Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains

, by Mazia, V. G.; Nazarov, Serguei; Plamenevskii, Boris; Heinig, Georg; Posthoff, Christian

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ISBN: 9783764363970 | 3764363975
Cover: Hardcover
Copyright: 5/1/2000

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For the first time in the mathematical literature, this two-volume work introduces a unified and general approach to the subject. To a large extent, the book is based on the authors work, and has no significant overlap with other books on the theory of elliptic boundary value problems.

Preface

xxi

Part I Boundary Value Problems for the Laplace Operator in Domains Perturbed Near Isolated Singularities

Dirichlet and Neumann Problems for the Laplace Operator in Domains with Corners and Cone Vertices

Boundary Value Problems for the Laplace Operator in a Strip

(5)

The Dirichlet problem

(2)

The complex Fourier transform

(1)

Asymptotics of solution of the Dirichlet problem

(1)

The Neumann problem

(1)

Final remarks

(1)

Boundary Value Problems for the Laplace Operator in a Sector

(3)

Relationship between the boundary value problems in a sector and a strip

(2)

The Dirichlet problem

(1)

The Neumann problem

(1)

The Dirichlet Problem in a Bounded Domain with Corner

(19)

Solvability of the boundary value problem

(2)

Particular solutions of the homogeneous problem

(2)

Asymptotics of solution

(2)

A domain with a corner outlet to infinity

(1)

Asymptotics of the solutions for particular right-hand sides

(4)

The Dirichlet problem for the operator Δ - 1

(4)

The Dirichlet problem in a domain with piecewise smooth boundary

(4)

The Neumann Problem in a Bounded Domain with a Corner

(4)

Boundary Value Problems for the Laplace Operator in a Punctured Domain and the Exterior of a Bounded Planar Domain

(3)

Dirichlet and Neumann problems in a punctured planar domain

(2)

Boundary value problems in the exterior of a bounded domain

(1)

Boundary Value Problems in Multi-Dimensional Domains

(7)

A domain with a conical point

(2)

A punctured domain

(1)

Boundary value problems in the exterior of a bounded domain

(4)

Dirichlet and Neumann Problems in Domains with Singularly Perturbed Boundaries

The Dirichlet Problem for the Laplace Operator in a Three-Dimensional Domain with Small Hole

(8)

Domains and boundary value problems

(1)

Asymptotics of the solution. The method of compound expansions

(2)

Asymptotics of the solution. The method of matched expansions

(4)

Comparison of asymptotic representations

(1)

The Dirichlet Problem for the Operator Δ - 1 in a Three-Dimensional Domain with a Small Hole

(2)

Mixed Boundary Value Problems for the Laplace Operator in a Three-Dimensional Domain with a Small Hole

(5)

The boundary value problem with Dirichlet condition at the boundary of the hole

(1)

First version of the construction of asymptotics

(2)

Second version of the construction of asymptotics

(2)

The boundary value problem with the Neumann condition at the boundary of the gap

(1)

Boundary Value Problems for the Laplace Operator in a Planar Domain with a Small Hole

(8)

Dirichlet problem

(4)

Mixed boundary value problems

(3)

The Dirichlet Problem for the Operator Δ - 1 in a Domain Perturbed Near a Vertex

(12)

Formulation of the problem

(1)

The first terms of the asymptotics

(3)

Admissible series

(1)

Redistribution of discrepancies

(1)

The set of exponents in the powers of ϵ, r, and ρ

(7)

Part II General Elliptic Boundary Value Problems in Domains Perturbed Near Isolated Singularities of the Boundary

Elliptic Boundary Value Problems in Domains with Smooth Boundaries, in a Cylinder, and in Domains with Cone Vertices

Boundary Value Problems in Domains with Smooth Boundaries

(13)

The operator of an elliptic boundary value problem

(1)

Elliptic boundary value problems in Sobolev and Holder spaces

(3)

The adjoint boundary value problem (the case of normal boundary conditions)

(1)

Adjoint operator in spaces of distributions

(1)

Elliptic boundary value problems depending on a complex parameter

(4)

Boundary value problems for elliptic systems

(3)

Boundary value problems in cylinders and cones

(14)

Solvability of boundary value problems in cylinders: the case of coefficients independent of t

(3)

Asymptotics at infinity of solutions to boundary value problems in cylinders with coefficients independent of t

(2)

Solvability of boundary value problems in a cone

(2)

Asymptotics of the solutions at infinity and near the vertex of a cone for boundary value problems with coefficients independent of r

(1)

Boundary value problems for elliptic systems in a cone

100

(4)

Asymptotics of the solution for the right-hand side given by an asymptotic expansion

104

(2)

Boundary Value Problems in Domains with Cone Vertices

106

(9)

Statement of the problem

106

(1)

Asymptotics of the solution near a cone vertex

107

(2)

Formulas for coefficients in the asymptotics of solution (under simplified assumptions)

109

(1)

Formula for coefficients in the asymptotics of solution (general case)

110

(3)

Index of the boundary value problem

113

(2)

Asymptotics of Solutions to General Elliptic Boundary Value Problems in Domains Perturbed Near Cone Vertices

Formulation of the Boundary Value Problems and some Preliminary Considerations

115

(5)

The domains

115

(1)

Admissible scalar differential operators

116

(1)

Limit operators

117

(1)

Matrices of differential operators

118

(1)

Boundary value problems

118

(1)

Function spaces with norms depending on the parameter ϵ

118

(2)

Transformation of the Perturbed Boundary Value Problem into a System of Equations and a Theorem about the Index

120

(10)

The limit operator

120

(1)

Reduction of the problem to a system

121

(3)

Reconstruction of the original problem from the system

124

(3)

Fredholm property for the operator of the boundary value problem in a domain with singularly perturbed boundary

127

(1)

On the index of the original problem

127

(3)

Asymptotic Expansions of Data in the Boundary Value Problem

130

(10)

Asymptotic expansion of the coefficients and the right-hand sides

131

(1)

Asymptotic formulas for solutions of the limit problems

132

(1)

Asymptotic expansions of operators of the boundary value problem

133

(1)

Preliminary description of algorithm for construction of the asymptotics of solutions

134

(3)

The set of exponents in asymptotics of solutions of the limit problems

137

(1)

Formal expansion for the operator in powers of small parameter

138

(2)

Construction and Justification of the Asymptotics of Solution of the Boundary Value Problem

140

(17)

The problem in matrix notation

140

(1)

Auxiliary operators and their properties

141

(1)

Formal asymptotics of the solution in the case of uniquely solvable limit problems

142

(2)

A particular basis in the cokernel of the operator Mo

144

(5)

Formal solution in the case of non-unique solvability of the limit problems

149

(5)

Asymptotics of the solution of the singularly perturbed problem

154

(3)

Variants and Corollaries of the Asymptotic Theory

Estimates of Solutions of the Dirichlet Problem for the Helmholtz Operator in a Domain with Boundary Smoothened Near a Corner

157

(4)

Sobolev Boundary Value Problems

161

(6)

General Boundary Value Problem in a Domain with Small Holes

167

(6)

Problems with Non-Smooth and Parameter Dependent Data

173

(9)

The case of a non-smooth domain

173

(2)

The case of parameter dependent auxiliary problems

175

(2)

The case of a parameter independent domain

177

(5)

Non-Local Perturbation of a Domain with Cone Vertices

182

(7)

Perturbations of a domain with smooth boundary

182

(2)

Regular perturbation of a domain with a corner

184

(2)

A non-local singular perturbation of a planar domain with a corner

186

(3)

Asymptotics of Solutions to Boundary Value Problems in Long Tubular Domains

189

(12)

The problem

189

(1)

Limit problems

190

(2)

Solvability of the original problem

192

(1)

Expansion of the right-hand sides and the set of exponents in the asymptotics

193

(2)

Redistribution of defects

195

(2)

Coefficients in the asymptotic series

197

(1)

Estimate of the remainder term

198

(2)

Example

200

(1)

Asymptotics of Solutions of a Quasi-Linear Equation in a Domain with Singularly Perturbed Boundary

201

(16)

A three-dimensional domain with a small gap

202

(5)

A planar domain with a small gap

207

(6)

A domain smoothened near a corner point

213

(4)

Bending of an Almost Polygonal Plate with Freely Supported Boundary

217

(11)

Boundary value problems in domains with corners

219

(1)

A singularly perturbed domain and limit problems

220

(1)

The principal term in the asymptotics

221

(2)

The principal term in the asymptotics (continued)

223

(5)

Part III Asymptotic Behaviour of Functionals on Solutions of Boundary Value Problems in Domains Perturbed Near Isolated Boundary Singularities

Asymptotic Behaviour of Intensity Factors for Vertices of Corners and Cones Coming Close

Dirichlet's Problem for Laplace's Operator

228

(4)

Statement of the problem

228

(1)

Asymptotic behaviour of the coefficient C+ϵ

229

(1)

Justification of the asymptotic formula for the coefficient C+ϵ

230

(1)

The case g ≠ 0

231

(1)

The two-dimensional case

231

(1)

Neumann's Problem for Laplace's Operator

232

(3)

Statement of the problem

232

(1)

Boundary value problems

232

(2)

The case of disconnected boundary

234

(1)

The case of connected boundary

235

(1)

Intensity Factors for Bending of a Thin Plate with a Crack

235

(8)

Statement of the problem

235

(1)

Clamped cracks (The asymptotic behaviour near crack tips)

236

(1)

Fixedly clamped cracks (Asymptotic behaviour of the intensity factors)

237

(1)

Freely supported cracks

238

(2)

Free cracks (The asymptotic behaviour of solution near crack vertices)

240

(1)

Free cracks (The asymptotic behaviour of intensity factors)

240

(3)

Antiplanar and Planar Deformations of Domains with Cracks

243

(8)

Torsion of a bar with a longitudinal crack

243

(2)

The two-dimensional problem of the elasticity theory in a domain with collinear close cracks

245

(6)

Asymptotic Behaviour of Energy Integrals for Small Perturbations of the Boundary Near Corners and Isolated Points

Asymptotic Behaviour of Solutions of the Perturbed Problem

251

(10)

The unperturbed boundary value problem

251

(3)

Perturbed problem

254

(1)

The second limit problem

254

(2)

Asymptotic behaviour of solutions of the perturbed problem

256

(3)

The case of right-hand sides localized near a point

259

(2)

Asymptotic Behaviour of a Bilinear Form

261

(6)

The asymptotic behaviour of a bilinear form (the general case)

261

(3)

Asymptotic behaviour of a bilinear form for right-hand sides localized near a point

264

(2)

Asymptotic behaviour of a quadratic form

266

(1)

Asymptotic Behaviour of a Quadratic Form for Problems in Regions with Small Holes

267

(10)

Statement of the problem

267

(1)

The case of uniquely solvable boundary problems

267

(3)

The case of the critical dimension

270

(7)

Asymptotic Behaviour of Energy Integrals for Particular Problems of Mathematical Physics

Dirichlet's Problem for Laplace's Operator

277

(14)

Perturbation of a domain near a corner or conic point

277

(3)

The case of right-hand sides depending on ξ

280

(1)

The case of right-hand sides depending on x and ξ

281

(1)

Dirichlet's problem for Laplace's operator in a domain with a small hole

282

(2)

Refinement of the asymptotic behaviour

284

(3)

Two-dimensional domains with a small hole

287

(1)

Dirichlet's problem for Laplace's operator in domains with several small holes

288

(3)

Neumann's Problem in Domains with one Small Hole

291

(2)

Dirichlet's Problem for the Biharmonic Equation in a Domain with Small Holes

293

(3)

Variation of Energy Depending on the Length of Crack

296

(6)

The antiplanar deformation

296

(3)

A problem in the two-dimensional elasticity

299

(3)

Remarks on the Behaviour of Solutions of Problems in the Two-dimensional Elasticity Near Corner Points

302

(6)

Statement of problems

302

(1)

The asymptotic behaviour of solutions of the antiplanar deformation problem

302

(1)

Asymptotic behaviour of solutions of the planar deformation problem

303

(3)

Boundary value problems in unbounded domains

306

(2)

Derivation of Asymptotic Formulas for Energy

308

(10)

Statement of problems

308

(1)

Antiplanar deformation

309

(1)

Planar deformation

310

(1)

Refinement of the asymptotic formula for energy

311

(2)

Defect in the material near vertex of the crack

313

(5)

Part IV Asymptotic Behaviour of Eigenvalues of Boundary Value Problems in Domains with Small Holes

Asymptotic Expansions of Eigenvalues of Classic Boundary Value Problems

Asymptotic Behaviour of the First Eigenvalue of a Mixed Boundary Value Problem

318

(13)

Statement of the problem

318

(1)

The three-dimensional case (formal asymptotic representation)

319

(3)

The planar case (formal asymptotic representation)

322

(4)

Justification of asymptotic expansions in the three-dimensional case

326

(3)

Justification of asymptotic expansions in the two-dimensional case

329

(2)

Asymptotic Expansions of Eigenvalues of Other Boundary Value Problems

331

(11)

Dirichlet's problem in a three-dimensional domain with a small hole

331

(3)

Mixed boundary value problem in domains with several small holes

334

(3)

Mixed boundary value problem with Neumann's condition on the boundary of small hole

337

(3)

Dirichlet's problem on a Riemannian manifold with a small hole

340

(2)

Asymptotic Representations of Eigenvalues of Problems of the Elasticity Theory for Bodies with Small Inclusions and Holes

342

(13)

Statement of the problem

342

(1)

Structure of the asymptotic representation

343

(1)

Particular solutions of the boundary layer problem

343

(4)

Perturbation of the eigenvalue Λ0

347

(2)

Problem in the two-dimensional elasticity (one hole with a free surface)

349

(6)

Homogeneous Solutions of Boundary Value Problems in the Exterior of a Thin Cone

Formal Asymptotic Representation

355

(8)

Statement of the problem

355

(1)

The case n - 1 > 2m

356

(4)

The case n - 1 = 2m

360

(3)

Inversion of the Principal Part of an Operator Pencil on the Unit Sphere with a Small Hole. An Auxiliary Problem with Matrix Operator

363

(11)

``Nearly inverse'' operator (the case 2m < n - 1)

363

(4)

``Nearly inverse'' operator (the case 2m = n - 1)

367

(4)

Reduction to a problem with a matrix operator (the case 2m < n - 1)

371

(2)

Reduction to a problem with a matrix operator (the case 2m = n - 1)

373

(1)

Justification of the Asymptotic Behaviour of Eigenvalues (The Case 2m < n - 1)

374

(5)

Justification of the Asymptotic Behaviour of Eigenvalues (The Case 2m = n - 1)

379

(9)

Examples and Corollaries

388

(3)

A scalar operator

388

(1)

Lame's and Stokes' systems

389

(1)

Continuity at the cone vertex of solution of Dirichlet's problem

390

(1)

Examples of Discontinuous Solutions to Dirichlet's Problem in Domains with a Conic Point

391

(4)

Equation of second order with discontinuous solutions

391

(2)

Dirichlet's problem for an elliptic equation of the fourth order with real coefficients

393

(2)

Singularities of Solutions of Neumann's Problem

395

(5)

Introduction

395

(1)

Formal asymptotic representation

396

(4)

Justification of the Asymptotic Formulas

400

(11)

Multiplicity of the spectrum near the point Λ = 2

400

(1)

Nearly inverse operator for Neumann's problem in Gϵ

401

(5)

Justification of asymptotic representation of eigenvalues

406

(5)

Comments on Parts I-IV

Comments on Part I

411

(1)

Chapter 1

411

(1)

Chapter 2

411

(1)

Comments on Part II

411

(1)

Chapter 3

411

(1)

Chapter 4

412

(1)

Chapter 5

412

(1)

Comments on Part III

412

(1)

Chapter 6

412

Chapter 7

408

(4)

Chapter 8

412

(1)

Comments on Part IV

412

(1)

Chapter 9

412

(1)

Chapter 10

412

(1)

List of Symbols

413

(4)

Basic Symbols

413

(1)

Symbols for function spaces and related concepts

414

(1)

Symbols for functions, distributions and related concepts

415

(1)

Other symbols

415

(2)

References

417

(16)

Index

433

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Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains

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