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Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains Volume II

Volume II

  • Book
  • © 2000

Overview

Authors:
  1. Vladimir Maz’ya

    1. Department of Mathematics, Linköping University, Linköping, Sweden

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  2. Serguei Nazarov

    1. Laboratory of Mathematical Methods in Mechanics of Solids Institute of Mathematics and Mechanics, St. Petersburg University, St. Petersburg, Russia

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    You can also search for this author in PubMed   Google Scholar

  3. Boris A. Plamenevskij

    1. Department of Mathematical Physics Faculty of Physics, St. Petersburg State University, St. Petersburg, Russia

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Part of the book series: Operator Theory: Advances and Applications (OT, volume 112)

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Table of contents (10 chapters)

  1. Front Matter

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

    1. Front Matter

      Pages 1-1
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    2. Boundary Value Problems in Domains with Edges on the Boundary

      • Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 3-21

    3. Asymptotics of Solutions to Classical Boundary Value Problems in a Domain with Thin Cavities

      • Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 23-74

    4. Asymptotics of Solutions to the Dirichlet Problem for High Order Equations in a Domain with a Thin Tube Excluded

      • Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 75-100

  3. Behaviour of Solutions of Boundary Value Problems in Thin Domains

    1. Front Matter

      Pages 101-101
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    2. The Dirichlet Problem in Domains with Thin Ligaments

      • Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 103-130

    3. Boundary Value Problems of Mathematical Physics in Thin Domains

      • Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 131-170

    4. General Elliptic Problems in Thin Domains

      • 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
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    2. Elliptic Boundary Value Problems with Rapidly Oscillating Coefficients

      • Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 211-235

    3. Paradoxes of Limit Passage in Solutions of Boundary Value Problems When Smooth Domains Are Approximated by Polygons

      • Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 237-258

    4. Homogenization of a Differential Operator on a Fine Periodic Net of Curves

      • Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 259-281

    5. Homogenization of Equations on a Fine Periodic Grid

      • Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij
      Pages 283-295

  5. Back Matter

    Pages 297-323
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, Linköping University, Linköping, Sweden

    Vladimir Maz’ya

  • Laboratory of Mathematical Methods in Mechanics of Solids Institute of Mathematics and Mechanics, St. Petersburg University, St. Petersburg, Russia

    Serguei Nazarov

  • Department of Mathematical Physics Faculty of Physics, St. Petersburg State University, St. Petersburg, Russia

    Boris A. Plamenevskij

Bibliographic Information

  • Book Title: Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains Volume II

  • Book Subtitle: Volume II

  • Authors: Vladimir Maz’ya, Serguei Nazarov, Boris A. Plamenevskij

  • Series Title: Operator Theory: Advances and Applications

  • DOI: https://doi.org/10.1007/978-3-0348-8432-7

  • Publisher: Birkhäuser Basel

  • eBook Packages: Springer Book Archive

  • Copyright Information: Birkh�user Verlag 2000

  • Hardcover ISBN: 978-3-7643-6398-7Published: 01 May 2000

  • Softcover ISBN: 978-3-0348-9564-4Published: 21 October 2012

  • eBook ISBN: 978-3-0348-8432-7Published: 06 December 2012

  • Series ISSN: 0255-0156

  • Series E-ISSN: 2296-4878

  • Edition Number: 1

  • Number of Pages: XXIII, 323

  • Additional Information: Original German edition published by Akademie Verlag Leipzig, 1991

  • Topics: Analysis

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