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

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.