Homogenization of Partial Differential Equations pp 105-135 | Cite as

# Strongly Connected Domains

## Abstract

In the preceding chapters we have shown that the solution *u*^{(s)}(x) of the Dirichlet boundary value problem in the domains ω^{(s)} ⊂ ω can be naturally extended into the complementary sets \(
F^{\left( s \right)} = \Omega \backslash \bar \Omega ^{\left( s \right)}
\) in such a way that the resulting sequence of functions *u*^{(s)} (x) ∈ W _{2} ^{1} (ω), *s*=1,2,… is bounded in the norm of *W* _{2} ^{1} (ω) and therefore is compact in *L*_{2} (ω). This allowed us to study the convergence of *u*^{(s)} (x) in a fixed domain ω, which, obviously, is much simpler than to study the convergence in varying (with *s*) domains ω^{ (s) }. This also explains the fact that all possible limits of solutions of the Dirichlet boundary problem are described by homogenized equations of the same type, regardless the topological structure of the sets *F*^{ (s) }.

## Keywords

Extension Condition Polygonal Line Straight Line Parallel Blue Node Strong Connectivity## Preview

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