Composite media and Dirichlet forms

Abstract

Some relevant “macroscopic” features of bodies with complicated “microscopic” structure are usually described, in the mathematical theory of composite media and homogenization, in terms of asymptotic properties of sequences of Dirichlet integrals

$$ {E_h} = \mathop{\smallint }\limits_{\Omega } \sum\limits_{{ij = 1}}^N {{\partial_i}u} {\partial_j}u \,a_h^{{ij}}(x)dx\;,h \in \mathbb{N}, $$

((1))

${\partial_i} = {{{\partial u}} \left/ {{\partial xi,{\partial_j} }} \right.} = {{{\partial u}} \left/ {{\partial xj}} \right.}$, by appropriately defining the “conductivity” coefficients $a_h^{{ij}}(x)$ on some open subset Ω of ℝ^N.