Layer potential methods for elliptic homogenization problems
Carlos E. Kenig, Zhongwei Shen · Communications on Pure and Applied Mathematics · 2010
Abstract In this paper we use the method of layer potentials to study L2 boundary value problems in a bounded Lipschitz domain Ω for a family of second‐order elliptic systems with rapidly oscillating periodic coefficients. Defining ${\cal L}_\varepsilon = - {\rm div}(A(\varepsilon ^{ - 1} X) abla )$ , under the assumption that A(X) is elliptic, symmetric, periodic, and Hölder‐continuous, we establish the solvability of the L2 Dirichlet, regularity, and Neumann problems for ${\cal L}_\varepsilon (u_\varepsilon ) = 0$ in Ω with optimal estimates uniform in ε > 0. © 2010 Wiley Periodicals, Inc.