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Green's Kernels and Meso-Scale Approximations in Perforated Domains

  • Vladimir Maz'ya
  • Alexander Movchan
  • Michael Nieves

Part of the Lecture Notes in Mathematics book series (LNM, volume 2077)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Green’s Functions in Singularly Perturbed Domains

    1. Front Matter
      Pages 1-1
    2. Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 59-73
    3. Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 75-81
    4. Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 83-94
  3. Green’s Tensors for Vector Elasticity in Bodies with Small Defects

    1. Front Matter
      Pages 95-95
    2. Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 97-137
    3. Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 139-167
    4. Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 169-188
  4. Meso-scale Approximations: Asymptotic Treatment of Perforated Domains Without Homogenization

    1. Front Matter
      Pages 189-189
    2. Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 191-219
    3. Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 221-247
  5. Back Matter
    Pages 249-260

About this book

Introduction

There are a wide range of applications in physics and structural mechanics involving domains with singular perturbations of the boundary. Examples include perforated domains and bodies with defects of different types. The accurate direct numerical treatment of such problems remains a challenge. Asymptotic approximations offer an alternative, efficient solution.
Green’s function is considered here as the main object of study rather than a tool for generating solutions of specific boundary value problems. The uniformity of the asymptotic approximations is the principal point of attention. We also show substantial links between Green’s functions and solutions of boundary value problems for meso-scale structures. Such systems involve a large number of small inclusions, so that a small parameter, the relative size of an inclusion, may compete with a large parameter, represented as an overall number of inclusions.
The main focus of the present text is on two topics: (a) asymptotics of Green’s kernels in domains with singularly perturbed boundaries and (b) meso-scale asymptotic approximations of physical fields in non-periodic domains with many inclusions. The novel feature of these asymptotic approximations is their uniformity with respect to the independent variables.
This book addresses the needs of mathematicians, physicists and engineers, as well as research students interested in asymptotic analysis and numerical computations for solutions to partial differential equations.

Keywords

4E10, 35B40, 35J08 Asymptotic analysis Green's functions Meso-scale approximations Perforated domains

Authors and affiliations

  • Vladimir Maz'ya
    • 1
  • Alexander Movchan
    • 2
  • Michael Nieves
    • 3
  1. 1.Department of MathematicsLinköping UniversityLinköpingSweden
  2. 2.Dept. Mathematical SciencesUniversity of LiverpoolLiverpoolUnited Kingdom
  3. 3.School of EngineeringLiverpool John Moores UniversityLiverpoolUnited Kingdom

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-00357-3
  • Copyright Information Springer International Publishing Switzerland 2013
  • Publisher Name Springer, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-00356-6
  • Online ISBN 978-3-319-00357-3
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site