On the homogenization of quasilinear divergence structure operators

Abstract

In the Allen-Cahn theory of phase transitions, minimizers partition the domain in subregions, the sets where a minimizer is near to one or to another of the zeros of the potential. These subregions that model the phases are separated by a tiny Diffuse Interface. Understanding the shape of this diffuse interface is an important step toward the description of the structure of minimizers.We assume Dirichlet data and present general conditions on the domain and on the boundary datum ensuring the connectivity of the diffuse interface.Then we restrict to the case of two dimensions and show that the phases can be separated, in a certain optimal way, by a connected network with a well defined structure. This network is contained in the diffuse interface and is a priori unknown.Under general assumption on the potential and on the Dirichlet datum, we show that, if we assume that the phase are connected, then we can obtain precise information on the shape of the network and in turn a detailed description of the fine structure of minimizers. In particular we can characterize the shape and the size of the various phases and also how they depend on the surface tensions.

Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems

“…The aim of homogenization theory is to provide a limiting equation for u 0 . The following result is shown in [14,Theorem 3.6] (see also [29]): there exists a subsequence of {a ε (·, s)} (again indexed by ε) such that the corresponding sequence of solutions {u ε } converges weakly to u 0 in H 1 (Ω), where u 0 is the solution of the so-called homogenized problem…”

“…Rigorously described by the mathematical homogenization theory [12], [31], coarse graining (or homogenization) aims at averaging the finest scales of a multiscale equation and deriving a homogenized equation that captures the essential macroscopic features of the problem as ε → 0. The mathematical homogenization of (1.1) has been developed in [14,10,29] where it is shown that the homogenized equation is of the same quasilinear type as the original equation, with a ε (x, u ε (x)) replaced by a homogenized tensor a 0 (x, u 0 (x)) c XXXX American Mathematical Society depending nonlinearly on a homogenized solution u 0 (the limit in a certain sense of u ε as ε → 0).…”

Numerical Homogenization of Nonlinear Random Parabolic Operators

“…The homogenization of linear parabolic operators with almost periodic and random coefficients has been studied in [23,22]. We would like also to mention several results on homogenization of nonlinear elliptic operators [2,5,10,11,15,16]. We would like to note that general elliptic operators in divergence form are considered in [15,16], including random homogenization, while articles [2,5,10,11] are devoted to the homogenization of monotone second-order elliptic operators.…”

“…We would like also to mention several results on homogenization of nonlinear elliptic operators [2,5,10,11,15,16]. We would like to note that general elliptic operators in divergence form are considered in [15,16], including random homogenization, while articles [2,5,10,11] are devoted to the homogenization of monotone second-order elliptic operators. For general references in the field of homogenization, we refer to [1,3,6,7,12,17].…”

An offline–online homogenization strategy to solve quasilinear two‐scale problems at the cost of one‐scale problems

“…Standard homogenization theory for such nonlinear problems [1,2,3] states that u ε converges as ε → 0 (up to a subsequence) weakly in H 1 (Ω) † toward a homogenized solution u 0 which is solution of a one-scale effective problem of the same form as (1),…”