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

Abstract

<p style='text-indent:20px;'>We investigate the homogenization through <inline-formula><tex-math id="M2">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence for the <inline-formula><tex-math id="M3">\begin{document}$ L^2({\Omega}) $\end{document}</tex-math></inline-formula>-weak topology of the conductivity functional with a zero-order term where the matrix-valued conductivity is assumed to be non strongly elliptic. Under proper assumptions, we show that the homogenized matrix <inline-formula><tex-math id="M4">\begin{document}$ A^\ast $\end{document}</tex-math></inline-formula> is provided by the classical homogenization formula. We also give algebraic conditions for two and three dimensional <inline-formula><tex-math id="M5">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-periodic rank-one laminates such that the homogenization result holds. For this class of laminates, an explicit expression of <inline-formula><tex-math id="M6">\begin{document}$ A^\ast $\end{document}</tex-math></inline-formula> 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.</p>