Periodic homogenization of elliptic systems with stratified structure

Abstract

This paper concerns with the quantitative homogenization of second-order elliptic systems with periodic stratified structure in Lipschitz domains. Under the symmetry assumption on coefficient matrix, the sharp O(ε)-convergence rate in L p 0 (Ω) with p 0 = 2d d−1 is obtained based on detailed discussions on stratified functions. Without the symmetry assumption, an O(ε σ )-convergence rate is also derived for some σ < 1 by the Meyers estimate. Based on this convergence rate, we establish the uniform interior Lipschitz estimate. The uniform interior W 1,p and Hölder estimates are also obtained by the real variable method.

Quantitative Estimates in Reiterated Homogenization

“…for 0 < ε < 1, where C depends only on d, m, µ, and Ω (see Lemma 4.1). Estimate (1.13) improves a similar estimate in [27], where a general case A ε (x) = A(x, ρ(x)/ε) was considered by the first and third authors. It also leads to the following theorem on the L 2 convergence rate for the operator L ε .…”

“…Also see [26] for the nonlinear case. The estimate (3.13) is sharper than the similar estimates in [27,26]. Lemma 3.2.…”

“…The following lemma was proved in [24] for the case A ε = A(x/ε). The case A ε = A(x, ρ(x)/ε) for stratified structures was considered in [27] by the first and third authors. Also see [26] for the nonlinear case.…”

Convergence rates in homogenization of parabolic systems with locally periodic coefficients

“…The earliest work on these kinds of scale-invariant results in homogenization should be attributed to Z. Shen who established the rate for elliptic systems with periodic coefficients in his noted book [25]. Later in [28,29], the scale-invariant error estimates were extended to elliptic systems with stratified coefficients A(x, ρ(x)/ε) under rather general smoothness assumptions, where ρ(x) could be simply assumed to be diffeomorphic. Stratified structure and locally periodic structure are quite similar and the keypoints of both concentrate on two scales.…”