Strongly Connected Domains
Progress in mathematical physics · 2006
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 L2 (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) .