G-convergence of monotone operators

Abstract

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Stochastic homogenization of a class of monotone eigenvalue problems

“…For a complete treatment of a large class of (possibly multivalued) elliptic operators we refer to [5] and [4]. However, in the present work we will only consider single-valued operators.…”

“…We refer to [4], [5], and [12] concerning G-convergence results for elliptic operators needed in this report. Here we show that the general theory also applies to the situation of multiple scales and multiscale stochastic homogenization of a class of nonlinear eigenvalue problems.…”

Reiterated homogenization of monotone parabolic problems

“…For a complete treatment of a large class of (possibly multi-valued) elliptic operators we refer to [4] and [3]. We consider the following sequence of Dirichlet boundary value problems −div(a h (x, Du h )) = f h in Ω, u h ∈ V. (5.1) Definition 5.1 Given 0 < α ≤ 1, 2 ≤ p < ∞ and three positive real constants c 0 , c 1 and c 2 , we define the class S E = S E (c 0 , c 1 , c 2 , α) of maps a : Ω × R n → R n such that…”

On the Local Representation of G-Closure

“…We only mention here that the properties (G1), (G3)-(G5) and (G2) with g = 0 are standard for the G-convergence of strongly monotone and Lipschitz continuous elliptic operators and we can obtain them for the case of systems in exactly the same way as for the case of a single equation, see, for instance, Chiado'Piat et al [1] or Pankov [7]. In turn, once the properties (G1), (G3), (G5) and (G2) with g = 0 are established, the property (G2) with g = 0 is a simple consequence from them and the separability of L 2 ( ; R nm ).…”