Periodic homogenization of elliptic systems with stratified structure

Yao Xu; Weisheng Niu

doi:10.3934/dcds.2019097

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2019, Volume 39, Issue 4: 2295-2323. Doi: 10.3934/dcds.2019097

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Periodic homogenization of elliptic systems with stratified structure

Yao Xu^1, and
Weisheng Niu^2, ,

1.
Department of Mathematics, Nanjing University, Nanjing 210093, China
2.
School of Mathematical Science, Anhui University, Hefei 230601, China

^* Corresponding author: Weisheng Niu
^* Corresponding author: Weisheng Niu

Received: September 2018

Revised: October 2018

Published: January 2019

The first author is supported by the NSF of China (11731005, 11801227, 11801228), and the second anthor is supported by the NSF of China (11701002) and NSF of Anhui Province (1708085MA02).

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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(\varepsilon) $-convergence rate in $ L^{p_0}(\Omega) $ with $ p_0 = \frac{2d}{d-1} $ is obtained based on detailed discussions on stratified functions. Without the symmetry assumption, an $ O(\varepsilon^\sigma) $-convergence rate is also derived for some $ \sigma<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.

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Mathematics Subject Classification: Primary: 35B27; Secondary: 35J47.

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