[Joint seminar on Probability and NLPDE] Stochastic homogenization of nonconvex discrete energies with degenerate growth

Date
2016/07/29 Fri 15:00 - 16:00
Room
3号館127大会議室
Speaker
Stefan Neukamm
Affiliation
TU Dresden
Abstract

The homogenization limit of discrete, nonconvex energy functionals defined on crystal lattices in dimensions $d \ge 2$ is well understood in the case of periodic or random pair interactions satisfying a uniform $p$-growth condition. In the talk I consider a degenerate situation, when the interactions obey a $p$-growth condition with a random growth weight $\lambda$. We show that if $\lambda$ satisfies the moment condition $ \mathbb{E}[\lambda^{\alpha} + \lambda^{- \beta}] < \infty $ for suitable values of $ \alpha $ and $ \beta $, then the discrete energy $ \Gamma $-converges to an integral functional with a non-degenerate energy density. In the scalar case (which covers the case of the random conductance model), it suffices to assume that $ \alpha \ge 1 $ and $ \beta \ge \frac{1}{p-1} $ (which is just the condition that ensures the non-degeneracy of the homogenized energy density). In the general, vectorial case, we additionally require that $ \alpha>1 $ and $ \frac{1}{\alpha}+\frac{1}{\beta}≤\frac{p}{d} $. The talk is based on joint work with M. Schäffner (TU Dresden) and A. Schlömerkemper (U Würzburg).