## Analysis & PDE

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

### Boundary estimates in elliptic homogenization

Zhongwei Shen

#### 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
Revised: 21 November 2016
Accepted: 22 January 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843452

Digital Object Identifier
doi:10.2140/apde.2017.10.653

Mathematical Reviews number (MathSciNet)
MR3641883

Zentralblatt MATH identifier
1368.35028

#### 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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