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

by Vladimir Maz'ya, B. Plamenevskij (Translator), Serguei Nazarov, Boris Plamenevskij

Overview

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.

Product Details

ISBN-13:	9783034895644
Publisher:	Birkhäuser Basel
Publication date:	10/21/2012
Series:	Operator Theory: Advances and Applications , #112
Edition description:	Softcover reprint of the original 1st ed. 2000
Pages:	323
Product dimensions:	7.01(w) x 10.00(h) x 0.03(d)

Table of Contents

V Boundary Value Problems in Domains Perturbed Near Multidimensional Singularities of the Boundary.- 11 Boundary Value Problems in Domains with Edges on the Boundary.- 12 Asymptotics of Solutions to Classical Boundary Value Problems in a Domain with Thin Cavities.- 13 Asymptotics of Solutions to the Dirichlet Problem for High Order Equations in a Domain with a Thin Tube Excluded.- VI Behaviour of Solutions of Boundary Value Problems in Thin Domains.- 14 The Dirichlet Problem in Domains with Thin Ligaments.- 15 Boundary Value Problems of Mathematical Physics in Thin Domains.- 16 General Elliptic Problems in Thin Domains.- VII Elliptic Boundary Value Problems with Oscillating Coefficients or Boundary of Domain.- 17 Elliptic Boundary Value Problems with Rapidly Oscillating Coefficients.- 18 Paradoxes of Limit Passage in Solutions of Boundary Value Problems When Smooth Domains Are Approximated by Polygons.- 19 Homogenization of a Differential Operator on a Fine Periodic Net of Curves.- 20Homogenization of Equations on a Fine Periodic Grid.- Comments on Parts V-VII.- Comments on Part V.- 11.- 12.- 13.- Comments on Part VI.- 14.- 15.- 16.- Comments on Part VII.- 17.- 18.- 19.- 20.- List of Symbols.- References.