G-Convergence and Homogenization of Nonlinear Partial Differential Operators: 422 (Mathematics and Its Applications, 422) - Hardcover

Pankov, A.A.

Hardcover

ISBN 10: 079234720X ISBN 13: 9780792347200

Publisher: Springer, 1997

Synopsis

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.

About this title

Synopsis

This volume deals with G-convergence and homogenization for various classes of nonlinear partial differential operators. Chapter 1 is devoted to some preliminary issues from nonlinear analysis as well as to G-convergence of abstract operators, including the case of abstract parabolic operators. Chapter 2 introduces details of the notion of strong G-convergence for nonlinear second order elliptic operators in divergence form, and in Chapter 3 the homogenization problem for rapidly oscillated nonlinear random homogenous elliptic operators is dealt with. On the basis of these results, almost periodic and periodic cases are studied. Finally, in Chapter 4, some of the previous results are extended to the case of nonlinear parabolic operators. The volume concludes with two appendices, one of which is devoted to homogenization of nonlinear difference schemes, while the other lists some open problems of relevance, and a bibliography.This work will be of interest to researchers whose work involves homogenization theory and its applications. It is also recommended for advanced courses in the fields of partial differential equations and nonlinear analysis.

Edition details

• Publisher: Springer
• Publication date: 1997
• Language: English
• ISBN 10: 079234720X
• ISBN 13: 9780792347200
• Binding: Hardcover
• Number of pages: 271