Elliptic Problems in Domains with Piecewise Smooth Boundaries 9783110848915, 3110135221, 9783110135220

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de Gruyter Expositions in Mathematics 13

Editors

Ο. H. Kegel, Albert-Ludwigs-Universität, Freiburg V. P. Maslov, Academy of Sciences, Moscow W. D. Neumann, Ohio State University, Columbus R.O.Wells, Jr., Rice University, Houston

de Gruyter Expositions in Mathematics

1

The Analytical and Topological Theory of Semigroups, Κ. H. Hofmann, J. D. Lawson, J. S. Pym (Eds J

2

Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues

3

The Stefan Problem, A. M. Meirmanov

4

Finite Soluble Groups, K. Doerk, T. O. Hawkes

5

The Riemann Zeta-Function, A.A.Karatsuba,

6

Contact Geometry and Linear Differential Equations, V. R. Nazaikinskii, V. E. Shatalov, B. Yu. Sternin

7

Infinite Dimensional Lie Superalgebras, Yu. A. Bahturin, A. A. Mikhalev, V. M. Petrogradsky, Μ. V. Zaicev

8

Nilpotent Groups and their Automorphisms, Ε. I. Khukhro

9

Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug

S.M. Voronin

10

The Link Invariants of the Chern-Simons Field Theory, E.Guadagnini

11

Global Affine Differential Geometry of Hypersurfaces, A.-M. Li, U. Simon, G. Zhao

12

Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions, K. Hulek, C. Kahn, S. H. Weintraub

Elliptic Problems in Domains with Piecewise Smooth Boundaries by Sergey A. Nazarov Boris A. Plamenevsky

W DE Walter de Gruyter · Berlin · New York 1994

Authors Sergey A . N a z a r o v D e p a r t m e n t o f Mathematics State Maritime A c a d e m y

Boris A. Plamenevsky Department o f Mathematical Physics State University o f St. Petersburg

199026, St. Petersburg, Russia

198904 St. Petersburg, Russia

1991 Mathematics Keywords:

©

Subject

Classification:

35-02; 35Jxx; 73Cxx; 7 3 M 2 5 ; 7 6 D x x , 7 8 - X X

Elliptic boundary value problems, conical points and edges, property, asymptotics, radiation condition, crack

Fredholm

Printed on acid-free paper which falls within the guidelines of the A N S I to ensure permanence and durability.

Library of Congress Cataloging-in-Publication

Data

Nazarov, S. A. Elliptic problems in domains with piecewise smooth boundaries / by S. A. Nazarov, Boris A. Plamenevsky. p. cm. — (De Gruyter expositions in mathematics ; 13) Includes bibliographical references and index. ISBN 3-11-013522-1 1. Differential equations, Elliptic. 2. Boundary value problems. I. Plamenevskn, Β. Α. II. Title. III. Series. QA377.N387 1994 515'.353 —dc20 94-7660 CIP

Die Deutsche Bibliothek

— Cataloging-in-Publication

Data

Nazarov, Sergej Α.: Elliptic problems in domains with piecewise smooth boundaries / by Sergey A. Nazarov ; Boris A. Plamenevsky. — Berlin ; New York : de Gruyter, 1994 (De Gruyter expositions in mathematics ; 13) ISBN 3-11-013522-1 NE: Plamenevskij, Boris A.:; G T

© Copyright 1994 by Walter de Gruyter & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typesetting: ASCO Trade Ltd. Hong Kong. Printing: Gerike G m b H , Berlin. Binding: Lüderitz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.

Contents

Introduction

1

Chapter 1 Elliptic boundary value problems in domains with smooth boundary

4

§ 1. Elliptic boundary value problems § 2. Elliptic problems with a parameter

4 11

Chapter 2 Dirichlet and Neumann problems for the Laplace operator in plane domains with corner points at the boundary

21

§ 1. Generalized solutions of the Dirichlet and Neumann problems in domains with corner points. Deviations from the "usual" properties of solutions § 2. Solvability and asymptotics of solutions of model problems in a strip and in an angle § 3. The Dirichlet problem in a bounded domain with a corner point § 4. The Neumann problem in a bounded domain with a corner point Chapter 3 General elliptic boundary value problems in a cylinder and in a cone

21 24 31 38

43

§ 1. Solvability and asymptotics of solutions of boundary value problems in a cylinder with the coefficients constant along the axis § 2. Calculation of the coefficients in the asymptotics of the solution § 3. Asymptotics of the solutions of boundary value problems in a cylinder with right-hand side of a special form § 4. Solvability and asymptotics of solutions of the boundary value problems with periodic coefficients § 5. Boundary value problems in a cone § 6. Estimates and asymptotics of solutions in Lp and Holder classes § 7. Fundamental solutions of the boundary value problem in a cone § 8. Two examples: The Dirichlet problem and the oblique derivative • problem

92

Chapter 4 General elliptic problems in domains with conical points

97

§ 1. The Fredholm property § 2. Asymptotics of solutions near conical points

43 49 53 55 63 74 82

97 105

vi

Contents

§ 3. Expressions for the coefficients in asymptotic formulas. Properties of the index §4. Asymptotics of fundamental solutions § 5. Boundary value problems in spaces with nonhomogeneous norms

109 117 127

Chapter 5 Self-adjoint problems in domains with outlets to infinity

144

§ 1. § 2. § 3. §4. § 5. § 6. § 7. § 8.

Self-adjoint problems in domains with cylindrical outlets to infinity Special choices of Jordan chains of self-adjoint operator pencils Waves, scattering matrices and intrinsic radiation conditions Energetic solutions and the polarization matrix The extensions of the symmetric operator On a problem of the scattering of electro-magnetic waves Elasticity problems in domains having cylindrical outlets to infinity Stokes and Navier-Stokes problems in domains with cylindrical outlets to infinity

Chapter 6 Self-adjoint problems in domains with conical points. Applications § 1. §2. § 3. § 4.

144 148 156 161 175 181 192 214

222

Self-adjoint problems. Properties of their pencils The generalized Green formula, radiation conditions Self-adjoint extensions in spaces with weighted norms Boundary value problems in domains with compact complement. The polarization matrix § 5. Asymptotic analysis of the problem in domains with small holes § 6. Two problems in plates theory

241 248 263

Chapter 7 Applications to crack theory

271

§1. § 2. § 3. § 4.

Stress singularities at a crack tip The rupture criterion The Cherepanov-Rice integral On the structure of the spectrum of the pencil generated by a self-adjoint problem in a plane domain with a crack

222 229 236

271 279 286 290

Chapter 8 Elliptic problems in domains with smooth edges

298

§ 1. § 2. § 3. §4. § 5. § 6.

298 302 308 311 319 326

Statement of the problem. Model problems Analysis of the model problems The Fredholm property of the elliptic problem in a bounded domain The Dirichlet and the Neumann problems for self-adjoint systems L p -estimates of solutions Estimates of solutions in weighted Holder classes

Contents

vii

Chapter 9 Elliptic problems on manifolds with intersecting edges

337

§1. § 2. §3. §4. § 5. § 6.

337 344 351 354 360 363

Manifolds of the class Φ Differential operators on class I) manifolds The function spaces Statement of the boundary problems. Model problems Boundary value problems on class D manifolds The Dirichlet and Neumann problems for self-adjoint systems

Chapter 10 Asymptotics of solutions of the Dirichlet problem for the Laplace operator in a three-dimensional domain having edges on the boundary

384

§1. Asymptotics of solution near smooth edges § 2. Asymptotics of solutions near the singular point having the type of the vertex of a polyhedron § 3. Formulas for coefficients in asymptotics of solutions near an edge

384 388 392

Chapter 11 The asymptotics of solutions of general problems near edges

396

§ 1. Guidelines for constructing asymptotics. Auxiliary assertions § 2. The asymptotics of solutions near an edge (in case of a smooth right-hand side) § 3. Asymptotics of solutions near an edge (general case) § 4. Corollaries of asymptotic formulas

404 410 430

Chapter 12 Self-adjoint problems with radiation conditions on the edges

441

§ 1. Setting of boundary value problems and information about the asymptotics of solutions § 2. The generalized Green formula. Fredholm property of the problem with radiation conditions § 3. Solvability conditions for the problem with radiation conditions in terms of the generalized Green formula § 4. Intrinsic radiation conditions on the edge § 5. Energetic radiation conditions. The polarization operator on the edge § 6. The scattering operator. The formula for the increment of the dimension of the kernel of the operator of the problem generated by the variation of the weight factor § 7. Sobolev problems § 8. The three-dimensional problem of crack mechanics Bibliographical notes References Index List of spaces

397

441 444 454 465 477

485 497 504 515 517 523 525

Introduction

In this book we study general elliptic boundary value problems in domains having "edges" of different dimensions on the boundary, e.g., polygons, cones, lenses and polyhedrons are domains of such kind. Discontinuities of coefficients of the operators along edges are allowed. We discuss solvability of the problems and obtain asymptotic formulas for solutions near singularities of the boundary and of the coefficients. Both results and methods find multiple applications in mechanics and solid electrodynamics, in various topics of the asymptotic theory of partial differential equations and in the calculus of approximations. In order to read the book one requires knowledge of material contained in standard university courses of functional analysis and differential equations. The theory to which this monograph is devoted was developed through two previous decades (we start our countdown from V.A. Kondrat'ev's fundamental paper [38]). We have selected the material for the book in order to be able to present the main "working" apparatus of the theory in a sufficiently detailed way. The solutions of elliptic problems lose their smoothness at the edges of the boundary. This circumstance plays a decisive role: questions appear about the behaviour of solutions near edges, about the choice of special function spaces (with weight norms) where the operator possesses "good" properties (i.e. is Fredholm). Essentially the matter reduces itself to the study of model problems with frozen coefficients in the v-dimensional cone IKV, in the wedge IKV χ R B-V , etc. The results are some concepts connected with the spectrum of some operator pencils CaAi—• 31(A) (i.e. polynomials with operator coefficients); the asymptotic formulas for solutions contain eigenvalues and associated vectors of the pencils. Aside from such local properties of solutions some global properties of the problem are discussed, namely the index of the operator of our problem, its dependence of the choice of function spaces, global influence of data of the problem on asymptotics of solutions. In the first chapter we give a summary of definitions and basic facts of the theory of elliptic boundary value problems in domains with smooth boundary. In addition we discuss here elliptic problems with a parameter, i.e. operator pencils, and we make a special choice of chains of eigen- and associated vectors of mutually adjoint pencils. Chapters 2 - 7 are devoted to problems in domains having "conic" points ("zero dimensional edges") on the boundary. In the second chapter several simple examples are used in order to motivate and explain some aspects of the general theory. Model problems in cones and cylinders for systems of equations are studied in Chapter 3. In Chapter 4 we prove the theorem on Fredholm property of operators of boundary value problems in bounded domains having conic points. In the same chapter the asymptotic formulas for solutions are derived. The general set-up of these formulas

2

Introduction

was essentially determined in Chapter 3, and in Chapter 4 the question is to find the coefficients in these asymptotics (functionals of the right-hand side of the problem). The significance of these computations particularly manifests itself later in index theorems and in deriving asymptotic formulas for fundamental solutions (Green functions). By addressing self-adjoint problems in Chapter 5, we consider domains having cylindrical outlets to infinity and give new problem settings that are connected with radiation conditions and extensions of symmetric operators. Similar questions for domains with conic points are discussed in Chapter 6. In § 5 of this chapter the theory given before is applied to justify the method of matched asymptotic expansions; in the same chapter we explain the choice of self-adjoint extension of symmetric elliptic operators, justified from a physical point of view. Chapter 7 is devoted to applications concerning crack theory. Using the results of the preceding chapters we derive known fracture criteria. An important role in Chapters 5 - 7 is played by coefficients in asymptotics of solutions, which are assigned a physical meaning in particular applications (scattering matrix in diffraction theory, stress intensity factors in crack mechanics, capacity and polarization tensor in electrostatics, adjoint mass tensor in fluid dynamics etc.). The second part of the book, Chapters 8-12, is devoted to boundary value problems in domains with multidimensional singularities. First we consider smooth closed edges and prove theorems on the Fredholm property of the problems and on properties of solutions (Chapter 8). The same questions for domains with intersecting edges of different dimensions are set forth in Chapter 9. In the general theory one has to introduce the condition of unique solvability of the model problems arising by localization (freezing coefficients); this condition is necessary and sufficient for the original boundary value problem to be Fredholm in a suitable weight class. In some cases it is possible to verify this condition explicitly. Thus, in §6 of Chapter 9 the study of Dirichlet and Neumann problems for a wide class of selfadjoint systems is treated completely without additional hypotheses. Chapter 10, where we give asymptotics near edges of solutions of Dirichlet problems for the Laplace operator, precedes the derivation of asymptotic expansions for general problems. One of the essential stages of studying asymptotics consists of the exact description of differential properties of coefficients in asymptotics, which are now functions on the edge. In general these coefficients turn out to be not sufficiently smooth (in order to keep for the remainder term the same smoothness that the solution possesses). One therefore has to change the form of expansions and use a special smoothing continuation of coefficients inside the domain. This is the topic of Chapter 11. In the final chapter we consider self-adjoint elliptic problems in domains with smooth edges. The correct problem setting (one leading to Fredholm operators) is ensured by imposing radiation conditions on the edges. The operators introduced describe scattering and polarization at the edges. The variation of the dimension of the kernel (or cokernel) of the original problem with change of the weight exponent is connected to properties of the spectrum of the corresponding scattering operator. The size of this book exceeds almost twice that of its Russian prototype [95] published in 1991. The list of literature given at the end of the book is far from

Introduction

3

comprehensive; in the review [41] and in the monographs [31,18,46] the reader can find additional bibliographical hints. The authors thank Professor Ilppo Simo Louhivaara for his friendly and effective help on the final stage of the preparation of this book.

Chapter 1

Elliptic boundary value problems in domains with smooth boundary

This introductory chapter contains material which will be constantly used throughout the book. Here we introduce the elliptic boundary value problems for systems of differential equations and state basic results about the operators of such problems. Then we discuss elliptic problems with a complex parameter, recall some definitions connected with the spectrum of operator pencils. Biorthogonality and normalization relations are proved for Jordan chains consisting of eigenvectors and associated vectors.

§ 1. Elliptic boundary value problems 1. The problem for an elliptic equation In the Euclidean space R" we consider a homogeneous linear differential operator L{DX) = Σα^Ω", |α| = I, with constant coefficients; here α = ( a j , . . . , ^ ) is a multiindex, |a| = ( * ! + · · · + a„, / = ordL, Dx = —idx, dx — (d/dxl,...,d/dx„). The operan tor L is called elliptic if and only if L(£) φ 0 for ξ e R \0. If η ^ 3 then ellipticity implies that I = 2m is even and the polynomial 1i—• L(£ + ίη) has m roots (counting their multiplicities) in the half-plane {t e C: Imi > 0} for all linearly independent vectors ξ, η e R". For n = 2 this assertion may be wrong and its validity is postulated in addition. An elliptic operator is called strongly elliptic if there exists a positive constant cL and a number 6L e [0,2π) such that Re{exp(/0 t )L(£)} ^ cL\ξ\2η> for all ξ e IR". If the coefficients of the operator are real, then ellipticity implies strong ellipticity. We denote by Ω a domain in IR" with compact closure Ω and smooth (of class C°°) boundary · ·· > is the operator of the elliptic boundary value problem in Ω while ord 36i ^ 2m - 1 and the system {3ix,...,3im} is normal at every point of