$ \Gamma $-convergence of quadratic functionals with non uniformly elliptic conductivity matrices
Lorenza D’Elia · Networks and Heterogeneous Media · 2021
We investigate the homogenization through \begin{document}$ \Gamma $\end{document} -convergence for the \begin{document}$ L^2({\Omega}) $\end{document} -weak topology of the conductivity functional with a zero-order term where the matrix-valued conductivity is assumed to be non strongly elliptic. Under proper assumptions, we show that the homogenized matrix \begin{document}$ A^\ast $\end{document} is provided by the classical homogenization formula. We also give algebraic conditions for two and three dimensional \begin{document}$ 1 $\end{document} -periodic rank-one laminates such that the homogenization result holds. For this class of laminates, an explicit expression of \begin{document}$ A^\ast $\end{document} 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.