Stochastic Homogenization of Nonconvex Discrete Energies with Degenerate Growth

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. In the present paper we prove stochastic homogenization results for nonconvex energy functionals with degenerate growth under moment conditions. In particular, we study the continuum limit of discrete, nonconvex energy functionals defined on crystal lattices in dimensions d≥ 2. We consider energy functionals with random (stationary and ergodic) pair interactions; thus our problem corresponds to a stochastic homogenization problem. In the nondegenerate 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 satisfy a uniform growth condition neither from above nor neither from below. We consider interaction potentials that obey a p-growth condition with a random growth weight λ. We show that if λ satisfies the moment condition E[λ^α+λ-β]<∞ for suitable values of α and β, then the discrete energy Γ-converges to an integral functional with a nondegenerate energy density. In the scalar case, it suffices to assume that α≥ 1 and β≥1/p-1 (which ensures the nondegeneracy of the homogenized energy density). In the general, vectorial case, we additionally require that α>1 and 1/α+1/β≤ p/d.