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Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains

Volume I

  • Vladimir Maz’ya
  • Serguei Nazarov
  • Boris A. Plamenevskij

Part of the Operator Theory book series (OT, volume 111)

Table of contents

  1. Front Matter
    Pages I-XXIII
  2. Boundary Value Problems for the Laplace Operator in Domains Perturbed Near Isolated Singularities

    1. Front Matter
      Pages 1-1
    2. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 3-41
    3. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 43-76
  3. General Elliptic Boundary Value Problems in Domains Perturbed Near Isolated Singularities of the Boundary

    1. Front Matter
      Pages 77-77
    2. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 115-155
    3. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 157-224
  4. Asymptotic Behaviour of Functionals on Solutions of Boundary Value Problems in Domains Perturbed Near Isolated Boundary Singularities

    1. Front Matter
      Pages 225-225
    2. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 227-250
    3. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 277-313
  5. Asymptotic Behaviour of Eigenvalues of Boundary Value Problems in Domains with Small Holes

    1. Front Matter
      Pages 315-315
    2. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 317-351
    3. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 353-409
  6. Back Matter
    Pages 411-435

About this book

Introduction

For the first time in the mathematical literature this two-volume work introduces a unified and general approach to the asymptotic analysis of elliptic boundary value problems in singularly perturbed domains. This first volume is devoted to domains whose boundary is smooth in the neighborhood of finitely many conical points. In particular, the theory encompasses the important case of domains with small holes. The second volume, on the other hand, treats perturbations of the boundary in higher dimensions as well as nonlocal perturbations.
The core of this book consists of the solution of general elliptic boundary value problems by complete asymptotic expansion in powers of a small parameter that characterizes the perturbation of the domain. The construction of this method capitalizes on the theory of elliptic boundary value problems with nonsmooth boundary that has been developed in the past thirty years.
Much attention is paid to concrete problems in mathematical physics, for example in elasticity theory. In particular, a study of the asymptotic behavior of stress intensity factors, energy integrals and eigenvalues is presented.
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.

Keywords

Boundary value problem Eigenvalue Laplace operator Partial differential equations differential equation differential operator distribution energy integral operator partial differential equation perturbation perturbation theory

Authors and affiliations

  • Vladimir Maz’ya
    • 1
  • Serguei Nazarov
    • 2
  • Boris A. Plamenevskij
    • 3
  1. 1.Department of MathematicsLinköping UniversityLinköpingSweden
  2. 2.Laboratory of Mathematical Methods in Mechanics of Solids Institute of Mathematics and MechanicsSt. Petersburg UniversitySt. PetersburgRussia
  3. 3.Department of Mathematical Physics, Faculty of PhysicsSt. Petersburg State UniversitySt. PetersburgRussia

Bibliographic information