Homogenization and field concentrations in heterogeneous media

Abstract

A multi-scale characterization of the field concentrations inside composite and polycrystalline media is developed. The analysis focuses on gradient fields associated with the intensive quantities given by the temperature and the electric potential. In the linear regime these quantities are modeled by the solution of a second order elliptic partial differential equation with oscillatory coefficients. Field concentrations are measured using the $L^p$ norm of the gradient of the solution for p>2. The analysis focuses on the case when the length scale of the heterogeneities are small relative to the domain containing them. Explicit lower bounds on the limit inferior of the sequence of $L^p$ norms are found in the fine scale limit These bounds provide a way to rigorously assess field concentrations generated by highly oscillatory microgeometies. Illustrative examples are provided that demonstrate the optimality of the lower bounds.