Variants and Corollaries of the Asymptotic Theory

Abstract

The aim of the present chapter is to facilitate the understanding of the asymptotic theory in Chapter 4, and, at the same time, to develop it. In Section 5.1 we demonstrate by a simple example transformation of the problem to a vector form, as it was used in 4.2, and show how this leads to exact coercive estimates. In Section 5.3 a special case of a singular peturbation of the boundary of a domain will be investigated, namely the case of a finite number of holes. The necessary information about the corresponding limit problem, i.e. about the problems in a punctured domain, will be collected in 5.2. In Sections 5.4–5.7, it will be shown that the theory in Chapter 4 can be generalized in different directions.