Layer potential methods for elliptic homogenization problems

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, arising in the theory of homogenization. Let L ε = −div A(ε −1 X)∇ . 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 L ε (u ε ) = 0 in Ω with optimal estimates uniform in ε > 0.

Periodic Homogenization of Green and Neumann Functions

“…The results rely on the uniform L p and Neumann function estimates obtained in [19,22] under the additional symmetry condition A D A.…”

Commutator estimates for the Dirichlet-to-Neumann map of Stokes systems in Lipschitz domains

“…We hope our results may be further applied to the study of fluid mechanics. For more knowledges on this subject, we refer the readers to [1,3,5,6,9,14,18,15,16] for more details and references therein.…”

Uniform boundary regularity in almost-periodic homogenization

“…[19]. In this subsection, we will show the uniform Rellich estimate for u ε in L 2 at large scale in Lipschitz domains without any assumption of smoothness.…”

“…In view of Lemma 4.3, we have 19) for all r ∈ [ε, 1/2]. Note that the function H(r) = H(r; u ε ) and h(r) satisfies the conditions (4.12), (4.13) and (4.14).…”

“…Indeed, it follows from [19 where C and c are independent of ε. Now using condition (1) and setting r = ω k,σ (ε) = Cε and combining (5.8), (5.12), (5.13) and (5.14), we obtain the usual well-known Rellich estimates:…”