Analysis & PDE

  • Anal. PDE
  • Volume 10, Number 3 (2017), 653-694.

Boundary estimates in elliptic homogenization

Zhongwei Shen

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Abstract

For a family of systems of linear elasticity with rapidly oscillating periodic coefficients, we establish sharp boundary estimates with either Dirichlet or Neumann conditions, uniform down to the microscopic scale, without smoothness assumptions on the coefficients. Under additional smoothness conditions, these estimates, combined with the corresponding local estimates, lead to the full Rellich-type estimates in Lipschitz domains and Lipschitz estimates in C1,α domains. The Cα , W1,p , and Lp estimates in C1 domains for systems with VMO coefficients are also studied. The approach is based on certain estimates on convergence rates. As a biproduct, we obtain sharp O(ε) error estimates in Lq(Ω) for q = 2d(d 1) and a Lipschitz domain Ω, with no smoothness assumption on the coefficients.

Article information

Source
Anal. PDE Volume 10, Number 3 (2017), 653-694.

Dates
Received: 9 August 2016
Revised: 21 November 2016
Accepted: 22 January 2017
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843452

Digital Object Identifier
doi:10.2140/apde.2017.10.653

Zentralblatt MATH identifier
1368.35028

Subjects
Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 35J55
Secondary: 74B05: Classical linear elasticity

Keywords
homogenization systems of elasticity convergence rates Rellich estimates Lipschitz estimates

Citation

Shen, Zhongwei. Boundary estimates in elliptic homogenization. Anal. PDE 10 (2017), no. 3, 653--694. doi:10.2140/apde.2017.10.653. https://projecteuclid.org/euclid.apde/1510843452


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