Stochastic homogenization of elliptic boundary‐value problems with L^p‐data

Article type: Research Article

Authors: Andrews, Kevin T. | Wright, Steve

Affiliations: Department of Mathematical Sciences, Oakland University, Rochester, MI 48309‐4485, USA

Abstract: Using stochastic two‐scale mean convergence, homogenization results are obtained for the Dirichlet problem for elliptic equations and systems in divergence form. The coefficients in the equations are assumed to have the bivariate form a(x, T(\varepsilon^{-1}x)\omega), where a is only assumed to be in L^\infty(Q \times {\varOmega}) and T is an n‐parameter dynamical system defined on a separable probability space {\varOmega} which induces random oscillations. The nonhomogeneous terms in the equations are assumed to be in L^p and the boundary data is taken to be in W^{1-(1/p),p}. Results are obtained for both unperforated and randomly perforated domains.

Keywords: Stochastic homogenization , random coefficients , elliptic equation , boundary‐value problem , randomly perforated domain

Journal: Asymptotic Analysis, vol. 17, no. 3, pp. 165-184, 1998