# The Dirichlet Boundary Value Problem in Strongly Perforated Domains with Fine-Grained Boundary

Chapter

## Abstract

In this chapter we consider the simplest type of domains with complex microstructure for which the homogenization approach appears to be fairly natural. Namely, we consider strongly perforated domains (domains with fine-grained boundary) having the following structure: \(
\Omega ^{\left( s \right)} = \Omega \backslash \cup _{i = 1}^s F_i^{\left( s \right)}
\) where Ω is a fixed domain in ℝ^{ n } and *F* _{i} ^{(s)} (i=1,2, …s) (“grains”) are disjoint closed sets of decreasing, as *s*→∞, diameter; see Figure 2.1.

## Keywords

Asymptotic Behavior Orthogonal Projection Dirichlet Problem Dirichlet Boundary Linear Span
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## Copyright information

© Birkhäuser Boston 2006