Stochastic homogenization of \Lambda -convex gradient flows

Abstract

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λ-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the p-Laplace operator with p ∈ (1, ∞). The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ-)convex functionals.

Relating a Rate-Independent System and a Gradient System for the Case of One-Homogeneous Potentials

“…The previous result can be seen as a special case of the "inverse energy-dissipation balance" also used in [8,Lem. 4.1].…”

Stochastic Homogenization on Irregularly Perforated Domains

“…In [ZP06], Zhikov and Piatnitsky reopened the case by introducing the stochastic twoscale convergence as a generalization of [Ngu89,All92,Zhi00] to the stochastic setting, particularly to random measures that comprise random perforations and random lower-dimensional structures in a natural way. The method was generalized to various applications in discrete and continuous homogenization [MP07,Fag08,FHS19] and recently also to an unfolding method [NV18,HNV21]. Concerning the homogenization on randomly perforated domains, there seem to be few results in the literature, with [GK15, FHL20, PP20] being the closest related work from the PDE point of view.…”

Stochastic two-scale convergence and Young measures

“…Introduction. In this paper we compare quenched stochastic two-scale convergence [38] with the notion of stochastic unfolding [30,19], which is equivalent to stochastic two-scale convergence in the mean [6]. In particular, we introduce the concept of stochastic two-scale Young measures to relate quenched stochastic two-scale limits with the mean limit and discuss examples of convex homogenization problems that can be treated with two-scale convergence in the mean, but not conveniently in the quenched setting of two-scale convergence.…”

“…In this work, we restrict our considerations to the simplest case where the random measure is the Lebesgue measure. Similarly to the periodic case, stochastic two-scale convergence in the mean can be rephrased with help of a transformation operator, see [30,19,34], where the stochastic unfolding operator…”

“…Stochastic unfolding and two-scale convergence in the mean. In the following we briefly introduce the stochastic unfolding operator and provide its main properties, for the proofs and detailed studies we refer to [30,19,34,31].…”