Calculus of Variations and Geometric Measure Theory

L. D'Elia

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

created by d'elia on 09 Sep 2021
modified on 04 Oct 2023


Published Paper

Inserted: 9 sep 2021
Last Updated: 4 oct 2023

Journal: Netw. Heterog. Media
Volume: 128
Year: 2022

ArXiv: 2106.06728 PDF


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.