e j φj (x)φj (x0 )  e−C0 /t + e μj , x ∈ ω, t ∈ (0, T ]. μj ≤μ

μj >μ

√ We choose αj = e−tμj φj (x0 ), and we take t = 1/ μ ≤ T . We have √  

−tμj 2k  αj φj (x)  e−C0 μ + e μj , x ∈ ω. μj ≤μ

μj >μ

To estimate the second term, we introduce Jμ = {l; μl ≤ μ}. The Weyl law given in Theorem 10.3 yields #Jμ  μd/2 . We write

−tμj 2k

e μj = e−tμj μ2k j n≥0 2n μμ



e−tμ2

n



 2n+1 μ ,

 = 2k + d/2.

n≥0

√ For μ ≥ 1 and t = 1/ μ, we obtain √ √ √

−(2n −1) (n+1)

−tμj 2k e μj  e− μ μ e 2  e− μ μ  e−C μ . μj >μ

n≥0

  √ We have thus obtained  μj ≤μ αj φj (x)  e−C μ , which yields √ 2  αj φj (x) dx  |ω|e−C μ . ∫

ω

μj ≤μ

We now conclude by proving μj ≤μ |αj |2 ≥ Cμd/4 ≥ 1, for μ sufficiently large with the choice of coefficients αj , j ∈ N, we have made above. In fact we find

−2tμj

−2tμj

|αj |2 = e |φj (x0 )|2 = k2t (y0 , y0 ) − e |φj (x0 )|2 . μj ≤μ

μj ≤μ

μj >μ

√ As here t = 1/ μ is small, Theorem 10.39, in Sect. 10.2.4, gives k2t (x0 , x0 )  (2t)−d/2  μd/4 . Finally, using (7.5.14), we obtain the following estimate: √

−2tμj 2k

−2tμj e |φj (x0 )|2  e μj  e−C μ , μj >μ

as proven above.

μj >μ



7.6. PARTIAL OBSERVABILITY AND PARTIAL CONTROL

267

7.6. Partial Observability and Partial Control For T > 0, we consider the following control system: d y + P0 y = 1ω v, 0 ≤ t ≤ T, y(0) = y 0 ∈ L2 (Ω). dt For μ ≥ 0, we define the finite-dimensional space

(7.6.1)

Eμ = span{φk ; μk ≤ μ}. Proposition 7.14 (Partial Control). Let μ ≥ 0. System (7.6.1) is null-controllable in Eμ : there exists a control function v that is such that ΠEμ y(T ) = 0. Moreover, 1

vL2 ((0,T )×ω) ≤ T − 2 eK

√

μ

y 0 L2 (Ω) ,

for y 0 ∈ L2 (Ω), where K is the constant in the spectral inequality of Theorem 7.10. For such a control function, we have √   (7.6.2) y(T )L2 (Ω) ≤ 1 + eK μ y 0 L2 (Ω) . For a ≥ 0, when we consider the time interval [a, a + T ] instead of [0, T ], we shall denote by Vμ (y 0 , a, T ) such a control satisfying (7.6.3)

1

Vμ (y 0 , a, T )L2 ((a,a+T )×Ω) ≤ T − 2 eK

√

μ

y 0 L2 (Ω) .

By Proposition 7.7, proving the first part of Proposition 7.14 is equivalent to the derivation of the following observability inequality. Lemma 7.15 (Partial Observability). For μ ≥ 0, we have 1

S(T )ΠEμ zL2 (Ω) ≤ T − 2 eK

√

μ

S(t)ΠEμ zL2 ((0,T )×ω) ,

z ∈ L2 (Ω),

where K is the constant in the spectral inequality of Theorem 7.10. 1

Proof. We prove that z(T )L2 (Ω) ≤ T − 2 eK z(t) = S(t)z 0 is the weak solution to d z + P0 z = 0, dt

√

μ z

L2 ((0,T )×ω) ,

where

z(0) = z 0 ∈ Eμ .

Observe that for all t ∈ (0, T ) we have z(t) ∈ Eμ since P0 and ΠEμ commute. We then have T

T z(T )2L2 (Ω) ≤ ∫ z(t)2L2 (Ω) dt ≤ e2K 0

√

μ

T

∫ ∫ |z(t)|2 dt dx, 0ω

because of the parabolic decay of the L2 -norm (see (10.2.4)) and from the spectral inequality of Theorem 7.10.  We now conclude the proof of Proposition 7.14.

268

7. CONTROLLABILITY OF PARABOLIC EQUATIONS

Proof of Estimate (7.6.2). We write y(T ) = y 1 (T ) + y 2 (T ) with T

y 1 (T ) = S(T )y 0 and y 2 (T ) = ∫ S(T − σ)1ω v(σ, .)dσ. As S(t)L (L2 (Ω) ≤ 1, 0

we have T

y 1 (T )L2 (Ω) ≤ y 0 L2 (Ω) ,

y 2 (T )L2 (Ω) ≤ ∫ v(σ, .)L2 (Ω) dσ. 0



The Cauchy–Schwarz inequality then yields the conclusion.

We finish this section by the following lemma on the natural exponential decay of the semigroup S(t) on the space Eμ⊥ (note that S(t) leaves that space invariant). Lemma 7.16. We have S(t)z 0 L2 (Ω) ≤ e−μt z 0 L2 (Ω) for t ≥ 0, if z 0 ∈ Eμ⊥ . Proof. With the spectral family (φj , μj ) introduced in Sect. 10.1, if

z 0 ∈ Eμ⊥ we have z 0 = μj >μ (z 0 , φj )φj yielding S(t)z 0 = μj >μ (z 0 , φj ) e−μj t φj and

2 |(z 0 , φj )|2 e−2μj t S(t)z 0 L2 (Ω) = μj >μ

≤ e−2μt

μj >μ

2

|(z 0 , φj )|2 = e−2μt z 0 L2 (Ω) . 

7.7. Construction of a Control Function for a Parabolic Equation For j ∈ N, we set Fj = E22j = span{φk ; μk ≤ 22j }.

! We split the time interval [0, T ] into sub-intervals, [0, T ] = j∈N [aj , aj+1 ], and Tj = L2−jρ with ρ ∈ (0, 1) with a0 = 0, aj+1 = aj + 2Tj , for j ∈ N

∞ and the constant L chosen such that 2 j=0 Tj = T . We now define the control function v(t, x) for system (7.6.1) according to the strategy exposed at the end of Sect. 7.4: we set, recalling the definition of the partial control operator Vμ given in (7.6.3), • if t ∈ (aj , aj + Tj ], then v(t, x) = V22j (ΠFj y(aj , .), aj , Tj ) and t

y(t, .) = S(t − aj )y(aj , .) + ∫ S(t − s)1ω v(s, .)ds; aj

• if t ∈ (aj + Tj , aj+1 ], then v(t, x) = 0 and y(t, .) = S(t − aj − Tj )y(aj + Tj , .), where, as above, S(t) denotes the parabolic semigroup.

7.7. CONSTRUCTION OF A CONTROL FUNCTION FOR A PARABOLIC. . .

269

We now follow the evolution of the L2 -norm of y(t) for the values of t = aj and t = aj + Tj , for j ∈ N. The choice of the control v in the time interval [aj , aj + Tj ], j ∈ N, and Proposition 7.14 yield, for some C0 > 0,  j j y(aj + Tj , .)L2 (Ω) ≤ 1 + eK2 y(aj , .)L2 (Ω) ≤ eC0 2 y(aj , .)L2 (Ω) , and ΠFj y(aj + Tj , .) = 0. During the passive mode, t ∈ [aj + Tj , aj+1 ], the solution is subject to an exponential decay as given by Lemma 7.16: 2j T

y(aj+1 , .)L2 (Ω) ≤ e−2

j

y(aj + Tj , .)L2 (Ω) .

j −22j T

We thus obtain y(aj+1 , .)L2 (Ω) ≤ eC0 2 have

y(aj+1 , .)L2 (Ω) ≤ e

j k=0

j

C0 2k −22k Tk

y(aj , .)L2 (Ω) , and hence we y 0 L2 (Ω) ,

j ∈ N.

We have 22k Tk = L2k(2−ρ) . We observe that 2 − ρ > 1, which yields lim

j

(C0 2k − (L − L )2k(2−ρ) ) = −∞,

j→∞ k=0

for 0 < L < L. We thus have (7.7.1)

 j(2−ρ)

y(aj+1 , .)L2 (Ω) ≤ e−L 2

y 0 L2 (Ω) ,

j ∈ N.

We conclude that limj→∞ y(aj , .)L2 (Ω) = 0, i.e. y(T, .) = 0 as y ∈ C 0 ([0, T ]; L2 (Ω)) since the r.h.s. of (7.6.1), namely 1ω v, is in L2 ((0, T ) × Ω) by construction as we shall now see. We have v2L2 ((0,T )×Ω) = j≥0 v2L2 ((aj ,aj +Tj )×Ω) . From the L2 -norm of the control given in Proposition 7.14 and (7.7.1), we deduce 

−1 2K2j −2L 2(j−1)(2−ρ)  0 2 y L2 (Ω) Tj e e v2L2 ((0,T )×Ω) ≤ T0−1 e2K + j≥1



−1 2K2j −C2j(2−ρ)  0 2 y L2 (Ω) . ≤ T0−1 e2K + Tj e j≥1

As 2−ρ>1 and Tj =L2−jρ , the series is convergent. We have vL2 ((0,T )×Ω) ≤ CT y 0 L2 (Ω) with CT < ∞. Hence, we have obtained the following nullcontrollability result. Theorem 7.17 (Null-Controllability of a Linear Parabolic Equation). Let T > 0. There exists CT > 0 such that for all initial conditions y 0 ∈ L2 (Ω), there exists v ∈ L2 ((0, T ) × Ω), with vL2 ((0,T )×Ω) ≤ CT y 0 L2 (Ω) , such that the weak solution y(t) ∈ C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) ∩ H 1 (0, T ; H −1 (Ω)) to d y + P0 y = 1ω v, dt satisfies y(T ) = 0.

y(0) = y 0 ,

270

7. CONTROLLABILITY OF PARABOLIC EQUATIONS

With Proposition 7.7 with E = L2 (Ω) (see Sect. 7.4), we have the following observability result. Theorem 7.17 (Observability of a Linear Parabolic Equation). There exists CT > 0 such that the weak solution z(t) ∈ C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) ∩ H 1 (0, T ; H −1 (Ω)) to the adjoint system d z + P0 z = 0, z(0) = z 0 , dt with z 0 ∈ L2 (Ω), satisfies the following observability inequality: z(T )L2 (Ω) ≤ CT 1ω zL2 ((0,T )×Ω) , or equivalently S(T )z 0 L2 (Ω) ≤ CT 1ω S(t)z 0 L2 ((0,T )×Ω) , for z 0 ∈ L2 (Ω). Recall that the value of the constant CT is the same in both result. We see that it estimates the cost of the null-controllability in time T > 0. Remark 7.18. Observe that the set of all controls that achieve nullcontrollability T   Wy0 = v ∈ L2 ((0, T ) × Ω); S(T )y 0 + ∫ S(T − σ)1ω v(σ)dσ = 0 0

L2 ((0, T ) × Ω).

is a nonempty affine subspace of The null control of minimal L2 -norm is then characterized as the orthogonal projection of 0 onto V Wy0 . See Proposition 7.8 in the case E = H = L2 (Ω). 7.8. Dual Approach for Observability and Control Cost Here, we provide a direct derivation of the observability inequality of Theorem 7.17 , meaning that only basic properties of the solutions of the adjoint parabolic equation (7.4.2) and the spectral inequality of Theorem 7.10 will be used. This approach can be extended to other cases (see Remark 7.21). Moreover, this approach allows one to derive an estimation of the observability constant CT . This yields in particular an estimate of the control cost, which is the minimal L2 -norm of the control function, as the control time T > 0 goes to zero. The key result is the following lemma. Lemma 7.19. Let Z, H be two Hilbert spaces. Let S(t) be a bounded semigroup on Z and B ∗ : Z → H be a bounded operator. Let T > 0. If, for some real function f : (0, T ] → R such that limt→0+ f (t) = 0 and some q ∈ (0, 1), we have the following inequality: t

(7.8.1)

f (t)S(t)z2Z − f (qt)z2Z ≤ ∫ B ∗ S(σ)z2H dσ, 0

t ∈ (0, T ],

for z ∈ Z, then the following observability inequality holds: T   2 2 z 0 ∈ Z. f (1 − q)T S(T )z 0 Z ≤ ∫ B ∗ S(t)z 0 H dt, 0

7.8. DUAL APPROACH FOR OBSERVABILITY AND CONTROL COST

271

The inequality (7.8.1) is sometimes called an approximate observability estimate. Proof. The proof is based on a telescoping series argument. We set k T

τk = q (1 − q)T , k ∈ N. We have

0 = T and Tk+1 = Tk − τk < Tk , where k∈N τk = T ; in particular, Tk = T − 0≤j≤k−1 τj goes to 0 when k goes to ∞. We apply (7.8.1) to z = S(Tk+1 )z 0 and t = τk , and we obtain, as qτk = τk+1 , 2

Tk

2

2

f (τk )S(Tk )z 0 Z − f (τk+1 )S(Tk+1 )z 0 Z ≤ ∫ B ∗ S(σ)z 0 H dσ. Tk+1

Summing these inequalities for k = 0, . . . j − 1, we obtain 2

2

T

2

f (τ0 )S(T )z 0 Z − f (τj )S(Tj )z 0 Z ≤ ∫ B ∗ S(σ)z 0 H dσ. Tj

The limit j → ∞ yields the result.



The following lemma provides an approximate observability estimate. Lemma 7.20. Let T > 0. There exists f : [0, T ] → R such that limt→0+ f (t) = 0 and (7.8.2) t

f (t)S(t)z2L2 (Ω) − f (t/2)z2L2 (Ω) ≤ ∫ 1ω S(σ)z2L2 (Ω) dσ, 0

t ∈ (0, T ].

A possible choice is

   t exp − K(K + K 2 + 2t ln 6)/t , 4 where K is the constant that occurs in the spectral inequality of Theorem 7.10. f (t) =

With Lemma 7.19, with Z = H = L2 (Ω) and B = 1ω , we recover the result of Theorem 7.17 and we give an estimation of the observability constant; that is, the L2 -norm of the control as T goes to s 0+ . Theorem 7.17 (Observability of a Linear Parabolic Equation). There exists CT such that, S(T )z 0 L2 (Ω) ≤ CT 1ω S(t)z 0 L2 ((0,T )×Ω) ,

z 0 ∈ L2 (Ω).

Moreover, we have CT2 ∼

8 4K 2 e T as T → 0+ . T

Proof of Lemma 7.20. As in Sect. 7.6, we set Eμ = span{φk ; μk ≤ μ}. Let z ∈ L2 (Ω), we write z = z + z⊥ , where z ∈ Eμ and z⊥ ∈ Eμ⊥ . The value of μ ≥ 0 will be chosen below. Note that we have z 2L2 (Ω) +

272

7. CONTROLLABILITY OF PARABOLIC EQUATIONS

z⊥ 2L2 (Ω) = z2L2 (Ω) and that the semigroup S(t) leaves both Eμ and Eμ⊥ invariant. Let t ∈ [0, T ]. We write, with Lemma 7.16, S(t)z2L2 (Ω) = S(t)z 2L2 (Ω) + S(t)z⊥ 2L2 (Ω) ≤ S(t)z 2L2 (Ω) + e−2tμ z2L2 (Ω) . For 0 ≤ σ ≤ t, one has S(t)z L2 (Ω) ≤ S(σ)z L2 (Ω) , as can be observed by writing S(t)z = S(t − σ) ◦ S(σ)z using the semigroup property and by using the contraction property of the semigroup given in Lemma 10.26. One can thus write 2 t 2 t S(t)z 2L2 (Ω) = ∫ S(t)z 2L2 (Ω) dσ ≤ ∫ S(σ)z 2L2 (Ω) dσ. t t/2 t t/2 With the spectral inequality of Theorem 7.10, one obtains t √ 2 S(t)z 2L2 (Ω) ≤ e2K μ ∫ 1ω S(σ)z 2L2 (Ω) dσ t t/2 t  √  4 ≤ e2K μ ∫ 1ω S(σ)z2L2 (Ω) + 1ω S(σ)z⊥ 2L2 (Ω) dσ. t t/2

With Lemma 7.16, we have t √ 4 4 2K √μ t e ∫ 1ω S(σ)z⊥ 2L2 (Ω) dσ ≤ e2K μ−tμ ∫ z⊥ 2L2 (Ω) dσ t t t/2 t/2

≤ 2e2K

√

yielding S(t)z2L2 (Ω)

≤ 3e

√ 2K μ−tμ

z2L2 (Ω)

4e2K + t

μ−tμ

√

z2L2 (Ω) ,

μ t

∫ 1ω S(σ)z2L2 (Ω) dσ, 0

which we write t 3t t −2K √μ e S(t)z2L2 (Ω) − e−tμ z2L2 (Ω) ≤ ∫ 1ω S(σ)z2L2 (Ω) dσ. 4 4 0 Note that this inequality holds for any μ ≥ 0. Below we shall pick μ as a function of t. To obtain the sought inequality (7.8.2), we wish to find a function f : [0, T ] → R such that √ 3t t g1 (t) := e−tμ ≤ f (t/2) and f (t) ≤ g2 (t) := e−2K μ , t ∈ [0, T ]. 4 4 A sufficient condition is g1 (2t) ≤ g2 (t) for t ∈ [0, T ]. One then picks g1 (2t) ≤ f (t) ≤ g2 (t), and the property limt→0+ f (t) = 0 is also fulfilled. The sufficient condition reads (7.8.3)

6e−2tμ ≤ e−2K

√

μ

,

t ∈ [0, T ],

which can be fulfilled for μ = μ(t) ≥ 0 properly chosen. In fact, this √ √ condition reads q( μ) = 2tμ − 2K μ − ln 6 ≥ 0. Hence, if one chooses

7.9. PROPERTIES OF THE REACHABLE SET AND GENERALIZATIONS

273

√   μ = K + K 2 + 2t ln 6 /(2t), that is, the larger root of q, the sufficient condition (7.8.3) holds and one can use the function f given in the statement.  √

Remark 7.21. An inspection of the proof of Lemma 7.20 shows that the above analysis applies to any semigroup t → S(t) for which the following properties hold: there exists C > 0 such that (1) S(t)u2L2 (Ω) ≤ e−2μt u2L2 (Ω) for u ∈ Eμ⊥ ; (2) u2L2 (Ω) ≤ CeC (3)

S(t)u2L2 (Ω)

≤

√

2 ω uL2 (Ω) for u ∈ t−1 ∫0t S(σ)u2L2 (Ω) dσ μ 1

Eμ ; for u ∈ L2 (Ω).

7.9. Properties of the Reachable Set and Generalizations If we set, for T > 0, LT : L2 ((0, T ) × Ω) → L2 (Ω) T

v → ∫ S(T − σ)1ω v(σ) dσ, 0

the null-controllability result at time T of Theorem 7.17 (and the equivalent observability inequalities of Theorems 7.17 and 7.17 ) reads in fact   Ran S(T ) ⊂ Ran(LT ), (7.9.1) following the proof of Proposition 7.7 in Sect. 7.3. If we further set FT : L2 (Ω) × L2 ((0, T ) × Ω) → L2 (Ω) T

v → S(T )y0 + ∫ S(T − σ)1ω v(σ) dσ. 0

From (7.9.1), we have Ran(FT ) = Ran(LT ). We refer to the range of LT as the reachable set. Proposition 7.22. Let 0 < T ≤ T  . We have Ran(LT ) = Ran(LT  ).   Proof. Let y T ∈ Ran(LT ). Then, there exists v ∈ L2 0, T ; L2 (Ω) such that y T = LT v. If we set  0 if 0 < t ≤ T  − T, v˜(t) = v(t + T − T  ) if T  − T < t ≤ T  ,   we have v˜ ∈ L2 0, T  ; L2 (Ω) and T

T

0

T  −T

LT  v˜ = ∫ S(T  − σ)1ω v˜(σ) dσ = ∫ S(T  − σ)1ω v˜(σ) dσ T

= ∫ S(T − σ)1ω v(σ) dσ = y T . 0

Thus

yT

∈ Ran(LT  ).

274

7. CONTROLLABILITY OF PARABOLIC EQUATIONS 

Conversely, let y T ∈ Ran(LT  ). Then, there exists v ∈ L2 ((0, T  ) × Ω) such that T



y T = ∫ S(T  − σ)1ω v(σ) dσ 0

T  −T

T

= ∫ S(T  − σ)1ω v(σ) dσ + ∫ S(T  − σ)1ω v(σ) dσ 0

T  −T

T  −T

= S(T ) ∫ S(T  − T − σ)1ω v(σ) dσ 0

T

+ ∫ S(T − σ)1ω v(T  − T + σ) dσ, 0 

meaning that y T = FT (y 0 , w) with T  −T

y 0 = ∫ S(T  − T − σ)1ω v(σ) dσ ∈ L2 (Ω), 0   w(t) = v(T  − T + t) ∈ L2 0, T ; L2 (Ω) , 

and thus y T ∈ Ran(FT ) = Ran(LT ).



The reachable set is thus independent of T . We simply denote it by R. With the null-controllability property, we have    Ran S(T ) ⊂ R. T >0

Let α > 0. The unbounded operator Pα0 : L2 (Ω) → L2 (Ω) with domain D(Pα0 ) = K 2α (Ω) (see Sect. 10.1.3). This operator generates a C0 -semigroup S α (t) on L2 (Ω). With the Hilbert basis used for the statement of the spectral inequality in Sect. 7.5, we have

α μj u j φ j , uj = (u, φj )L2 (Ω) , u ∈ K 2α (Ω), Pα0 u = j∈N

and following Sect. 10.2 we have

−μα t e j yj φ j , S α (t)y =

yj = (y, φj )L2 (Ω) , y ∈ L2 (Ω).

j∈N

  Theorem 7.23. There exists T0 > 0 such that Ran S 1/2 (T0 ) ⊂ R.   ! Corollary 7.24. Let α > 1/2. We have T >0 Ran S α (T ) ⊂ R. Note that the case α = 1 is the case treated in the previous sections. For T > 0, as R = Ran(L  T ) = Ran(F  T ). Thus, the result of Theorem 7.23 shows solution that any y 1 ∈ Ran S 1/2 (T0 ) can be reached by the controlled   of 0 2 2 (7.1.3) at time T : for all y ∈ L (Ω), there exists v ∈ L (0, T ) × Ω such that y 1 = y(T ).

7.9. PROPERTIES OF THE REACHABLE SET AND GENERALIZATIONS

275

  Proof of Theorem 7.23. The property Ran S 1/2 (T0 ) ⊂ R holds if   one has Ran S 1/2 (T0 ) ⊂ Ran(LT ) for one value of T > 0 by Proposition 7.22. By Lemma 11.19, this is equivalent to having S 1/2 (T0 )∗ zL2 (Ω)  L∗T zL2 ((0,T )×Ω) ,

z ∈ L2 (Ω).

1/2

Since P0 is self-adjoint, one has S 1/2 (T0 )∗ = S 1/2 (T0 ). By the computation made in the proof of Proposition 7.6, one has L∗T zL2 ((0,T )×Ω) = 1ω S(T )zL2 ((0,T )×Ω) . The result thus follows if one finds T0 and T > 0 such that (7.9.2)

S 1/2 (T0 )zL2 (Ω)  1ω S(T )zL2 ((0,T )×Ω) ,

z ∈ L2 (Ω).

By Theorem 7.17 , one has, for some C0 > 0, t

1

0

0

e−C0 /t S(t)z2L2 (Ω)  ∫ 1ω S(s)z2L2 (Ω) ds  ∫ 1ω S(s)z2L2 (Ω) ds, if 0 < t ≤ 1. Integration with respect to t ∈ [0, 1] yields 1

1

∫ e−C0 /t S(t)z2L2 (Ω) dt  ∫ 1ω S(t)z2L2 (Ω) dt. 0

The function z reads z =

−2μj t |z |2 yielding j j∈N e

j∈N

0

j∈N zj φj

with (zj )j∈N ∈ 2 (C) and S(t)z2L2 (Ω) =

1

1

0

0

|zj |2 ∫ e−C0 /t−2μj t dt  ∫ 1ω S(t)z2L2 (Ω) dt.

With Lemma 7.25 below, there exist T0 > 0 and C > 0 such that 1

∫ e−C0 /t−2μj t dt ≥ Ce−2T0 0

√

μj

,

for all j ∈ N. This yields 2

S 1/2 (T0 )zL2 (Ω) =

e−2T0

j∈N

√

μj

1

|zj |2  ∫ 1ω S(t)z2L2 (Ω) dt, 0

z ∈ L2 (Ω), 

that is, (7.9.2) for T = 1. Lemma 7.25. Let C0 > 0. There exists C1 , C2 > 0 such that 1

Is = ∫ e−C0 /t−st dt ≥ C1 e−C2

√

s

0

Proof. With the change of variable t → t λ/C0

Is = λ−1 C0 ∫ e−λ(1/t+t) dt, 0

,

s ≥ 0.

 C0 /s, one finds

with λ =

 C0 s.

276

7. CONTROLLABILITY OF PARABOLIC EQUATIONS

Since it suffices to obtain the result for s large, we may assume that λ/C0 ≥ 2 and thus 2

Is ≥ λ−1 C0 ∫ eλφ(t) dt,

with φ(t) = −1/t − t.

0

With the Laplace method, one obtains (see, e.g., [134]) 2

∫ eλφ(t) dt 0

π 1/2 −2λ e , λ→+∞ 2λ1/2 ∼



which yields the result. 7.10. Boundary Null-Controllability for Parabolic Equations

Let T > 0 and Ω be a smooth bounded connected open set of Rd . For some Γ nonempty open subset of ∂Ω, we consider the following control system: ⎧ ⎪ in (0, T ) × Ω, ⎨∂t y + P0 y = 0 (7.10.1) on (0, T ) × ∂Ω, y = 1Γ v ⎪ ⎩ in Ω. y(0) = y 0 The function v is the control. It can only act on Γ as a Dirichlet boundary condition, and we wish to achieve null-controllability at arbitrary time T > 0. We shall seek the control function v ∈ L2 (0, T ; L2 (∂Ω)). For such a regularity, a natural type of solution is given by Definition 10.60 in Chap. 10: for y 0 ∈ H −1 (Ω) and v ∈ L2 (0, T ; L2 (∂Ω)), a so-called weak solution is a function:       y ∈ C 0 [0, T ]; K −1 (Ω) ∩ L2 0, T ; L2 (Ω) ∩ H 1 (0, T ; K −2 (Ω) such that 0 = y(t), ψH −1 (Ω),H01 (Ω) − y 0 , ψH −1 (Ω),H01 (Ω) + (y, P0 ψ)L2 ((0,t)×Ω) + (1Γ v, ∂ν ψ|∂Ω )L2 ((0,T )×∂Ω) , ] for all t ∈ (0, T ) and for all ψ ∈ H 2 (Ω) ∩ H01 (Ω), since such a solution exists and is unique, as shown in Theorem 10.61. In particular, this unique weak solution y is given by the following Duhamel formula: t

(7.10.2)

y(t) = S(t)y 0 + ∫ S(t − σ)M 1Γ v(σ)dσ. 0

  Here, S(t) is the C0 -semigroup on K −2 (Ω) = K 2 (Ω) , with K 2 (Ω) = D(P0 ) = H 2 (Ω) ∩ H01 (Ω), generated by the operator P−2 : K −2 (Ω) → K −2 (Ω) with domain D(P−2 ) = L2 (Ω), which is an extension of the operator (P0 , D(P0 )) to K −2 (Ω). This semigroup S(t) is an extension of the C0 semigroup on L2 (Ω) generated by (P0 , D(P0 )), used in the above sections, where it is also denoted by S(t). For details, see Sects. 10.1.3 and 10.2. The bounded operator M : H −1/2 (∂Ω) → K −2 (Ω) is given by M = P−2 ◦ D, where D is the Dirichlet lifting map introduced in Definition 10.56 in

7.10. BOUNDARY NULL-CONTROLLABILITY FOR PARABOLIC EQUATIONS

277

Sect. 10.5. In the semigroup setting this shows that y is solution to the equation: d y + P−2 y = M 1Γ v, dt which holds in L2 (0, T ; K −2 (Ω)). Definition 7.26 (Boundary Null-Controllability). We say that we have boundary null-controllability for system (7.10.1) if for all y 0 ∈ H −1 (Ω), there exists v ∈ L2 (0, T ; ∂Ω) such that the weak solution y given by (7.10.2) satisfies y(T ) = 0. Similarly to what is done in Sect. 7.3, we introduce the operator: T

LT v = ∫ S(T − σ)M 1Γ v(σ)dσ. 0

By Theorem 10.42 (in the case r = −2), the operator LT maps continuously L2 (0, T ; ∂Ω) into H −1 (Ω). Remark 7.27. Observe that the solution of the parabolic equation lies in the space C ([0, T ]; H −1 (Ω)) here. However, the range of the control operator M is K −2 (Ω). One thus says that the control operator is unbounded. Compare with the general setting of Sect. 7.3. This is however not an obstruction to obtaining well-behaved solutions, as the range of LT is precisely H −1 (Ω). In fact, the regularization effect of the Duhamel term associated with the parabolic semigroup compensates for the lower regularity of the control operator. Boundary null-controllability is equivalent to having an observability estimate. Proposition 7.28. The boundary null-controllability for system (7.10.1) is equivalent to having, for some Cobs > 0, the following observability inequality: z(T, .)H 1 (Ω) ≤ Cobs |1Γ ∂ν z|(0,T )×∂Ω |L2 ((0,T )×∂Ω) , 0       0 where z(t, .) = S(t)z ∈ C 0 [0, T ]; H01 (Ω) ∩L2 0, T ; D(P0 ) ∩H 1 0, T ; L2 (Ω) is the unique strong solution to   d z(0) = z 0 , z + P0 z = 0 in L2 (0, T ) × Ω , dt for z 0 ∈ H01 (Ω). Strong solutions of the homogeneous parabolic semigroup equation are given in Corollary 10.35. Proof. Boundary null-controllability is equivalent to having (7.10.3)

Ran(S(T )) ⊂ Ran(LT ).

278

7. CONTROLLABILITY OF PARABOLIC EQUATIONS

Here S(t) is the C0 -semigroup on H −1 (Ω) = K −1 (Ω) generated by (P−1 , D(P−1 )), that is, extension to H −1 (Ω) of that generated by (P0 , D(P0 )), or restriction to H −1 (Ω) of that generated by (P−2 , D(P−2 )). With Lemma 11.19, the inclusion (7.10.3) is equivalent to S(T )∗ z 0 H 1 (Ω) ≤ CL∗T z 0 L2 ((0,T )×∂Ω) ,

(7.10.4)

0

z 0 ∈ H01 (Ω),

for some C > 0, with L2 (Ω) playing the rˆ ole of a pivot space. The operator S(T )∗ is simply the restriction of S(T ) to H01 (Ω) as one can readily check: it is thus the C0 -semigroup generated on H01 (Ω) by the operator (P1 , D(P1 )), with D(P1 ) = K 3 (Ω); see Sects. 10.1.3 and 10.2. As P1 is itself the restriction of P0 to H01 (Ω), we may use either P1 or P0 here. We now compute the adjoint operator of LT and its action on z 0 ∈ 1 H0 (Ω). At first, we consider z 0 ∈ K 2 (Ω) = H 2 (Ω) ∩ H01 (Ω), and we write, for v ∈ L2 ((0, T ) × ∂Ω), LT v, z 0 H −1 (Ω),H01 (Ω) = LT v, z 0 K −2 (Ω),K 2 (Ω) T

= ∫ S(T − σ)M 1Γ v(σ), z 0 K −2 (Ω),K 2 (Ω) dσ 0 T

= ∫ M 1Γ v(σ), S(T − σ)z 0 K −2 (Ω),K 2 (Ω) dσ 0

= (v, 1Γ M ∗ S(T − .)z 0 )L2 ((0,T )×∂Ω) , using Proposition 10.30. If z 0 ∈ H01 (Ω), then S(T − t)z 0 ∈ L2 (0, T ; K 2 (Ω)) by Corollary 10.35. Observe then that M ∗ S(T − t) : H01 (Ω) → L2 (0, T ; H 1/2 (Ω)) is bounded as M ∗ ∈ L (K 2 (Ω), H 1/2 (Ω)). By density, we thus obtain LT v, z 0 H −1 (Ω),H01 (Ω) = (v, 1Γ M ∗ S(T − .)z 0 )L2 ((0,T )×∂Ω) ,

z 0 ∈ H01 (Ω).

This gives L∗T u(t) = 1Γ M ∗ S(T − t)z 0 ,

z 0 ∈ H01 (Ω), 0 ≤ t ≤ T.

Inequality (7.10.4) then reads T

S(T )z 0 H 1 (Ω) ≤ C ∫ |1Γ M ∗ S(T − t)z 0 |L2 (∂Ω) dt 0

0 T

= C ∫ |1Γ M ∗ S(t)z 0 |L2 (∂Ω) dt, 0

z0

H01 (Ω).

for ∈ With the characterization of M ∗ given in Proposition 10.62, we then obtain the result.  We now prove the boundary null-controllability of the parabolic equation given in (7.10.1). Theorem 7.29 (Boundary Null-Controllability of a Linear Parabolic Equation). Let Ω be a smooth bounded connected open set of Rd in the sense recalled in Sect. 1.7, and let Γ be nonempty open subset of ∂Ω. The boundary null-controllability for system (7.10.1) holds true.

7.10. BOUNDARY NULL-CONTROLLABILITY FOR PARABOLIC EQUATIONS

279

By Proposition 7.28, Theorem 7.29 is equivalent to the following result. Theorem 7.29 (Boundary Observability of a Linear Parabolic Equation). Let Ω be a smooth bounded connected open set of Rd in the sense recalled in Sect. 1.7, and let Γ be nonempty open subset of ∂Ω. There exists CT > 0, such that z(T, .)H 1 (Ω) ≤ CT |1Γ ∂ν z|(0,T )×∂Ω |L2 ((0,T )×∂Ω) , 0

S(t)z 0

∈ C 0 ([0, T ]; H01 (Ω)) ∩ L2 (0, T ; H 2 (Ω) ∩ H01 (Ω)) ∩ where z(t, .) = H 1 (0, T ; L2 (Ω)) is the unique strong solution to d z + P0 z = 0, dt

z(0) = z 0 ,

for z 0 ∈ H01 (Ω). The value of the constant CT is of importance. As in the case of the null-controllability result with an inner control, the constant CT allows one to estimate the control cost. In fact, if y 0 ∈ H −1 (Ω), there exists a control function v ∈ L2 ((0, T ) × ∂Ω) such that |v|L2 ((0,T )×∂Ω) ≤ CT y 0 H −1 (Ω) such that the weak solution y of (7.10.1) satisfies y(T ) = 0. Proof of Theorem 7.29. Using the construction described at the be˜ to be a smooth open ginning of the proof of Proposition 3.39, we choose Ω 0 ˜ ˜ is an set that extends Ω near a point x of Γ, that is, Ω = Ω ∪ W and Ω 0 open neighborhood of x , and (∂Ω \ Γ) ∩ W = ∅. An illustration is given in Fig. 7.3.

ω x Ω

0

W Γ

∂Ω

˜= Figure 7.3. Geometry of the locally extended open set Ω ˜ Ω ∪ W and ω ⊂ Ω \ Ω ˜ \ Ω. We also extend smoothly We pick ω an open set such that ω ⊂ Ω ij ˜ requiring the coefficients p to Ω

˜ ξ ∈ Rd , pij (x)ξi ξj ≥ C|ξ|2 , x ∈ Ω, 1≤i,j≤d

280

7. CONTROLLABILITY OF PARABOLIC EQUATIONS

˜ R) with all derivatives bounded and such that pij = pji , and pij ∈ C ∞ (Ω; 1 ≤ i, j ≤ d.

We denote by P˜0 the second-order differential operator 1≤i,j≤d Di (pij (x) ˜ 0 the associated unbounded operator on L2 (Ω) ˜ and by P ˜ Dj ) defined on Ω, 2 1 2 ˜ ˜ ˜ ˜ with domain D(P0 ) = H (Ω)∩H0 (Ω). It generates on L (Ω) a C0 -semigroup ˜ that we denote by S(t). We apply the following control strategy. Let 0 < T1 < T . We set the boundary control function v to zero for t ∈ (0, T1 ). From (7.10.2), the weak solution is then simply given by y(t) = S(t)y 0 . From Proposition 10.32, we ˜ we extend y 1 by zero outside of Ω to obtain have y 1 = y(T1 ) ∈ H01 (Ω). In Ω, 1 ˜ a function in H0 (Ω). We denote this function by y˜1 . We now apply the null-controllability result of Theorem 7.17 to the parabolic semigroup equation: d ˜ 0 y˜ = 1ω v˜ for T1 ≤ t ≤ T, (7.10.5) y˜(T1 ) = y˜1 . y˜ + P dt ˜ that acts in ω only such that It yields a control function v˜ ∈ L2 ((T1 , T ) × Ω) the weak solution y˜ to (7.10.5) given by t

˜ − σ)1ω v˜(σ) dσ, ˜ − T1 )˜ y 1 + ∫ S(t y˜(t) = S(t T1

T1 ≤ t ≤ T,

˜ by Corolsatisfies y˜(T ) = 0. The function y˜ belongs to C ([T1 , T ]; H01 (Ω)) 2 2 lary 10.46, implying that y˜|(T1 ,T )×∂Ω ∈ L (T1 , T ; L (∂Ω)) with a support only located in [T1 , T ] × Γ. On the time interval (T1 , T ), we may thus set the boundary control function v to be equal to y˜|(T1 ,T )×∂Ω . From the uniqueness part of Theorem 10.61, we find that the weak solution to ⎧ ⎪ in (T1 , T ) × Ω, ⎨∂t y + P0 y = 0 on (T1 , T ) × ∂Ω, y = 1Γ v ⎪ ⎩ 1 in Ω, y(T1 ) = y is precisely given by y(t) = y˜(t)|Ω for T1 ≤ t ≤ T . With the boundary control v ∈ L2 ((0, T ) × ∂Ω) built in this two-step process, we have hence obtained y(T ) = 0.  7.11. Notes For general aspects of controllability for evolution equations and systems, we suggest the expositions by J.-L. Lions [235], J.-M. Coron [109], and M. Tucsnak and G. Weiss [321]. The unique continuation property for parabolic operators through a time-like hypersurface goes back to the work of S. Mizohata [261]. See also J.-C. Saut and B. Scheurer [300]. With such a property, one deduces the approximate controllability of the associated parabolic equation: any target in the energy space, here L2 , can be approached arbitrary close. A proof of this consequence can be done by means of duality; see, for instance,

7.11. NOTES

281

[235, 321]. This duality argument is also used to prove the equivalence between null-controllability and observability, the latter providing a quantification of the unique continuation property; see the work of S. Dolecki and D.L. Russell [121]. In the framework of parabolic equations, lots of details in the description of controllability properties in connection with observability inequalities can also be found in F. Boyer [84]. The first null-controllability result for the heat equation was obtained by H.O. Fattorini and D.L. Russell [141]. Their proof relies on the moment method. The proof of the null-controllability of the heat equation in arbitrary dimension is due to [218] and the work of F. Fursikov and O. Yu. Imanuvilov [156]. In [296], D.L. Russell showed that one can deduce the null-controllability of the heat equation from the exact controllability of the associated wave equation. This idea was further developed by K.-D. Phung [271] and L. Miller [253] by means of a transformation introduced by Y. Kannai [191]. This approach allowed the authors to estimate the behavior of the observability constant as the control time goes to zero; a result of this form is given in Theorem 7.17 : CT2 ∼

α β e T as T → 0+ . T

Some lower bound of the constant β can be found in L. Miller [253, 256] and the work of G. Tenenbaum and M. Tucsnak [319]. A recent reference on the estimation of the observability constant is the work of C. Laurent and M. L´eautaud [204]. Here, we provide several proofs of the null-controllability or equivalently the observability inequality. All the approaches we describe make use of a spectral inequality as in Theorem 7.10 that is proven by means of Carleman estimates for the associated elliptic operator. This strategy appears as a simplification of the method introduced in [218] for the proof of the nullcontrollability of the heat equation. The actual spectral inequality in the form given here can be found in joint works with E. Zuazua [220] and D. Jerison [186]. The proof of controllability given in Sect. 7.7 where a control is built by the iterative action on a growing part of the spectrum originates from [218]. The more direct proof of observability given in Sect. 7.8 can be found in the work of L. Miller [259]. In this book, we do not cover the important contribution of F. Fursikov and O. Yu. Imanuvilov [156]. In their approach, observability for a parabolic equation is deduced from a Carleman estimate derived for the parabolic operator. This makes use of a well-adapted choice of weight function that depends on both time and space coordinates. This approach allows one to consider parabolic semigroups that are not generated by self-adjoint operators. Moreover, it allows one to estimate the cost of the control with respect to coefficients in the lower-order parts of the operators as in the work of E. Fernandez-Cara and E. Zuazua [148]. This turns out to be

282

7. CONTROLLABILITY OF PARABOLIC EQUATIONS

crucial in the proof of the null-controllability of semilinear parabolic equations by means of a fixed-point argument as in the work of V. Barbu [52], E. Fernandez-Cara and E. Zuazua [149], and A. Doubova, E. FernandezCara, M. Gonz´ alez-Burgos, and E. Zuazua [126] and when considering equations like the Navier–Stokes equation; we refer to F. Fursikov and O. Yu. Imanuvilov [157], E. Fernandez-Cara, S. Guerrero, O. Yu. Imanuvilov, and J.-P. Puel [145, 146], E. Fernandez-Cara, M. Gonz´ alez-Burgos, S. Guerrero, and J.-P. Puel [144], O. Yu. Imanuvilov [178], and S. Ervedoza, O. Glass, S. Guerrero, and J.-P. Puel [137]. In Sect. 7.9 we consider the set of functions that can be reached by a controlled parabolic equation. We show that this reachable set contains the image of L2 by the semigroup S 1/2 (T ) generated by P 1/2 for T > 0 sufficiently large. Such result can be found in the works of E. FernandezCara and E. Zuazua [148] and L. Miller [257]. In dimension one, sharper descriptions are made of the reachable set in the works of H.O. Fattorini and D.L. Russel [141], Ph. Martin, P. Rouchon, and L. Rosier [243], J. Dard´e and S. Ervedoza [111], and A. Hartmann, K. Kellay, and M. Tucsnak [166]. The operators we consider have principal parts with smooth coefficients. With the analysis of Sect. 3.7.3, one can easily extend the results of this chapter to the case of principal parts with Lipschitz coefficients. In dimension one a spectral inequality can be obtained if only assuming the coefficients to be L∞ as shown in G. Alessandrini and L. Escauriaza [15], where the doubling property is used rather than the derivation of a Carleman estimate. A Carleman estimate in the case of coefficients with jumps is given in a joint work with A. Benabdallah and Y. Dermenjian [73]. The case of coefficients with bounded variations is further treated in [209]. In these two latter references, Carleman estimates are derived for a parabolic operator following [156], thus allowing for the treatment of the null-controllability of semilinear equations. This was extended in dimension greater than two by the derivation of Carleman estimates for elliptic transmission problems. References are given in Sect. 3.8. For the derivation of Carleman estimates for parabolic transmission problems, we refer to the work of A. Doubova, A. Osses, and J.-P. Puel [127] and [214]. This chapter is concerned with parabolic equations. If coupled, several parabolic equations form a system and its controllability is a relevant question. This question is readily treated if each equation is subject to its own control function. The analysis is much more involved if one wishes to only act directly on some equations and indirectly on the others through the coupling terms. For such questions, Carleman estimates have given positive answers but that have also shown their limitation. We refer to the works of F. Ammar-Khodja, A. Benabdallah, C. Dupaix, and M. Gonz´ alezBurgos [29, 30], F. Ammar-Khodja, A. Benabdallah, M. Gonz´ alez-Burgos, and L. de Teresa [31], F. Boyer, A. Benabdallah, M. Gonz´ alez-Burgos, and G. Olive [72], and F. Boyer and G. Olive [89].

7.11. NOTES

283

The elliptic operators we consider are uniformly elliptic. Degeneracy of the ellipticity property has been a source of studies in connection with the null-controllability property for the associated parabolic problem. We refer to the works by P. Cannarsa, P. Martinez, and J. Vancostenoble [102–104] and by F. Alabau-Boussouira, P. Cannarsa, and G. Fragnelli [9]. Grushintype equations are studied in the works of K. Beauchard, P. Cannarsa, and R. Guglielmi [62] and K. Beauchard, L. Miller, and M. Morancey [64]. Cases are exhibited where a minimal time is required for null-controllability to hold, a very different behavior from regular parabolic equation where nullcontrollability is achieved in an arbitrary small time. Hypoelliptic equations like the Kolmogorov equation are studied in the works of K. Beauchard and E. Zuazua [66], K. Beauchard [61], joint work with K. Beauchard, B. Helffer, and R. Henry [63], and joint work with I. Moyano [212]. Other classes of hypoelliptic equations are covered in the work of K. Beauchard and K. PravdaStarov [65]. Result on the controllability of the heat equation in the case of an unbounded operator of the form 1/|x|2 can be found in the works of J. Vancostenoble and E. Zuazua [322], S. Ervedoza [135], and R. Buffe [91]. The case of unbounded domains is treated in the works of P. Cannarsa, P. Martinez, and J. Vancostenoble [101], L. Miller [255], and [212]. For the study of the controllability of a parabolic equation associated with a fractional Laplace operator, we refer to the work of L. Miller [258]. For parabolic operators with complex coefficients, we refer to Fu [153]. For the controllability properties of stochastic parabolic equation, we refer to V. Barbu, A. R˘a¸scanu, and G. Tessitore [53], S. Tang, and X. Zhang [311], X.A. Zhang [328], Q. L¨ u [239], and H. Li and Q. L¨ u [232]. The control functions we consider here act on the parabolic equation in a nonempty open set. The natural question of the treatment of control sets with only positive measure finds answers in the case of analytic coefficients in the works of J. Apraiz and L. Escauriaza [38], K.-D. Phung, L. Wang, and C. Zhang [274], J. Apraiz, L. Escauriaza, G. Wang, and C. Zhang [39]. In the present chapter, we only consider the case of Dirichlet boundary conditions. For other types of boundary condition, Robin–Fourier-type conditions, see the works of F. Fursikov and O. Yu. Imanuvilov [156], A. Doubova, E. Fern´andez-Cara, M. Gonz´alez-Burgos [125], and E. FernandezCara, M. Gonz´ alez-Burgos, S. Guerrero, and J.-P. Puel [143, 144]. For dynamic Ventcel-type condition, see the work of L. Maniar, M. Meyries, and ˇ R. Schnaubelt [242]. Some general conditions of Lopatinski˘ı-Sapiro type are covered in Chap. 12 in Volume 2 for the derivation of a spectral inequality and the resulting null-controllability property. For Zaremba mixed conditions, see the joint work with P. Cornilleau [108] and the works of T. Ali Ziane, H. Ouzzane, and O. Zair [17, 18]. For some mixed conditions with equivalued boundary condition, see the work of Q. L¨ u and Z. Yin [240]. For results on the controllability of discretized version of parabolic equations, we refer to the work of S. Labb´e and E. Tr´elat [201], to joint works with F. Boyer and F. Hubert [85–87], the work of E. Fern´andez-Cara and A. M¨ unch [147], F. Boyer [84], and joint work with F. Boyer [88].

Part 3

Background Material: Analysis and Evolution Equations

CHAPTER 8

A Short Review of Distribution Theory Contents 8.1. Distributions on an Open Set of Rd and on a Manifold 8.1.1. Test Functions 8.1.2. Definition of Distributions and Basic Properties 8.1.3. Composition by Diffeomorphisms, Distributions on a Manifold 8.2. Temperate Distributions on Rd and Fourier Transformation 8.2.1. The Schwartz Space and Temperate Distributions 8.2.2. The Fourier Transformation on S (Rd ), S  (Rd ), and L2 (Rd ) 8.3. Distributions on a Product Space 8.3.1. Tensor Products of Functions 8.3.2. Tensor Products of Distributions 8.3.3. Convolution 8.3.4. The Kernel Theorem (Various Forms) 8.4. Notes

287 287 288 291 293 294 295 296 296 296 297 298 298

8.1. Distributions on an Open Set of Rd and on a Manifold 8.1.1. Test Functions. Let Ω be an open set in Rd . We recall that we denote by Cc∞ (Ω) the space of functions C ∞ , compactly supported in Ω. We do not describe the topology1 adapted to Cc∞ (Ω) but we rather define converging sequences in Cc∞ (Ω). First, we define seminorms on Cc∞ (Ω). For K a compact subset in Ω and N ∈ N we set ϕK,N =

sup x∈K,|α|≤N

|∂ α ϕ(x)|,

ϕ ∈ Cc∞ (Ω) with supp(ϕ) ⊂ K.

1See Sect. 8.4 for references on the precise topology of the various test function and distribution spaces we review here.

© 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 I, PNLDE Subseries in Control 97, https://doi.org/10.1007/978-3-030-88674-5 8

287

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Definition 8.1. Let (ϕn )n be a sequence in Cc∞ (Ω). We say that ϕn converges to ϕ in Cc∞ (Ω) if there exists a compact K of Ω such that ϕn , for all n, and ϕ are supported in K and for all N , ϕn − ϕK,N goes to 0 as n goes to ∞. Remark 8.2. The definition is natural. Note that it is important that the compact set K be independent of n. The space Cc∞ (Ω) is then complete in the following sense. Let (ϕn )n be a sequence in Cc∞ (Ω) and K be a compact subset of Ω such that ϕn , for all n, are supported in K. We assume ∀ε>0, ∀N >0, ∃n0 ∈N, such that ∀k, n∈N, k, n ≥ n0 ⇒ ϕn −ϕk K,N < ε. Then there exits ϕ ∈ Cc∞ (Ω) such that (ϕn )n converges to ϕ in Cc∞ (Ω). The assumptions imply that the limit ϕ is supported in K. For further comments on the topology of the space of test functions we refer to the notes at the end of this chapter. 8.1.2. Definition of Distributions and Basic Properties. A distribution on Ω is defined as an element of the dual of Cc∞ (Ω) with a continuity estimate. For a distribution T and for ϕ ∈ Cc∞ (Ω), we denote by T, ϕ ∈ C the image of ϕ, that is T, ϕ = T (ϕ). Definition 8.3. We say that T ∈ D  (Ω) if T is a linear form on Cc∞ (Ω) satisfying, for all K compact subset in Ω, there exist N ≥ 0 and C > 0 such that |T, ϕ| ≤ CϕK,N for all ϕ ∈ Cc∞ (Ω) supported in K. Remark 8.4. In the previous definition the constants N and C depend on the compact set K. Let f ∈ L1loc (Ω), we can define a canonical distribution Tf associated with f , given by Tf , ϕ = ∫Ω f (x)ϕ(x)dx for ϕ ∈ Cc∞ (Ω). One can check that Tf , · satisfies the properties of Definition 8.3 and the map f → Tf from L1loc (Ω) into D  (Ω) is injective. We shall thus simply denote Tf by f by abuse of notation. More generally, for μ a locally finite positive measure on Ω, we define μ, ϕ = ∫Ω ϕdμ. If μ is a signed measure we can also define an associated canonical distribution by an analogous formula. In particular, the Dirac measure δx0 , for x0 ∈ Ω, given by δx0 , ϕ = ϕ(x0 ), is a distribution. If we want to make precise the variable in which a distribution acts we shall write Tx , ϕ(x) in place of T, ϕ. This can be useful if the test function depends on several variables and if the distribution only acts with respect to one of them, say Tx , ϕ(x, y). See for instance Sects. 8.2.2 and 8.3.2. By duality, differentiation and multiplication by smooth function can be defined on distributions. Definition 8.5. Let T ∈ D  (Ω) and α ∈ Nd , we associate a distribution ∂ α T by the following formula ∂ α T, ϕ = (−1)|α| T, ∂ α ϕ,

ϕ ∈ Cc∞ (Ω).

8.1. DISTRIBUTIONS ON AN OPEN SET OF Rd AND ON A MANIFOLD

289

Let φ ∈ C ∞ (Ω) we associate a distribution φT by the following formula φT, ϕ = T, φϕ,

ϕ ∈ Cc∞ (Ω).

Remark 8.6. One can check that ∂ α T and φT are well defined distributions. Moreover if f is a sufficiently smooth function the usual differentiation or product yields the same distribution, that is, T∂ α f = ∂ α Tf and Tφf = φTf . For the product it is important to assume φ smooth. In fact, one cannot define a natural product on D  (Ω). Transformations such as translations and dilations can also be extended to distributions. For the sake of simplicity we only consider the case Ω = Rd but these operations can also be defined in the case of a general open set Ω that changes by the transformation. Let h ∈ Rd , λ ∈ R, and φ ∈ C ∞ (Rd ), we define τh φ(x) = φ(x − h),

Hλ φ(x) = φ(λx).

By duality, we extend these two transformations to distributions. We then obtain a definition that is consistent with the usual translations and dilations for locally integrable functions. Definition 8.7. Let h ∈ Rd and λ ∈ R∗ . Let T ∈ D  (Rd ), we define τh T and Hλ T by τh T, ϕ = T, τ−h ϕ,

Hλ T, ϕ = |λ|−d T, H1/λ ϕ,

ϕ ∈ Cc∞ (Rd ).

Remark 8.8. One can then define periodic distributions, translation invariant distributions, in one or several directions, homogeneous distributions etc. The restriction of a distribution to an open set is done as follows. Definition 8.9. Let Ω be an open set in Rd and U ⊂ Ω an open set. Let T ∈ D  (Ω) we define T|U by T|U , ϕ = T, ϕ for all ϕ ∈ Cc∞ (U ). Remark 8.10. The assumption that U is open is important. In general we cannot define the restriction on a closed space. In particular the restriction to a sub-manifold requires some smoothness assumption on T . Similarly to the space of test functions, here, we shall not define a topology on the space D  (Ω) but we rather give a definition of converging sequences of distributions. Definition 8.11 (Sequential Topology). Let (Tn )n be a sequence of elements of D  (Ω) we say that Tn converges to T ∈ D  (Ω) if Tn , ϕ goes to T, ϕ when n goes to ∞ for all ϕ ∈ Cc∞ (Ω). Remark 8.12. We can prove by an adapted Banach-Steinhaus theorem (uniform boundedness principle) that if Tn , ϕ converges for all ϕ ∈ Cc∞ (Ω) then the limit defines a distribution. This notion of convergence is thus to be understood as a weak convergence. This notion is sufficient for our purpose (above we made a similar choice for the space Cc∞ (Ω)).

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As for many functional spaces, approximation by a smooth function can be useful. The following proposition allows one to often use test functions instead of distributions and then invoke density. Proposition 8.13. The space Cc∞ (Ω) is dense in D  (Ω) (in the sense of the sequential topology given above). Above, in Definition 8.3, we pointed out that the number of derivatives N in the continuity estimate depends on the compact set K. Fixing this number allows one to introduce classes of distributions. Definition 8.14. We say that T ∈ D  (Ω) is a distribution of order N ∈ N if for all K compact subset of Ω there exist C > 0 such that |T, ϕ| ≤ CϕK,N for all ϕ ∈ Cc∞ (Ω) supported in K. The space of distributions of order N is denoted by D N (Ω). Distributions of order 0 are called Radon measures. Remark 8.15. Observe that if T ∈ D  (Ω) is a distribution of order N ∈ N then it can be uniquely extended as a continuous linear form on Cck (Ω) (also equipped with a sequential topology). One then sees that positive locally finite measures and signed measures are Radon measures. A positive distribution is defined as follows. Definition 8.16. We say that T ∈ D  (Ω) is a positive distribution if T, ϕ ≥ 0 for all ϕ ∈ Cc∞ (Ω) such that ϕ(x) ≥ 0 for all x ∈ Ω. The next result shows that positive distributions are in fact positive locally finite measures and thus distributions of order zero. Proposition 8.17. Let F : Cc∞ (Ω) → C be such that F (ϕ) ≥ 0 if ϕ ≥ 0. Then F is positive locally finite measure on Ω. Note that this latter result does not require F to be a priori continuous on Cc∞ (Ω) 8.1.2.1. Localization and Support. We first recall the notion of partition of unity subordinated to an open set covering. Proposition 8.18. Let Ω be an open set in Rd and Ωj , j ∈ J, be relatively compact open subsets of Ω such

that ∪j∈J Ωj = Ω. Then, there exist ϕj ∈ Cc∞ (Ωj ) such that ϕj ≥ 0 and j∈J ϕj (x) = 1 for all x ∈ Ω. The functions ϕj can be chosen to yield a locally finite sum. One says that the open sets Ωj , j ∈ J, form an open covering of Ω. For a proof we refer for instance to [176]. This proposition is also proven in Sect. 15.2 of Volume 2 in a more general setting. In particular, this is useful to prove the following important localization result.

8.1. DISTRIBUTIONS ON AN OPEN SET OF Rd AND ON A MANIFOLD

291

Lemma 8.19. Let Ωj , j ∈ J, be an open covering of Ω. If T|Ωj = 0 for all j ∈ J then T = 0. Note that we shall say that T vanishes on an open subset U if T|U = 0. We can define the support of a distribution as follows. Definition 8.20. Let T ∈ D  (Ω). We say that x0 ∈ Ω is not in supp(T ) if there exist an open neighborhood U of x0 where T vanishes. By definition, supp(T ) is a closed set of Ω. With Lemma 8.19 we find that supp(T ) is the complement of the largest open subset of Ω where T vanishes. In particular, we find that if supp(T ) ∩ supp(ϕ) = ∅ then T, ϕ = 0. Proposition 8.21. Let K be a compact subset in Ω, and U a neighborhood of K such that U  Ω. There exist χ ∈ Cc∞ (Ω) such that χ ≡ 1 in a neighborhood of K and χ vanishes outside U . We say that χ is a cut-off function. If χ ∈ Cc∞ (Ω) is a cut-off function such that χ ≡ 1 in a neighborhood of x0 the local properties of T and χT coincide in a neighborhood of x0 . This motivates the study of distributions with a compact support. 8.1.2.2. Distributions with Compact Support. Definition 8.22. We say that T ∈ D  (Ω) is compactly supported if supp(T ) is compact. We denote by E  (Ω) the set of compactly supported distribution in Ω. For K a compact set, one also denotes by E  (K) the space of distributions in D  (Rd ) supported in K. Remark 8.23. Let T ∈ E  (Ω) and K = supp(T ). If χ ∈ Cc∞ (Ω) is a cut-off function such that χ ≡ 1 in K, one has T = χT . We can then extend ˜ with Ω ˜ an open set the action of the distribution T to functions in C ∞ (Ω), d ˜ ˜ With of R such that Ω ⊂ Ω, by writing T, ϕ = T, χϕ for all ϕ ∈ C ∞ (Ω).  ∞ ˜ = Ω, we can see E (Ω) as the dual of C (Ω). In fact, the space C ∞ (Ω) Ω can be equipped with a Fr´echet topology with seminorms of the form pk,j (u) = sup sup |∂xα u(x)|, x∈Kj |α|≤k

k, j ∈ N,

where Kj is an exhaustive sequence of compact sets in Ω. Then, E  (Ω) is precisely with the dual of C ∞ (Ω). Proposition 8.24. Let T ∈ E  (Ω), then T is a distribution of finite order. Remark 8.25. In particular, any distribution is locally of finite order. 8.1.3. Composition by Diffeomorphisms, Distributions on a Manifold. We present the action of a change of variables on a distribution. Let then X and Y be two open sets of Rd . Let κ be a smooth diffeomorphism

292

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from X onto Y . If f is a smooth function on Y , the pullback of f by κ, is the smooth function on X given by κ∗ f = f ◦ κ. For ϕ ∈ Cc∞ (X), we have Tκ∗ f , ϕ= ∫ f ◦κ(x)ϕ(x)dx= ∫ f (y)ϕ(κ−1 (y))Jκ−1 (y) dy=Tf , Jκ−1 (κ−1 )∗ ϕ, X

Y −1  | det(κ ) (y)|

where Jκ−1 (y) = = | det κ (κ−1 (y)|−1 . This justifies the following definition, that extends the pullback of functions to distributions. Definition 8.26. Let T ∈ D  (Y ) we define κ∗ T ∈ D  (X) by the following formula κ∗ T, ϕ = T, Jκ−1 (κ−1 )∗ ϕ for all ϕ ∈ Cc∞ (X). In the case f ∈ L1loc (Y ) we thus have κ∗ Tf = Tκ∗ f .

(8.1.1)

As an example in the case of a distribution that does not identifies with an L1loc -function, consider u = δy0 with y 0 ∈ Y . If y0 = κ(x0 ) for x0 ∈ X we find κ∗ δy0 = | det κ (x0 )|−1 δx0 . Let now M be a smooth d-dimensional manifold2 and A = {(Oi , κi ); i ∈ i I} be an altlas of M. For a function f on M, if f C are the local representatives for f in C i , i ∈ I, then f C = (κij )∗ f C i

j

on κi (Oij ),

with Oij = Oi ∩ Oj and κij = κj ◦ (κi )−1 . Similarly, a distribution (resp. i Radon measure) u on M can be given by local representatives uC ∈ D  (int κi (Oi )) (resp. D 0 (int κi (Oi ))), i ∈ I, such that uC = (κij )∗ uC i

j

on κi (Oij ),

with the pullback as in Definition 8.26. We denote the space of distributions (resp. Radon measures) on M by 0 D  (M) (resp. 0 D 0 (M)). Note that the interior of k i (Oi ), int κi (Oi , is to be understood here as the interior of a subset of Rd . Yet, distributions (resp. Radon measures) so defined do not form the dual space of compactly supported smooth (resp. continuous) functions on M, with support away from the boundary3 ∂M. Denote the latter space by 0 Dc∞ (M) (resp. 0 Dc0 (M)). i A density function f on M can be given by local representatives f C , i ∈ I, such that f C = Jκij (κij )∗ f C i

j

on κi (Oij ),

where Jκij (x) = | det(κij ) (x)|. In particular, this allows one to define the integral of density functions on a manifold since they naturally “carry” the 2For the notion of manifold the reader is referred to any expository book on differential geometry. A small exposition is given in Sect. 15.1 of Volume 2. 3Note that since ∂M ⊂ M for a manifold with boundary if f is function with compact support we may have supp(f ) ∩ ∂M = ∅.

8.2. TEMPERATE DISTRIBUTIONS ON Rd AND FOURIER TRANSFORMATION 293

Jacobian. More details are given in Sect. 16.2 of Volume 2. The reader is also referred to [176]. The space of compactly supported smooth (resp. continuous) densities on M, with support away from the boundary is denoted by 1 Dc∞ (M) (resp. 1 0 Dc (M)). Density distributions (resp. Radon measures) on M can be defined similarly expect that each representative is a distribution (resp. Radon measure). The associated space is denoted by 1 D  (M) (resp. 1 D 0 (M)). On the one hand, one finds that 1 D  (M) (resp. 1 D 0 (M)) forms the dual of 0 Dc∞ (M) (resp. 0 Dc0 (M)). On the other hand, one finds that the space of distributions (resp. Radon measures) on M, 0 D  (M) (resp. 0 D 0 (M)) forms the dual of 1 Dc∞ (M) (resp. 1 Dc0 (M)). This is further explained in Sect. 16.2 in Volume 2. More generally, f is a a-density function if  a i j f C = Jκij (κij )∗ f C on κi (Oij ). The space of compactly supported smooth (resp. continuous) a-densities on M, with support away from the boundary is denoted by a Dc∞ (M) (resp. a 0 Dc (M)). Similarly, one defines a-density distributions and Radon measures with respective spaces denoted by a D  (M) (resp. a D 0 (M)). One finds that a  D (M) (resp. a D 0 (M)) is the dual of 1−a Dc∞ (M) (resp. 1−a Dc0 (M)). In the duality product u, ϕ where for instance u ∈ a D  (M) and ϕ ∈ 1−a ∞ Dc (M), the distribution a-density “carries” the Jacobian to the power a while the test smooth (1−a)-density ϕ “carries” the Jacobian to the power 1 − a. The product then u, ϕ appears geometrically invariant and can be understood by means of local charts and a partition of unity. Functions (resp. distributions, resp. Radon measures) on M can be understood as 0-density functions (resp. distributions, resp. Radon measures). Note that a natural choice can be half-densities for both test densities, that 1 1 is, 2 Dc∞ (M), and distribution densities, that is, 2 D  (M). 8.2. Temperate Distributions on Rd and Fourier Transformation We recall the definition of the Fourier transform of a function in L1 (Rd ): (8.2.1)

F ϕ(ξ) = ϕ(ξ) ˆ = ∫ e−ix·ξ ϕ(x) dx. Rd

This is a continuous function by the Lebesgue dominated-convergence theorem. Moreover ϕ(ξ) ˆ → 0 as |ξ| → ∞ by the Riemann-Lebesgue lemma. On the one hand, properties of distributions in D  (Rd ) do not make them suitable candidates for Fourier transformation. In fact, even for functions in local spaces, e.g., Lploc (Rd ), one faces this difficulty. Smoothness is no answer to this issue: the Fourier transform of (smooth) functions such as x → ex cannot be defined for instance. On the other hand, one wishes to find functional spaces that are invariant by Fourier transformation. To that purpose, compactly supported distributions are not well suited: the

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Fourier transform of distributions in E  (Rd ) is analytic and, thus, cannot be compactly supported. Here also, smoothness is not the answer, since there are smooth functions whose Fourier transform is not smooth, e.g., the Fourier transform of x → (1+x2 )−1 is ξ → π exp(−|ξ|). The space L2 (Rd ) is invariant under (some extension of) the Fourier transformation; however, it does not contain many distributions for which a Fourier transformation can be computed, e.g., Dirac masses or, more generally, compactly supported distributions, and, unfortunately, the usual duality method, as developed above, does not yield a larger space. Function and distribution spaces well adapted for Fourier transformation need to be introduced. 8.2.1. The Schwartz Space and Temperate Distributions. The Schwartz space of smooth functions on Rd is given by S (Rd ) = {ϕ ∈ C ∞ (Rd ); ∀n, k ∈ N, pn,k (ϕ) < ∞}, with pn,k (ϕ) := sup{xn |Dα ϕ(x)|; α ∈ Nd , |α| ≤ k, x ∈ Rd }. These seminorms pn,k , n, k ∈ N, yield a Frechet space topology on S (Rd ). The space Cc∞ (Rd ) is dense in S (Rd ). One defines S  (Rd ) as the dual space of S (Rd ). Definition 8.27. One says that T ∈ S  (Rd ) if T is a linear form on S (Rd ) and there exist n, k ∈ N and C > 0 such that |T, ϕ| ≤ Cpn,k (ϕ) for all ϕ ∈ S (Rd ). One says that T is a temperate distribution on Rd . Remark 8.28. We have the injections E  (Rd ) ⊂ S  (Rd ) ⊂ D  (Rd ) in accordance with the dense injections Cc∞ (Rd ) ⊂ S (Rd ) ⊂ C ∞ (Rd ). In particular, all operations defined above on D  (Rd ) also apply to S  (Rd ). Upon localisation, elements of D  (Rd ) and E  (Rd ) coincide. This extends to S  (Rd ). Sequential weak convergence can be introduced in S  (Rd ) as was done above in D  (Rd ). Definition 8.29 (Sequential Topology). Let (Tn )n ⊂ S  (Rd ). We say that Tn converges to T ∈ S  (Rd ) if Tn , ϕ → T, ϕ as n → ∞ for all ϕ ∈ S (Rd ). Remark 8.30. Note that a sequence that weakly converges in the sense given above in fact converges for the strong topology on S  (Rd ). Note that for a sequence in S  (Rd ), convergence in D  (Rd ) does not imply convergence in S  (Rd ). The weak topology on S  (Rd ) is finer than that on D  (Rd ) since the space of test functions used is much larger. This occurs in spite of the density of Cc∞ (Rd ) in S (Rd ). An example is the sequence Tn = χ[−n,n] ex that converges to exp(x) in D  (Rd ), whereas it does not converge in S  (Rd ), since exp(x) ∈ / S  (Rd ). Similarly to D  (Rd ), we have the following useful density result.

8.2. TEMPERATE DISTRIBUTIONS ON Rd AND FOURIER TRANSFORMATION 295

Proposition 8.31. The space Cc∞ (Rd ) is dense in S  (Rd ) (in the sense of the sequential topology given above). 8.2.2. The Fourier Transformation on S (Rd ), S  (Rd ), and L2 (Rd ). As motivated above, it is convenient to consider the Fourier transformation on the Schwartz space. For ϕ ∈ S (Rd ), the Fourier transform F ϕ is defined according to (8.2.1). Proposition 8.32. Let ϕ ∈ S (Rd ) then F ϕ = ϕˆ ∈ S (Rd ). The Fourier transformation is an isomorphism on S (Rd ) and moreover the inverse transformation is given by F −1 ψ(x) = (2π)−d ∫ eix·ξ ψ(ξ) dξ, Rd

ψ ∈ S (Rd ),

yielding ϕ(x) = (2π)−d ∫Rd eix·ξ ϕ(ξ) ˆ dξ. Observe that we have α ϕ(ξ) = ξ α ϕ(ξ),  ˆ D

α ϕ = (−1)|α| D α ϕ, x" ˆ

α ∈ Nd .

We have by an elementary computation, (8.2.2)

ˆ ϕ = φ,

∫

Rd ×Rd

e−ix·ξ φ(x)ϕ(ξ)dxdξ = φ, ϕ, ˆ

ϕ, φ ∈ S (Rd ).

Similarly, with Proposition 8.32 we have the formula ˆ ϕ) ˆ L2 (Rd ) , (φ, ϕ)L2 (Rd ) = (2π)−d (φ,

ϕ, φ ∈ S (Rd ),

implying the Plancherel equality (8.2.3)

ˆ L2 (Rd ) , ϕL2 (Rd ) = (2π)−d/2 ϕ

ϕ ∈ S (Rd ).

The identity (8.2.2) motivates the following definition of the Fourier transform of a temperate distribution by means of transposition, similarly to the many operations on distributions that were defined this way in what precedes. Definition 8.33. Let T ∈ S  (Rd ). We define F T = Tˆ ∈ S  (Rd ) by the following formula Tˆ, ϕ = T, ϕ, ˆ

ϕ ∈ S (Rd ).

The Fourier transformation F is continuous isomorphism on S  (Rd ). Its inverse is defined by duality from F −1 given above. Remark 8.34. Observe that for f ∈ L1 (Rd ) we have f ∈ S  (Rd ); yet, the definition of F f in (8.2.1) yields a continuous function that coincides with the Fourier transform of f as a temperate distribution.

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As L2 (Rd ) ⊂ S  (Rd ), the Fourier transform of L2 functions on Rd is well defined. However, the Plancherel equality (8.2.3) provides a way to uniquely extend F : S (Rd ) → S (Rd ) ⊂ L2 (Rd ) to an isometry on L2 (Rd ) (up to the constant (2π)−d/2 ), as S (Rd ) is dense in L2 (Rd ). One sees that the two definitions of the Fourier transformation on L2 (Rd ) coincide. Proposition 8.35. If T ∈ E  (Rd ) then its Fourier transform is an entire function given by Tˆ(ξ) = Tx , e−ix·ξ , for ξ ∈ Cd . 8.3. Distributions on a Product Space We consider firstly tensor products of functions, and secondly tensor products of distributions. 8.3.1. Tensor Products of Functions. Let X ⊂ Rd and Y ⊂ Rq be two open sets. For two functions ϕ and φ, defined on X and Y respectively, we define the function ϕ ⊗ φ on X × Y by ϕ ⊗ φ(x, y) = ϕ(x)φ(y). We say that ϕ ⊗ φ is the tensor product of ϕ and φ. Observe that ϕ ⊗ φ ∈ Cc∞ (X × Y ) if ϕ ∈ Cc∞ (X) and φ ∈ Cc∞ (Y ) and ϕ ⊗ φ ∈ S (Rd+q ) if ϕ ∈ S (Rd ) and φ ∈ S (Rq ). We have the following two density results. Proposition 8.36. The linear space Cc∞ (X) ⊗ Cc∞ (Y ) spanned by the functions ϕ⊗φ, where ϕ ∈ Cc∞ (X) and φ ∈ Cc∞ (Y ), is dense in Cc∞ (X ×Y ). Proposition 8.37. The linear space S (Rd ) ⊗ S (Rq ) spanned by the functions ϕ ⊗ φ, where ϕ ∈ S (Rd ) and φ ∈ S (Rq ) is dense in S (Rd+q ). These two results allow one to simplify the analysis on product spaces. Proposition 8.38. Any two distributions on X × Y are equal if they coincide on Cc∞ (X) ⊗ Cc∞ (Y ). Proposition 8.39. Any two temperate distributions on Rd+q are equal if they coincide on S (Rd ) ⊗ S (Rq ). 8.3.2. Tensor Products of Distributions. Let T ∈ D  (X) and S ∈ ). If ψ ∈ Cc∞ (X × Y ) we have x → Sy , ψ(x, y) ∈ Cc∞ (X). This allows one to define the linear map on Cc∞ (X × Y ) D  (Y

R : ψ → Tx , Sy , ψ(x, y). Proposition 8.40. We have R ∈ D  (X × Y ) and R, ψ = R(ψ) = Tx , Sy , ψ(x, y) = Sy , Tx , ψ(x, y), Moreover, if ϕ ∈ Cc∞ (X) and φ ∈ Cc∞ (Y ), we have (8.3.1)

R, ϕ ⊗ φ = T, ϕS, φ.

ψ ∈ Cc∞ (X × Y ).

8.3. DISTRIBUTIONS ON A PRODUCT SPACE

297

We denote the distribution R by T ⊗ S; it is called the tensor product of T and S. By Proposition 8.36 we see that R is the unique distribution on X × Y that satisfies (8.3.1). Note that if f and g are two L1loc functions, the tensor product f ⊗ g introduced in the previous section coincides with the tensor product of f and g viewed as distributions: with the notation introduced in Sect. 8.1.2 we have Tf ⊗g = Tf ⊗ Tg . Proposition 8.41. Let T ∈ S  (Rd ) and S ∈ S  (Rq ). We have T ⊗ S ∈ and more over T ⊗ S, φ ⊗ ϕ = T, φS, ϕ for all φ ∈ S (Rd ) and ϕ ∈ S (Rq ). S  (Rd+q )

8.3.3. Convolution. For two L1 -functions f and g, their convolution product is given by f ∗ g(x) = ∫Rd f (x − y)g(y) dy, for almost every x, and it yields an L1 -function. Letting f ∗ g act on a test function ϕ we find, with the Fubini theorem, and a change of variables, f ∗ g, ϕ = ∫ f (x − y)g(y)ϕ(x) dydx = ∫ f (x)g(y)ϕ(x + y) dydx R2d

R2d

= f ⊗ g, ϕ(x + y). As (x, y) → ϕ(x + y) is not compactly supported if ϕ = 0, we see that a support requirement is necessary for an extension to distributions. If L, L are closed subsets of Rd , one says that they are convolutive if for any compact set K ⊂ Rd the set {(x, y) ∈ L × L ; x + y ∈ K} is a compact set of R2d . Then, if T, S ∈ D  (Rd ) have convolutive supports we may define their convolution product by T ∗ S, ϕ = T ⊗ S, ϕ(x + y). One sees that the convolution product is commutative. Yet, it is not associative. Note that T ∈ E  (Rd ) can be convolved with any S ∈ D  (Rd ). The Dirac measure at 0 is the unit distribution for the convolution product. We have the following result on the support of a convolution. Proposition 8.42. Let T, S ∈ D  (Rd ) have convolutive supports. We have supp(T ∗ S) ⊂ supp(T ) + supp(S). 



We observe that ∂xα (T ∗ S) = ∂xα T ∗ ∂xα S if α + α = α. Moreover, the convolution of T ∈ D  (Rd ) with a smooth function with compact support yields a smooth function. Finally, we state results in connection with the Fourier transformation. Proposition 8.43. Let T ∈ S  (Rd ) and S ∈ E  (Rd ). We have T ∗ S ∈ S  (Rd ) and F (T ∗ S) = F T F S. Note that the product F T F S is well defined as F S is a smooth (entire) function (see Proposition 8.35).

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8.3.4. The Kernel Theorem (Various Forms). Let Ω1 and Ω2 be two open sets of Rd and Rq respectively. Let K ∈ D  (Ω1 × Ω2 ). For all φ ∈ Cc∞ (Ω2 ), the map ϕ → K, ϕ ⊗ φ ∈ D  (Ω1 ), in the sense of Definition 8.3. We denote by Kφ this distribution on Ω1 , thus defining a map K : Cc∞ (Ω2 ) → D  (Ω1 ). In fact, the map K is a continuous map. The following result referred to as the kernel theorem states that this identification between a distribution on Ω1 × Ω2 and a continuous map from Cc∞ (Ω2 ) to D  (Ω1 ) is in fact an isomorphism. Theorem 8.44 (Schwartz’ Kernel Theorem). We have   ∼ L C ∞ (Ω2 ), D  (Ω1 ) . D  (Ω1 × Ω2 ) = c   This isomorphism is to be understood as follows: if K ∈ L Cc∞ (Ω2 ), D  (Ω1 ) , there exists a unique K ∈ D  (Ω1 × Ω2 ) such that Kφ, ϕ = K, ϕ ⊗ φ,

ϕ ∈ Cc∞ (Ω1 ), φ ∈ Cc∞ (Ω2 ).

The distribution K is called the (Schwartz) kernel of the operator K. This terminology can be easily understood if one uses the lousy yet useful notation Kφ, ϕ =

∫

Ω1 ×Ω2

K(x1 , x2 )ϕ(x1 )φ(x2 ) dx1 dx2 ,

or even Kφ(x1 ) = ∫ K(x1 , x2 )φ(x2 ) dx2 . Ω2

Note that in the present book, we shall use this latter notation, yet in the case of particular kernels K defined by means of oscillatory integrals. In such case, the use of integral symbols is fully justified; see Chap. 2 and in particular Sect. 2.4. In the framework of Schwartz functions and temperate distributions, the following kernel theorem also holds. Theorem 8.45 (Kernel Theorem for Temperate Distributions). We have   S  (Rd × Rq ) ∼ = L S (Rq ), S  (Rd ) .   This isomorphism is to be understood as follows: if K ∈ L S (Rq ), S  (Rd ) , there exists a unique K ∈ S  (Rd × Rq ) such that Kφ, ϕ = K, ϕ ⊗ φ,

ϕ ∈ S (Rd ), φ ∈ S (Rq ).

8.4. Notes For an introduction to distribution theory we refer to the book of F. G. Friedlander [152] and the course notes of J. J. Duistermaat and J. A. C. Kolk [129]. We refer also to the book of C. Zuily [331] for many exercises. For more advanced subjects we refer to the first volume of L. H¨ormander’s treatise [176]. For a topological vector space perspective of distribution theory we refer to the book of F. Treves [320] (see also below). The reader

8.4. NOTES

299

wishing to read the seminal contribution of L. Schwartz to the subject is referred to [303]. In this expository section on distribution theory, we have chosen to not present the adapted topologies on Cc∞ (Ω) and D  (Ω). This choice calls for the following comment. The adapted topology on Cc∞ (Ω) is that of a LF space, a countable strict limit of Fr´echet spaces indexed by a exhaustive sequence of compact sets [320, Chapter 13]. With this topology D  (Ω) is defined as the topological dual of Cc∞ (Ω) and is given the associated strong topology. However, one can prove that both Cc∞ (Ω) and D  (Ω) are then Montel spaces [320, Section 34.4]: in particular they are reflexive and furthermore weak converging sequences do strongly converge [320, Corollary 2 page 358]. Above we defined the product of a distribution with a smooth function. There is however an obstruction to the definition of a general multiplication of distributions [302]. Distribution theory is essentially a linear theory. Some products on restricted classes of distributions can be introduced. For such a product, based on the spectral analysis of the singularities of the distributions, see for instance [176, Section 8.2]. An application is given in the following remark. Concerning the convolution product and the Fourier transform of a convolution products, we can make the following remarks. Remark 8.46. (1) If T and S are in S  (Rd ), even if their supports are convolutive and if T ∗ S ∈ S  (Rd ), the product F T F S may not classically defined. Take for instance T = S = Y , the Heaviside function Y (x) = 1(0,+∞) ; we have Y ∈ S  (R) and Y ∗ Y = xY ∈ S  (Rd ). Yet, F Y = πδ0 − ivp(1/ξ) and the product of F Y with itself is not well defined with the standard notions we gathered above. However, one can defined the product F Y F Y within a hierarchy of distributional products. One of them, the H¨ormander product (see [176, Theorem 8.2.10]) which is based on the spectral analysis of the singularities of the distributions, allows one to define this product and, then, one has F Y F Y = F (Y ∗ Y ). (2) Above, the convolution T ∗S was defined exploiting a joint property of the supports of the two distributions T and S. However, as the L1 -function case shows, such a condition may not be necessary. The following result shows that T ∗ f can be well defined in T ∈ S  (Rd ) and f ∈ S (Rd ). Moreover, the Fourier transform of T ∗ f can be easily obtained from F T and F f . Proposition 8.47. Let T ∈ S  (Rd ) and f ∈ S (Rd ). We have T ∗ f ∈ S  (Rd ) ∩ C ∞ (Rd ) and F (T ∗ S) = F T F S. This is in fact contained in the following proposition (see e.g. [320, Theorem 30.4]).

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Proposition 8.48. Let T ∈ S  (Rd ) and S ∈ Oc (Rd ). We have T ∗ S ∈ S  (Rd ) and F (T ∗ S) = F T F S. The space Oc (Rd ) is the space of rapidly decreasing distributions at infinity. We refer to [320, Chapter 30] for a description. Note that the case T = Y and S = Y discussed above does not enter the scope of this last result. Here, we stated two versions of the kernel theorem in Theorems 8.44 and 8.45. Many other results, based on the theory of nuclear spaces originating from the work of A. Grothendieck [163], can be found in [320].

CHAPTER 9

Invariance Under Change of Variables Contents 9.1. A Review of the Actions of Change of Variables 9.1.1. Pullbacks and Push-Forwards 9.1.2. Action of a Change of Variables on a Differential Operator 9.2. Action on Symplectic Structures 9.2.1. The Symplectic Two-Form 9.2.2. Hamiltonian Vector Fields 9.2.3. Poisson Bracket 9.3. Invariance of the Sub-ellipticity Condition 9.3.1. Action of a Change of Variables on the Conjugated Operator 9.3.2. The Sub-ellipticity Condition 9.4. Normal Geodesic Coordinates

302 302 304 304 305 306 307 308 308 309 309

In this chapter, we show that some of the notions introduced in Chap. 3 are independent of the choice of coordinates. Such notions include the joint sub-ellipticity property of a differential operator and a weight function (see Definition 3.2.2). This allows one to freely choose local coordinates for special purposes, as is done in Sect. 3.4 for the proof of local Carleman estimates at boundary points. The analysis described below borrows a lot from the analysis of manifolds, through the notions of tangent and cotangent vectors. However, we make the choice to only consider open sets of Rd here, and to restrict our analysis to the understanding of the effect of a change of variables on some object: vector fields, one-forms, and differential operators. The reader who is not interested in the analysis on manifolds will find here a more straightforward path toward the invariance properties that we wish to put forward. The reader interested in the extension of this analysis to manifold is, for © 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 I, PNLDE Subseries in Control 97, https://doi.org/10.1007/978-3-030-88674-5 9

301

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9. INVARIANCE UNDER CHANGE OF VARIABLES

instance, referred to Chap. 15 in Volume 2. That chapter uses some of the results described here. 9.1. A Review of the Actions of Change of Variables Let X and Y be two open subsets of Rd with κ : X → Y a C ∞ diffeomorphism. We denote by Tx X ∼ = Rd the tangent space of X at x. The differential or the tangent map of κ at x ∈ X, dκ(x) = κ (x) maps linearly and continuously tangent vectors at x into tangent vectors at κ(x): κ (x) : Tx X → Tκ(x) Y.  ∗ Setting Tx∗ X = Tx X ∼ = Rd , for the adjoint of κ (x) we then have t 

∗ κ (x) : Tκ(x) Y → Tx∗ X.

The vector space Tx∗ X is the cotangent space at x. We define T X = ∪x∈X {x} × Tx X = X × Rd and T ∗ X = ∪x∈X {x} × Tx∗ X = X × Rd . A typical example of cotangent vector at a point x is df (x), the differential at x of a function X → R, that acts linearly on tangent vectors at x. A vector field v on X is a section of T X, which is a smooth map X → T X such that for all x ∈ X, v(x) = (x, vx ) with vx ∈ Tx X. By abuse of notation, we shall often confuse v(x) and vx and write v(x) ∈ Tx X. Let v be a vector field on X. The action of a vector field v on a function f : X → R is given by v(f )(x) = df (x)(v(x)),

or in a coordinate form v(f )(x) = j v j (x)∂j f (x), in accordance with the

j usual notation v(x) = j v (x)∂j . A one-form ω on X is a section of T ∗ X, which is a smooth map X → T ∗ X such that for all x ∈ X, ω(x) = (x, ωx ) with ωx ∈ Tx∗ X. By abuse of notation, we shall often confuse ω(x) and wx and write ω(x) ∈ Tx∗ X. A one-form acts on a vector field as follows: ω, v(x) = ω(x), v(x), yielding a real-valued function on X. An important example of one-form is the map x → (x, df (x)), for f : X → R smooth. 9.1.1. Pullbacks and Push-Forwards. We recall here how functions, vector fields, and one-form can be transferred from one open set to another by means of the diffeomorphism κ. Let f ∈ C ∞ (Y ). The pullback of f to X by κ is given by κ∗ f = f ◦ κ. The push-forward of a vector field v on X is a vector field on Y given by κ∗ v(y) = κ (x)v(x),

y = κ(x).

9.1. A REVIEW OF THE ACTIONS OF CHANGE OF VARIABLES

303

  For f ∈ C ∞ (Y ) and v a vector field on X, we have v(κ∗ f )(x) = (κ∗ v)f (y) for y = κ(x). Indeed, we write       (9.1.1) v(κ∗ f )(x) = d(f ◦ κ)(x) v(x) = df (y) κ (x)v(x) = df (y) κ∗ v(y)   = (κ∗ v)f (y). The action of vector fields on functions thus appears independent of the coordinates used. We say that it is invariant under the change of variables. Let ω be a one-form on Y . The pullback of ω is a one-form on X given by y = κ(x). κ∗ ω(x) = t κ (x)ω(y), ∗ Observe that we then have ω, κ∗ v(y) = κ ω, v(x). Indeed, we write ω, κ∗ v(y) = ω(y), κ∗ v(y) = ω(y), κ (x)v(x) = t κ (x)ω(y), v(x) = κ∗ ω, v(x). The action of a one-form on a vector field is thus invariant under the change of variables. Let g ∈ C ∞ (Y ) and v be a vector field on X. In the following formula we compute the action of v on the pullback of g, resulting in the action of v on the pullback of the differential of g. Note that dg is a one-form on Y . For y = κ(x), we have   (9.1.2) v(κ∗ g)(x) = (κ∗ v)g (y) = dg, κ∗ v(y) = κ∗ dg, v(x) = t κ (x)dg(y), v(x). This yields the following lemma. Lemma 9.1. If f = κ∗ g ∈ C ∞ (X), then df = κ∗ dg; that is,   df (x) = t κ (x)dg κ(x) .

Proof. Let v be a vector field on X. We have df, v(x) = v(f )(x) = v(κ∗ g)(x). We conclude with (9.1.2).



We shall now use ξ to denote the coordinates used for Tx∗ X = Rd . Starting with the diffeomorphism κ : X → Y , we then naturally introduce a diffeomorphism ψ : T ∗ X → T ∗ Y in the following way:   (9.1.3) ψ(x, ξ) = κ(x), t κ (x)−1 ξ .   A function p ∈ C ∞ T ∗ Y can then be pulled back to T ∗ X:   ψ ∗ p(x, ξ) = p ◦ ψ(x, ξ) = p κ(x), t κ (x)−1 ξ . It is sometimes more elegant to write (9.1.4)

  ψ ∗ p(x, t κ (x)η) = p κ(x), η .

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9. INVARIANCE UNDER CHANGE OF VARIABLES

9.1.2. Action of a Change of Variables on a Differential Operator. We consider the particular case v = Dj = −i∂j in formula (9.1.2). We find

t  

κ (x) jk (Dk f )(κ(x)) = ∂j κk (x)Dk f (κ(x)). Dj , κ∗ f (x) = 1≤k≤d

1≤k≤d

With α ∈ Nd , we then have Dα (f ◦ κ)(x) =

d  d  # t j=1

κ (x)

k=1



(Dk f )(κ(x))) jk

 αj

+ lower-order derivatives of f . Through the change of variables, the operator P (Dx ) = Dxα , with symbol p(ξ) = ξ α , is thus transformed in Q(y, Dy ), meaning  P (Dx )(f ◦ κ) = Q(y, Dy )f ) ◦ κ, with principal symbol given q(κ(x), η) = (t κ (x)η)α . More generally we see that, for a differential operator of order m,

aα (x)Dxα , P (x, Dx ) = with principal symbol p(x, ξ) =

|α|≤m

|α|=m aα (x)ξ

α,

we have

 P (x, Dx )(f ◦ κ) = Q(y, Dy )f ) ◦ κ, where Q(y, Dy ) is a differential operator of order m with principal symbol given by q(κ(x), η) = p(x, t κ (x)η). Comparing with (9.1.4), we thus find that the principal symbol p(x, ξ) follows the natural transformation through change of variables for functions defined on T ∗ X: (9.1.5)

ψ ∗ q = p.

In that sense, the principal symbol of a differential operator is invariant under the change of variables. 9.2. Action on Symplectic Structures As above, X and Y denote open sets of Rd . Let ρ ∈ T ∗ X, with ρ = (x, ξ). Note that     Tρ T ∗ X = Tx X × Tξ Tx∗ X . As the tangent space at a point of a vector space can be identified with the vector space, we find that   Tρ T ∗ X ∼ = Tx X × Tx∗ X ∼ = Rd × Rd .

9.2. ACTION ON SYMPLECTIC STRUCTURES

305

9.2.1. The Symplectic Two-Form. For ρ = (x, ξ) ∈ T ∗ X, we define the following nondegenerate alternating bilinear form:     σρ : Tρ T ∗ X × Tρ T ∗ X → R,   (v; w) = (vx , vξ ); (wx , wξ ) → vξ , wx  − wξ , vx , where the duality brackets correspond to the duality between Tx X and Tx∗ X. Letting ρ vary, this yields a two-form on T ∗ X acting bilinearly on vector fields, which is denoted by σ and referred to as the symplectic two-form, or the symplectic form for simplicity. If needed, we shall write σX to emphasize that it is associated with T ∗ X. Note that using the exterior product, dξj ∧ dxj = dξj ⊗ dxj − dxj ⊗ dξj , we have

(9.2.1) dξj ∧ dxj . σ= j

If θ is a diffeomorphism T ∗ X → T ∗ Y , we then define the pullback of σY as



 θ∗ σY (v; w)(ρ) = σY (θ∗ v; θ∗ v)(θ(ρ))

for v, w two vector fields on T ∗ X. σX

Definition 9.2. A diffeomorphism θ : T ∗ X → T ∗ Y that satisfies θ∗ σY = is called a symplectomorphism or a canonical transform.

The natural diffeomorphism between T ∗ X and T ∗ Y that we introduced in (9.1.3) is in fact a symplectomorphism. Proposition 9.3. Let κ : X → Y be a diffeomorphism and set ψ : T ∗ X → T ∗ Y,   (x, ξ) → κ(x), t κ (x)−1 ξ . Then ψ is a symplectomorphism. ∗ Y given by Proof. We set ψ1 : T ∗ X → Y and ψ2 : T ∗ X → Tκ(x)

ψ2 (x, ξ) = t κ (x)−1 ξ.   Let ρ = (x, ξ) ∈ T ∗ X and v = (vx , vξ ) ∈ Tρ T ∗ X . We have ψ1 (x, ξ) = κ(x),

(9.2.2) ∂x ψ1 (ρ)vx = κ (x)vx ,

∂ξ ψ1 (ρ) = 0,

∂ξ ψ2 (ρ)vξ = t κ (x)−1 vξ .

To compute ∂x ψ2 (ρ)vx , we proceed as follows. Consider u a vector field on X locally constant and set f (x, ξ) = ψ2 (x, ξ), u. We have   f (x, ξ) = ξ, κ (x)−1 u = ξ, κ−1 (κ(x))u. We thus find

  (9.2.3) ∂x f (x, ξ)(vx ) = ∂x ψ2 (x, ξ)vx , u = ξ, κ−1 (κ(x))(u, κ (x)vx ).

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9. INVARIANCE UNDER CHANGE OF VARIABLES

Let v, w be two vector fields on T ∗ X and ρ = (x, ξ) ∈ T ∗ X. We write v(ρ) = (vx , vξ ) and w(ρ) = (wx , wξ ) for concision. We compute ψ ∗ σY (v; w)(ρ) = σY (ψ∗ v; ψ∗ w)(ψ(ρ)) = σY (ψ  (ρ)v(ρ); ψ  (ρ)w(ρ)) = ∂x ψ2 (ρ)vx + ∂ξ ψ2 (ρ)vξ , ∂x ψ1 (ρ)wx  − ∂x ψ2 (ρ)wx + ∂ξ ψ2 (ρ)wξ , ∂x ψ1 (ρ)vx    = ξ, κ−1 (κ(x))(κ (x)wx , κ (x)vx ) + t κ (x)−1 vξ , κ (x)wx    − ξ, κ−1 (κ(x))(κ (x)vx , κ (x)wx ) − t κ (x)−1 wξ , κ (x)vx  = vξ , wx  − wξ , vx  = σX (v; w)(ρ), by (9.2.2)–(9.2.3) and the Schwarz theorem. We have thus found ψ ∗ σY =  σX . A second example of symplectomorphism is the following one. Proposition 9.4. Let ϕ ∈ C ∞ (X; R) and set ψ : T ∗ X → T ∗ X,   (x, ξ) → x, ξ + dϕ(x) . Then ψ is a symplectomorphism. Proof. Let v, w be two vector fields on T ∗ X and ρ = (x, ξ) ∈ T ∗ X. We write v(ρ) = (vx , vξ ) and w(ρ) = (wx , wξ ) for concision. We compute ψ ∗ σX (v; w)(ρ) = σX (ψ∗ v; ψ∗ w)(ψ(ρ))   = σX (vx , vξ + d2 ϕ(x)(vx )); (wx , wξ + d2 ϕ(x)(wx )) (ρ) = vξ + d2 ϕ(x)(vx ), wx (ρ) − vx , wξ + d2 ϕ(x)(wx )(ρ)   = σX (vx , vξ ); (wx , wξ ) (ρ) + d2 ϕ(x)(vx , wx ) − d2 ϕ(x)(wx , vx ). With the Schwarz theorem, we conclude that ψ ∗ σX = σX .



9.2.2. Hamiltonian Vector Fields. For a function f on T ∗ X, we define a vector field Hf on T ∗ X as the unique vector field such that (9.2.4)

v(f ) = σ(v, Hf ),

for all vector fields v on T ∗ X. One calls Hf the Hamiltonian vector field of f . In local coordinates, this yields Hf = ∂ ξ f ∂ x − ∂ x f ∂ ξ =

d

j=1

∂ ξj f ∂ xj −

d

j=1

∂ xj f ∂ ξj .

We have the following transformation rule for Hamiltonian vector fields under symplectomorphisms. Proposition 9.5. Let θ : T ∗ X → T ∗ Y be a symplectomorphism and f be a function on T ∗ Y . We then have θ∗ H θ ∗ f = Hf .

9.2. ACTION ON SYMPLECTIC STRUCTURES

307

In particular, this states that Hamiltonian vector fields are invariant under change of variables by Proposition 9.3. Proof. Set g = θ∗ f : T ∗ X → R. Let v be a vector field on T ∗ X. By (9.2.4) and (9.1.1), we have σX (v; Hg )(ρ) = v(g)(ρ) = (θ∗ v)(f )(θ(ρ)) = σY (θ∗ v; Hf )(θ(ρ)). Since θ is a symplectomorphism, we have σX (v; Hg )(ρ) = σY (θ∗ v; θ∗ Hg )(θ(ρ)), which yields the result since v, and thus θ∗ v is arbitrary.



9.2.3. Poisson Bracket. The Poisson bracket of two functions f, g defined on T ∗ X is given by {f, g} =

d

j=1

∂ ξj f ∂ xj g −

d

j=1

∂xj f ∂ξj g,

which can be defined through the Hamiltonian vector fields associated with f and g {f, g} = Hf g = −Hg f = σ(Hf ; Hg ). As for the symplectic form, we shall write {f, g}X to explicitly express that the Poisson bracket concerns functions on T ∗ X. Symplectomorphisms yield a natural transformation for the Poisson bracket. The following proposition in particular applies to the natural diffeomorphism between T ∗ X and T ∗ Y introduced in (9.1.3). Proposition 9.6. Let θ : T ∗ X → T ∗ Y be a symplectomorphism. Then for f and g two functions on T ∗ Y , we have θ∗ {f, g}Y = {θ∗ f, θ∗ g}X .

Proof. We introduce the function p = θ∗ f and q = θ∗ g on T ∗ X. For ρ ∈ T ∗ X, we write {f, g}Y (θ(ρ)) = σY (Hf , Hg )(θ(ρ)) = σX ((θ−1 )∗ Hf , (θ−1 )∗ Hg )(ρ) = σX (Hp , Hq )(ρ) = {p, q}X (ρ), as θ∗ σY = σX and using Proposition 9.5. This yields the result.



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9. INVARIANCE UNDER CHANGE OF VARIABLES

9.3. Invariance of the Sub-ellipticity Condition As above, X and Y denote open sets of Rd . In X, we consider a difp(x, ξ), which ferential operator P (x, Dx ) of order m with

principal symbol α is homogeneous of order m: p(x, ξ) = |α|=m aα (x)ξ . As above, we define Q(y, Dy ) as the operator P under the change of variables κ : X → Y , namely,  f ∈ C ∞ (Y ), P (x, Dx )(f ◦ κ) = Q(y, Dy )f ) ◦ κ, which can also be written as P (x, Dx ) ◦ κ∗ = κ∗ ◦ Q(y, Dy ). By (9.1.5), the principal symbol of Q(y, Dy ) is given by ψ ∗ q = p, which reads q(κ(x), η) = p(x, t κ (x)η),

∗ x ∈ X, η ∈ Tκ(x) Y.

9.3.1. Action of a Change of Variables on the Conjugated Operator. Let P (x, Dx ) be a differential operator of order m on X, and let ϕ ∈ C ∞ (X) be a weight function. As in Chap. 3, we consider the conjugated operator: Pϕ (x, Dx , τ ) = eτ ϕ(x) P (x, Dx )e−τ ϕ(x) ,

τ ≥ 1,

which we understand as a differential operator with a large parameter as τϕ −τ ϕ = D +iτ ∂ ϕ ∈ D 1 . introduced in Chap. j τ

j

2. Observe αthat we have e Dj e If P reads P = |α|≤m aα (x)Dx , we then obtain Pϕ = |α|≤m aα (x)(Dx + iτ dϕ(x))α . Its principal symbol pϕ (x, ξ, τ ) is given by the homogeneous terms in (ξ, τ ) of order m in the full symbol: pϕ (x, ξ, τ ) = p(x, ξ + iτ dϕ(x)). The same consideration can for the operator Q(y, Dy ) on Y  be made ∗ −1 with the weight function φ = κ ϕ ∈ C ∞ (Y ). We set Qφ (y, Dy , τ ) = eτ φ(y) Q(y, Dy )e−τ φ(y) and its principal symbol is given by qφ (y, η, τ ) = q(y, η + iτ dφ(y)).   As ϕ = κ∗ φ, we have dϕ = κ∗ dφ, that is, dϕ(x) = t κ (x)dφ κ(x) , by Lemma 9.1, yielding     pϕ (x, t κ (x)η, τ ) = p x, t κ (x)η + iτ dϕ(x) = p x, t κ (x)(η + iτ dφ(κ(x)))   = q κ(x), η + iτ dφ(κ(x)) = qφ (κ(x), η, τ ). As a consequence, the homogeneous principal symbol of the conjugated operators is invariant under the change of variables: pϕ = ψ ∗ qφ , recalling that ψ is the symplectomorphism associated with κ (see (9.1.3)). In other words

9.4. NORMAL GEODESIC COORDINATES

309

the following diagram is commutative: κ∗

φ −−−−−−−−−→ ϕ ψ∗

Q −−−−−−−−−→ ⏐ ⏐ conjugation

P ⏐ ⏐conjugation 

ψ∗

qφ −−−−−−−−−→ pϕ 9.3.2. The Sub-ellipticity Condition. Let V be an open set of X. The sub-ellipticity condition in V for P and the weight function ϕ reads ∀(x, ξ) ∈ T ∗ X, with x ∈ V , ∀τ > 0, pϕ (x, ξ, τ ) = 0

⇒

{Re pϕ , Im pϕ }(x, ξ, τ ) > 0

(see Definition 3.2). As above, we define the operator Q(y, Dy ) on Y , with symbol satisfying p = ψ ∗ q and the weight function φ(y) on Y with ϕ = ψ ∗ φ. Using that pϕ = ψ ∗ qφ and Proposition 9.6, applied with the symplectomorphism ψ, we find that the operator Q(y, Dy ) and the weight function φ(y) satisfy the sub-ellipticity condition in W , with W = κ(V ) open subset of Y : ∀(y, η) ∈ T ∗ Y, with y ∈ W , ∀τ > 0, qφ (y, η, τ ) = 0

⇒

{Re qφ , Im qφ }(y, η, τ ) > 0.

Hence, the sub-ellipticity property for P and ϕ is a property invariant by the change of variables. We have thus proven Proposition 3.10 in Sect. 3.2.3. 9.4. Normal Geodesic Coordinates We recall some notation. We denote by P (x, D) a general second-order elliptic operator with a principal part of the form: $

Di (pij (x)Dj ), with pij (x)ξi ξj ≥ C|ξ|2 , (9.4.1) P0 = 1≤i,j≤d

1≤i,j≤d

where pij ∈ C ∞ (Rd ; R) is such that pij = pji , 1 ≤ i, j ≤ d. The elliptic operator under consideration is then

i b (x)Di + c(x), P = P0 + 1≤i≤d

where bi , c ∈ L∞ (Rd ), 1 ≤ i ≤ d. We denote by p the principal symbol of P given by

pij (x)ξi ξj . p(x, ξ) = 1≤i,j≤d

We now consider a smooth function ψ defined in a neighborhood V of x0 , and we assume that dψ(x0 ) = 0 in V . We set S = {x ∈ V ; ψ(x) = ψ(x0 )},

V+ = {x ∈ V ; ψ(x) ≥ ψ(x0 )}.

310

9. INVARIANCE UNDER CHANGE OF VARIABLES

The hypersurface S is smooth in V . Theorem 9.7. There exist X an open neighborhood of x0 , Y an open neighborhood of 0 in Rd , and a C ∞ -diffeomorphism κ : X → Y , such that: (1) We have κ(x0 ) = 0 and κ(V+ ∩ X) = {y ∈ Y ; yd ≥ 0} and κ(S ∩ X) = {y ∈ Y ; yd = 0}. (2) In the local coordinates y = κ(x), the operator P takes the form: P = Dd2 +

d−1

p˜ij (y)Di Dj +

i,j=1

d

˜bi (y)Di + c˜(y), i=1

C∞

where the coefficients are in Y and the coefficients ˜bi and c˜ are in L∞ (Y ). (3) Moreover there exists C > 0 such that p˜ij

d−1

p˜ij (y)ηi ηj ≥ C|η  |2 ,

η  = (η1 , . . . , ηd−1 ), y ∈ Y.

i,j=1

Coordinates as described in this theorem are often referred to as normal geodesic coordinates. In fact, the Riemannian metric induced by the principal part P0 of the elliptic operator is given by

gij (x) dxi dxj , g(x) = 1≤i,j≤d

with the matrix g = (gij (x))1≤i,j≤d given as the inverse of (pij (x))1≤i,j≤d . We refer the reader to any text on Riemannian geometry for details. This is covered in Chap. 17 in Volume 2. In particular the Laplace–Beltrami operator associated with g is  

∂i (det g)1/2 pij ∂j u . Δg u = (det g)−1/2 1≤i,j≤d

It has the same principal symbol as −P . In the coordinates y provided by Theorem 9.7 we have gid (y) = δid ,

i = 1, . . . , d.

Hence locally, the variable yd coincides with the Riemannian distance to the boundary and for any y  ∈ Rd the map s → (y  , s) yields a curve that is orthogonal to the boundary. The coordinates y  ∈ Rd−1 locally parameterize the boundary ∂Ω. This is illustrated in Fig. 9.1 in the case of a flat Laplace operator Δ = ∂12 + · · · + ∂d2 , i.e., the Euclidean metric. Here, normal geodesic coordinates are given locally. In Sect. 17.6 in Volume 2, we shall see how this can be transposed in the neighborhood of a point of the boundary of a Riemannian manifold. We shall also show how such coordinates can be chosen over a neighborhood of a bounded part of the boundary. Proof of Theorem 9.7. The proof is made of several steps.

9.4. NORMAL GEODESIC COORDINATES

311

y

X

x0 yd

S

Figure 9.1. Local normal geodesic coordinates for the Laplace operator Δ = ∂12 + · · · + ∂d2 in Rd Preliminary Remarks. Each step of the proof is associated with the construction of a change of variables, which is a smooth diffeomorphism. For simplicity, at each step, the original and final variables will be denoted by x and y, respectively. At each step, we shall start from an open neighborhood V of x0 . The diffeomorphism κ that will be built for that step will then map X, a possibly smaller open neighborhood of x0 , onto an open set Y of Rd . If x → y = κ(x) = (κ1 (x), . . . , κd (x)) is the built change of variables, we

then have the following relation Dxi = dj=1 (∂xi κj (x))Dyj . We denote by κ (x) = dκ(x) the differential of κ that can be identified with its Jacobian matrix of κ at x. If Q denotes the differential operator P after the action of the change of variables,1 that is, P (f ◦ κ) = (Qf ) ◦ κ,

f ∈ C ∞ (Y ),

then if q(y, η) denotes the principal symbol of Q we have q(κ(x), η) = p(x, t κ (x)η),

x ∈ X, η ∈ Rd .

We refer to Sect. 9.1.2 where the action of change of variables on differential operators is reviewed. (Here, we have identified Tx∗ X and Ty∗ Y with Rd .) In particular q(y, η) is a positive quadratic form, uniformly w.r.t. y ∈ Y . In the course of the proof, at every step we shall ignore first- and zero-order terms as they are only required to have bounded coefficients, which is, and as they do not appear in the smooth principal symbols. At each step, P0 will denote the principal part of the operator obtained at the previous step, and Q0 will denote the principal part of P after the change of variables. Step 1. By reordering the variables, we can assume ∂d ψ(x0 ) = 0 in V . We then define the following change of variables y = κ(x) by  for j = 1, . . . , d − 1, yj = xj − x0j 0 yd = ψ(x) − ψ(x ). 1In the course of the proof we shall not use the same letter for P and Q, as is commonly done, to avoid any confusion.

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9. INVARIANCE UNDER CHANGE OF VARIABLES

With the local diffeomorphism theorem, there exits an open neighborhood X of x0 such that κ is a smooth diffeomorphism of X onto Y = κ(X). Moreover we have κ(x0 ) = 0 and κ(V+ ∩ X) = {y ∈ Y ; yd ≥ 0} and κ(S ∩ X) = {y ∈ Y ; yd = 0}. With this step, we have preserved the assumptions of the theorem and we have achieved the first point in the statement of the theorem. The next two steps will not affect this property. Step 2. We now have with x0 = 0, S = {x ∈ V ; xd = 0} and ψ(x) = xd . For this second step, we aim to write P0 in the new variables under the form: d−1 d−1

id

ij q (y)Dyi Dyd + q (y)Dyi Dyj . Q0 = Dy2d + 2 i=1

i,j=1

If compared to the previous step, we thus want to also enforce q(y, ed ) = 1, for all y ∈ Y , where ed = (0, . . . , 0, 1), i.e., (9.4.2) d

pij (x)(∂i κd (x))(∂j κd (x)) = p(x, dκd (x)) = 1, x ∈ X. p(x, t κ (x)ed ) = i,j=1

We thus obtain an equation that solely involves the coordinate function κd , in the form of an Eikonal equation. Then, an admissible change of variables is, for example, yj = κj (x) = xj for j = 1, . . . , d − 1,

yd = κd (x),

with κd solution to (9.4.2) and such that κd (x) = 0 if xd = 0 and ∂xd κd (0) = pdd (0)−1/2 . The existence of a local smooth solution is given in Proposition 9.8 below, using that p(0, pdd (0)−1/2 ed ) = 1 and ∂ξd p(0, ed ) = 0 as pdd (0) > 0. As ∂d κd (0) = 0, this holds in a neighborhood of 0. This implies, as in the first step, with the local diffeomorphism theorem, that there exits an open neighborhood X of x0 such that κ is a smooth diffeomorphism of X onto Y = κ(X). Note that the sets {xd = 0} and {xd ≥ 0} are changed into {yd = 0} and {yd ≥ 0}, respectively. Note that this second step has preserved the assumptions of the theorem. Step 3. Now we have x0 = 0, S = {xd = 0}, ψ(x) = xd , and moreover P0 takes the form: P0 = Dx2d + 2

d−1

i=1

pid (x)Dxi Dxd +

d−1

i,j=1

pij (x)Dxi Dxj .

We then write P0 in the following form: (9.4.3)

P0 = (Dxd +

d−1

i=1

pid (x)Dxj )2 +

d−1

i,j=1

q ij (x)Dxi Dxj ,

9.4. NORMAL GEODESIC COORDINATES

313

where the coefficients q ij are related to the coefficients pij in a smooth way that need not be made explicit here. From the positivity of p(x, ξ) uniformly w.r.t. x in V , we find that there exists C > 0 such that d−1

(9.4.4)

q ij (x)ξi ξj ≥ C|ξ  |2 .

i,j=1

Now build a diffeomorphism x → κ(x) = y so as to have D yd = D x d +

d−1

i=1

pid (x)Dxj ,

and Dyi as a linear combination of Dx1 , . . . , Dxd−1 for i = 1, . . . , d − 1. In fact, we shall build κ−1 by considering the following differential system:  x˙ i = pid (x), xj (0) = yj for i = 1, . . . , d − 1, (9.4.5) x˙ d = 1, xd (0) = 0. We denote the solution of System (9.4.5) by x(y  , t) with y  = (y1 , . . . , yd−1 ). We define κ−1 by κ−1 (y) = x(y  , yd ), which is a local diffeomorphism that maps an open neighborhood Y of 0 into an open neighborhood X ⊂ V of 0 by the Cauchy–Lipschitz theorem. Through this change of variables, we have D yi =

d

(∂yi κ−1 j (y))Dxj ,

j=1

which yields D yd =

d

(∂yd κ−1 j (y))Dxj =

j=1

d

j=1

∂yd xj (y  , yd )Dxj = Dxd +

d−1

i=1

pid (x)Dxj ,

and D yi =

d

(∂yi xj (y  ; yd ))Dxj =

j=1

d−1

j=1

(∂yi xj (y  ; yd ))Dxj ,

i = 1, . . . , d − 1,

as ∂yi xd (y  , t) = 0 for i = 1, . . . , d − 1 since xd (y  , t) = t. Note that the matrix (∂yi xj (y  , yd ))1≤i,j≤d−1 is the identity for yd = 0 according to the initial conditions in System (9.4.5). Thus, for Y chosen sufficiently small it remains invertible. The vector fields Dxi , i = 1, . . . , d−1,

are then transformed into d−1 j=1 cij (y)Dyj . Following (9.4.3), in the new variable y, P0 takes the form: Q0 = Dy2d +

d−1

i,j=1

p˜ij (y)Dyi Dyj .

Point 2 of the statement of the theorem is achieved. From the positivity of (q ij (x))1≤i,j≤d−1 in (9.4.4), we deduce that a similar positivity holds for (˜ pij (y))1≤i,j≤d−1 , which gives point 3 of the statement of the theorem .

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9. INVARIANCE UNDER CHANGE OF VARIABLES

Finally, observe that since x˙ d = 1 > 0 and xd (y  , 0) = 0, we see that the sets {xd = 0} and {xd > 0} are transformed into {yd = 0} and {yd > 0}, respectively, in a neighborhood of 0.  As above, we write x = (x , xd ) ∈ Rd−1 × R and similarly ξ = (ξ  , ξd ) the associated cotangent vectors. Proposition 9.8. Let q(x, ξ) be a smooth real function defined in a neighborhood of (0, η) in Rd × Rd such that q(0, η) = 0 and ∂ξd q(0, η) = 0. Let f ∈ C ∞ (Rd−1 ) be real valued and such that dx f (0) = η  . Then, there exists a neighborhood U of (0, η) and g ∈ C ∞ (U ), real valued, such that q(x, dg(x)) = 0, for x ∈ U , and the boundary condition: g(x , 0) = f (x ), for (x , 0) ∈ U,

and dg(0) = η.

For a proof, we refer to [176, Theorem 6.4.5].

CHAPTER 10

Elliptic Operator with Dirichlet Data and Associated Semigroup Contents 10.1. Resolvent and Spectral Properties of Elliptic Operators 10.1.1. Basic Properties of Second-Order Elliptic Operators 10.1.2. Spectral Properties 10.1.3. A Sobolev Scale and Operator Extensions 10.2. The Parabolic Semigroup 10.2.1. Spectral Representation of the Semigroup 10.2.2. Well-Posedness: An Elementary Proof 10.2.3. Additional Properties of the Parabolic Semigroup 10.2.4. Properties of the Parabolic Kernel 10.3. The Nonhomogeneous Parabolic Cauchy Problem 10.3.1. Properties of the Duhamel Term 10.3.2. Abstract Solutions of the Nonhomogeneous Semigroup Equations 10.3.3. Strong Solutions 10.3.4. Weak Solutions 10.4. Elementary Form of the Maximum Principle 10.5. The Dirichlet Lifting Map 10.6. Parabolic Equation with Dirichlet Boundary Data

315 316 318 319 327 328 329 331 335 340 340 342 343 344 347 348 351

10.1. Resolvent and Spectral Properties of Elliptic Operators On a smooth bounded open set Ω of Rd , we consider elliptic second-order operator P0 given by

Di (pij (x)Dj ), with pij (x)ξi ξj ≥ C|ξ|2 , (10.1.1) P0 = 1≤i,j≤d

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 I, PNLDE Subseries in Control 97, https://doi.org/10.1007/978-3-030-88674-5 10

315

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10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

where pij ∈ C ∞ (Rd ; R) with furthermore pij = pji , 1 ≤ i, j ≤ d. In addition we shall impose Dirichlet boundary conditions, that is, the trace of the solution at the boundary ∂Ω. At first, we consider a homogeneous Dirichlet boundary condition, that is, a vanishing trace at the boundary ∂Ω for the solution. We postpone the study of nonhomogeneous Dirichlet boundary conditions to Sect. 10.5. The study of Neumann boundary conditions can be found in classical texts. We provided it in Sect. 18.7 in Volume 2. The study of general Lopatinski˘ıˇ Sapiro boundary conditions can be found in Chap. 3 in Volume 2. 10.1.1. Basic Properties of Second-Order Elliptic Operators. Here we recall some well-known facts on elliptic operators such as P0 . In particular, in the case of homogeneous Dirichlet boundary conditions, we recall that P0 is maximal monotone and has a spectral decomposition with a Hilbert basis of eigenfunctions. We consider the following problem: P0 u + λu = f,

for λ ∈ R.

Assume first that f is smooth and there exists a smooth solution with u|∂Ω = 0. Picking a second function v, also satisfying v|∂Ω = 0, upon multiplying the equation by v, integrating over Ω, and performing integrations by part, we find a(u, v) = (f, v)L2 (Ω) with the sesquilinear form a(., .) given by

a(u, v) =

1≤i,j≤d

(pij Di u, Dj v)L2 (Ω) + λ(u, v)L2 (Ω) .

Invoking first the ellipticity of p and second the Poincar´e inequality on H01 (Ω), we have

(10.1.2) (pij Di v, Dj v)L2 (Ω)  Dv2L2 (Ω)  v2H 1 (Ω) . 1≤i,j≤d

Thus there exists λ0 < 0 such that (u, v) → a(u, v) is coercive on H01 (Ω) for λ > λ0 . The value λ0 is given by the best possible constant in the following Poincar´e inequality:

(10.1.3) (pij Di v, Dj v)L2 (Ω) ≥ Cv2L2 (Ω) . 1≤i,j≤d

If now f ∈ H −1 (Ω), as v → f, vH −1 (Ω),H01 (Ω) is continuous on H01 (Ω), the Lax–Milgram theorem (see, e.g., [90, 161]), yields the existence and the uniqueness of a solution u ∈ H01 (Ω) such that (10.1.4)

a(u, v) = (f, v)L2 (Ω) ,

∀v ∈ H01 (Ω),

and we have uH 1 (Ω)  f H −1 (Ω) , 0

10.1. RESOLVENT AND SPECTRAL PROPERTIES OF ELLIPTIC OPERATORS 317

with the H01 -norm defined in (6.1.3). One says that u is a weak solution of the elliptic problem: (10.1.5)

P0 u + λu = f,

for λ > λ0 , f ∈ H −1 (Ω).

Observe that (10.1.4) is the Euler–Lagrange equation associated with the minimization of the functional: 1 J(u) = a(u, u) − (f, u)L2 (Ω) , 2 1 over H0 (Ω). The weak formulation (10.1.4) is thus also called the variational formulation of the elliptic problem. If now f ∈ L2 (Ω) in (10.1.4)–(10.1.5), and if the boundary is C 2 (which is the case here), the solution u ∈ H01 (Ω) given above is in fact in H 2 (Ω). Moreover, (10.1.6)

uH 2 (Ω)  f L2 (Ω) .

Hence, in the case f ∈ L2 (Ω) the weak solution is in fact classical and satisfies P0 u + λu = f in L2 (Ω). Finally, if m ∈ N, if the boundary is C m+2 , and if f ∈ H m (Ω), then u ∈ H m+2 (Ω) and we have (10.1.7)

uH m+2 (Ω)  f H m (Ω) .

We refer to [161, Section 8.4] and [90, Section 9.6] for proofs. We define the unbounded operator P0 : L2 (Ω) → L2 (Ω), with domain D(P0 ) = H 2 (Ω) ∩ H01 (Ω), given by P0 u = P0 u,

u ∈ D(P0 ).

From the elements reviewed above, we see that the H 2 -norm, viz.,  1/2

2 2 uH 2 (Ω) = u2L2 (Ω) + ∇u2L2 (Ω) + Djk uL2 (Ω) , j,k



1/2 , or simply the the graph norm, viz., uD(P0 ) = u2L2 (Ω) + P0 u2L2 (Ω) norm P0 uL2 (Ω) are all equivalent on the space D(P0 ), that is, (10.1.8)

uH 2 (Ω)  uD(P0 )  P0 uL2 (Ω) ,

u ∈ D(P0 ),

and they make it a Hilbert space. In particular, D(P0 ) is a closed subspace of H 2 (Ω). Observe that Cc∞ (Ω) ⊂ D(P0 ). However, the closure of Cc∞ (Ω) for the H 2 -norm is the space H02 (Ω), which is the H 2 -functions u on Ω such that the (well-defined) traces u|∂Ω and ∂ν u|∂Ω vanish. Since H02 (Ω) is strictly included in D(P0 ), we see that the space Cc∞ (Ω) is not dense in D(P0 ). We recall that yet the space Cc∞ (Ω) is dense in H01 (Ω) (see, e.g., [2]). In particular this implies that D(P0 ) is dense in H01 (Ω). We observe that we have (P0 u, u)L2 (Ω) ≥ 0 for all u ∈ D(P0 ). From the properties gathered above, we have the following result. Proposition 10.1. The operator (P0 , D(P0 )) is maximal monotone.

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10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

The definition of monotone operators is recalled in Definition 12.22 in the case of Hilbert spaces. That of maximal monotone operators is given in Definition 12.10. Note that the domain of P0 is dense in L2 (Ω) from general results on Sobolev space. In fact, a general argument can also be invoked, as a maximal monotone operator on a Hilbert space has a dense domain (see Remark 12.12). Note also that P0 is a closed operator from the estimation (10.1.6). In fact, a maximal monotone operator is always closed; see Lemma 12.14. 10.1.2. Spectral Properties. From the symmetry of P0 , viz., (P0 u, v)L2 (Ω) = (u, P0 v)L2 (Ω) ,

u, v ∈ D(P0 ),

we have the following result by Proposition 12.28. Lemma 10.2. The operator (P0 , D(P0 )) is self-adjoint on L2 (Ω). From the elliptic results recalled above, the value 0 is in the resolvent set ρ(P0 ) of P0 : P0 is a bijection from D(P0 ) ⊂ L2 (Ω) onto L2 (Ω) and the −1 2 map P−1 0 : L (Ω) → D(P0 ) is bounded. We then set R0 = ι ◦ P0 where ι 2 2 is the natural injection of H (Ω) into L (Ω). As ι is a compact map by the Rellich–Kondrachov theorem1 [90, Theorem 9.16], so is R0 . We say that P0 has a compact resolvent on L2 (Ω). In what follows we shall often omit to write the map ι explicitly. From the symmetry of P0 , we conclude that R0 is self-adjoint on L2 (Ω). As R0 is injective, the spectral decomposition of compact self-adjoint operators on separable Hilbert spaces yields the existence of a nonincreasing sequence of real eigenvalues (mj )j∈N ⊂ (0, +∞) (counted with their multiplicity) that converges to 0 and an associated sequence of eigenfunctions, denoted by (φj )j∈N , that forms a Hilbert basis of L2 (Ω):

uj φj in L2 (Ω). ∀u ∈ L2 (Ω), ∃(uj )j ∈ 2 (C), u = Moreover uj =

(u, φj )L2 (Ω) , u2L2 (Ω)

(u, v)L2 (Ω) =

uj v j ,

=

j∈N j∈N |uj |

2,

and for u, v ∈ L2 (Ω),

uj = (u, φj )L2 (Ω) , vj = (v, φj )L2 (Ω) .

j∈N

We have R0 (φj ) = mj φj . In particular φj ∈ D(P0 ), and equivalently we have φj = mj P0 φj . Note that we concluded above that the eigenvalues of R0 are positive because of the injectivity of R0 and the nonnegativity of P0 , viz., (P0 u, u)L2 (Ω) ≥ 0. We have

uj mj φj , for u = uj φj ∈ L2 (Ω). R0 u = j∈N

j∈N

1The Rellich–Kondrachov theorem requires the boundary of Ω to be at least Lipschitz, which is the case here. A version on smooth Riemannian manifolds is given in Theorem 18.7 in Volume 2.

10.1. RESOLVENT AND SPECTRAL PROPERTIES OF ELLIPTIC OPERATORS 319

As D(P0 ) is the range of R0 setting μj = m−1 j , j ∈ N, the domain D(P0 ) is characterized by  

vj φj ; (μj vj )j ∈ 2 (C) , D(P0 ) = H 2 (Ω) ∩ H01 (Ω) = v = j∈N

and, with this characterization and the fact that P0 is closed, we then have

(10.1.9) μj v j φ j , v = vj φj ∈ D(P0 ). P0 v = j∈N

j∈N

As a summary, we have here obtained the classical result of the existence of a Hilbert basis (φj )j∈N ⊂ L2 (Ω), formed by eigenfunctions of the operator P0 , associated with the eigenvalues (μj )j∈N , sorted here as an nondecreasing sequence: (10.1.10)

P 0 φ j = μj φ j ,

j ∈ N,

with 0 < μ0 ≤ μ1 ≤ · · · ≤ μk ≤ · · ·

The following asymptotic result is known as the Weyl law for the sequence of eigenvalues (μj )j∈N . Theorem 10.3 (Weyl law). Define Jμ = #{j ∈ N; μj ≤ μ}. We have Jμ ∼ (2π)−d ωd |Ω| μd/2 ,

as r → ∞,

where ωd is the volume of the Euclidean unit ball, that is, ωd = π d/2 /Γ(1 + d/2), with Γ the gamma function. We refer, for example, to [110, Chapter 6, Theorems 16 and 18] or to [292, Theorem 8.16]. Remark 10.4. An equivalent formulation is the following asymptotic formula: (10.1.11)  −2/d 2/d  2/d 2/d j = 4π Γ(1 + d/2)/|Ω| j , as j → ∞. μj ∼ (2π)2 ωd |Ω| 10.1.3. A Sobolev Scale and Operator Extensions. For the analysis of the semigroup generated by the operator (P0 , D(P0 )) (and some of its extensions) carried out below, the proper functional framework needs to be introduced. To that purpose, we define some adapted spaces of Sobolev type. With the above spectral family, the space H01 (Ω) is characterized by the following proposition. Proposition 10.5. We have the following equivalence: u ∈ H01 (Ω)

1/2

u ∈ L2 (Ω) and (μj uj )j ∈ 2 (C), uj = (u, φj )L2 (Ω) .

In particular, the inner product (u, v) → j∈N μj uj vj gives the usual Hilbert space structure on H01 (Ω), with vj = (v, φj )L2 (Ω) . We also have ⇔

(10.1.12) Du2L2 (Ω) 

1≤i,j≤d

(pij Di u, Dj u)L2 (Ω) = u2H 1 (Ω) = 0

j∈N

μj |uj |2 ,

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10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

recalling (6.1.3). Proof. As recalled in (6.1.4), the H 1 -norm is given by uH 1 (Ω) =

(u2L2 (Ω) +Du2L2 (Ω) )1/2 . On H01 (Ω), an equivalent norm is simply DuL2 (Ω) by the Poincar´e inequality. In (6.1.3) we defined the following norm:

(pij Di u, Dj u)L2 (Ω) u2H 1 (Ω) = 0

1≤i,j≤d

H01 (Ω)

that is equivalent to DuL2 (Ω) . Above, we recalled that Cc∞ (Ω) ⊂ D(P0 ) ⊂ H01 (Ω), implying the density of D(P0 ) in H01 (Ω). If u ∈ D(P0 ), one has P0 u = j∈N μj uj φj and

(10.1.13) (pij Di u, Dj u)L2 (Ω) = (P0 u, u)L2 (Ω) = μj |uj |2 . on

1≤i,j≤d

j∈N

Hence, on D(P0 ), the norm uH 1 (Ω) is equivalent to that associated with 0

the inner product (u, v) → j∈N uj vj μj . Consider v ∈ H01 (Ω) and (v (n) ) ⊂ D(P0 ) is such that v (n) → v in H01 (Ω). One has

(n)

v j φj , v (n) = v j φj , v= j∈N

j∈N

1/2 (n)

with V (n) = (μj vj )j∈N ∈ 2 (C), n ∈ N. With (10.1.13), we see that (V (n) )n∈N is a Cauchy sequence in 2 (C). Thus, there exists (wj )j∈N such 1/2 that (μj wj )j∈N ∈ 2 (C) and

(n) μj |vj − wj |2 → 0 as n → ∞. j∈N

(n)

(n)

In particular for each j ∈ N, one has vj → wj as n → ∞. Yet, vj = (v (n) , φj )L2 (Ω → (v, φj )L2 (Ω = vj meaning that vj = wj . Hence, v is such 1/2

that (μj vj )j∈N ∈ 2 (C). 1/2

Let now v ∈ L2 (Ω) be such that (μj vj )j∈N ∈ 2 (C). For n ∈ N, set

v (n) = j≤n vj φj . One has v (n) ∈ D(P0 ). With the norm equivalence given in (10.1.13), one finds that m

2 μj |vj |2 , n ≤ m, v (n) − v (m) H 1 (Ω)  0

j=n+1

(v (n) )

1 implying that n∈N is a Cauchy sequence in H0 (Ω). Since it converges  to v in L2 (Ω), it shows that v ∈ H01 (Ω). √ Remark 10.6. Note that μ0 coincides with the optimal constant in the Poincar´e inequality since we have

μj |uj |2 ≥ μ0 u2L2 (Ω) , u2H 1 (Ω) = 0

j∈N

with equality in the case u = φ0 .

10.1. RESOLVENT AND SPECTRAL PROPERTIES OF ELLIPTIC OPERATORS 321

With the description of H01 (Ω) by means of the spectral family (φj )j , we can recover classical characterizations of H01 (Ω) functions. Proposition 10.7. Let u ∈ L2 (Ω) be such that |(u, P0 v)L2 (Ω) | ≤ LvH 1 (Ω) , 0

for some L > 0 and all v ∈ D(P0 ). Then, u ∈ H01 (Ω) and uH 1 (Ω) ≤ L. 0

2 Proof. We have u = j∈N uj φj with (uj )j ∈  (C). For v ∈ D(P0 ),

v = j∈N vj φj , with (μj vj )j ∈ 2 (C) we have (u, P0 v)L2 (Ω) = j∈N μj uj v j . Letting N ∈ N and choosing vj = uj for j ∈ {0, . . . , N }, and vj = 0 for j ≥ N + 1, we find 1/2 

μj |uj |2 ≤ LvH 1 (Ω) = L μj |uj |2 . 0

0≤j≤N

This gives



0≤j≤N

μj |uj |

 2 1/2

0≤j≤N

≤ L, which yields the conclusion.



The space H −1 (Ω) denotes the dual space of H01 (Ω). Instead of identifying H01 (Ω) and H −1 (Ω) by the Riesz theorem through the inner product on H01 (Ω), one usually uses the space L2 (Ω) as a pivot space. This is possible because H01 (Ω) is dense in L2 (Ω). Then, we have H01 (Ω) → L2 (Ω) → H −1 (Ω), where both injections have a dense range. We may then write, for u ∈ L2 (Ω),

uj v j , v ∈ H01 (Ω). u, vH −1 (Ω),H01 (Ω) = (u, v)L2 (Ω) = j∈N

H −1 (Ω),

we set uj = u, φj H −1 (Ω),H01 (Ω) . Considering the norm given If u ∈ 1 on H0 (Ω) by Proposition 10.5, this leads to the following characterization of the space H −1 (Ω). −1/2

Proposition 10.8. If u ∈ H −1 (Ω), then (μj u = lim

n

n→∞ j=0

uj φj

uj )j ∈ 2 (C) and

in H −1 (Ω). −1/2

Conversely, if (wj )j ⊂ C is such that (μj wj )j ∈ 2 (C), then the sequence  

n −1 (Ω) and w = of L2 -functions j j=0 wj φj n∈N converges to some u in H u, φj H −1 (Ω),H01 (Ω) . We thus write  

−1/2 uj φj ; (μj uj )j ∈ 2 (C) . H −1 (Ω) = u = j∈N

With the pivot space L2 (Ω), the duality between H −1 (Ω) and H01 (Ω) reads

uj v j , u, vH −1 (Ω),H01 (Ω) = j∈N

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10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

for u = −1/2

(μj

uj )j

j∈N uj φj ∈ ∈ 2 (C) and

H −1 (Ω) and v =

1/2

j∈N vj φj

∈ H01 (Ω), that is,

(μj vj )j ∈ 2 (C).

With the characterizations of H01 (Ω) and H −1 (Ω) given above through the spectral family, we can introduce the unbounded operator P−1 : H −1 (Ω) → H −1 (Ω), with domain D(P−1 ) = H01 (Ω), given by

P−1 u = μj u j φ j , u = uj φj ∈ H01 (Ω). j∈N

j∈N

From (10.1.9), P−1 is an extension of P0 to H −1 (Ω). We have (10.1.14)

P−1 uH −1 (Ω) = uH 1 (Ω) , 0

u ∈ H01 (Ω).

For u ∈ H01 (Ω), the action of P0 makes no sense in general. However, P0 u is well-defined in H −1 (Ω) in the sense of distributions. Proposition 10.9. Let u ∈ H01 (Ω). We have P0 u = P−1 u in H −1 (Ω). Proof. The proof uses that H −1 (Ω) is a space of distributions since is dense in H01 (Ω). Let ϕ ∈ Cc∞ (Ω). On the one hand, we naturally have Cc∞ (Ω)

P0 u, ϕD  (Ω),Cc∞ (Ω) = u, P0 ϕD  (Ω),Cc∞ (Ω) by the symmetry of P0 . On the other hand, we write P−1 u, ϕD  (Ω),Cc∞ (Ω) = P−1 u, ϕH −1 (Ω),H01 (Ω) = u, P−1 ϕH01 (Ω),H −1 (Ω) = u, P0 ϕL2 (Ω),L2 (Ω) = u, P0 ϕD  (Ω),Cc∞ (Ω) ,

since ϕ ∈ D(P0 ). Hence, P0 u = P−1 u in D  (Ω), and thus this equality holds  in H −1 (Ω). With the above duality, we may then write

(u, v)H01 (Ω) = (10.1.15) (pij Di u, Dj v)L2 (Ω) = μj u j v j 1≤i,j≤d

j∈N

= P−1 u, vH −1 (Ω),H01 (Ω) = P0 u, vH −1 (Ω),H01 (Ω) , using Proposition 10.9. Proposition 10.10. Let u ∈ H −1 (Ω) be such that |u, vH −1 (Ω),H01 (Ω) | ≤ LvL2 (Ω) , for some L > 0 and all v ∈ H01 (Ω). Then, u ∈ L2 (Ω) and uL2 (Ω) ≤ L. The proof can be adapted from that of Proposition 10.7. For s ≥ 0, we introduce the unbounded operator Ps0 : L2 (Ω) → L2 (Ω), with domain  

D(Ps0 ) = u = uj φj ; (μsj uj )j ∈ 2 (C) ⊂ L2 (Ω), j∈N

10.1. RESOLVENT AND SPECTRAL PROPERTIES OF ELLIPTIC OPERATORS 323

given by Ps0 u =

μsj uj φj ,

u=

j∈N

uj φj ∈ D(Ps0 ).

j∈N

D(Ps0 )

with the following inner product and We naturally equip the space associated norm that endows it with a Hilbert space structure:

s

s μj u j v j , u2D(Ps ) = μj |uj |2 . (u, v)D(Ps0 ) = 0

j∈N

j∈N

This norm is equivalent to the graph norm on D(Ps0 ). We have P00 = IdL2 (Ω) with D(P00 ) = L2 (Ω), and the case s = 1 is consistent with the domain of 1/2 the operator P0 on L2 (Ω). Note that D(P0 ) = H01 (Ω) and (10.1.16) 1/2

1/2

(u, v)H01 (Ω) = P0 u, vH −1 (Ω),H01 (Ω) = (P0 u, P0 v)L2 (Ω) ,

u, v ∈ H01 (Ω).

using (10.1.15), and 1/2

2

(10.1.17) u2H 1 (Ω) = P0 u, uH −1 (Ω),H01 (Ω) = P0 uL2 (Ω) , 0

u ∈ H01 (Ω).

Note that this is precisely the norm defined in (6.1.3) by Proposition 10.5. Note also that we have (10.1.18)

(u, v)H01 (Ω) = (P0 u, v)L2 (Ω)

if u ∈ D(P0 ), v ∈ H01 (Ω).

In the case s = k ∈ N, Pk0 and D(Pk0 ) correspond to the iterated operators and domains for the elliptic operator P0 , that is, D(Pk+1 0 ) = {u ∈ k k D(P0 ); P0 u ∈ D(P0 )}. Note that for the Hilbert basis (φj )j introduced above, we have φj ∈ ∩s≥0 D(Ps0 ),

j ∈ N.

For s < 0, we can define the following bounded operator on L2 (Ω):

−s

Ps0 u = μj u j φ j , u= uj φj ∈ L2 (Ω). j∈N

j∈N

In fact, if s < 0, the operator Ps0 is compact. With this notation, we have2 −1/2 1/2 R0 = P−1 maps L2 (Ω) onto H01 (Ω) = D(P0 ) 0 . We observe that P0 −1/2 1/2 −1/2 = (P0 )−1 . Noting that P0 uL2 (Ω) = isometrically, and we have P0 −1/2 can be (uniquely) extended to an isometry uH −1 (Ω) , we see also that P0 from H −1 (Ω) onto L2 (Ω).

Arguing as we did above for the H01 (Ω)-H −1 (Ω) duality with L2 (Ω) as a pivot space, we may then obtain the following result.

2Here, as we omit the operator ι defined above, in the case s = −1, we can identify defined at the beginning of Sect. 10.1.2. Ps0 and the operator P−1 0

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10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

Proposition 10.11. Let s ≥ 0. We denote by D(Ps0 ) the dual of D(Ps0 ). We have D(Ps0 ) → L2 (Ω) → D(Ps0 ) , where both injections have a dense range. For u ∈ D(Ps0 ) , if we set uj = u, φj D(Ps0 ) ,D(Ps0 ) , the space D(Ps0 ) is characterized as follows: and u =

u ∈ D(Ps0 ) j∈N uj φj

2 (μ−s j uj )j ∈  (C),

⇔

in D(Ps0 ) . Finally, we have

u, vD(Ps0 ) ,D(Ps0 ) = uj v j , j∈N

s  for u = j∈N uj φj ∈ D(P0 ) (Ω) and v = 2 s 2 (μ−s j uj )j ∈  (C) and (μj vj )j ∈  (C).

j∈N vj φj

∈ D(Ps0 ), that is,

s/2

Definition 10.12. For s ≥ 0, we set K s (Ω) = D(P0 ), and, for s < 0, −s/2 we set K s (Ω) = D(P0 ) . We have



K s (Ω) → K s (Ω) for s ≥ s , where the injection has a dense range. From what is presented above, for all s ∈ R, we have

s/2 uj φj ; (μj uj )j ∈ 2 (C)}. K s (Ω) = {u = j∈N

On

K s (Ω),

the following inner product and associated norm:

s

s (u, v)K s (Ω) = μj uj vj , u2K s (Ω) = μj |uj |2 < ∞, j∈N

j∈N

with uj = u, φj K s (Ω),K −s (Ω) and vj = v, φj K s (Ω),K −s (Ω) , yield a Hilbert space structure. For s ≥ 0, if u ∈ L2 (Ω) and v ∈ K s (Ω), we recover the pivot rˆ ole played by L2 (Ω):

uj v j . u, vK −s (Ω),K s (Ω) = (u, v)L2 (Ω) = j∈N

For r ∈ R and s ≥ 0, with the spectral family (φj )j , we defined the unbounded operator Psr : K r (Ω) → K r (Ω), with domain D(Psr ) = K r+2s (Ω) given by

s

(10.1.19) μj u j φ j , u = uj φj ∈ K r+2s (Ω). Psr u = j∈N

j∈N

We have (10.1.20)

Psr uK r (Ω) = uK r+2s (Ω) ,

u ∈ K r+2s (Ω).

If r ≥ 0, the operator Psr is a restriction of Ps0 to K r (Ω) ⊂ K 0 (Ω) = L2 (Ω). If r < 0, the operator Psr is an extension of Ps0 to K r (Ω) ⊃ L2 (Ω).

10.1. RESOLVENT AND SPECTRAL PROPERTIES OF ELLIPTIC OPERATORS 325

For r ∈ R and s < 0, we define the bounded operator Psr : K r (Ω) → also given by (10.1.19). We have

K r (Ω)

Psr uK r+2|s| (Ω) = uK r (Ω) . We may then state the following results whose proof is elementary from what precedes. Proposition 10.13. Let r, s, σ ∈ R. (1) If s ≥ 0 and r + 2s, we have K σ (Ω) ⊂ D(Psr ) = K r+2s (Ω) and  σ ≥σ−2s s σ Pr K (Ω) = K (Ω) ⊂ K r (Ω). Moreover, we have Psr (u)K σ−2s (Ω) = uK σ (Ω) ,

u ∈ K σ (Ω).

(2) If s < 0 and σ ≥ r, we have K σ (Ω) ⊂ D(Psr ) = K r (Ω) and Psr K σ (Ω) = K σ+2|s| (Ω) ⊂ K σ (Ω). Moreover, we have Psr (u)K σ+2|s| (Ω) = uK σ (Ω) ,

u ∈ K σ (Ω).

The self-adjointness property further extends to Ps0 . Lemma 10.14. Let s ∈ R, and let u, v ∈ D(Ps0 ). We have (Ps0 u, v)L2 (Ω) = (u, Ps0 v)L2 (Ω) . We also have the following results. Lemma 10.15. Let r, s ∈ R, and let u ∈ K r+2s (Ω) and v ∈ K −r (Ω). We have Psr u, vK r (Ω),K −r (Ω) = u, Ps−r−2s vK r+2s (Ω),K −r−2s (Ω) . Lemma 10.16. Let u ∈ K r+2 (Ω). We have u2K r+1 (Ω) = (Pr u, u)K r (Ω) . We finish this section by further analyzing the properties of the functions in K k (Ω). k/2

Proposition 10.17. Let k ∈ N. We have K k (Ω) = D(P0 ) ⊂ H k (Ω) and K k (Ω) = {u ∈ H k (Ω); Pj0 u ∈ H01 (Ω), j = 0, . . . , E[(k − 1)/2]}. Moreover, there exists C > 0 such that C −1 uH k (Ω) ≤ uK k (Ω) ≤ C uH k (Ω) . In fact, K k (Ω) is a closed linear subspace of H k (Ω). Here, we denote by E[.] the integer part of a real number. Proof. The property holds for k = 0, 1, 2. We proceed by induction and assume that the property holds for k − 1 and k for some k ∈ N, with k ≥ 2. Let then u ∈ K k+1 (Ω). We thus have P0 u ∈ K k−1 (Ω) from the results given 1 above. We thus have Pj+1 0 u ∈ H0 (Ω) for j = 0, . . . , E[(k−2)/2] = E[k/2]−1, that is Pj0 u ∈ H01 (Ω) for j = 1, . . . , E[k/2]. We also have u ∈ H01 (Ω), since

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10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

u ∈ K k (Ω) ⊂ K 1 (Ω) as k ≥ 2. Moreover, we have P0 u ∈ H k−1 (Ω). By (10.1.7), we have u ∈ H k+1 (Ω) and uH k+1 (Ω)  P0 uH k−1 (Ω)  P0 uK k−1 (Ω) = uK k+1 (Ω) . We have thus found K k+1 (Ω) ⊂ {u ∈ H k+1 (Ω); Pj0 u ∈ H01 (Ω), j = 0, . . . , E[k/2]}. Let now u ∈ H k+1 (Ω) be such that Pj0 u ∈ H01 (Ω), for j = 0, . . . , E[k/2]. Thus u ∈ D(P0 ) and v = P0 u ∈ H k−1 (Ω). As Pj0 v ∈ H01 (Ω), for j = 0, . . . , E[k/2] − 1 = E[(k − 2)/2], we find that P0 u ∈ K k−1 (Ω). This implies that u ∈ K k+1 (Ω) from the results given above. This concludes the proof.  Proposition 10.18. Let α ∈ C ∞ (Ω). If u ∈ K k (Ω), then αu ∈ K k (Ω) for k = 0, 1, 2. This is a consequence of the following lemma that follows from the smoothness of the coefficients of the operator P0 . ∞ Lemma 10.19. Let and let k ∈ N. If u ∈ H k+2 (Ω),

α ∈ C (Ω), ij k we then have v = 1≤i,j≤d Di (p (x)Dj )(αu) ∈ H (Ω) and vH k (Ω) ≤ CuH k+2 (Ω) , where the constant C only depends on α and the coefficients pij .

Next, we observe that if u ∈ K 3 (Ω) and α as above, we have αu ∈ D(P0 ) = K 2 (Ω) by Proposition 10.18, and P0 (αu) = P0 (αu) = αP0 u +



1≤i,j≤d

 Di (pij Dj )α u + 2

1≤i,j≤d

pij (x)(Di α)(Dj u).

While the first two terms are in H01 (Ω), the last sum is in H 1 (Ω) but not / K 3 (Ω) by Proposition 10.17. in H01 (Ω) in general, meaning then that αu ∈ Yet, we note that, at the boundary, ∇u is colinear to the normal vector to ∂Ω, n = (n1 , . . . , nd ), as u vanishes at ∂Ω. Consequently, if we have

pij (x)Di α|Ω nj = 0, 1≤i,j≤d

we find that P0 (αu) vanishes at the boundary. Hence, in this case we have αu ∈ K 3 (Ω) by Proposition 10.17. For higher orders in the Sobolev scale, we may simply write the following result. Lemma 10.20. If α ∈ C ∞ (Ω) and if α is flat at all orders at ∂Ω, then, for any k ∈ N and u ∈ K k (Ω) we have αu ∈ K k (Ω) and αuK k (Ω) ≤ CuK k (Ω) , where the constant C > 0 is only dependent upon the function α.

10.2. THE PARABOLIC SEMIGROUP

327

10.2. The Parabolic Semigroup The unbounded operator P0 on L2 (Ω) with dense domain D(P0 ) = ∩ H01 (Ω) is maximal monotone by Proposition 10.1. Then, with the Lumer–Phillips theorem (Theorem 12.11) we have the following result that states the well-posedness of the parabolic equation associated with the operator P0 .

H 2 (Ω)

Theorem 10.21. Let T ∈ R+ ∪ {+∞}. The operator P0 generates C0 semigroup of contraction S(t) = e−tP0 on L2 (Ω). If y 0 ∈ D(P0 ), then y(t) = S(t)y 0 is the unique solution in C 0 ([0, T ]; D(P0 )) ∩ C 1 ([0, T ]; L2 (Ω)), such that y(0) = y 0 and d y(t) + P0 y(t) = 0 dt holds in L2 (Ω) for all 0 ≤ t ≤ T . Here, [0, T ] means [0, +∞) if T = +∞. We recall that Sobolev spaces K s (Ω) as introduced in Sect. 10.1.3 are s/2 −s/2 given by K s (Ω) = D(P0 ) for s ≥ 0 and K s (Ω) = D(P0 ) for s < 0. As P0 is moreover self-adjoint by Lemma 10.2, then the stronger version of the Lumer–Phillips theorem adapted to Hilbert spaces given in Theorem 12.26 yields the following result (see also Corollary 12.27). Theorem 10.22. Let T ∈ R+ ∪ {+∞}. The semigroup S(t) is analytic, and for y 0 ∈ L2 (Ω), the function y(t) = S(t)y 0 is in C 0 ([0, T ]; L2 (Ω)) ∩ C ∞ ((0, T ]; K s (Ω)),

s ∈ R,

and is such that (10.2.1) d y(t) + P0 y(t) = 0 holds in L2 (Ω) for 0 < t ≤ T. dt Moreover, y(t) = S(t)y 0 is the unique solution of (10.2.1) in y(0) = y 0 and

C 0 ([0, T ]; L2 (Ω)) ∩ C 1 ((0, T ]; L2 (Ω)) ∩ C 0 ((0, T ]; D(P0 )). Here, [0, T ] (resp.., (0, T ]) means [0, +∞) (resp.., (0, +∞)) if T = +∞. Observe that the C0 -semigroup S(t) is self-adjoint by Corollary 12.25. The above theorems are consequences of general results on semigroups. Here, in the particular case of the operator P0 and of the semigroup S(t) it generates, using the spectral representation S(t) given in Sect. 10.2.1 by means of the Hilbert basis introduced in Sect. 10.1.2, we can recover all the results of Theorem 10.22 in a quite elementary way, only invoking few aspects of semigroup theory that are exposed at the beginning of Sect. 12. This is exposed in Sect. 10.2.2 below. A reader experienced with semigroup theory can readily skip this section. In Sect. 10.2.1 we give however a spectral

328

10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

representation of the semigroup S(t). This point of view is important in the framework of the method used to prove the null controllability of parabolic equations in Chap. 7 that is precisely based on the use of the spectral family (φj )j∈N . 10.2.1. Spectral Representation of the Semigroup. We recall that the Hilbert basis of L2 (Ω) introduced in Sect. 10.1.2 is comprised of eigenfunctions (φj )j∈N of P0 , with (μj )j∈N ⊂ R for associated eigenvalues. In particular, if u ∈ L2 (Ω), we have u = j∈N uj φj , with uj = (u, φj )L2 (Ω) . It is quite simple to obtain the form of the semigroup S(t) within this spectral family according to the following lemma. Lemma 10.23. Let T ∈ R+ ∪ {+∞}. Let y 0 ∈ L2 (Ω), and let t → y(t) ∈ C 0 ([0, T ]; L2 (Ω)) ∩ C 1 ((0, T ]; L2 (Ω)) ∩ C 0 ((0, T ]; D(P0 )) be such that y(0) = y 0 and such that d y(t) + P0 y(t) = 0 holds in L2 (Ω) for 0 < t ≤ T. (10.2.2) dt If we set yj (t) = e−tμj (y 0 , φj )L2 (Ω) , for t ≥ 0 and j ∈ N, then (yj (t))j ∈ C 0 ([0, T ], 2 (C)) and

y(t) = yj (t)φj , t ≥ 0, j∈N

with convergence in

L2 (Ω).

The action of the semigroup S(t) generated by P0 on L2 (Ω) and given by Theorems 10.21 and 10.22 is thus given by

−tμj (10.2.3) e (u, φj )L2 (Ω) φj , u ∈ L2 (Ω), t ≥ 0, S(t)u = j∈N

where the series convergence is to be understood in L2 (Ω). Note that the result of Lemma 10.23 is also to be understood as a uniqued y(t) + P0 y(t) = 0. ness result for the semigroup equation dt Proof. Let t > 0. As P0 y(t) ∈ L2 (Ω), we have y(t) ∈ D(P0 ). We set zj (t) = (y(t), φj )L2 (Ω) , for t ≥ 0. Then (zj (t))j ∈ C 0 ([0, T ], 2 (C)). We have, for t, t > 0       (t − t)−1 zj (t ) − zj (t) = (t − t)−1 y(t ) − y(t) , φj L2 (Ω) . As y ∈ C 1 ((0, T ]; L2 (Ω)), letting t → t, we find that zj (t) is differentiable for t > 0 and d d zj (t) = ( y(t), φj )L2 (Ω) = −(P0 y(t), φj )L2 (Ω) = −(y(t), P0 φj )L2 (Ω) , dt dt by Lemma 10.2, as φj ∈ D(P0 ). Since P0 φj = μj φj , we obtain d zj (t) = −μj (y(t), φj )L2 (Ω) = −μj zj (t). dt  Consequently zj (t) = yj (t) for any t ≥ 0, which concludes the proof.

10.2. THE PARABOLIC SEMIGROUP

329

10.2.2. Well-Posedness: An Elementary Proof. As mentioned above, we provide here a simple proof of Theorems 10.21 and 10.22, based on the decomposition (10.2.3) of the semigroup S(t) in the spectral family (φj )j∈N . Lemma 10.23 is to be treated as the uniqueness part of both theorems. With Lemma 10.23, for t ≥ 0, for u ∈ L2 (Ω), we define the map:

−μj t e uj φj , uj = (u, φj )L2 (Ω) . Σ(t)u = j∈N

As (uj )j ∈

2 (C),

so is

(e−μj t u

j )j ,

implying that Σ(t)u ∈ L2 (Ω).

Lemma 10.24. The map Σ(t) is the strongly continuous contraction semigroup S(t) generated by the unbounded operator (P0 , D(P0 )) on L2 (Ω).

Proof. Let u ∈ L2 (Ω). We write u = j∈N uj φj with (uj ) ∈ 2 (C).

−2μj t |u |2 ≤ 2 Observe that we have Σ(t)u2L2 (Ω) = j j∈N e j∈N |uj | = u2L2 (Ω) , implying that Σ(t) is in L (L2 (Ω)) and moreover of contraction type. Observe also that Σ(t) satisfies the following semigroup properties:

Σ(t) ◦ Σ(t ) = Σ(t + t ).

With u ∈ L2 (Ω) as above, we write Σ(t)u−u = j∈N (e−μj t −1)uj φj , yielding

−μj t )2 |u |2 . As for each j ∈ N, we have Σ(t)u − u2L2 (Ω) = j j∈N (1 − e e−μj t − 1 → 0 as t → 0+ , and as 0 ≤ 1 − e−μj t ≤ 1, the Lebesgue dominatedconvergence theorem (for the counting measure) implies that Σ(t)u → u in L2 (Ω) as t → 0+ for all u ∈ L2 (Ω). Considering the definition of a C0 semigroup, as recalled in Definition 12.1, we have obtained that Σ(t) is such a semigroup. We now prove that P0 with domain D(P0 ) = K 2 (Ω) is the generator of Σ(t). As the map (12.1.9) that associates a semigroup to its generator is injective, this allows one to conclude that Σ(t) is the C0 -semigroup generated by (P0 , D(P0 )). For the time being, we denote by A : L2 (Ω) → L2 (Ω), with D(A) ⊂ L2 (Ω), the generator of Σ(t). Let u ∈ L2 (Ω) such that, moreover, the limit − Σ(t)u)/t exists in L2 (Ω). We denote by v this limit, and we limt→0+ (u have v = j∈N vj φj with (vj = (v, φj ))j ∈ 2 (C). Then, u ∈ D(A) and Au = v (see the beginning of Sect. 12.1.1). We have 2

 (1 − e−μj t )uj /t − vj  . (u − Σ(t)u)/t − v2L2 (Ω) = Σ(0) = IdL2 (Ω) ,

j∈N

For all j ∈ N, we thus have (1 − e−μj t )uj /t − vj → 0 as t → 0+ , meaning that vj = μj uj . Hence, if u ∈ D(A), then u ∈ D(P0 ) and Au = P0 u. Conversely, let us consider u ∈ D(P0 ); we have (μj uj )j ∈ 2 (C). Then,

P0 u = j∈N μj uj φj . We then have 2

 (u − Σ(t)u)/t − P0 u2L2 (Ω) = (1 − e−μj t )uj /t − μj uj  . j∈N

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10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

As for each j ∈ N, we have (1 − e−μj t )/t − μj → 0 as t → 0+ and as (1 − e−μj t )/t ≤ μj for t > 0, the Lebesgue dominated-convergence theorem implies (u − Σ(t)u)/t → P0 u in L2 (Ω) as t → 0+ for all u ∈ D(P0 ). We thus conclude that the domain of the generator A of Σ(t) is precisely D(P0 ) and  that A coincides with P0 . From Proposition 12.2, if y 0 ∈ D(P0 ) and y(t) = Σ(t)y 0 = S(t)y 0 , then for t ≥ 0, and y(t) ∈ C 0 ([0, T ]; D(P0 )) concludes the second proof of Theorem 10.21. The next lemma concludes the proof of Theorem 10.22.

d 2 dt y(t)+P0 y(t) = 0 is satisfied in L (Ω) d and dt y(t) ∈ C 0 ([0, T ]; L2 (Ω)). This

Lemma 10.25. Let y 0 ∈ L2 (Ω) and y(t) = S(t)y 0 ∈ C ([0, +∞); L2 (Ω)). d For any s ≥ 0, we have y(t) ∈ C ∞ ((0, +∞); K s (Ω)) and dt y(t) = −P0 y(t)   k d s k s in K (Ω) for t > 0. Moreover, dt y(t) = (−P0 ) y(t) in K (Ω) for t > 0.

Proof. We write y 0 = j∈N yj0 φj with (yj0 )j ∈ 2 (C). We pick s ≥ 0.

First, let us consider t > 0. We have y(t)2K s (Ω) = j∈N μsj e−2μj t |yj0 |2 ≤

2 Ct j∈N |yj0 |2 = Ct y 0 L2 (Ω) , implying that y(t) ∈ ∩r∈R K r (Ω). Second, let t > 0 and h ∈ R such that t + h > 0. We write y(t + h) − y(t) =

−μj h − 1)eμj t y 0 φ . As y(t + h) − y(h) ∈ K s (Ω), we find j j j∈N (e

y(t + h) − y(t)2K s (Ω) = (1 − e−μj h )2 μsj e2μj t |yj0 |2 . j∈N

As (1 − e−μj h )2 μsj e2μj t converges to zero as h → 0 and is bounded by some constant Cs,t independent of j, the Lebesgue dominated-convergence theorem (for the counting measure) implies that y(t + h) − y(t)K s (Ω) → 0 as h → 0. We thus have y ∈ C 0 ((0, +∞); K s (Ω)) for any s ≥ 0. We now proceed by induction and assume that y ∈ C k ((0, +∞); K s (Ω)), for any s ≥ 0, for some k ∈ N. For t > 0, and h ∈ R such that t + h > 0, we write, in K s (Ω), for some s > 0, 2

h−1 (y(t + h) − y(t)) + P0 y(t)K s (Ω) 2

 −1 −μj h = − 1) + μj  μsj e−2μj t |yj0 |2 . h (e j∈N

Note that P0 y(t) ∈ K s (Ω) as y(t) ∈ K s+2 (Ω) ⊂ D(P0 ) by the induction hypothesis and Proposition 10.13. As we have |h−1 (e−μj h − 1) − μj |  μj , d the Lebesgue dominated-convergence theorem yields that dt y(t) + P0 y(t) = s 0 in K (Ω) if t > 0. With the induction hypothesis, we have P0 y ∈ C k ((0, +∞); K s (Ω)) for any s ≥ 0, implying that y ∈ C k+1 ((0, +∞); K s (Ω)). We thus have y ∈ C ∞ ((0, +∞); K s (Ω)). Similarly, we prove that h−1 (Pk0 y(t + h) − Pk0 y(t)) + Pk+1 0 y(t)K s (Ω) → 0 as h → 0, for any s ≥ 0, implying that

k+1 d k dt P0 y(t) + P0 y(t)

= 0 in K s (Ω),

10.2. THE PARABOLIC SEMIGROUP

which allows one to conclude that t > 0.

 d k dt

331

y(t) = (−P0 )k y(t), in K s (Ω), for 

10.2.3. Additional Properties of the Parabolic Semigroup. We have the following bounds for the semigroup, expressing in particular the natural decay of the L2 -norm of the solution. Proposition 10.26. The semigroup S(t) maps L2 (Ω) into L2 (Ω) with S(t)L (L2 (Ω)) ≤ e−μ0 t ,

(10.2.4)

and moreover, for some C > 0, if t > 0, √ S(t)L (L2 (Ω),D(P0 )) ≤ C/t. S(t)L (L2 (Ω),H 1 (Ω)) ≤ C/ t, 0

−1 In addition, S(t) can be uniquely extended √ to H (Ω) and there exists C > 0 such that S(t)L (H −1 (Ω),L2 (Ω)) ≤ C/ t if t > 0.

Proof. Let u ∈ L2 (Ω), with u = j∈N uj φj . We have

|uj |2 e−2μj t ≤ e−2μ0 t |uj |2 = e−2μ0 t u2L2 (Ω) . S(t)u2L2 (Ω) = j∈N

j∈N

We also have S(t)u2H 1 (Ω) = 0

j∈N

μj |uj |2 e−2μj t ≤

1 sup (xe−2x )u2L2 (Ω) . t [0,+∞)

The other operator norm estimates can be proven similarly.



More generally, we have the following result. Proposition 10.27. For r < 0, the semigroup S(t) can be uniquely extended as a map from K r (Ω) into itself. For r ≥ 0, the restriction of S(t) to K r (Ω) ⊂ L2 (Ω) maps K r (Ω) into itself.

r/2 For any r ∈ R, if u = j∈N uj φj ∈ K r (Ω), that is, with (μj uj )j ∈

−μj t u φ ∈ K r (Ω) for t ≥ 0 and S(t)u ∈ 2 (C), then S(t)u = j j j∈N e ∩s∈R K s (Ω) for t > 0. Moreover, if s ≥ 0, there exists Cs,r > 0 such that S(t)L (K r (Ω),K r+s (Ω)) ≤ Cs,r t−s/2 ,

t > 0.

If s = 0, then one has S(t)L (K r (Ω)) ≤ e−μ0 t , for all r ∈ R. To avoid cumbersome notation, the extension or restriction of the semigroup S(t) to K r (Ω) is also denoted by S(t), for all values of r ∈ R. Proof. We only prove that S(t) extended to K r (Ω) if r < 0

can−μbe j tu φ and that, in this case, S(t)u = j j if u = j∈N e j∈N uj φj with (μj uj )j ∈ 2 (C). The rest of the proof is similar to that of Proposition 10.26. r/2

332

10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

Let thus r < 0. If u ∈ K 0 (Ω) = L2 (Ω), we have u = j∈N uj φj , with

uj = (u, φj )L2 (Ω) , and S(t)u = j∈N e−μj t uj φj ∈ K 0 (Ω) for t ≥ 0. Observe that we have uj = u, φj K r (Ω),K −r (Ω) and αj (t) = (S(t)u, φj )L2 (Ω) = S(t)u, φj K r (Ω),K −r (Ω) = e−μj t uj . We thus have (μj αj (t))j ∈ 2 (C) for t ≥ 0 and

r

r μj |αj (t)|2 ≤ μj |uj |2 = u2K r (Ω) . S(t)u2K r (Ω) = r/2

j∈N

j∈N

As K 0 (Ω) is dense in K r (Ω) (since r ≤ 0 here), we see that S(t) can be r uniquely extended to K r (Ω) and, if u = j∈N uj φj in K (Ω), we have

−μ t r  S(t)u = j∈N e j uj φj , with convergence occurring in K (Ω). With the above results, we see that the C0 -semigroup S(t) is differentiable for t > 0. Moreover, with the estimate S(t)L (L2 (Ω),D(P0 )) ≤ C/t, for t > 0, we have the analyticity of the semigroup t → S(t); we refer to Theorem 12.19. Arguing as in the proof of Lemma 10.24, we obtain the following result. Lemma 10.28. Let r ∈ R. The bounded operator S(t) : K r (Ω) → K r (Ω) is a C0 -semigroup. It is generated by the unbounded operator (Pr , D(Pr )) on K r (Ω). We can state an equivalent version of Lemma 10.23 and Theorem 10.22. Theorem 10.29. Let r ∈ R and y 0 ∈ K r (Ω). Let also T ∈ R+ ∪ {+∞}. The function y(t) = S(t)y 0 is in C 0 ([0, T ]; K r (Ω)) ∩ C ∞ ((0, T ]; K s (Ω)),

s ∈ R,

and is such that (10.2.5) d y(t) + Pr y(t) = 0 holds in K r (Ω) for 0 < t ≤ T. dt Moreover, y(t) = S(t)y 0 is the unique solution of (10.2.5) in y(0) = y 0 and

C 0 ([0, T ]; K r (Ω)) ∩ C 1 ((0, T ]; K r (Ω)) ∩ C 0 ((0, T ]; K r+2 (Ω)). If we set yj (t) = e−tμj y 0 , φj K r (Ω),K −r (Ω) , for t ≥ 0 and j ∈ N, then (μj yj (t))j ∈ C 0 ([0, T ], 2 (C)) and

yj (t)φj , y(t) = r/2

t ≥ 0,

j∈N

with convergence in K r (Ω). We recall that [0, T ] (resp.., (0, T ]) means [0, +∞) (resp.., (0, +∞)) if T = +∞. Observe that if r, s ∈ R, then we have (10.2.6)

Psr S(t)u = S(t)Psr u,

u ∈ D(Psr ), t ≥ 0.

10.2. THE PARABOLIC SEMIGROUP

333

Above it was mentioned that S(t) is self-adjoint on L2 (Ω) by Corollary 12.25. Similarly, using L2 (Ω) as a pivot space, with Proposition 10.27 we obtain the following result. Proposition 10.30. Let s ∈ R, u ∈ K s (Ω), and v ∈ K −s (Ω). We have, for t ≥ 0, S(t)u, vK s (Ω),K −s (Ω) = u, S(t)vK s (Ω),K −s (Ω) . If moreover t > 0, then for s, r ∈ R, u ∈ K s (Ω), and v ∈ K r (Ω), we have S(t)u, vK −r (Ω),K r (Ω) = u, S(t)vK s (Ω),K −s (Ω) . The second statement makes perfect sense by Proposition 10.27. If S(t) is some semigroup on a Banach space X, for every x ∈ X, we have S(t)x → x in X as t → 0+ . Note that we do not have S(t) − IdX L (X) → 0 in general, as this is equivalent to having a bounded generator. In the present case of the parabolic semigroup, we however have S(t) − Id L (D(P0 ),L2 (Ω)) = O(t) for t > 0. This is stated in the following proposition in a more general form. Proposition 10.31. Let r, s ∈ R with 0 ≤ s ≤ 2. There exists C > 0 such that S(t) − Id L (K r+s (Ω),K r (Ω)) ≤ Cts/2 , for t ≥ 0.

(r+s)/2 uj )j ∈ Proof. Let u ∈ K r+s (Ω). Then u = j∈N uj φj with (μj

2 (C). For t > 0, we write S(t)u − u = j∈N (e−μj t − 1)uj φj . Thus, we have

r+s

(1 − e−μj t )2 μrj |uj |2 ≤ ts μj |uj |2 = ts u2K r+s , S(t)u − u2K r (Ω) = j∈N

j∈N

as 0 ≤ 1 − e−α ≤ αs/2 for α ≥ 0, as 0 ≤ s/2 ≤ 1.



Further regularity results and bounds are given by the following proposition. Proposition 10.32. Let r ∈ R. If y 0 ∈ K r (Ω) and y(t) = S(t)y 0 , then y ∈ C ([0, +∞); K r (Ω)) ∩ C ∞ ((0, +∞); K s (Ω)), s ∈ R, and y ∈ L2 (0, +∞; K r+1 (Ω)) ∩ H 1 (0, ∞; K r−1 (Ω)). Moreover, there exists C > 0 such that yL2 (0,+∞;K r+1 (Ω)) + yH 1 (0,+∞;K r−1 (Ω)) ≤ Cy 0 K r (Ω) . In particular, the equation

d dt y

+ Pr−1 y = 0 holds in L2 (0, +∞; K r−1 (Ω)).

Proof. One way to prove this result is to use the spectral representa 0 r 0 tion (10.2.3) of the semigroup. If y ∈ K (Ω) with y = j∈N yj0 φj where (μj yj0 )j∈N ∈ 2 (C), we have r/2

y(t) = S(t)y 0 =

j∈N

e−μj t yj0 φj ,

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10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

and by (10.1.12) we have

+∞ r+1 −2μj t 0 2 1 r 02 1 0 2 ∫ μj e |yj | dt = μ |y | = y K r (Ω) , y2L2 (0,+∞;K r+1 (Ω)) = 2 j∈N j j 2 j∈N 0 yielding y ∈ L2 (0, T ; K r+1 (Ω)). Alternatively, for t > 0, as y ∈ C ∞ ((0, +∞), K s (Ω)) for any s ∈ R, we can compute  d 1 d y(t) + Pr y(t), y(t) K r (Ω) = y(t)2K r (Ω) + (P0 y(t), y(t))K r (Ω) . 0= dt 2 dt For T > 0, integrating for t ∈ (0, T ), we find T 1 1 2 y(T )2K r (Ω) + ∫ (Pr y(t), y(t))K r (Ω) dt = y 0 K r (Ω) . 2 2 0 By Lemma 10.16, we write 1 1 2 y(T )2K r (Ω) + y2L2 (0,T ;K r+1 (Ω)) = y 0 K r (Ω) . 2 2 We then let T → +∞ and, by the K r -norm decay given by Proposition 10.27, we obtain the same equality as above.

Now we write, as

d dt y(t)

+ Pr y(t) ∈ K r (Ω) for t > 0,

 d 2 y2H 1 (0,+∞;K r−1 (Ω)) = y2L2 (0,+∞;K r−1 (Ω)) +  y  dt L2 (0,+∞;K r−1 (Ω))  y2L2 (0,+∞;K r+1 (Ω)) + Pr y2L2 (0,+∞;K r−1 (Ω))  y2L2 (0,+∞;K r+1 (Ω)) + Pr−1 y2L2 (0,+∞;K r−1 (Ω))  y2L2 (0,+∞;K r+1 (Ω)) by (10.1.20), which gives the second estimation from the previous one. We conclude that y ∈ H 1 (0, +∞; K r−1 (Ω)).  d dt y

Remark 10.33. By abuse of notation, one often writes that the equation + P0 y = 0 holds in L2 (0, +∞; K r−1 (Ω)).

Particular and important cases that are often used in practice are the following ones (r = 0 and r = 1). Corollary 10.34. If y 0 ∈ L2 (Ω) and y(t) = S(t)y 0 , then y ∈ C ([0, +∞); L2 (Ω)) ∩ C ∞ ((0, +∞); K s (Ω)), s ∈ R, and y ∈ L2 (0, +∞; H01 (Ω)) ∩ H 1 (0, +∞; H −1 (Ω)). Moreover, there exists C > 0 such that y(t)L2 (0,+∞;H 1 (Ω)) + y(t)H 1 (0,+∞;H −1 (Ω)) ≤ Cy 0 L2 (Ω) . 0

This implies in particular that the equation H −1 (Ω)).

d dt y+P−1 y

= 0 holds in L2 (0, +∞;

10.2. THE PARABOLIC SEMIGROUP

335

Corollary 10.35. If y 0 ∈ H01 (Ω) and y(t) = S(t)y 0 , then y ∈ C ([0, +∞); H01 (Ω)) ∩ C ∞ ((0, +∞); K s (Ω)), s ∈ R, and y ∈ L2 (0, +∞; D(P0 )) ∩ H 1 (0, ∞; L2 (Ω)). Moreover, there exists C > 0 such that yL2 (0,+∞;D(P0 )) + yH 1 (0,+∞;L2 (Ω)) ≤ Cy 0 L2 (Ω) . In particular the equation:

d dt y

+ P0 y = 0 holds in L2 ((0, +∞) × Ω).

We conclude this section with the following uniqueness result. Proposition 10.36. Let T ∈ R+ ∪ {+∞}, and let r ∈ R. If y is in C 0 ([0, T ]; K r+1 (Ω)) ∩ L2 (0, T ; K r+2 (Ω)) ∩ H 1 (0, T ; K r (Ω)) and satisfies y(0) = 0 and d y + Pr y = 0 in L2 (0, T ; K r (Ω)), dt then y = 0. We recall that [0, T ] (resp.., (0, T ]) means [0, +∞) (resp.., (0, +∞)) if T = +∞. Proof. We have Pr y ∈ L2 (0, T ; K r (Ω)) ⊂ L2 (0, T ; K r−1 (Ω)). Note d y∈ that we have Pr y = Pr−1 y ∈ C 0 ([0, T ]; K r−1 (Ω)). We thus find that dt 0 r−1 1 r−1 C ([0, T ]; K (Ω)). This implies that y ∈ C ([0, T ]; K (Ω)). As y ∈ C 1 ([0, T ]; K r−1 (Ω)) ∩ C 0 ([0, T ]; K r+1 (Ω)), the equation: d y(t) + Pr−1 y(t) = 0 dt holds in K r−1 (Ω) for 0 < t ≤ T . Since y(0) = 0, by the second part of Theorem 10.29 with r − 1 in place of r, we obtain the result.  10.2.4. Properties of the Parabolic Kernel. The parabolic kernel kt (x, x ) is defined as the kernel of the semigroup S(t) for t ≥ 0. Evidently, for t = 0, as S(t) = IdL2 (Ω) , we may say, with some abuse, that k0 (x, x ) = δ(x − x ); hence, the parabolic kernel associated with S(t) is a measure for t = 0. However, for t > 0, as we have

−μj t 0 e (y , φj )φj , y 0 ∈ L2 (Ω), S(t)y 0 = j∈N

by Lemma 10.23, where convergence occurs in L2 (Ω) but also in any space K s (Ω), s ∈ R, by Proposition 10.27. This reads 

−μj t  0  ∫ y (x )φj (x ) dx φj (x). e S(t)y 0 (x) = j∈N

Ω

336

10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

As φj ⊂ ∩s∈R K s (Ω) ⊂ ∩s∈R H s (Ω) (see Proposition 10.17), by the Sobolev imbedding theorem [2, Theorem 4.12] we find that, for k > d/2, (10.2.7)

k/2

k/2

φj L∞ (Ω)  φj H k (Ω)  φj K k (Ω) = P0 φj L2 (Ω) = μj .

Hence, we have |φj (x)φj (x )e−μj t y 0 (x )|  μkj e−μj t |y 0 (x )|. For t > 0, this function is L1 with respect to the Lebesgue measure in x in Ω (using that |Ω| < ∞) and the counting measure in j using that the sequence μj grows to +∞ sufficiently fast by the Weyl law given in (10.1.11). Then, the Fubini theorem yields  −μ t  e j φj (x)φj (x ) y 0 (x ) dx , S(t)y 0 (x) = ∫ Ω

j∈N

and we may write S(t)y 0 (x) = ∫Ω kt (x, x )y 0 (x ) dx , with the parabolic kernel given by

−μj t (10.2.8) e φj (x)φj (x ). kt (x, x ) = j∈N

Observing that the series converges uniformly with respect to x and x if t > 0, we see that kt (x, x ) is smooth, as φj ∈ C ∞ (Ω) by the Sobolev imbedding theorem [2, Theorem 4.12]. Note that we have (10.2.9)

kt (x, x ) = kt (x , x),

t > 0, x, x ∈ Ω,

as can be readily observed by the form of (10.2.8). It also follows from the fact that S(t)∗ = S(t) by Corollary 12.25. For y 0 , z 0 ∈ L2 (Ω), we have, by the Fubini theorem,   ∫∫ kt (x, x )y 0 (x)z 0 (x ) dxdx = ∫ ∫ kt (x, x )z 0 (x )dx y 0 (x)dx Ω2

Ω

Ω 0

= (y , S(t)z 0 )L2 (Ω) → (y 0 , z 0 )L2 (Ω) , as t → 0+ , using the strong convergence of the semigroup to the identity map on L2 (Ω). Taking z 0 , y 0 ∈ Cc∞ (Ω), we thus see that kt (x, x ) converges to δ(x − x ) as t → 0+ in the sense of distributions (resp., measures), by Proposition 8.38. The following result provides a Gaussian upper bound for the heat kernel. Theorem 10.37. There exist C > 0 and δ > 0 such that |kt (x, x )| ≤ Ct−d/2 eCt−

δ|x−x |2 t

,

x, x ∈ Ω.

We refer to Theorem 6.10 in [266] for a proof. Note that for the Laplace operator, that is in the case pij = δij a Gaussian upper bound can be obtained (in a sharper form) by simply using the maximum principle. We give this simple proof here.

10.2. THE PARABOLIC SEMIGROUP

337

 Proposition 10.38. semigroup gen Denote2 by pt (x, x ) the kernel of the erated by B = −Δ = 1≤j≤d Dj with domain D(B) = H 2 (Ω) ∩ H01 (Ω), that

is, the Dirichlet Laplace operator. We have |pt (x, x )| ≤ (4πt)−d/2 e−

|x−x |2 4t

.

|x−x |2

Proof. The heat kernel in Rd is given by p0,t (x, x ) = (4πt)−d/2 e− 4t if t > 0. Consider y 0 ∈ Cc∞ (Ω) such that y 0 ≥ 0. We also denote by y 0 its zero extension to Rd . We consider the following parabolic problems: ⎧  ⎪ ⎨∂t y − Δy = 0 in (0, T ) × Ω, ∂t z − Δz = 0 in (0, T ) × Rd , y=0 on (0, T ) × ∂Ω, ⎪ in Rd . z|t=0 = y 0 ⎩ in Ω, y|t=0 = y 0 For t > 0, the solutions z and y are smooth functions given by y(t, x) = (pt (x, .), y 0 (.))L2 (Ω) and z(t, x)=(p0,t (x, .), y 0 (.))L2 (Rd ) =(p0,t (x, .), y 0 (.))L2 (Ω) . In particular z(t, x) ≥ 0 if x ∈ ∂Ω. Thus z − y ≥ 0 in (0, T )× ∂Ω. The difference of the two solutions thus satisfies a parabolic problem of the following form: ⎧ ⎪ ⎨∂t (z − y) − Δ(z − y) = 0 in (0, T ) × Ω, z−y ≥0 on (0, T ) × ∂Ω, ⎪ ⎩ in Ω. (z − y)|t=0 = 0 The maximum principle of Theorem 10.52 gives z − y ≥ 0 in (0, T ) × Ω. If y 0 ∈ Cc∞ (Ω) with y 0 ≥ 0, it follows that (p0,t (x, .), y 0 (.))L2 (Ω) ≥ (pt (x, .), y 0 (.))L2 (Ω) ,

t > 0. 

This yields the result. We also have a lower bound for the heat kernel on the diagonal.

Theorem 10.39. Let O  Ω, and let T > 0. There exists C > 0 such that |kt (x, x)| ≥ Ct−d/2 ,

x ∈ O, t ∈ (0, T ].

Proof. We adapt the proof of Proposition 7.28 in [266]. We let φ1 be the first eigenfunction for the operator P0 , associated with the eigenvalue μ1 > 0. We can choose φ1 > 0 in Ω by Theorem 8.38 in [161]. We then have φ1 ≥ C0 > 0 in O, as φ1 is smooth. We also have φ1 ∈ L∞ (Ω) by (10.2.7). For α > 0 and x ∈ O, by Theorem 10.37, we compute   δ|x−x |2   kt (x, x )φ1 (x )dx   t−d/2 ∫ e− t dx ∫  Ω\B(x,αt1/2 )

Ω\B(x,αt1/2 )

 e−δα

2 /2

 e−δα

2 /2

t−d/2

∫

e−

δ|x−x |2 2t

Ω\B(x,αt1/2 )

t−d/2 ∫ e− Rd

δ|x−x |2 2t

dx .

dx

338

10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

As t−d/2 ∫Rd e−

δ|x−x |2 2t

  

dx ≤ Cδ , we obtain ∫

Ω\B(x,αt1/2 )

 2  kt (x, x )φ1 (x )dx  ≤ C1 e−δα /2 .

We now write, for x ∈ O, ∫

B(x,αt1/2 )

kt (x, x )φ1 (x )dx = e−tμ1 φ1 (x) −

∫

Ω\B(x,αt1/2 )

≥ C0 e−T μ1 − C1 e−δα

2 /2

kt (x, x )φ1 (x )dx

≥ C2 > 0,

for α chosen sufficiently large. We have, by the semigroup property and the symmetry of the kernel (10.2.9), k2t (x, x) = ∫ kt (x, x )kt (x , x) dx = ∫ kt (x, x )2 dx ≥ Ω

Ω

∫

B(x,αt1/2 )

kt (x, x )2 dx .

We now write, by the Cauchy–Schwarz inequality,  2 0 < C22 ≤ ∫ kt (x, x )φ1 (x )dx B(x,αt1/2 )

≤

∫

B(x,αt1/2 )

kt (x, x )2 dx

∫

B(x,αt1/2 )

φ1 (x )2 dx

 |B(x, αt1/2 )| k2t (x, x), 

which yields the result.

We now consider some regularity aspects of the heat kernel. The bounds we obtain below are far from optimal but will suffice to our purpose in Chap. 7 in the understanding of the obstruction to exact controllability for the heat equation in general. / ω. We then consider B 0 = Let ω ⊂ Ω, such that ω = Ω, and let x0 ∈ 0 0 0 B(x , r0 ) with r0 > 0 such that d = dist(B , ω) > 0 and B 0 ⊂ Ω. We have the following result. Proposition 10.40. Let T > 0, and let α ∈ Nd+1 . There exist Cα , Cα > 0 such that 

α kt (x, x )| ≤ Cα e−Cα /t , |∂t,x

x ∈ B 0 , x ∈ ω, t ∈ (0, T ].

Proof. Here, for convenience we set L(t, x, x ) = kt (x, x ). We consider = B(x0 , r1 ) with r1 > r0 and B 1 fulfilling the same properties as B 0 . By Lemma 10.41 below, we have for all N ∈ N, B1

(10.2.10)



L(., ., x )H N (It ×B 1 ) ≤ KN t−KN ,

x ∈ ω,

 > 0 and for I = (t/2, 3t), with t ∈ (0, T ]. for some KN , KN t From Theorem 10.37, as B 0 is bounded, we may write, for C, C  > 0,

(10.2.11)



L(., ., x )L2 (It ×B 1 )) ≤ Ce−C /t ,

x ∈ Ω.

10.2. THE PARABOLIC SEMIGROUP

339

If  ∈ N, an interpolation of (10.2.10)–(10.2.10) (see e.g. [74, 236]) with N > , say N = 2, yields, for some C , C  > 0, (10.2.12)



L(., ., x )H (It ×B 1 ) ≤ C e−C /t ,

x ∈ ω.

This yields for any α ∈ Nd+1 , for some Cα , Cα > 0, (10.2.13)



α ∂t,x L(., ., x )L∞ ((t,2t)×B 0 ) ≤ Cα e−Cα /t ,

x ∈ ω,

which yields the conclusion. If fact, with ψ1 ∈ Cc∞ (1/2, 3) such that ψ1 ≡ 1 in a neighborhood of [1, 2], and ψ2 ∈ Cc∞ (B 1 ) such that ψ2 ≡ 1 in a neighborhood of B 0 , if we set ψt (s, x) = ψ1 (s/t)ψ2 (x), we have, for n ∈ N fixed, with n > (d + 1)/2, α α L(., ., x )L∞ ((t,2t)×B 0 )  ψt (., .)∂s,x L(., ., x )L∞ (Rd+1 ) ∂t,x α  ψt (., .)∂s,x L(., ., x )H n (Rd+1 ) α  t−n ∂s,x L(., ., x )H n (I

t ×B

1)

,

for x ∈ ω, by Sobolev injection yielding (10.2.13) using (10.2.12) with  = n + |α|.  Lemma 10.41. Set L(t, x, x ) = kt (x, x ). Let It = (t/2, 3t), and let = B(x0 , r1 ) with r1 > 0 such that dist(B 1 , ω) > 0 and B 1 ⊂ Ω. Let  > 0 such that T > 0, for N ∈ N, there exist KN , KN

B1



L(., ., x )H N (It ×B 1 ) ≤ KN t−KN ,

x ∈ ω,

t ∈ (0, T ].

Proof. We write, for , n ∈ N  n+ −μ t  |∂tn P0 L(t, x, x )| =  μj e j φj (x)φj (x ) j∈N



j∈N

μn+ +k e−μj t j

using (10.2.7), with k > d/2. Using the Weyl law (10.1.11), we obtain

n+ +k+d −2 −μj t (10.2.14) μj j e |∂tn P0 L(t, x, x )|  j∈N



t−(n+ +k+d) j −2  t−(n+ +k+d) ,

j∈N

as μM e−μt  t−M . We consider B 2 = B(x0 , r2 ) with r2 > r1 and B 2 fulfilling the same properties as B 1 . By Proposition 2.35, adapted to standard pseudo-differential operators, that is, fixing, e.g., τ = 1, we have L(., ., x )H 2N (It ×B 1 )  (P0 + Dt2 )N )L(., ., x )L2 (It ×B 2 ) + L(., ., x )L2 (It ×B 2 ) .

This estimate and (10.2.14) yield the result as It × B 2 is bounded.



340

10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

10.3. The Nonhomogeneous Parabolic Cauchy Problem We now consider the nonhomogeneous parabolic equation: d y + P0 y = f, y|t=0 = y 0 . (10.3.1) dt The mild solution (see Definition 12.21) is given by the Duhamel formula: t

y(t) = S(t)y 0 + ∫ S(t − σ)f (σ)dσ, 0

if f ∈ L1loc (0, ∞; D(P0 )). The second term is called the Duhamel term. 10.3.1. Properties of the Duhamel Term. Let r ∈ R. For f ∈ L2 (0, T ; K r (Ω)), with the properties of the semigroup S(t) on K r (Ω) we can define t

Ψr (f )(t) = ∫ S(t − σ)f (σ) dσ,

(10.3.2)

t ≥ 0.

0

Theorem 10.42. Let T ∈ R+ ∪ {+∞}. The map Ψr maps linearly and continuously L2 (0, T ; K r (Ω)) into C 0 ([0, T ]; K r+1 (Ω)) ∩ L2 (0, T ; K r+2 (Ω)) ∩ H 1 (0, T ; K r (Ω)). If f ∈ L2 (0, T ; K r (Ω)), then d Ψr (f ) + Pr Ψr (f ) = f. dt Proof. We set G = L∞ (0, T ; K r+1 (Ω)) ∩ L2 (0, T ; K r+2 (Ω)) ∩ H 1 (0, T ; equipped with the norm:

K r (Ω))

z → zL∞ (0,T ;K r+1 (Ω)) + zL2 (0,T ;K r+2 (Ω)) + zH 1 (0,T ;K r (Ω)) . We first consider f ∈ C ([0, T ]; K r (Ω)), as this space is dense in L2 (0, T ; K r (Ω)). Then, for all t ∈ [0, T ] we have f (t) = j∈N fj (t)φj with fj (t) = f (t), φj K r (Ω),K −r (Ω) and (μj fj (t))j ∈ C ([0, T ]; 2 (C)). For N ∈ N, we

N = f in L2 (0, T ; K r (Ω)). set f N (t) = N j=0 fj (t)φj and we have limN →∞ f Hence, the space N  

r/2 gj (t)φj ; N ∈ N, (μj gj (t))j ⊂ C ([0, T ]; 2 (C)) E := g = r/2

j=0

is dense in

L2 (0, T ; K r (Ω)).

Note that we have  C ([0, T ]; K s (Ω)), E ⊂ s∈R

recalling that φj ∈ ∩s∈R (see Sect. 10.1.3). We consider g = φj ∈ E , and we set, for any s ∈ R, K s (Ω) t

N

0

j=0 0

z(t) = Ψs (g)(t) = ∫ S(t − σ)g(σ)dσ =

t

N

j=0 gj (t)

∫ e(σ−t)μj gj (σ) dσ φj .

10.3. THE NONHOMOGENEOUS PARABOLIC CAUCHY PROBLEM

341

With the uniform continuity property of Proposition 10.31, we have S(t + h) − S(t) = O(h) in L (K s+2 (Ω), K s (Ω)) for any s ∈ R. We then see that z ∈ C ∞ ([0, T ]; K s (Ω)) for any s ∈ R. Moreover, we find d z(t) + Ps z(t) = g(t), dt

t ≥ 0,

for any s ∈ R. We may thus write 1d z(t)2K r+1 (Ω) + z(t)2K r+2 (Ω) 2 dt  d z(t), z(t) K r+1 (Ω) + (Pr+1 z(t), z(t))K r+1 (Ω) = dt = (g(t), z(t))K r+1 (Ω) ≤ g(t)K r (Ω) z(t)K r+2 (Ω) , yielding, after integration with respect to time t, using that z(0) = 0, t 1 z(t)2K r+1 (Ω) + z2L2 (0,t;K r+2 (Ω)) ≤ ∫ g(σ)K r (Ω) z(σ)K r+2 (Ω) dσ 2 0 ≤ gL2 (0,t;K r (Ω)) zL2 (0,t;K r+2 (Ω)) , t ∈ [0, T ].

With the Young inequality, we obtain (10.3.3)

zL∞ (0,T ;K r+1 (Ω)) + zL2 (0,T ;K r+2 (Ω))  gL2 (0,T ;K r (Ω)) .

From the equation satisfied by z, (10.3.4)

d z(t) + Pr z(t) = g(t), dt

t ≥ 0,

we also have zH 1 (0,T ;K r (Ω))  gL2 (0,T ;K r (Ω)) . From the density of E in L2 (0, T ; K r (Ω)), these estimates show that Ψr maps L2 (0, T ; K r (Ω)) continuously into G . If (f N )N ⊂ E converges to f ∈ L2 (0, T ; K r (Ω)), then Ψr (f N ) converges to Ψr (f ) in G . From (10.3.4), one finds that d Ψr (f ) + Pr Ψr (f ) = f in L2 (0, T ; K r (Ω)). dt As Ψr (f N ) ∈ C 0 ([0, T ]; K r+1 (Ω)), the estimate in (10.3.3) shows that Ψr (f )  ∈ C 0 ([0, T ]; K r+1 (Ω)), by uniform convergence. We state the regularity result in the case r = 0. Corollary 10.43. Let T ∈ R+ ∪ {+∞}. The map Ψ0 given in (10.3.2) in the case r = 0 maps linearly and continuously L2 ((0, T ) × Ω) into C 0 ([0, T ]; H01 (Ω)) ∩ L2 (0, T ; H 2 (Ω) ∩ H01 (Ω)) ∩ H 1 (0, T ; L2 (Ω)).

342

10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

10.3.2. Abstract Solutions of the Nonhomogeneous Semigroup Equations. Observe that the regularity of the Duhamel term Ψr (f ) for f ∈ L2 (0, T ; K r (Ω)) given by Theorem 10.42 coincides with that of the free evolution term S(t)y 0 if y 0 ∈ K r+1 (Ω) according to Theorem 10.29 and Proposition 10.32: S(t)y 0 ∈ C ([0, +∞); K r+1 (Ω)) ∩ L2 (0, +∞; K r+2 (Ω)) ∩ H 1 (0, ∞; K r (Ω)). However, the Duhamel term does not exhibit the same degree of regulariza/ C ∞ ((0, +∞); K s (Ω)). tion as does the term S(t)y 0 . In general, Ψr (f ) ∈ This observation gives a natural regularity level for both the initial condition y 0 and the source term to state an existence and uniqueness result d y + Pr y = f , for some r ∈ R. for a solution of the equation dt Theorem 10.44. Let T ∈ R+ ∪{+∞} and r ∈ R. Let f ∈ L2 (0, T ; K r (Ω)) and y 0 ∈ K r+1 (Ω). There exists a unique function y ∈ C 0 ([0, T ]; K r+1 (Ω))∩ L2 (0, T ; K r+2 (Ω)) ∩ H 1 (0, T ; K r (Ω)) that is solution of the parabolic equation: d y + Pr y = f dt in L2 (0, T ; K r (Ω)) 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 (Ω)) + yL2 (0,T ;K r+2 (Ω)) +  y  dt L2 (0,T ;K r (Ω))   ≤ C y 0 K r+1 (Ω) + f L2 (0,T ;K r (Ω)) . 

Remark 10.45. Let s ∈ R and s > 0. If y 0 ∈ K s+s +1 (Ω) and f ∈  2 L ([0, T ]; K s+s (Ω)), then Theorem 10.44 applies both in the cases r = s and r = s + s . Uniqueness shows that the two obtained solutions coincide. If fact, both are given by the same Duhamel formula. Proof of Theorem 10.44. First, we address uniqueness. Assume that there are two solutions in C 0 ([0, T ]; K r+1 (Ω))∩L2 (0, T ; K r+2 (Ω))∩H 1 (0, T ; K r (Ω)). Then, their difference z(t) lies in that space and is solution to d 2 r dt z + Pr z = 0 in L (0, T ; K (Ω)) and z(0) = 0. By Proposition 10.36, we find that z = 0. Second, we address existence. If we set y(t) = S(t)y 0 + Ψr (f )(t), we see by Theorem 10.29, Proposition 10.32, and Theorem 10.42, using the linearity of the equation, that y ∈ C 0 ([0, T ]; K r+1 (Ω)) ∩ L2 (0, T ; K r+2 (Ω)) ∩ H 1 (0, T ; K r (Ω))

10.3. THE NONHOMOGENEOUS PARABOLIC CAUCHY PROBLEM

343

and that d y + Pr y = f dt holds in L2 (0, T ; K r (Ω)).



10.3.3. Strong Solutions. In general, one calls a strong solution a function y ∈ C 0 ([0, T ]; H01 (Ω))∩L2 (0, T ; D(P0 ))∩H 1 (0, T ; L2 (Ω)) that solves the equation: d y + P0 y = f dt 2 in  Lr ((0, T ) × Ω). Its definition does not require the use of the Sobolev scale K (Ω) r∈R of Sect. 10.1.3. All terms in the equation are functions in Ω. The uniqueness and the existence of such strong solution under regularity assumptions for the initial condition y 0 and the source term f are given by Theorem 10.44 in the case r = 0, which we write explicitly in the following corollary.

Corollary 10.46 (Strong Solutions—First Version). Let T ∈ R+ ∪ {+∞}. Let f ∈ L2 ((0, T ) × Ω) and y 0 ∈ H01 (Ω). There exists a unique function y ∈ C 0 ([0, T ]; H01 (Ω)) ∩ L2 (0, T ; D(P0 )) ∩ H 1 (0, T ; L2 (Ω)) that is solution of the parabolic equation: d y + P0 y = f dt in L2 ((0, T ) × Ω) 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 ;H 1 (Ω)) + yL2 (0,T ;D(P0 )) +  y  0 dt L2 ((0,T )×Ω)   ≤ C y 0 H 1 (Ω) + f L2 ((0,T )×Ω) . 0

The term “strong solution” is sometimes used for more regular solutions, namely solutions that lie in C 0 ([0, T ]; D(P0 )). They are given by Theorem 10.44 in the case r = 1. Corollary 10.47 (Strong Solutions—Second Version). Let T ∈ R+ ∪ {+∞}. Let f ∈ L2 (0, T ; H01 (Ω)) and y 0 ∈ D(P0 ). There exists a unique function y ∈ C 0 ([0, T ]; D(P0 )) ∩ L2 (0, T ; K 3 (Ω)) ∩ H 1 (0, T ; H01 (Ω)) that is solution of the parabolic equation: d d y + P0 y = y + P1 y = f dt dt

344

10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

in L2 (0, T ; H01 (Ω)) 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 ;D(P0 )) + yL2 (0,T ;K 3 (Ω)) +  y  dt L2 (0,T ;H01 (Ω))   ≤ C y 0 D(P0 ) + f L2 (0,T ;H 1 (Ω)) . 0

If one further assumes that f ∈ C 0 ((0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)), since y ∈ C 0 ([0, T ]; D(P0 )) the semigroup equation further gives that y ∈ C 1 ((0, T ]; L2 (Ω)). We then obtain a classical solution in the sense introduced in Sect. 12.3 for an abstract nonhomogeneous semigroup equation. 10.3.4. Weak Solutions. For a regularity lower than that of strong solutions as introduced in Sect. 10.3.3, with y 0 ∈ L2 (Ω) and f ∈ L2 (0, T ; H −1 (Ω)), Theorem 10.44 for r = −1 yields the existence and uniqueness of solution in C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) ∩ H 1 (0, T ; H −1 (Ω)). One is often inclined to use a weak formulation to characterize these solutions. Definition 10.48. Let T ∈ R+ ∪ {+∞}. Let y 0 ∈ L2 (Ω) and f ∈ One says that y ∈ C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) is a weak solution to the parabolic equation:

L2 (0, T ; H −1 (Ω)).

d y + P0 y = f, dt

y(0) = y 0 ,

if we have (y(t), ψ)L2 (Ω) + (y, ψ)L2 (0,t;H01 (Ω)) t

= (y 0 , ψ)L2 (Ω) + ∫ f (σ), ψH −1 (Ω),H01 (Ω) dσ, 0

for all ψ ∈ H01 (Ω) and for all t ∈ [0, T ]. We recall that the H01 -norm is given by (10.1.17), yielding t

1/2

1/2

(y, ψ)L2 (0,t;H01 (Ω)) = ∫ (P0 y(σ), P0 ψ)L2 (Ω) dσ 0 t

= ∫ P0 y(σ), ψH −1 (Ω),H01 (Ω) dσ 0 t

= ∫ y(σ), P0 ψH −1 (Ω),H01 (Ω) dσ. 0

10.3. THE NONHOMOGENEOUS PARABOLIC CAUCHY PROBLEM

345

In fact, it is equivalent if one chooses ψ ∈ D(P0 ) yielding the form: (y(t), ψ)L2 (Ω) + (y, P0 ψ)L2 ((0,t)×Ω) t

= (y 0 , ψ)L2 (Ω) + ∫ f (σ), ψH −1 (Ω),H01 (Ω) dσ, 0

for all t ∈ [0, T ]. Theorem 10.49. Let T ∈ R+ ∪ {+∞}. Let y 0 ∈ L2 (Ω) and f ∈ There exists a unique weak solution y to the parabolic equation:

L2 (0, T ; H −1 (Ω)).

d y(0) = y 0 , y + P0 y = f, dt in the sense of Definition 10.48. It coincides with the solution of the semigroup equation: d y + P−1 y = f, y(0) = y 0 , dt given by Theorem 10.44 in the case r = −1. In particular we have y ∈ C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) ∩ H 1 (0, T ; H −1 (Ω)). By Remark 10.45, we see that a weak solution associated with data with the following regularity, y 0 ∈ H01 (Ω) and f ∈ L2 ((0, T ) × Ω), is in fact a strong solution as given by Corollary 10.46. Proof. First, we address uniqueness. Assume that there are two solutions in C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)). Then, their difference z(t) lies in that space and is solution to (z(t), ψ)L2 (Ω) + (z, ψ)L2 (0,t;H01 (Ω)) = 0,

ψ ∈ H01 (Ω), t ∈ [0, T ],

which we write t

0 = (z(t), ψ)L2 (Ω) + ∫ P−1 z(σ), ψH −1 (Ω),H01 (Ω) dσ %

0

t

= z(t) + ∫ P−1 z(σ) dσ, ψ

&

0

H −1 (Ω),H01 (Ω)

.

As a result, we have, in H −1 (Ω), t

z(t) + ∫ P−1 z(σ) = 0, 0

0 ≤ t ≤ T.

d z+ As P−1 z ∈ L2 (0, T ; H −1 (Ω)), we find that z ∈ H 1 (0, T ; H −1 (Ω)) and dt 2 −1 P−1 z = 0 holds in L (0, T ; H (Ω)). With Proposition 10.36, we conclude that z = 0. Second, we address existence. Let

y ∈ C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) ∩ H 1 (0, T ; H −1 (Ω))

346

10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

d be the solution to dt y + P−1 y = f in L2 (0, T ; H −1 (Ω)) and y(0) = y 0 , as given by Theorem 10.44. We then see that d y(t), ψ H −1 (Ω),H 1 (Ω) + P−1 y(t), ψH −1 (Ω),H01 (Ω) = f (t), ψH −1 (Ω),H01 (Ω) 0 dt 2 holds in L (0, T ). As y ∈ L2 (0, T ; H01 (Ω)), we observe that for almost every t ∈ (0, T )

P−1 y(t), ψH −1 (Ω),H01 (Ω) = (y(t), ψ)H01 (Ω) . d ∈ H 1 (0, T ) and dt y, ψH −1 (Ω),H01 (Ω) Since y, ψH −1 (Ω),H01 (Ω) d = dt y(t), ψ H −1 (Ω),H 1 (Ω) , we find that, for all t ∈ (0, T ), 0

t

y(t), ψH −1 (Ω),H01 (Ω) − y(0), ψH −1 (Ω),H01 (Ω) + ∫ (y(σ), ψ)H01 (Ω) dσ 0 t

= ∫ f (σ), ψH −1 (Ω),H01 (Ω) dσ. 0

C ([0, T ], L2 (Ω)),

we find that y(t), ψH −1 (Ω),H01 (Ω) = Since y(t) ∈ (y(t), ψ)L2 (Ω) , implying that y is a weak solution in the sense of Definition 10.48.  We observe that one can simply use solutions in the sense of distributions to define weak solutions. Proposition 10.50. Let T ∈ R+ ∪ {+∞}. Let y 0 ∈ L2 (Ω) and f ∈ There exists a unique y ∈ C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) such that

L2 (0, T ; H −1 (Ω)). (10.3.5)

∂t y + P0 y = f in D  ((0, T ) × Ω),

y(0) = y 0 .

It coincides with the unique solution of the semigroup equation: d y + P−1 y = f, y(0) = y 0 , dt given by Theorem 10.44 (in the case r = −1) and thus with the unique weak solution given in Theorem 10.49. Note that in (10.3.5) the occurrence of the operator P0 acting on y in the sense of distribution should not be confused with the unbounded operator P0 . Proof. We first treat uniqueness. Let y ∈ C 0 ([0, T ]; L2 (Ω))∩L2 (0, T ; H01 (Ω)) be such that d y + P0 y = 0 in D  ((0, T ) × Ω), y(0) = 0. dt As we have y ∈ L2 (0, T ; H01 (Ω)), we have P0 y = P−1 y ∈ L2 (0, T ; H −1 (Ω)), yielding ∂t y ∈ L2 (0, T ; H −1 (Ω)) and thus y ∈ H 1 (0, T ; H −1 (Ω)). Thus d dt y + P−1 y = 0. As y|t=0 = 0, the uniqueness part of Theorem 10.44 in the case r = −1 gives y ≡ 0.

10.4. ELEMENTARY FORM OF THE MAXIMUM PRINCIPLE

347

d Conversely, if y is the solution given by Theorem 10.44, we have dt y+ 2 −1 P−1 y = f ∈ L (0, T ; H (Ω)). As P−1 y(t) = P0 y(t) for almost all t ∈ (0, T ),  we have ∂t y + P0 y = f in D  ((0, T ) × Ω).

Weak solutions are also called solutions by transposition because of the alternative formulation given in the following proposition. Proposition 10.51. Let T ∈ R+ ∪ {+∞}. Let y 0 ∈ L2 (Ω) and f ∈ L2 (0, T ; H −1 (Ω)). There exists a unique y ∈ C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) such that t

(10.3.6) (y(t), ϕ(t))L2 (Ω) + ∫ y(σ), (−∂σ + P−1 )ϕ(σ)H01 (Ω),H −1 (Ω) dσ 0

t

= (y 0 , ϕ(0))L2 (Ω) + ∫ f (σ), ϕ(σ)H −1 (Ω),H01 (Ω) dσ, 0

C 0 ([0, T ]; L2 (Ω))∩L2 (0, T ; H01 (Ω))∩H 1 (0, T ; H −1 (Ω))

and for all for all ϕ ∈ t ∈ [0, T ]. It coincides with the unique solution of the semigroup equation: d y + P−1 y = f, y(0) = y 0 , dt given by Theorem 10.44 (in the case r = −1) and thus with the unique weak solution given in Theorem 10.49. Proof. First, we treat uniqueness. Let y ∈ C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; be such that (10.3.6) holds. But choosing ϕ constant with respect to t, we find that the property of Definition 10.48 is fulfilled. Hence, y is the weak solution given in Theorem 10.49. Conversely, let y be the solution given by Theorem 10.44 in the case r = −1. We then see that d y(t), ϕ(t) H −1 (Ω),H 1 (Ω) + P−1 y(t), ϕ(t)H −1 (Ω),H01 (Ω) 0 dt = f (t), ϕ(t)H −1 (Ω),H01 (Ω) H01 (Ω))

holds in L2 (0, T ). As the identity: d y(t), ϕ(t) H −1 (Ω),H 1 (Ω) 0 dt  d d y(t), ϕ(t) L2 (Ω),L2 (Ω) − y(t), ϕ(t) H 1 (Ω),H −1 (Ω) = 0 dt dt holds in L2 (0, T ). We see that an adaptation of the proof of Theorem 10.49 shows that y is indeed a solution of (10.3.6).  10.4. Elementary Form of the Maximum Principle For heat-like equations of the form: ∂t y + P0 y = f,

y|t=0 = y 0 ,

y|(0,T )×∂Ω = g,

348

10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

where P0 is as in (10.1.1), the (weak) maximum principle roughly states that for  f ≥ 0,the maximum of a solution is reached at the “parabolic” boundary {0} × Ω ∪ [0, T ] × ∂Ω. Consequently nonpositive data yield nonpositive solutions. Here, we only need the following two results. Theorem 10.52. Let y 0 ∈ L2 (Ω), and let y ∈ C ([0, T ], L2 (Ω)) be the weak solution y(t) = S(t)y 0 to the homogeneous parabolic equation with Dirichlet data: ∂t y + P0 y = 0,

y|t=0 = y 0 ,

y|(0,T )×∂Ω = 0,

given by Theorem 10.49. We then have min(0, inf y 0 ) ≤ y(t, x) ≤ max(0, sup y 0 ), Ω

Ω

t ∈ [0, T ], x ∈ Ω a.e.

We refer to [90, Theorem 10.3] for a proof. Theorem 10.53. Let y ∈ C ([0, T ] × Ω) ∩ C 1 ((0, T ); C (Ω)) ∩ C ((0, T ); C 2 (Ω)) be solution to ∂t y + P0 y ≤ 0,

(t, x) ∈ (0, T ) × Ω.

We then have max y ≤ max y,

[0,T ]×Ω

Z

 where Z = {0} × Ω ∪ [0, T ] × ∂Ω. 

We refer to [90, Theorem 10.6] for a proof. 10.5. The Dirichlet Lifting Map We introduce the following subspace of L2 (Ω): W(Ω) = {u ∈ L2 (Ω); P0 u ∈ L2 (Ω)}. Here, the action of P0 on u is to be understood in the sense of distributions and not as that of the unbounded operator (P0 , D(P0 )) on L2 (Ω) or one of its extension or restriction (Pr , D(Pr )) on K r (Ω). In particular, there is no a priori assumption made on the traces of functions in W(Ω). One can readily check that W(Ω) is a Hilbert space if equipped with the inner product: (u, v)W(Ω) = (u, v)L2 (Ω) + (P0 u, P0 v)L2 (Ω) ,

u, v ∈ W(Ω),

and the associated norm: u2W(Ω) = u2L2 (Ω) + P0 u2L2 (Ω) ,

u ∈ W(Ω).

The following lemma shows that functions in W(Ω) admit traces on the boundary ∂Ω.

10.5. THE DIRICHLET LIFTING MAP

349

Lemma 10.54. The Dirichlet trace map γ D : u → u|∂Ω and the Neumann

ij trace map γ N : u → ∂ν u|∂Ω = 1≤i,j≤d νi p ∂j u|∂Ω , both well-defined on H 2 (Ω), admit a unique extension to W(Ω) that we still denote by γ D and γ N . If u ∈ W(Ω), we have γ D (u) ∈ H −1/2 (∂Ω)

and γ N (u) ∈ H −3/2 (∂Ω),

and the linear maps γ D : W(Ω) → H −1/2 (∂Ω) and γ N : W(Ω) → H −3/2 (∂Ω) are both bounded. Finally, the Leibniz rule is compatible with these trace extensions, that is, for a ∈ C ∞ (Ω), we have γ N (au) = γ D (a)γ N (u) + γ N (a)γ D (u). A proof of this result requires a proper definition of Sobolev spaces on a hypersurface. This is, for instance, done in Chap. 18 in Volume 2 for a Riemannian manifold. In fact, a hypersurface of Rd is a Riemannian manifold with the metric induced by the Euclidean metric. The counterpart result is Lemma 18.32 in Volume 2, and in Sect. 18.9 of Volume 2, we show how results obtained for the Laplace–Beltrami operator on a Riemannian manifold can be transferred to operators such as P0 on an open set of Rd . All the remaining results on this section are in fact proven in Sect. 18.6.3 in Volume 2 in a more geometrical setting and can be adapted to the present case (see Sect. 18.9 in Volume 2). We are interested in defining a map that acts as a solution operator for the following elliptic problem: P0 u = 0 in Ω

and γ D (u) = g,

for some g ∈ H −1/2 (Ω). The Hilbert space W(Ω) appears as a natural space for solutions of this problem. In fact, we have the following result. Proposition 10.55. Let g ∈ H −1/2 (∂Ω). There exists a unique u ∈ W(Ω) such that P0 u = 0 and γ D (u) = g. Moreover, for all ψ ∈ H 2 (Ω) ∩ H01 (Ω) we have (10.5.1)

(u, P0 ψ)L2 (Ω) + g, γ N (ψ)H −1/2 (∂Ω),H 1/2 (∂Ω) = 0.

Definition 10.56 (Dirichlet Lifting Map). The bounded map D : h → u from H −1/2 (∂Ω) into W(Ω) that yields the unique u ∈ L2 (Ω) such that P0 u = 0 and γ D (u) = h is called the Dirichlet lifting map.

Using (10.5.1) with ψ = φj , j ∈ N, we find that u = Dh = j∈N uj φj with N uj = −μ−1 j h, γ (φj )H −1/2 (∂Ω),H 1/2 (∂Ω) .

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10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

If g is more regular than H −1/2 , then additional regularity is transferred to D(g), with consistency with the trace formula.3 Proposition 10.57. Let m ∈ N. If g ∈ H m−1/2 (∂Ω), then D(g) ∈ H m (Ω). Moreover, for some C > 0, we have (10.5.2)

D(g)H m (Ω) ≤ C|g|H m−1/2 (∂Ω) ,

g ∈ H m−1/2 (∂Ω).

We also consider the following subspace of L2 (Ω): (10.5.3)

W−1 (Ω) = {u ∈ L2 (Ω); P0 u ∈ H −1 (Ω)}.

We have W(Ω) ∪ H 1 (Ω) ⊂ W−1 (Ω). The inner product: (u, v)W−1 (Ω) = (u, v)L2 (Ω) + (P0 u, P0 v)H −1 (Ω) ,

u, v ∈ W−1 (Ω),

and the associated norm: u2W−1 (Ω) = u2L2 (Ω) + P0 u2H −1 (Ω) ,

u ∈ W−1 (Ω)

yield a Hilbert space structure on W−1 (Ω). Lemma 10.58. The Dirichlet trace map γ D : u → u|∂Ω well-defined on H 1 (Ω) admits a unique extension to W−1 (Ω) that we still denote by γ D . If u ∈ W−1 (Ω), we have γ D (u) ∈ H −1/2 (∂Ω) and the linear map γ D : W−1 (Ω) → H −1/2 (∂Ω) is bounded. For any w ∈ H 2 (Ω) ∩ H01 (Ω), we have P0 u, wH −1 (Ω),H01 (Ω) = (u, P0 w)L2 (Ω) + γ D (u), γ N (w)H −1/2 (∂Ω),H 1/2 (∂Ω) . The previous result allows one to fully treat nonhomogeneous Dirichlet problems for the elliptic operator P0 . Theorem 10.59. Let m ∈ N. The map: D

L : H m+1 (Ω) → H m−1 (Ω) ⊕ H m+1/2 (∂Ω)   u → P0 u, γ D (u)

is an isomorphism. Moreover, the map: W−1 (Ω) → H −1 (Ω) ⊕ H −1/2 (∂Ω)   u → P0 u, γ D (u) is an isomorphism.

3The reader is referred to standard texts; the trace formula is, for instance, proved in Theorem 18.25 in Volume 2.

10.6. PARABOLIC EQUATION WITH DIRICHLET BOUNDARY DATA

351

10.6. Parabolic Equation with Dirichlet Boundary Data In this section we are interested in solving a parabolic equation of the form: d y(t) + P0 y(t) = 0 in (0, T ) × Ω, (10.6.1) y(0) = y 0 in Ω, dt for some T ∈ R+ ∪ {+∞}, in addition with the boundary condition: y|(0,T )×∂Ω = g,

(10.6.2)

for some function g defined in (0, T ) × ∂Ω. To properly formulate the type of solutions we seek, we first assume that the solution is sufficiently regular for all terms in the equations to make sense in a classical way. Choosing ψ ∈ C ∞ (Ω) such that ψ|∂Ω = 0, we compute with an integration by parts  d y(t) + P0 y(t), ψ L2 (Ω) dt

d = (y(t), ψ)L2 (Ω) + (pij Di y(t), Dj ψ)L2 (Ω) , dt 1≤i,j≤d

0=

using that ψ vanishes at the boundary. A second integration by parts yields d 0 = (y(t), ψ)L2 (Ω) + (y(t), P0 ψ)L2 (Ω) + (g(t), ∂ν ψ|∂Ω )L2 (∂Ω) , dt where we have used the Dirichlet boundary condition for y(t). We recall that ∂ν ψ|∂Ω = 1≤i,j≤d νi pij ∂j ψ|∂Ω , where ν = (ν1 , . . . , νd ) is the unitary outward-pointing (co-)normal vector to ∂Ω. If we now integrate with respect to the time variable, we find, for 0 ≤ t ≤ T , t

0 = (y(t), ψ)L2 (Ω) − (y 0 , ψ)L2 (Ω) + ∫ (y(σ), P0 ψ)L2 (Ω) dσ 0

t

+ ∫ (g(t), ∂ν ψ|∂Ω )L2 (∂Ω) dσ. 0

Note that all terms in this equation make in fact sense if ψ ∈ H 2 (Ω)∩H01 (Ω),     y ∈ C 0 [0, T ]; H −1 (Ω) ∩ L2 0, T ; L2 (Ω) , and g ∈ L2 (0, T ; H −1/2 (∂Ω)). With this formal derivation, we are led to define the following notion of weak solutions. Definition 10.60. Let T ∈ R+ ∪ {+∞}, y 0 ∈ H −1 (Ω),  and2 g ∈ 2 −1/2 0 −1 2 (∂Ω)). A function y ∈ C [0, T ]; H (Ω) ∩ L 0, T ; L (Ω) L (0, T ; H is said to be a weak solution to (10.6.1)–(10.6.2) if we have (10.6.3) 0 = y(t), ψH −1 (Ω),H01 (Ω) − y 0 , ψH −1 (Ω),H01 (Ω) + (y, P0 ψ)L2 ((0,t)×Ω) t

+ ∫ g, ∂ν ψ |∂Ω H −1/2 (∂Ω),H 1/2 (∂Ω) , 0

for all t ∈ (0, T ) and for all ψ ∈ D(P0 ) = H 2 (Ω) ∩ H01 (Ω).

352

10. ELLIPTIC OPERATORS AND ASSOCIATED SEMIGROUP

Observe that the regularity required for the solution y in Definition 10.60 coincides with that given for Theorem 10.44 in the case r = −2. 2 −1/2 Theorem 10.61. Lety 0 ∈ H −1 (Ω) and  g 2∈ L (0, T2 ; H  (∂Ω)). There 0 −1 exists a unique y ∈ C [0, T ]; H (Ω) ∩ L 0, T ; L (Ω) that is a weak solution to (10.6.1)–(10.6.2) in the sense of Definition 10.60. The solution y is given by t

y(t) = S(t)y 0 + ∫ S(t − σ)M g(σ)dσ, 0

with M : H −1/2 (∂Ω) → K −2 (Ω) bounded and given by M = P−2 ◦ D. This solution coincides with the solution given by Theorem 10.44 in the case r = −2 of the nonhomogeneous semigroup equation: d y + P−2 y = M g, dt   Moreover y ∈ H 1 0, T ; K −2 (Ω) .

y(0) = y 0 .

We recall that D denotes the Dirichlet map introduced in Sect. 10.5 and that (P−2 , D(P−2 )) is the extension of the operator (P0 , D(P0 )) to K −2 (Ω) with D(P−2 ) = L2 (Ω) (see Sect. 10.1.3). Note that from the properties of the Dirichlet map, if h ∈ H −1/2 (∂Ω), we have P0 Dh = 0 in D  (Ω). However, P−2 Dh does not vanish in general in K −2 (Ω). Theorem 10.61 shows that the treatment of the parabolic equation with boundary Dirichlet data (10.6.1)–(10.6.2) can be recast into the semigroup equation: d y + P−2 y = M g, dt that holds in L2 (0, T ; K −2 (Ω)). Proof. First, we address uniqueness. By linearity, it suffices to prove that a solution y in the sense of Definition 10.60 vanishes if y 0 = 0 and g = 0. In such a case we have y(t), ψH −1 (Ω),H01 (Ω) + (y, P0 ψ)L2 ((0,t)×Ω) = 0, which we write, with Lemma 10.15, & % t y(t) + ∫ P−2 y(σ) dσ, ψ 0

t ≥ 0, ψ ∈ K 2 (Ω) = D(P0 ),

K −2 (Ω),K 2 (Ω)

= 0,

implying that, for all 0 < t < T , we have y(t) + ∫0t P−2 y(σ) dσ = 0 in K −2 (Ω). This gives y ∈ H 1 (0, T ; K −2 (Ω)) and d y(t) + P−2 y(t) = 0 in L2 (0, T ; K −2 (Ω)) dt

and y(0) = 0.

By Proposition 10.36, we conclude that y(t) vanishes for t ≥ 0.

10.6. PARABOLIC EQUATION WITH DIRICHLET BOUNDARY DATA

353

Second, we address existence. For y0 ∈ H −1 (Ω) and g ∈ L2 (0, T ; H −1/2 (∂Ω)), we consider the parabolic equation: d y + P−2 y = M g, (10.6.4) y(0) = y 0 . dt As M g ∈ L2 (0, T ; K −2 (Ω)), by Theorem 10.44, there exists a unique solution: y ∈ C 0 ([0, T ]; H −1 (Ω)) ∩ L2 (0, T ; L2 (Ω)) ∩ H 1 (0, T ; K −2 (Ω)) such that (10.6.4) holds in L2 (0, T ; K −2 (Ω)). Moreover, we have t

y(t) = S(t)y 0 + ∫ S(t − σ)M g(σ)dσ. 0

K 2 (Ω).

Let ψ ∈ We carry again the computation made at the beginning of this section, making precise now the functional spaces and duality brackets we use. The following computations are to be understood for almost every 0 < t < T . We have d (10.6.5)  y(t), ψK −2 (Ω),K 2 (Ω) + P−2 y(t), ψK −2 (Ω),K 2 (Ω) dt = M g(t), ψK −2 (Ω),K 2 (Ω) . From the regularity of y, we have y(t), ψH −1 (Ω),H01 (Ω) = y(t), ψK −2 (Ω),K 2 (Ω) ∈ H 1 (0, T ), which yields d d y(t), ψH −1 (Ω),H01 (Ω) =  y(t), ψK −2 (Ω),K 2 (Ω) . dt dt By Lemma 10.15, we have P−2 y(t), ψK −2 (Ω),K 2 (Ω) = (y(t), P0 ψ)L2 (Ω) . We also have, using (10.5.1), M g(t), ψK −2 (Ω),K 2 (Ω) = (Dg(t), P0 ψ)L2 (Ω) = −g(t), ∂ν ψH −1/2 (Ω),H 1/2 (Ω) . Now, integrating (10.6.5), using the above computations, we then obtain (10.6.3). This proves the existence of a solution in the sense of Definition 10.6.3 that coincides with the solution of the unique semigroup Eq. (10.6.4).  Proposition 10.62. The adjoint of operator M maps K 2 (Ω) = D(P0 ) = ∩ H01 (∂Ω) into H 1/2 (∂Ω) continuously and is given by

H 2 (Ω)

M ∗ v = −∂ν v|∂Ω ,

v ∈ K 2 (Ω).

Proof. Let u ∈ H −1/2 (∂Ω) and v ∈ K 2 (Ω). We compute M u, vK −2 (Ω),K 2 (Ω) = (Du, P0 v)L2 (Ω) = −u|∂Ω , ∂ν v |∂Ω H −1/2 (∂Ω),H 1/2 (∂Ω) , by Lemma 10.15 and (10.5.1). This yields the result.



CHAPTER 11

Some Elements of Functional Analysis Contents 11.1. 11.2. 11.3. 11.4. 11.5. 11.5.1. 11.6.

Linear Operators in Banach Spaces Continuous and Bounded Operators Spectrum of a Linear Operator in a Banach Space Adjoint Operator Fredholm Operators Characterization of Bounded Fredholm Operators Linear Operators in Hilbert Spaces

355 356 357 358 358 359 363

Here, X and Y will denote Banach spaces with their norms denoted by .X , .Y , or simply . when there is no ambiguity. 11.1. Linear Operators in Banach Spaces An operator A from X to Y is a linear map on its domain, a linear subspace of X, to Y . One denotes by D(A) the domain of this operator. An operator from X to Y is thus characterized by its domain and how it acts on this domain. Operators defined this way are usually referred to as unbounded operators. One writes (A, D(A)) to denote the operator along with its domain. The set of linear operators from X to Y is denoted by L (X, Y ). If D(A) is dense in X, the operator is said to be densely defined. If D(A) = X, one says that the operator A is on X to Y . The range of the operator is denoted by Ran(A), that is, Ran(A) = {Ax; x ∈ D(A)} ⊂ Y, and its kernel, ker(A), is the set of all x ∈ D(A) such that Ax = 0.

© 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 I, PNLDE Subseries in Control 97, https://doi.org/10.1007/978-3-030-88674-5 11

355

356

11. SOME ELEMENTS OF FUNCTIONAL ANALYSIS

The graph of A, G(A), is given by G(A) = {(x, Ax); x ∈ D(A)} ⊂ X × Y. We naturally endow X × Y with the norm (x, y)2X×Y = x2X + y2Y , which makes X × Y a Banach space. One says that A is a closed operator if its graph G(A) is a closed subset of X × Y for this norm. The so-called graph norm on D(A) is given by x2D(A) = x2X + Ax2Y = (x, Ax)2X×Y . The operator A is closed if and only if the space D(A) is complete for the graph norm .D(A) . If a linear operator A from X to Y is injective, one can define the operator A−1 from Y to X such that D(A−1 ) = Ran(A), Ran(A−1 ) = D(A), A−1 A = IdD(A) , AA−1 = IdRan(A) . One says that A is invertible and A−1 is called the inverse operator. If (A1 , D(A1 )), (A2 , D(A2 )) are two linear operators from X to Y , one defines that the operator B = A1 + A2 with domain D(A1 ) ∩ D(A2 ). 11.2. Continuous and Bounded Operators A linear operator A from X to Y is said to be continuous if it is continuous at every x ∈ D(A) or equivalently if it is continuous at x = 0. This is equivalent to having M > 0 such that AxY ≤ M xX for all x ∈ D(A). One says that A is a bounded operator. The positive number M = sup x∈D(A) x =0

AxY xX

is called the bound of A and denoted by AL (X,Y ) or simply A. Note that linear operators from X to Y that fails to be continuous are such that AxY = +∞. sup x∈D(A) xX x =0

This justifies the name unbounded for general linear operators from X to Y . Theorem 11.1 (Closed-Graph Theorem). Let A be such that D(A) is a closed linear subspace in X. Then, A is bounded if and only if A is a closed operator. For a proof, see, for instance, [192]. Remark 11.2. While one aspect of the proof of the closed-graph theorem is involved and based on the Baire lemma, one can also easily prove the following statements: if A is closed and A is bounded, then D(A) is a closed

11.3. SPECTRUM OF A LINEAR OPERATOR IN A BANACH SPACE

357

linear subspace in X. Hence, if an linear operator A is densely defined, closed, and bounded, then D(A) = X: the operator is bounded on X to Y . Note also that any bounded operator A with domain D(A) can be uniquely extended to D(A), as a bounded operator with the same bound, thus leading to a closed operator. We shall denote by B(X, Y ) the set of bounded operators A on X to Y , that is, such that D(A) = X. In the main text, if we speak of a bounded operator A : X → Y without any mention of its domain, this means that D(A) = X; that is, A is on X to Y . Remark 11.3. Following the above remark, assume that A is a closed linear operator from X to Y that is invertible and such that A−1 is bounded. As A−1 is also closed for obvious reasons, we find that Ran(A) = D(A−1 ) is a closed subset of Y . 11.3. Spectrum of a Linear Operator in a Banach Space We consider here a linear operator from X to itself. One says that λ ∈ C is in the resolvent set ρ(A) of an linear operator A from X to X if −1 the operator λ Id −A  is injective,  and the inverse operator (λ Id −A) has −1 a dense domain D (λ Id −A) = Ran(λ Id −A) in X and is bounded. If λ ∈ ρ(A), then we set the resolvent operator as Rλ (A) = (λ Id −A)−1 . The spectrum is then simply the complement set of ρ(A) in C. We denote it by sp(A). The spectrum of a linear operator is often separated into three disjoint sets: (1) The point spectrum that gathers all λ ∈ C such that the operator λ Id −A is not injective. Such a complex number λ is called an eigenvalue of A, and the (finite or infinite) dimension of the kernel ker(λ Id −A) is the geometric multiplicity associated with this eigenvalue. An element of ker(λ Id −A) is called an eigenvector or, often, an eigenfunction in the case that the Banach space X is a function space. (2) The continuous spectrum that gathers all λ ∈ C such that the operator λ Id −A is injective, has a dense image, but its inverse (λ Id −A)−1 is not bounded. (3) The residual spectrum that gathers all λ ∈ C such that the operator λ Id −A is injective but does not have a dense image. In the case that A is a closed operator, if λ ∈ ρ(A), then D(Rλ (A)) = Ran(λ Id −A) = X (see Remark 11.3). Hence, in this case, λ ∈ ρ(A) if and only if λ Id −A is injective and Ran(λ Id −A) = X because of the closedgraph theorem (Theorem 11.1). For λ0 ∈ ρ(A), if we set L0 = (λ0 Id −A)−1 , then L0 is a bounded operator on X and we may write   λ Id −A = (λ0 Id −A) Id +(λ − λ0 )L0 .

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For |λ − λ0 | < L0 −1 , one then finds that Id +(λ − λ0 )L0 is itself invertible with a bounded inverse. Consequently, the resolvent set is an open set in C and the spectrum is closed. Moreover, one finds that on ρ(A), the map λ → Rλ (A) is holomorphic. We refer the reader, for instance, to Chapter 3.6 in [192]. 11.4. Adjoint Operator If X  is the dual space of a Banach space X, that is, the linear space of bounded linear forms on X, we equip X  with the strong topology associated with the norm: x∗ X  = sup |x∗ , x|. x∈X

x X ≤1

With this topology, X  is a Banach space. If A is a linear operator from X to Y densely defined, one sets   D(A∗ ) = y ∗ ∈ Y  ; ∃C > 0, ∀x ∈ D(A), |y ∗ , AxY  ,Y | ≤ CxX . If y ∗ ∈ D(A∗ ), there exists a unique x∗ ∈ X  such that y ∗ , AxY  ,Y = x∗ , xX  ,X ,

x ∈ D(A).

Uniqueness follows from the density of D(A) in X. One then sets A∗ y ∗ = x∗ , which defines a linear operator A∗ from Y ∗ to X ∗ with domain D(A∗ ). Proposition 11.4. Let A be a densely defined operator from X to Y . Then, the operator (A∗ , D(A∗ )) is a closed operator. Proposition 11.5. If the operator A is bounded on X to Y , that is, D(A) = X and A ∈ B(X, Y ), then D(A∗ ) = Y  and A∗ is a bounded operator on Y  to X  . Moreover AL (X,Y ) = A∗ L (Y  ,X  ) . 11.5. Fredholm Operators Let A be a linear closed operator from X to Y . The nullity of A, nul A, is defined as the dimension of ker(A). The deficiency of A, def A, is defined as the dimension of Y /Ran(A). Both nul A and def A take value in N ∪ {∞}. Definition 11.6. A linear operator A from X to Y is said to be Fredholm if (1) it is closed; (2) Ran(A) is closed; (3) both nul A and def A are finite. One then sets the index of A as ind(A) = nul A − def A.

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11.5.1. Characterization of Bounded Fredholm Operators. We denote by F B(X, Y ) the space of Fredholm operators that are bounded on X into Y . The following result states that those operators are the operators in B(X, Y ) that have an inverse up to remainder operators that are compact. Theorem 11.7. Let A ∈ B(X, Y ). It is Fredholm if and only if there exists S ∈ B(Y, X) such that (11.5.1)

SA = IdX +K ,

AS = IdY +K r ,

where K ∈ B(X, X) and K r ∈ B(Y, Y ) are compact operators. In particular, S is Fredholm and ind(A) = − ind(S). For the proof, we shall need the following lemma. Lemma 11.8. Let A ∈ B(X, Y ) and K ∈ B(X, X1 ) be compact, with X, Y , and X1 Banach spaces, and C > 0 such that   xX ≤ C AxY + KxX1 , (11.5.2) for x ∈ X. Then, Ran(A) is closed. Proof. Let (yn )n ⊂ Ran(A) be a converging sequence in Y . Set y = lim yn and consider a sequence (xn )n ⊂ X such that Axn = yn . Set also X0 = ker A. First, assume that dn = dist(xn , X0 ) is bounded, say dn ≤ R. Thus, for ˜n X ≤ R + 1. Replacing any n ∈ N there exists x ˜n ∈ X0 such that xn − x ˜n we have found (xn )n ⊂ X such that Axn = yn with (xn )n xn by xn − x bounded. Then, (Kxϕ(n) )n converges in X1 , for some increasing function ϕ : N → N. With (11.5.2), we have xϕ(n) − xϕ(m) X  A(xϕ(n) − xϕ(m) )Y + K(xϕ(n) − xϕ(m) )X , 1

implying that (xϕ(n) )n is a Cauchy sequence in X complete. We denote by x its limit and, as A is bounded, we find Ax = lim Axϕ(n) = y. Second, we assume that dn = dist(xn , X0 ) is unbounded. By contradiction, we prove that this second case does not occur, which yields the conclusion. In fact, up to a subsequence we have dn ≥ 1 and lim dn = +∞. For any ˜n X ≤ dn + 1 and we set n ∈ N, there exists x ˜n ∈ X0 such that dn ≤ xn − x xn . Naturally, we have dist(zn , X0 ) = dn . If we set un = zn /zn X , zn = xn −˜ we have dist(un , X0 ) = dn /zn X , yielding dist(un , X0 ) ≥ dn /(dn +1). Using that t → t/(t + 1) is increasing on [0, +∞), we find that dist(un , X0 ) ≥ 1/2. We now see that Aun = yn /zn X converges to 0 as lim zn X = +∞ and that (Kuψ(n) )n converges in X1 , for some increasing function ψ : N → N. With (11.5.2), we have uψ(n) − uψ(m) X  A(uψ(n) − uψ(m) )Y + K(uψ(n) − uψ(m) )X , 1

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implying that (uψ(n) )n is a Cauchy sequence in X. Set u = lim uψ(n) . By continuity, we have Au = 0, meaning that u ∈ X0 in contradiction with  dist(un , X0 ) ≥ 1/2 obtained above. Lemma 11.9. Let X be a Banach space and K ∈ B(X, X) be compact. Then ker(Id +K) is finite dimensional. Proof. If x ∈ ker(Id +K), we have x = −Kx. In particular, the unit ball in ker(Id +K) is the image of bounded set by the compact operator K. It follows that the unit ball of ker(Id +K) is compact and thus, by the Riesz theorem, ker(Id +K) is finite dimensional.  Proof of Theorem 11.7. First, assume that (11.5.1) holds. The first identity gives ker(A) ⊂ ker(Id +K ), with the latter space finite dimensional by Lemma 11.9. From the first equality in (11.5.1), we deduce xX  AxY + K xX .

By Lemma 11.8, this implies that Ran(A) is closed. As Y / Ran(A) ∼ = Ran(A)⊥ , proving codim Ran(A) < ∞ amounts to proving that Ran(A)⊥ is finite dimensional. From the second equality in (11.5.1), we have Ran(IdY + K r ) ⊂ Ran(A) and thus Ran(A)⊥ ⊂ Ran(IdY +K r )⊥ . By Corollary 2.18 in [90], we have Ran(IdY +K r )⊥ = ker(IdY  +(K r )∗ ) and the latter space is finite dimensional by Lemma 11.9. ˜a Second, assume that A is Fredholm. As dim ker A < ∞, there exists X closed linear subspace of X that is a complementary subspace of ker A, that ˜ ⊕ ker A = X in the algebraic sense and moreover the projections assois, X ciated with this direct sum are continuous. Similarly, as codim Ran(A) < ∞ and as Ran(A) is closed, there exists also Z complementary subspace of Ran(A) in Y . We refer, for instance, [90, Section 2.4]. Observe that the projections Πker A onto ker A and ΠZ onto Z associated with the above direct sums are compact since dim ker A < ∞ and dim Z < ∞. ˜ → Ran(A) given by Ax ˜ = Ax. We consider the bijective map A˜ : X ˜ As X and Ran(A) are Banach spaces if equipped with the norms inherited from X and Y , the open map theorem shows that A˜ is an isomorphism. We ˜ Y −ΠZ ) We then find that denote by S˜ its inverse map, and we set S = S(Id ˜ Y −ΠZ ) = A˜S(Id ˜ Y −ΠZ ) = IdY −ΠZ . AS = AS(Id We also write ˜ = SA(Id ˜ ˜˜ SA = SA X −Πker A ) = S A(IdX −Πker A ) = IdX −Πker A , which concludes the proof.



Proposition 11.10. Let A ∈ B(X, Y ). It is Fredholm if and only if there exist K1 ∈ B(X, Z1 ) and K2 ∈ B(Y  , Z2 ) both compact, with Z1 and Z2 Banach spaces, and C > 0 such that     xX ≤ C AxY + K1 xZ1 , y ∗ Y  ≤ C A∗ y ∗ X  + K2 y ∗ Z2 , for x ∈ X and y ∗ ∈ Y  .

11.5. FREDHOLM OPERATORS

361

Remark 11.11. The first part of the proof shows that one can use the compact operators K1 = K ∈ B(X, X) and K2 = (K r )∗ ∈ B(Y  , Y  ) that are given by Theorem 11.7. Then one has Z1 = X and Z2 = Y  . Proof. By Theorem 11.7, if A is Fredholm, there exists S bounded from Y to X such that SA = IdX +K ,

AS = IdY +K r ,

with K : X → X and K r : Y → Y both compact operators. With the first identity, we obtain xX  AxY + K xX . With the second identity, we compute S ∗ A∗ = IdY  +(K r )∗ , yielding y ∗ Y   A∗ y ∗ X  + (K r )∗ y ∗ Y  . Conversely, if xX  AxY + K1 xZ1 , for some K1 : X → Z1 compact, we consider a sequence (xn )n ⊂ ker(A) such that xn X = 1. Then, up to a subsequence, (K1 xn )n converges in Z1 . Writing xn − xm X  K1 xn − K1 xm Z1 , we find that (xn )n is a Cauchy sequence and thus converges as X is a complete. The unit ball of ker(A) is thus compact; ker(A) is thus finite dimensional by the Riesz theorem. Similarly we find that ker(A∗ ) is finite dimensional. As Ran(A) is closed by Lemma 11.8, we have Ran(A) = Ran(A) = ker(A∗ )⊥ by Corollary 2.18 in [90] implying that codim Ran(A) < ∞ as codim Ran(A) =  dim(X/ Ran(A)) = dim ker(A∗ ). Corollary 11.12. Let A ∈ F B(X, Y ) and F be a closed subspace of X. Then A(F ) is closed. Proof. As A ∈ F B(X, Y ), we have the estimations of Proposition 11.10 and the first one applies to A|F . By Lemma 11.8, we conclude that Ran(A|F ) = A(F ) is closed in Y .  The set F B(X, Y ) of bounded Fredholm operators has some important topological properties. Theorem 11.13. The set F B(X, Y ) is open in B(X, Y ). Proof. Let A ∈ F B(X, Y ). By Proposition 11.10 and Remark 11.11, we have xX  AxY + K xX ,

y ∗ Y   A∗ y ∗ X  + (K r )∗ y ∗ Y  .

with the compact operators K ∈ B(X, X) and K r ∈ B(Y, Y ) given by Theorem 11.7. With these two inequalities, we see that there exists ε > 0

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such that xX  (A + B)xY + K xX , + (K r )∗ y ∗ Y  ,

y ∗ Y   (A + B)∗ y ∗ X 

for B ∈ (X, Y ) such that BL (X,Y ) ≤ ε. By Proposition 11.10, we then find that A + B ∈ F B(X, Y ).  Theorem 11.14. The maps F B(X, Y ) → N nul : A → dim ker(A) and def : A → codim Ran(A) are both upper semicontinuous. Moreover, the index map, ind = nul − def, is constant in each connected component of F B(X, Y ). Proof. Let A ∈ F B(X, Y ). As we have nul A = dim ker(A) < ∞, ˜ and Y˜ def A = codim Ran(A) < ∞, and Ran(A) is closed, there exist X that are complementary ker(A) and Ran(A) in X and Y , respectively, that is, ˜ ⊕ ker(A) = X and Ran(A) ⊕ Y˜ = Y, (11.5.3) X ˜ and Y˜ closed (see Section 2.4 in [90]). Set Z = X ˜ ⊕ Y˜ . with moreover X For T ∈ B(X, Y ), we define κT ∈ B(Z, Y ) given by ˜ y ∈ Y˜ . κT (x + y) = T x + y, x ∈ X, Observe that κA is bijective. Note that κT1 − κT2 L (Z,Y ) ≤ T1 − T2 L (X,Y ) . Thus, for T ∈ B(X, Y ) chosen such that T − AL (X,Y ) ≤ ε, the operator κT is also bijective, since the set of bounded invertible operators from Z into Y is open in B(Z, Y ), and T ∈ F B(X, Y ) by Theorem 11.13, for ε > 0 chosen sufficiently small. ˜ × Below, T is chosen such that T − AL (X,Y ) ≤ ε. We have κT (X ˜ and, as κT is an isomorphism, we have {0}) = T (X) (11.5.4)

˜ = codim κT (X ˜ × {0}) = dim Y˜ . codim T (X)

˜ ⊂ Ran(T ), we thus find that As T (X) (11.5.5)

def T = codim Ran(T ) ≤ dim Y˜ = codim Ran(A) = def A,

meaning that the map T → def T is upper semicontinuous at A. ˜ = {0}. Since ker(T ) As κT is an isomorphism, we find that ker(T ) ∩ X ˜ ˜ is finite dimensional, codim X < ∞, and X closed, there exists a finiteˆ of X ˜ ⊕ ker(T ), that is, dimensional complementary subspace X ˆ ⊕X ˜ ⊕ ker(T ). (11.5.6) X=X ˆ ⊕ X) ˜ = T (X) ˆ ⊕ T (X) ˜ = T (X), yielding Note that T (X ˆ = codim T (X), ˜ codim T (X) + dim T (X) ˆ = def A, since X ˆ ∩ ker T = {0} and using (11.5.4)– that is, def T + dim X (11.5.5). In turn, from (11.5.3) and (11.5.6) we conclude that nul T +

11.6. LINEAR OPERATORS IN HILBERT SPACES

363

ˆ = nul A. Together, these last two equalities show that ind(T ) = dim X ind(A). Finally, as we now have nul A − nul T = def A − def T , we also conclude that the map T → nul T is upper semicontinuous at A.  Given a bounded operator A ∈ F B(X, Y ), we saw above that small perturbations B ∈ B(X, Y ) affect neither its Fredholm property nor its index as one remains in the connected component of A in F B(X, Y ). The following proposition shows if K ∈ B(X, Y ) is compact, then A + K also remains in the connected component of A, without any size assumption on K. Theorem 11.15. Let A ∈ F B(X, Y ) and K ∈ B(X, Y ) be compact. Then A + K ∈ F B(X, Y ) and ind(A + K) = ind(A). Proof. As A is Fredholm, by Proposition 11.10 there exist K1 ∈ B(X, Z1 ) and K2 ∈ B(Y  , Z2 ) both compact, with Z1 and Z2 Banach spaces, and C > 0 such that xX  AxY + K1 xZ1 ,

y ∗ Y   A∗ y ∗ X  + K2 y ∗ Z2 .

We thus have xX  (A + K)xY + KxY + K1 xZ1 ,

y ∗ Y   (A + K)∗ y ∗ X  + K ∗ y ∗ X  + K2 y ∗ Z2 . We then define ˜ 2 : Y  → X  ⊕ Z2 K

˜ 1 : X → Y ⊕ Z1 K

˜ 1 x = Kx + K1 x and K ˜ 2 (y ∗ ) = K ∗ (y ∗ ) + K2 (y ∗ ). Both are given by K compact, and we have ˜ 1 x , x  (A + K)x + K X

∗

Y ∗ ∗

y Y   (A + K) y X 

Y ⊕Z1

˜ 2y∗  + K X ⊕Z2 .

We thus find that A + K is Fredholm by the converse part of Proposition 11.10. Next, for the same reason A + tK ∈ F B(X, Y ) for any t ∈ [0, 1]. This implies that A+K and A lie in the same connected component of F B(X, Y ). Their index is thus the same by Theorem 11.14.  11.6. Linear Operators in Hilbert Spaces Let H be a Hilbert space. By the Riesz theorem, there exists an isomorphism J : H  → H such that u∗ , uH  ,H = (Ju∗ , u)H ,

u∗ ∈ H  , u ∈ H,

which allows one to identify H  with H. Let H1 and H2 be two Hilbert spaces. For an unbounded operator A from H1 to H2 densely defined, its adjoint operator, as defined in Sect. 11.4,

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can then be uniquely identified with an operator from H2 to H1 that we also denote by A∗ , with domain   D(A∗ ) = v ∈ H2 ; ∃C > 0, ∀u ∈ D(A), |(v, Au)H2 | ≤ CuH1 ⊂ H2 , and such that (Au, v)H2 = (u, A∗ v)H1 ,

u ∈ D(A) ⊂ H1 , v ∈ D(A∗ ) ⊂ H2 .

As a Hilbert space is reflexive, we have the following result (see, e.g., Theorem III.5.29 in [192]). Proposition 11.16. Let (A, D(A)) be a closed and densely defined operator from H1 to H2 . Then, the operator (A∗ , D(A∗ )) from H2 to H1 is also closed and densely defined. Moreover A∗∗ = A. We also have the following result (see Theorem III.5.30 in [192]). Proposition 11.17. Let (A, D(A)) be a closed and densely defined operator from H1 to H2 . If A−1 exists and is bounded on H2 to H1 , then (A∗ )−1 exists and is bounded on H1 to H2 and (A∗ )−1 = (A−1 )∗ . Moreover, (A∗ )−1 L (H1 ,H2 ) = A−1 L (H2 ,H1 ) . If A is an unbounded operator from a Hilbert space H into itself, the operator is said to be symmetric if one has (Au, v)H = (u, Av)H ,

u, v ∈ D(A).

If its domain is dense, A∗ is well-defined, D(A) ⊂ D(A∗ ) and A∗ coincides with A on D(A). One usually writes (A, D(A)) ⊂ (A∗ , D(A∗ )). The operator A is furthermore said to be self-adjoint if D(A) = D(A∗ ): we then have (A, D(A)) = (A∗ , D(A∗ )). Observe that a symmetric operator may not be self-adjoint. Consider ,for instance, the operator A given by Au = Δu with domain D(A) = Cc∞ (Ω) ⊂ H = L2 (Ω), for Ω a bounded open set in Rd . The operator A is symmetric as one has (Au, v)L2 = (u, Av)L2 , for all u, v ∈ D(A). One sees readily that H 2 (Ω) ⊂ D(A∗ ). The operator is not self-adjoint. A useful criterion is the following result; we refer to [282, Theorem 8.3] for a proof. Theorem 11.18. Let (A, D(A)) densely defined be a symmetric linear operator on a Hilbert space H. The following three statements are equivalent: (1) (A, D(A)) is self-adjoint. (2) A is closed and ker(A∗ + i IdH ) = ker(A∗ − i IdH )) = {0}. (3) Ran(A + i IdH )) = Ran(A − i IdH )) = H. For a bounded operator A on a Hilbert space H, its adjoint operator yields a bounded operator on H by what precedes and Proposition 11.5. The following lemma due to [121, 128] is based on the closed-graph theorem and allows one to quantify the inclusion of the ranges of two operators.

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Lemma 11.19. Let K1 : H1 → H and K2 : H2 → H with H, H1 , and H2 Hilbert spaces and K1 and K2 linear and bounded. The following statements are equivalent: (1) We have Ran(K1 ) ⊂ Ran(K2 ). (2) There exists a bounded linear map Φ : H1 → H2 such that K1 = K2 ◦ Φ. (3) There exists C0 ≥ 0 such that K1∗ zH1 ≤ C0 K2∗ zH2 ,

(11.6.1)

z ∈ H.

Moreover, if the second statement holds, then one can choose C0 = ΦL (H1 ,H2 ) in (11.6.1). Conversely, if C0 ≥ 0 is the best possible constant for which (11.6.1) holds, then there exists Φ as in the second statement such that ΦL (H1 ,H2 ) = C0 . Proof. We start by proving that the first statement implies the second statement. Note that the converse is obvious. We thus assume that Ran(K1 ) ⊂ Ran(K2 ). Let u1 ∈ H1 . Then L(u1 ) = {u ∈ H2 ; K1 (u1 ) = K2 (u)} is a nonempty closed affine subspace of H2 . We denote by Φ(u1 ) the orthogonal projection of 0 onto L(u1 ), characterized as the unique element w of L(u1 ) such that ∀v ∈ L(u1 ), (w, v − w)H2 = 0.

(11.6.2)

Observe that L(u1 ) = Φ(u1 ) + ker(K2 ) and (11.6.2) means that Φ(u1 ) is orthogonal to ker(K2 ). The operator Φ : H1 → H2 is linear, and we prove that the graph of Φ (n) (n) is closed. In fact, consider two sequences (u1 )n ⊂ H1 , (u2 )n ⊂ H2 such that (n)

u2

(n)

= Φ(u1 ),

(n)

(n)

u1

→ u1 in H1 ,

n→∞

(n)

u2

→ u2 in H2 .

n→∞

(n)

Then K1 (u1 ) = K2 (u2 ) giving in the limit K1 (u1 ) = K2 (u2 ). Moreover (n) u2 is orthogonal to ker(K2 ) giving in the limit u2 orthogonal to ker(K2 ). Hence u2 = Φ(u1 ), which is the graph of Φ is closed. The closed-graph theorem then implies that Φ is a bounded operator. There exists C0 > 0 such that Φ(u1 )H2 ≤ C0 u1 H1 . Having proven the equivalence of the first two statements, we now show that they imply the inequality of the third statement. Note that having K2 (u2 ) = K1 (u1 ) implies (u1 , K1∗ z)H1 = (u2 , K2∗ z)H2 ,

z ∈ H.

For z ∈ H, we set u1 = K1∗ z and set u2 = Φ(u1 ), with Φ as defined above. This gives K1∗ z2H1 ≤ Φ(u1 )H2 K2∗ zH2 ≤ C0 K1∗ zH1 K2∗ zH2 , which gives K1∗ zH1 ≤ C0 K2∗ zH2 for z ∈ H.

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Finally, we prove that the third statement implies the second one. We thus assume that there exists C0 > 0 such that (11.6.3)

∀z ∈ H, K1∗ zH1 ≤ C0 K2∗ zH2 .

Let u1 ∈ H1 . We define the linear map: Ψ : Ran(K2∗ ) ⊂ H2 → C K2∗ z → (K1∗ z, u1 )H1 . This map is well-defined. In fact if w = K2∗ z = K2∗ z  , then K1∗ z = K1∗ z  by (11.6.3). We have, for w = K2∗ z, |Ψ(w)| ≤ K1∗ zH1 u1 H1 ≤ C0 wH2 u1 H1 , that is, the map Ψ is bounded and Ψ ≤ C0 u1 H1 . We then denote by ˜ the map Ran(K ∗ ) → C that uniquely extends Ψ to the Hilbert space Ψ 2 ˜ = Ψ ≤ Ran(K2∗ ) endowed with the inner product on H2 . We have Ψ C0 u1 H1 . By the Riesz theorem, there exists u2 ∈ Ran(K2∗ ) such that u2 H2 = Ψ ≤ C0 u1 H1 and ˜ Ψ(w) = (w, u2 )H2 ,

w ∈ Ran(K2∗ ).

We define the map Φ : H1 → H2 by u2 = Φ(u1 ). It is linear and bounded. For z ∈ H, we set w = K2∗ z and we obtain (K1∗ z, u1 )H1 = (K2∗ z, Φ(u1 ))H2 , and thus for all z ∈ H we have (z, K1 (u1 ))H = (z, K2 (Φ(u1 )))H . That is,  K1 (u1 ) = K2 (Φ(u1 )). The proof is complete. A corollary is the following result that characterizes the surjectivity of a bounded operator. Corollary 11.20. Let K : H1 → H with H1 , H Hilbert spaces and K linear and bounded. The following statements are equivalent: (1) We have Ran(K) = H. (2) There exists a bounded linear map Φ : H → H1 such that IdH = K ◦ Φ. (3) There exists C0 ≥ 0 such that (11.6.4)

xH ≤ C0 K ∗ xH1 .

Moreover, if the second statement holds, then one can choose C0 = ΦL (H,H2 ) in (11.6.4). Conversely, if C0 ≥ 0 is the best possible constant for which (11.6.4) holds, then there exists Φ as in the second statement such that ΦL (H,H2 ) = C0 . Remark 11.21. Note that if we decide to not identify the Hilbert spaces H and H1 with their respective dual spaces, we then obtain the characterization: (11.6.5)

xH  ≤ C0 K ∗ xH  . 1

We may also decide to identify H1 with its dual and not H and vice versa.

CHAPTER 12

Some Elements of Semigroup Theory Contents 12.1. 12.1.1. 12.1.2. 12.1.3. 12.2. 12.3. 12.4.

Strongly Continuous Semigroups Definition and Basic Properties The Hille–Yosida Theorem The Lumer–Phillips Theorem Differentiable and Analytic Semigroups Mild Solution of the Inhomogeneous Cauchy Problem The Case of a Hilbert Space

368 368 371 371 374 375 376

Semigroup theory is at the heart of the understanding of many evolution equations that can be put in the form:

(12.0.6)

d x(t) + Ax(t) = f (t), dt

t > 0,

x(0) = x0 ,

with x(t) and x0 in a proper function space, usually a Banach space, denoted by X below, if not a Hilbert space, with A an unbounded operator on X, with dense domain, and f a function of the time variable t taking values in X. First, in Sects. 12.1 and 12.2, we review the case of a homogeneous equation, that is, f ≡ 0. Second, in Sect. 12.3 we consider the more general case of an inhomogeneous equation. In particular, we provide the necessary form of the solution, given by the so-called mild solution based on the Duhamel formula. Third, in Sect. 12.4 we show how some results improve in the case the function space X is a Hilbert space. For general references on semigroups, we refer the reader to the books of E. Hille and R. S. Phillips [169], E.B. Davies [112], A. Pazy [270], and K.-J. Engel and R. Nagel [133]. © 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 I, PNLDE Subseries in Control 97, https://doi.org/10.1007/978-3-030-88674-5 12

367

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12.1. Strongly Continuous Semigroups Consider the homogeneous equation associated with the evolution problem (12.0.6), that is, d x(t) + Ax(t) = 0, t > 0, x(t) = x0 . (12.1.1) dt Under proper assumptions on A, we can write the solution in the form x(t) = S(t)x0 , where S(t) : X → X is a bounded operator. As some sort of differentiation with respect to time is expected in (12.1.1), a minimal assumption is then that S(0)x = x

(12.1.2)

and

t → S(t)x be continuous for all x ∈ X.

With t → x(t) solution to (12.1.1), if the evolution problem is well-posed, we expect from uniqueness that solving the following problem, for some t0 ≥ 0: d y(t) + Ay(t) = 0, t > t0 , y(t0 ) = x(t0 ), (12.1.3) dt yield a solution that satisfies y(t) = x(t) for t ≥ t0 . In particular, this implies the following property: S(t + t ) = S(t) ◦ S(t ), for t, t ∈ [0, +∞).

(12.1.4)

Properties (12.1.2) and (12.1.4) are precisely the starting point of semigroup theory in Banach spaces. 12.1.1. Definition and Basic Properties. Let X be a Banach space. Definition 12.1. A family S(t) of bounded operators on X, indexed by t ∈ [0, +∞), is called a semigroup if: (12.1.5)

S(0) = IdX

and

S(t + t ) = S(t) ◦ S(t ) for t, t ∈ [0, +∞).

The semigroup is called strongly continuous if, moreover, for all x ∈ X we have limt→0+ S(t)x = x. One says that S(t) is a C0 -semigroup for short. For each C0 -semigroup, there exist M ≥ 1 and ω ∈ R such that (12.1.6)

S(t)L (X) ≤ M eωt ,

by the uniform boundedness principle [270, Theorem 1.2.2]. It follows that the map (t, x) → S(t)x is continuous from [0, +∞) × X into X. A C0 semigroup S(t) is said to be bounded if there exists M ≥ 1 such that S(t)L (X) ≤ M , for t ≥ 0. In the case M = 1 one says that the C0 semigroup is of contraction. We define the unbounded linear operator A from X to X, with domain (12.1.7)

D(A) = {x ∈ X; lim t−1 (x − S(t)x) exists}, t→0+

and given by (12.1.8)

Ax = lim t−1 (x − S(t)x), t→0+

x ∈ D(A).

12.1. STRONGLY CONTINUOUS SEMIGROUPS

369

For the basic aspects of unbounded operators, we refer to Chap. 11. The domain D(A) is equipped with the graph norm: xD(A) = xX + A(x)X . Since A is closed, one finds that (D(A), .D(A) ) is complete. This operator (A, D(A)) is called the generator of the C0 -semigroup. One can prove that the generator of a C0 -semigroup has a dense domain in X and is a closed operator [270, Corollary 1.2.5] (see Chap. 11 for the notion of closed operators). Note that the map: (12.1.9)

S(t) → A

is injective [270, Theorem 1.2.6]. The following proposition shows that computing S(t)x yields a solution of an evolution equation. Proposition 12.2. Let T ∈ R+ ∪ {+∞} and let x ∈ D(A). We have u(t) = S(t)x ∈ C 0 ([0, T ], D(A)) ∩ C 1 ([0, T ], X) and d u(t) + Au(t) = 0, 0 ≤ t ≤ T, u(0) = x. dt Moreover, S(t)x is the unique solution to (12.1.10) in C 0 ([0, T ], D(A)) ∩ C 1 ([0, T ], X). In addition, we have AS(t)x = S(t)Ax. (12.1.10)

Here, [0, T ] means [0, +∞) if T = +∞. Proof. Let x ∈ D(A). We write, for h > 0,     h−1 IdX −S(h) S(t)x = S(t)h−1 IdX −S(h) x →h→0+ S(t)Ax, as S(t) is bounded on X. Hence, we have S(t)x ∈ D(A) and AS(t)x = S(t)Ax by (12.1.7)–(12.1.8). Moreover, we have the following right derivative d+ dt S(t)x = −AS(t)x. To compute the left derivative, we write, for h > 0, h−1 (S(t)x − S(t − h)x) = S(t − h)h−1 (S(h)x − x). Since h−1 (S(h)x − x) → −Ax as h → 0+ and since (t, x) → S(t)x is continuous on [0, T ] × X, we find h−1 (S(t)x − S(t − h)x) → −S(t)Ax. h→0+

d S(t)x + AS(t)x = 0. From AS(t)x = S(t)Ax and the contiWe thus obtain dt nuity of t → S(t)y for all y ∈ X, we conclude that S(t)x ∈ C 0 ([0, T ], D(A)) and finally using the equation we have S(t)x ∈ C 1 ([0, T ], X). Uniqueness. Let u ∈ C 0 ([0, T ], D(A)) ∩ C 1 ([0, T ], X) be a solution to (12.1.10) satisfying u(0) = x. Let 0 < s ≤ T . For t ∈ [0, s], we set v(t) = S(s − t)u(t). With the first part, we have

d d v(t) = S(s − t) u(t) + S(s − t)Au(t) = 0. dt dt We thus find S(s)x = S(s)u(0) = v(0) = v(s) = u(s).



370

12. SOME ELEMENTS OF SEMIGROUP THEORY

We provide also the following result that shows that integration with respect to time yields a gain of regularity. Lemma 12.3. For x ∈ X and T > 0, we have ∫0T S(t)x dt ∈ D(A) and T

S(T )x − x + A ∫ S(t)x dt = 0. 0

Proof. For h > 0, we compute T T  Fh = h−1 (IdX −S(h)) ∫ S(t)x dt = h−1 ∫ S(t) − S(t + h) x dt 0

0

h

 = h−1 ∫ S(t)x − S(t)S(T )x dt. 0

With the continuity of t → S(t)x and t → S(t)S(T )x, the fundamental theorem of calculus yields the result, by (12.1.7)–(12.1.8).  We shall use the following version in which a smooth window function is introduced. Lemma 12.4. For x ∈ X and χ ∈ Cc∞ (0, ∞), we have ∞

∫ χ(t)S(t)x dt ∈ D(A)

and

0

∞

∞

0

0

A ∫ χ(t)S(t)x dt = ∫ χ (t)S(t)x dt.

We see that the result of Lemma 12.3 formally coincides with that of Lemma 12.4 in the case χ = 1(0,T ) . Proof. For h > 0, we compute ∞ ∞   Fh = h−1 (IdX −S(h)) ∫ χ(t)S(t)x dt = h−1 ∫ χ(t) S(t) − S(t + h) x dt. 0

0

Observe that we have ∞

∞

∞

0

h

0

∫ χ(t)S(t + h)x dt = ∫ χ(t − h)S(t)x dt = ∫ χ(t − h)S(t)x dt,

because of the support of χ. We thus obtain ∞  Fh = h−1 ∫ χ(t) − χ(t − h) S(t)x dt. 0

With the continuity of t → S(t)x, the Lebesgue dominated-convergence theorem yields ∞

lim Fh = ∫ χ (t)S(t)x dt.

h→0+

0

Consequently, by (12.1.7)–(12.1.8), we obtain the result.



Observe that if S(t) is a C0 -semigroup and z ∈ C, then ezt S(t) satisfies (12.1.5). The following proposition is then clear from what precedes. Proposition 12.5. Let S(t) be a C0 -semigroup and z ∈ C. Then ezt S(t) is also a C0 -semigroup and its generator is A − z IdX .

12.1. STRONGLY CONTINUOUS SEMIGROUPS

371

Note that, because of the uniqueness of the generator of a C0 -semigroup [270, Theorem 1.2.6], conversely, if A generates a C0 -semigroup, then A − z IdX is the generator of a C0 -semigroup, namely, ezt S(t). 12.1.2. The Hille–Yosida Theorem. The next natural question is to wonder if an unbounded operator on X is the generator of a C0 -semigroup. The Hille–Yosida theorem is central in the semigroup theory, providing a clear answer to this question. We refer to [270, Theorem 1.3.1] for a proof. Theorem 12.6. Let (A, D(A)) be a linear unbounded operator on a Banach space X. It generates a C0 -semigroup of contraction if and only if: (1) A is closed and D(A) is dense in X. (2) The resolvent set ρ(A) of A contains (−∞, 0), and we have the following estimate: Rλ (A)L (X) ≤ 1/|λ|,

λ < 0,

Rλ (A) = (λ IdX −A)−1 .

For the notions of closed operators, resolvent set, spectrum, and resolvent operator Rλ (A), we refer to Chap. 11. The previous result is limited to contraction C0 -semigroups. The following corollary provides a characterization of all generators of C0 -semigroups; we refer to [270, Theorem 1.5.3] for a proof. Corollary 12.7. Let (A, D(A)) be a linear unbounded operator on a Banach space X. It generates a C0 -semigroup S(t) such that S(t)L (X) ≤ M eωt , for some M ≥ 1 and ω ∈ R, if and only if: (1) A is closed and D(A) is dense in X. (2) The resolvent set ρ(A) of A contains (−∞, −ω), and we have the following estimate: Rλ (A)n L (X) ≤ M/|ω + λ|n , λ < −ω, n ∈ N∗ , Rλ (A) = (λ IdX −A)−1 . The Hille–Yosida theorem has the following simple consequence. Corollary 12.8. Let (A, D(A)) be the generator of a bounded C0 -semigroup S(t), that is, S(t)L (X) ≤ M , for t ≥ 0, for some M > 0. Then, its spectrum satisfies sp(A) ⊂ {z ∈ C; Re z ≥ 0}. Proof. Let b ∈ R and the C0 -semigroup eibt S(t) is generated by A − ib IdX . As eibt S(t) satisfies eibt S(t)L (X) ≤ M , for t ≥ 0, the conclusion follows from Corollary 12.7 in the case ω = 0.  12.1.3. The Lumer–Phillips Theorem. The Lumer–Phillips theorem provides another characterization of generators of contraction semigroups. Let X  be the dual space of X equipped with the strong topology (see Sect. 11.4 in Chap. 11). For x ∈ X, we set   (12.1.11) F (x) = φ ∈ X  ; φ(x) = φ, xX  ,X = x2X = φ2X  ,

372

12. SOME ELEMENTS OF SEMIGROUP THEORY

which is not empty by the Hahn–Banach theorem. Definition 12.9. A linear unbounded operator (A, D(A)) on X is said to be monotone (or accretive) if for all x ∈ D(A), x = 0, there exists φ ∈ F (x) such that Reφ, AxX  ,X ≥ 0. Definition 12.10. A linear unbounded operator (A, D(A)) on X is said to be maximal monotone if it is monotone and if moreover there exists λ0 > 0 such that the range of λ0 IdX +A, Ran(λ0 IdX +A), is equal to X. The Lumer–Phillips theorem reads as follows. Theorem 12.11. Let (A, D(A)) be a linear unbounded operator. It generates a C0 -semigroup of contraction if and only if: (1) A has a dense domain. (2) A is maximal monotone. A proof based on the Hille–Yosida theorem directly follows from Lemmata 12.13 and 12.14 given below. Remark 12.12. Observe that there is no need to assume that the operator A is closed in the converse part of the Lumer–Phillips theorem as in the Hille–Yosida theorem. In fact, as proven below, a maximal monotone operator is closed (see Lemma 12.14). In the case of a reflexive Banach space, the dense domain assumption may be dropped in the converse part of the Lumer–Phillips theorem: a maximal monotone operator has a dense domain; see [270, Theorem 1.4.6] (see also [90, Proposition 7.1] for the Hilbert space case). The next lemma gives a characterization of monotone operators. Lemma 12.13. An unbounded operator (A, D(A)) on X is monotone if and only if (12.1.12)

(λ IdX +A)xX ≥ λxX ,

x ∈ D(A) and λ > 0.

Proof. First, we assume that A is monotone. Let λ > 0 and x ∈ D(A). We may then write, for some φ ∈ F (x), λx2X ≤ λ Reφ, xX  ,X + Reφ, AxX  ,X = Reφ, λx + AxX  ,X ≤ φX  λx + AxX = xX λx + AxX , yielding (12.1.12). Second, we assume that (12.1.12) holds. Let x ∈ D(A) with x = 0. For λ > 0, we let φλ ∈ F (λx + Ax). By (12.1.12), we have λx + Ax = 0 and thus φλ = 0. We normalize it by setting ψλ = φλ /φλ X  . We then have λx + AxX = ψλ , λx + AxX  ,X . We may thus write, with (12.1.12), λxX ≤ (λ IdX +A)xX = ψλ , λx + AxX  ,X = λψλ , xX  ,X + ψλ , AxX  ,X .

12.2. DIFFERENTIABLE AND ANALYTIC SEMIGROUPS

373

As ψλ X  = 1, the conclusion is twofold: ψλ , AxX  ,X ≥ 0,

ψλ , xX  ,X ≥ xX − AxX /λ,

λ > 0.

1

X

is compact for the weak star topology by the Banach– As the unit ball of Alaoglu theorem, there exists ψ ∈ X  with ψX  ≤ 1 and an increasing ∗ sequence (λn )n that diverges to +∞ such that ψλn  ψ, implying ψ, AxX  ,X ≥ 0,

ψ, xX  ,X ≥ xX .

This yields ψX  = 1. We set φ = xX ψ, and we have φ ∈ F (x) and φ, AxX  ,X ≥ 0. As x = 0 is arbitrary in D(A), this yields that A is monotone.  The value of λ0 > 0 in Definition 12.10 is not of great significance. In fact, we have the following result. Lemma 12.14. Let A be a maximal monotone operator on X. Then, A is closed, and for all λ > 0, the operator λ IdX +A is bijective from D(A) onto X. Moreover, if λ > 0, its inverse, (λ IdX +A)−1 , is a bounded operator and we have the following estimation (λ IdX +A)−1 L (X) ≤ λ−1 . Proof. Let λ > 0. The injectivity of λ IdX +A follows from Lemma 12.13. As A is maximal monotone, there exists λ0 > 0 such that λ0 IdX +A is also surjective. Its inverse (λ0 IdX +A)−1 is thus well-defined on X. By Lemma 12.13, we have (λ0 IdX +A)−1 L (X) ≤ λ−1 0 . By the closed-graph theorem (see Theorem 11.1), the graph of (λ0 IdX +A)−1 is closed in X × X and thus so is the graph of A. We now prove that if λ IdX +A is surjective, then so is λ IdX +A for any λ such that 0 < λ < 2λ. By induction, starting with λ = λ0 we then reach the conclusion that λ IdX +A is onto for any λ > 0 and then the boundedness of its inverse follows from Lemma 12.13. Let λ, λ > 0 be such that λ IdX +A is onto and 0 < λ < 2λ. Let y ∈ X. We wish to find x ∈ X such that λ x + Ax = y. This reads λx + Ax = y + (λ − λ )x and thus we have x = (λ IdX +A)−1 (y + (λ − λ )x), meaning that we seek a fixed point for the bounded affine map H : x → (λ IdX +A)−1 (y + (λ − λ )x). By the computation above, we have (λ IdX +A)−1 L (X) ≤ 1/λ, we thus find H(x) − H(x )X ≤ |1 − λ /λ| x − x X . As 0 < |1 − λ /λ| < 1, the Banach contraction fixed point theorem applies. 

1The weak star topology is often referred to as the σ(E  , E) topology in E  (see, e.g.,

[90, 320]).

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12. SOME ELEMENTS OF SEMIGROUP THEORY

12.2. Differentiable and Analytic Semigroups Above, we have considered C0 -semigroups. If the continuity assumption of a semigroup S(t) is reinforced, say by assuming uniform continuity instead of strong continuity: S(t) − IdX L (X) → 0 as t → 0+ . Then, one can prove that the generator A of S(t)

is a bounded operator on X and that one simply has S(t) = exp(−tA) = n≥0 (−tA)n /n! (see, e.g., [270, Section 1.1]). Other regularity assumptions with respect to t can be made, yet not making the semigroup becoming a trivial exponential of a bounded operator. Definition 12.15. A C0 -semigroup S(t) is called differentiable for t > t0 if for all x ∈ X the map t → S(t)x is differentiable for t > t0 . d S(t)x = If x ∈ D(A), then t → S(t)x is differentiable for t ≥ 0 and dt −AS(t)x by Proposition 12.2. If the semigroup is differentiable for t > t0 , this means that this property extends to all x ∈ X if t > t0 . The following proposition states the properties of differentiable semigroups; see [270, Lemma 2.4.2] for a proof.

Proposition 12.16. Let S(t) be a differentiable semigroup for t > t0 . n Then for t > nt0 , n ∈ N, S(t) maps X in D(A  d n), and for all nx ∈ X, the map t → S(t)x is n times differentiable, and dt S(t)  = (−A) S(t) ∈ L (X).  n Moreover, t → S(t) ∈ C ((n + 1)t0 , +∞), L (X) . Note that if t0 = 0, then S(t) ∈ C k ((0, +∞), L (X, D(A ))), for any k,  ∈ N. Like many other semigroup properties, the differentiability of a semigroup can be characterized through a resolvent estimate; see [270, Theorem 2.4.8] for a proof. Theorem 12.17. Let S(t) be a C0 -semigroup that satisfies S(t)L (X) ≤ with A as its generator. The semigroup S(t) is differentiable for t > 0 if and only if for every b > 0 there exist a > 0 and C0 > 0 such that M eωt

Σb = {z ∈ C; Re z ≤ −a + b log | Im z|} ⊂ ρ(A), and z IdX −AL (X) ≤ C0 | Im z|

z ∈ Σb , Re z ≥ −ω.

We now consider analytic semigroups. For 0 < θ < π/2, we set Σθ = {z; | arg(z)| ≤ θ} ∪ {0}. Definition 12.18. A map S : Σθ → L (X) is said to be an analytic semigroup on Σθ if (1) z → S(z) is analytic on Σθ \ {0} in the topology of L (X). (2) S(0) = IdX and lim S(z)x = x for all x ∈ X. z→0 z∈Σθ \{0}

12.3. MILD SOLUTION OF THE INHOMOGENEOUS CAUCHY PROBLEM

375

(3) S(z1 + z2 ) = S(z1 )S(z2 ), for all z1 , z2 ∈ Σθ . The following result yields a characterization of an analytic semigroup; we refer to [270, Theorem 2.5.2] for a proof. Theorem 12.19. Let S(t) be a C0 -semigroup on X and let A be its generator such that 0 ∈ ρ(A) and S(t)L (X) ≤ M , for some M > 0. The following statements are equivalent. (1) There exists 0 < θ < π/2 such that S(t) can be extended as an analytic semigroup on Σθ and is such that S(z)L (X) ≤ M  for z ∈ Σθ for some M  > 0. (2) There exists C > 0 such that σ + iτ ∈ ρ(A)

and (σ + iτ − A)−1 L (X) ≤ C/|τ |,

σ < 0, τ = 0.

(3) There exist 0 < a < π/2 and C > 0 such that z ∈ ρ(A)

and (z − A)−1 L (X) ≤ C/|z|,

if z = 0 and π/2 − a < arg(z) < 3π/2 + a. (4) The semigroup S(t) is differentiable for t > 0 and there exists C > 0 such that AS(t)L (X) ≤ C/t,

t > 0.

12.3. Mild Solution of the Inhomogeneous Cauchy Problem In what precedes, we have seen that semigroups can be used to solve the abstract homogeneous Cauchy problem in a Banach space X: d u(t) + Au(t) = 0 for t > 0 u(0) = u0 . dt In fact, if A generates a semigroup on X and if u0 ∈ D(A), then the unique solution in C 0 ([0, +∞), D(A)) is given by u(t) = S(t)u0 by Proposition 12.2. Note that u(t) ∈ C 1 ([0, +∞), X) and moreover the equation is even satisfied for t ≥ 0. In the case of a self-adjoint generator on a Hilbert space this can be extended to x ∈ X by Corollary 12.27. Then, the unique solution is in C 0 ([0, +∞), X) ∩ C k ((0, ∞), D(A )), k,  ∈ N, and note that the equation is only satisfied for t > 0. We are now interested into solving a nonhomogeneous abstract Cauchy problem of the form: d u(t) + Au(t) = f (t) ∈ X for t > 0 and u(0) = u0 . dt Here, A is assumed to generate a C0 -semigroup on X. A classical solution is a function

(12.3.1)

u ∈ C ([0, ∞), X) ∩ C 1 ((0, ∞); X),

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12. SOME ELEMENTS OF SEMIGROUP THEORY

such that u(t) ∈ D(A) for t > 0 and that satisfies (12.3.1), that is, both the equation for t > 0 and the initial condition. If u(t) is such a solution, we choose T > 0 and we set w(t) = S(T − t)u(t). We have, for t > 0, d  d w(t) = S(T − t) u(t) + Au(t) = S(T − t)f (t). dt dt 1 If f|(0,T ) ∈ L (0, T ; X), we find T

T

0

0

u(T ) = w(T ) = w(0) + ∫ S(T − t)f (t) dt = S(T )u0 + ∫ S(T − t)f (t) dt. This is precisely the Duhamel formula in an abstract setting. We thus have the following uniqueness result. Proposition 12.20. If f ∈ L1loc (0, ∞; X) and if u is a classical solution to (12.3.1), it is given by t

u(t) = S(t)u0 + ∫ S(t − s)f (s) ds, 0

t ≥ 0.

We are thus naturally led to the following definition. Definition 12.21. Let f ∈ L1loc (0, ∞; X) and u0 ∈ X. The function: t

u(t) = S(t)u0 + ∫ S(t − s)f (s) ds, 0

t≥0

is such that u ∈ C ([0, ∞); X) and is called the mild solution of the inhomogeneous abstract Cauchy problem (12.3.1). If f ∈ L1loc (0, ∞; D(A)) ∩ C ((0, ∞); X) and u0 ∈ D(A), then the mild solution is a classical solution as can be readily checked. In Sect. 10.2, we are interested in the solution of inhomogeneous Cauchy problems in the case of parabolic equations. In fact, in such case, we provide the proper functional framework to give a sense to the mild solution and to address the question of uniqueness. Therefore, we do not provide further material on the general solution of inhomogeneous abstract Cauchy problems. 12.4. The Case of a Hilbert Space In the case the space X is a Hilbert space. It can be identified with its dual space X  by the Riesz theorem. Out of habit, we shall denote by H the Hilbert rather than by X. Then if x ∈ H, we have, for the set defined in (12.1.11), F (x) = {x}. We then have the following definition in agreement with Definition 12.9. Definition 12.22. A linear unbounded operator (A, D(A)) on H is said to be monotone (or accretive) if for all x ∈ D(A) one has Re(Ax, x)H ≥ 0. In the Hilbert case the Lumer–Phillips theorem reads, in agreement with Remark 12.12.

12.4. THE CASE OF A HILBERT SPACE

377

Theorem 12.23. Let A be a linear unbounded operator. It generates a C0 -semigroup of contraction if and only if it is maximal monotone. We recall that a maximal monotone operator is maximal such that Ran(λ0 IdH +A) = H for some λ0 > 0 (see Definition 12.10). Proposition 12.24. Let (A, D(A)) be an unbounded operator on H that generates a C0 -semigroup S(t). Then, the operator (A∗ , D(A∗ )) generates a semigroup Σ(t) and Σ(t) = S(t)∗ . Recall that if (A, D(A)) generates a semigroup, then it is closed and densely defined. This allows one to properly define its adjoint operator (see Sects. 11.4 and 11.6). Proof. By Proposition 11.16, (A∗ , D(A∗ )) is closed and densely defined. By Corollary 12.7, consequence of the Hille–Yosida theorem (Theorem 12.6), there exist M ≥ 1 and ω ∈ R such that, for λ < −ω, the operator Rλ (A) = (λ IdX −A)−1 exists and is bounded on H and moreover Rλ (A)n L (H) ≤ M/|ω + λ|n . n  By Proposition 11.17, the operator Rλ (A∗ )n = (λ IdX −A∗ )−1 exists and we have Rλ (A∗ )n L (H) = Rλ (A)n L (H) ≤ M/|ω + λ|n . Hence, by Corollary 12.7, the operator (A∗ , D(A∗ )) generates a C0 -semigroup. We denote by Σ(t) the semigroup generated by A∗ . Let t > 0, and let x ∈ D(A) and x ∈ D(A∗ ). For s ∈ [0, t], we introduce the function: h(s) = (S(t − s)x, Σ(s)x )H . We compute h (s) = −(AS(t − s)x, Σ(s)x )H + (S(t − s)x, A∗ Σ(s)x )H = 0, as S(t − s)x ∈ D(A) and Σ(s)x ∈ D(A∗ ). We thus have h(t) = h(0), that is, (x, Σ(t)x )H = (S(t)x, x )H ,

x ∈ D(A), x ∈ D(A∗ ).

Since the domains of the two operators are dense in H, we conclude that we have (x, Σ(t)x )H = (S(t)x, x )H , which yields the conclusion.

x, x ∈ H, 

In the case of a self-adjoint operator (A, D(A)), that is, D(A∗ ) = D(A) and A = A∗ , we have the following consequence. Corollary 12.25. Let (A, D(A)) be a self-adjoint operator on H. If it generates a C0 -semigroup S(t) on H, then S(t)∗ = S(t). If the operator (A, D(A)) is self-adjoint, the conclusions of the Lumer– Phillips theorem are even stronger; we refer to [90, Theorem 7.7] for a proof.

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12. SOME ELEMENTS OF SEMIGROUP THEORY

Theorem 12.26. Let A be a linear unbounded self-adjoint operator on a Hilbert space H. It generates an analytic semigroup S(t) of contraction if and only if it is maximal monotone. From Proposition 12.16 and Theorem 12.19, we have the following properties. Corollary 12.27. Let S(t) be the analytic semigroup of contraction generated by A unbounded and self-adjoint on H a Hilbert space. Then: (1) We have t → S(t) ∈ C k ((0, ∞); L (H, D(A ))), for any k,  ∈ N. (2) There exists C > 0 such that AS(t)L (H) ≤ C/t. (3) For all x ∈ H, there exists a unique solution in C 0 ([0, ∞); H) ∩ C 1 ((0, ∞); H) ∩ C 0 ((0, ∞); D(A)) d to dt u(t) + Au(t) = 0, for t > 0. It is given by u(t) = S(t)x. (4) Moreover, t → S(t)x ∈ C 0 ([0, ∞); H)∩C k ((0, ∞), D(A )), k,  ∈ N.

The important property, as compared to the result of Theorems 12.11 and 12.23, lies in the fact that here the function u(t) = S(t)x solves the d u(t) + Au(t) = 0 for t > 0 not only for x ∈ D(A) but also for equation dt x ∈ H, as the semigroup is differentiable for t > 0. The following result is often handy to assess that an operator is selfadjoint in the framework of semigroup theory. We refer to [90, Proposition 7.6]. Proposition 12.28. Let A be an unbounded operator on H a Hilbert space that is symmetric, in the sense that (Ax, y)H = (x, Ay)H for x, y ∈ D(A). If A is maximal monotone, then A is self-adjoint, that is, D(A∗ ) = D(A) and A = A∗ . The definitions of the adjoint operator and of its domain for an unbounded operator A with a dense domain are recalled in Sects. 11.4 and 11.6.

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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, 40 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 I, PNLDE Subseries in Control 97, https://doi.org/10.1007/978-3-030-88674-5

401

402

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, 409 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

403

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

404

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

405

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, 269 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, 407 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, 40 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 I, PNLDE Subseries in Control 97, https://doi.org/10.1007/978-3-030-88674-5

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408

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), 409 C  Trs M, 411 G Geometry differential geometry ∧, 407 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

409

410

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

411