We study the behavior of second-order degenerate elliptic systems in divergence form with random coefficients which are stationary and ergodic. Assuming moment bounds like Chiarini and Deuschel [Arxiv preprint 1410[Arxiv preprint .4483, 2014 on the coefficient field a and its inverse, we prove an intrinsic largescale C 1,α -regularity estimate for a-harmonic functions and obtain a first-order Liouville theorem for subquadratic a-harmonic functions.

It is well‐known that an elastic sheet loaded in tension will wrinkle and that the length scale of the wrinkles tends to 0 with vanishing thickness of the sheet [Cerda and Mahadevan, Phys. Rev. Lett. 90, 074302 (2003)]. We give the first mathematically rigorous analysis of such a problem. Since our methods require an explicit understanding of the underlying (convex) relaxed problem, we focus on the wrinkling of an annular sheet loaded in the radial direction [Davidovitch et al., Proc. Natl. Acad. Sci. 108 (2011), 18227]. Our main achievement is identification of the scaling law of the minimum energy as the thickness of the sheet tends to 0. This requires proving an upper bound and a lower bound that scale the same way. We prove both bounds first in a simplified Kirchhoff‐Love setting and then in the nonlinear three‐dimensional setting. To obtain the optimal upper bound, we need to adjust a naive construction (one family of wrinkles superimposed on a planar deformation) by introducing a cascade of wrinkles. The lower bound is more subtle, since it must be ansatz‐free. © 2014 Wiley Periodicals, Inc.

We are concerned with the homogenization of second-order linear elliptic equations with random coefficient fields. For symmetric coefficient fields with only short-range correlations, quantified through a logarithmic Sobolev inequality for the ensemble, we prove that when measured in weak spatial norms, the solution to the homogenized equation provides a higher-order approximation of the solution to the equation with oscillating coefficients. In the case of nonsymmetric coefficient fields, we provide a higher-order approximation (in weak spatial norms) of the solution to the equation with oscillating coefficients in terms of solutions to constant-coefficient equations. In both settings, we also provide optimal error estimates for the two-scale expansion truncated at second order. Our results rely on novel estimates on the second-order homogenization corrector, which we establish via sensitivity estimates for the second-order corrector and a large-scale L p theory for elliptic equations with random coefficients. Our results also cover the case of elliptic systems.

We study the wrinkling of a thin elastic sheet caused by a prescribed non-Euclidean metric. This is a model problem for the patterns seen, for example, in torn plastic sheets and the leaves of plants. Following the lead of other authors we adopt a variational viewpoint, according to which the wrinkling is driven by minimization of an elastic energy subject to appropriate constraints and boundary conditions. We begin with a broad introduction, including a discussion of key examples (some well-known, others apparently new) that demonstrate the overall character of the problem. We then focus on how the minimum energy scales with respect to the sheet thickness h, for certain classes of displacements. Our main result is that when the deformations are subject to certain hypotheses, the minimum energy is of order h 4/3 . We also show that when the deformations are subject to more restrictive hypotheses, the minimum energy is strictly larger -of order h; it follows that energy minimization in the more restricted class is not a good model for the applications that motivate this work. Our results do not explain the cascade of wrinkles seen in some experimental and numerical studies; and they leave open the possibility that an energy scaling law better than h 4/3 could be obtained by considering a larger class of deformations.