# G-Convergence and Homogenization of Nonlinear Partial Differential Operators

• Alexander Pankov Book

Part of the Mathematics and Its Applications book series (MAIA, volume 422)

1. Front Matter
Pages i-xiii
2. Alexander Pankov
Pages 1-44
3. Alexander Pankov
Pages 45-130
4. Alexander Pankov
Pages 131-172
5. Alexander Pankov
Pages 173-212
6. Back Matter
Pages 213-258

### Introduction

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.

### Keywords

DEX Scheme Volume convergence differential equation differential operator equation form homogenization online partial differential equation

#### Authors and affiliations

• Alexander Pankov
• 1
1. 1.Vinnitsa Polytechnical InstituteVinnitsaUkraine

### Bibliographic information

• DOI https://doi.org/10.1007/978-94-015-8957-4 