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Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains

Volume II

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

Part of the Operator Theory book series (OT, volume 112)

Table of contents

  1. Front Matter
    Pages I-XXIII
  2. Boundary Value Problems in Domains Perturbed Near Multidimensional Singularities of the Boundary

    1. Front Matter
      Pages 1-1
    2. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 3-21
    3. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 23-74
  3. Behaviour of Solutions of Boundary Value Problems in Thin Domains

    1. Front Matter
      Pages 101-101
    2. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 103-130
    3. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 131-170
    4. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 171-207
  4. Elliptic Boundary Value Problems with Oscillating Coefficients or Boundary of Domain

    1. Front Matter
      Pages 209-209
    2. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 211-235
    3. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 259-281
    4. Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 283-295
  5. Back Matter
    Pages 297-323

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. While the first volume is devoted to perturbations of the boundary near isolated singular points, this second volume treats singularities of the boundary in higher dimensions as well as nonlocal perturbations.
At the core of this book are solutions of elliptic boundary value problems by asymptotic expansion in powers of a small parameter that characterizes the perturbation of the domain. In particular, it treats the important special cases of  thin domains, domains with small cavities, inclusions or ligaments, rounded corners and edges, and problems with rapid oscillations of the boundary or the coefficients of the differential operator. The methods presented here capitalize on the theory of elliptic boundary value problems with nonsmooth boundary that has been developed in the past thirty years.
Moreover, a study on the homogenization of differential and difference equations on periodic grids and lattices is given. Much attention is paid to concrete problems in mathematical physics, particularly elasticity theory and electrostatics.
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 Partial differential equations difference equation differential equation differential operator homogenization 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