Layer potential methods for elliptic homogenization problems

✍ Scribed by Carlos Kenig; Zhongwei Shen

• Publisher: John Wiley and Sons
• Year: 2010
• Tongue: English
• Weight: 355 KB
• Volume: 64
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.20343

✦ Synopsis

Abstract

In this paper we use the method of layer potentials to study L^2^ boundary value problems in a bounded Lipschitz domain Ω for a family of second‐order elliptic systems with rapidly oscillating periodic coefficients. Defining
${\cal L}_\varepsilon = - {\rm div}(A(\varepsilon ^{ - 1} X)\nabla )$, under the assumption that A(X) is elliptic, symmetric, periodic, and Hölder‐continuous, we establish the solvability of the L^2^ 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.