G-Convergence and Homogenization of Nonlinear Partial Differential Operators / Edition 1

by A.A. Pankov

Overview

Various applications of the homogenization theory of partial differential equations resulted in the further development of this branch of mathematics, attracting an increasing interest of both mathematicians and experts in other fields. In general, the theory deals with the following: Let Ak be a sequence of differential operators, linear or nonlinepr. We want to examine the asymptotic behaviour of solutions uk to the equation Auk = f, as k ~ =, provided coefficients of Ak contain rapid oscillations. This is the case, e. g. when the coefficients are of the form a(e/x), where the function a(y) is periodic and ek ~ 0 ask~=. Of course, of oscillation, like almost periodic or random homogeneous, are of many other kinds interest as well. It seems a good idea to find a differential operator A such that uk ~ u, where u is a solution of the limit equation Au = f Such a limit operator is usually called the homogenized operator for the sequence Ak . Sometimes, the term "averaged" is used instead of "homogenized". Let us look more closely what kind of convergence one can expect for uk. Usually, we have some a priori bound for the solutions. However, due to the rapid oscillations of the coefficients, such a bound may be uniform with respect to k in the corresponding energy norm only. Therefore, we may have convergence of solutions only in the weak topology of the energy space.

Product Details

ISBN-13:	9780792347200
Publisher:	Springer Netherlands
Publication date:	09/30/1997
Series:	Mathematics and Its Applications , #422
Edition description:	1997
Pages:	258
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1 G-convergence of Abstract Operators.- 2 Strong G-convergence of Nonlinear Elliptic Operators.- 3 Homogenization of Elliptic Operators.- 4 Nonlinear Parabolic Operators.- A Homogenization of Nonlinear Difference Schemes.- A.1 Mesh Functions.- A.2 G-convergence.- A.3 Homogenization.- B Open Problems.- References.

Editorial Reviews

Deals with -convergence and homogenization for various classes of nonlinear partial differential operators. Chapter One is devoted to preliminary issues from nonlinear analysis as well as - convergence of abstract operators, including the case of abstract parabolic operators. Chapter Two introduces details of the notion of strong -convergence for nonlinear second order elliptic operators in divergence form, and Chapter Three discusses the homogenization problem for rapidly oscillated nonlinear random homogeneous elliptic operators, with periodic and almost periodic case studies. Chapter Four extends previous results to the case of nonlinear parabolic operators. Appendices cover homogenization of nonlinear difference schemes and list some open problems. Of interest to researchers whose work involves homogenization theory and applications. Can be used for advanced courses in partial differential equations and nonlinear analysis. Annotation c. by Book News, Inc., Portland, Or.

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