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 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 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 non-degenerate energy density. In the scalar case, it suffices to assume that α ≥ 1 and β ≥ 1 p−1 (which ensures the non-degeneracy of the homogenized energy density). In the general, vectorial case, we additionally require that α > 1 and 1 α + 1 β ≤ p d .

Abstract. We study the effective elastic behavior of incompatibly prestrained plates, where the prestrain is independent of thickness as well as uniform through the thickness. We model such plates as three-dimensional elastic bodies with a prescribed pointwise stress-free state characterized by a Riemannian metric G with the above properties, and seek the limiting behavior as the thickness goes to zero.We first establish that the Γ-limit is a Kirchhoff-type bending theory when the energy per volume scales as the second power of thickness. We then show the somewhat surprising result that there are metrics which are not immersible, but have zero bending energy. This implies that there are a hierarchy of plate theories for such prestrained plates. We characterize the non-immersible metrics that have zero bending energy (if and only if the Riemann curvatures R 3 112 , R 3 221 and R1212 of G do not identically vanish), and illustrate them with examples. Of particular interest is an example where G2×2, the two-dimensional restriction of G is flat, but the plate still has non-trivial energy that scales similar to Föppl -von Kármán plates. Finally, we apply these results to a model of nematic glass, including a characterization of the condition when the metric is immersible, for G = Id3 + γ n ⊗ n given in terms of the inhomogeneous unit director field distribution n ∈ R 3 .

We study local regularity properties for solutions of linear, non-uniformly elliptic equations. Assuming certain integrability conditions on the coefficient field, we prove local boundedness and Harnack inequality. The assumed integrability assumptions are essentially sharp and improve upon classical results by Trudinger [ARMA 1971]. We then apply the deterministic regularity results to the corrector equation in stochastic homogenization and establish sublinearity of the corrector.

We study quantitative periodic homogenization of integral functionals in the context of non-linear elasticity. Under suitable assumptions on the energy densities (in particular frame indifference; minimality, non-degeneracy and smoothness at the identity; p ≥ d-growth from below; and regularity of the microstructure), we show that in a neighborhood of the set of rotations, the multi-cell homogenization formula of non-convex homogenization reduces to a single-cell formula. The latter can be expressed with help of correctors. We prove that the homogenized integrand admits a quadratic Taylor expansion in an open neighborhood of the rotations -a result that can be interpreted as the fact that homogenization and linearization commute close to the rotations. Moreover, for small applied loads, we provide an estimate on the homogenization error in terms of a quantitative two-scale expansion.

We consider the random conductance model in a stationary and ergodic environment. Under suitable moment conditions on the conductances and their inverse, we prove a quenched invariance principle for the random walk among the random conductances. The moment conditions improve earlier results of Andres, Deuschel and Slowik [Ann. Probab.] and are the minimal requirement to ensure that the corrector is sublinear everywhere. The key ingredient is an essentially optimal deterministic local boundedness result for finite difference equations in divergence form.