Stochastic Homogenization of Nonconvex Discrete Energies with Degenerate Growth

Abstract

We study the continuum limit of discrete, nonconvex energy functionals defined on crystal lattices in dimensions $d\geq 2$. Since we are interested in energy functionals with random (stationary and ergodic) pair interactions, our problem corresponds to a stochastic homogenization problem. In the non-degenerate case, when the interactions satisfy a uniform $p$-growth condition, the homogenization problem is well-understood. In this paper, we are interested in a degenerate situation, when the interactions neither satisfy a uniform growth condition from above nor from below. We consider interaction potentials that 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 it suffices to assume that $\alpha\geq 1$ and $\beta\geq\frac{1}{p-1}$ (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 $\frac{1}{\alpha}+\frac{1}{\beta}\leq \frac{p}{d}$. Recently, there has been considerable effort to understand periodic and stochastic homogenization of elliptic equations and integral functionals with degenerate growth, as well as related questions on the effective behavior of conductance models in degenerate, random environments. The results in the present paper are to our knowledge the first stochastic homogenization results for nonconvex energy functionals with degenerate growth under moment conditions.