Elliptic Boundary Value Problems with Rapidly Oscillating Coefficients

Abstract

Up to now, we have studied elliptic boundary value problems with perturbations located near isolated points or submanifolds of positive dimension. The last part of the book is devoted to rapidly oscillating perturbations extended over the whole domain or its boundary. At first glance the problems collected here essentially differ from those studied in the preceding chapters. Nevertheless, one can follow an analogy in the iterative procedures for the asymptotic expansions. It suffices to mention that as before the coefficients in the asymptotic series are found with the help of limit problems. Here the role of limit problems is played by a problem in the periodicity cell; a problem on a semicylinder with periodic boundary conditions on the lateral area that describes a boundary layer; a homogenized boundary value problem that arises to provide the compatibility conditions for the above limit problems.