Quantitative homogenization of elliptic partial differential equations with random oscillatory boundary data

Abstract

We study the averaging behavior of nonlinear uniformly elliptic partial differential equations with random Dirichlet or Neumann boundary data oscillating on a small scale. Under conditions on the operator, the data and the random media leading to concentration of measure, we prove an almost sure and local uniform homogenization result with a rate of convergence in probability.© 2014 Elsevier Masson SAS. All rights reserved.r é s u m éOn étudie le comportement homogénéisant d'équations aux dérivées partielles elliptiques nonlinéaires, avec conditions au bord de Dirichlet ou de Neumann aléatoires, oscillantes à petite échelle. Sous certaines contraintes sur l'opérateur, telles que les données et les milieux aléatoires conduisent à une concentration de la mesure, on démontre un résultat presque sûr d'homogénéisation locale uniforme, avec un taux de convergence en probabilité.

Boundary Layers in Periodic Homogenization of Neumann Problems

“…In recent years there has been considerable interest in the homogenization of boundary value problems with oscillating boundary data [15,16,22,1,19,2,12,10,13,3,5] (also see related earlier work in [23,24,20,21,4]. In the case of Dirichlet problem, L ε (u ε ) = 0 in Ω and u ε = g(x, x/ε) on ∂Ω, (1.10) where g(x, y) is assumed to be periodic in y, major progress was made in [16] and more recently in [5].…”

Uniform Estimates in Homogenization: Compactness Methods and Applications

“…This problem has attracted a great deal of attention in recent years: see [24,23,36] and the related papers [13,17,18] for non divergence form equations. Its diculty lies in the following two facts: on the one hand the a priori bound (1.4) is singular in ε, on the other hand the boundary breaks the periodic microstructure, making the behavior of u ε bl very sensitive to the interaction between the periodic lattice and ∂Ω.…”