Gamma-convergence of quadratic functionals with non uniformly elliptic conductivity matrices

TL;DR

This paper studies the homogenization of quadratic conductivity functionals with non-uniform ellipticity using Gamma-convergence, providing conditions for classical homogenization formulas and explicit results for certain laminates, along with a counter-example.

Contribution

It establishes the Gamma-convergence of the conductivity functional with non-elliptic matrices and derives explicit formulas for specific laminate structures, extending classical homogenization results.

Findings

Homogenized matrix given by classical formula under certain assumptions.

Explicit homogenization formula for 1- and 3-dimensional rank-one laminates.

Counter-example showing anomalous asymptotic behavior in 2D.

Abstract

We investigate the homogenization through Gamma-convergence for the L^2(\Omega)-weak topology of the conductivity functional with a zero-order term where the matrix-valued conductivity is assumed to be non strongly elliptic. Under proper assumptions, we show that the homogenized matrix A^\ast is provided by the classical homogenization formula. We also give algebraic conditions for two and three dimensional 1-periodic rank-one laminates such that the homogenization result holds. For this class of laminates, an explicit expression of A^\ast is provided which is a generalization of the classical laminate formula. We construct a two-dimensional counter-example which shows an anomalous asymptotic behaviour of the conductivity functional.