Boundary estimates in elliptic homogenization

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Abstract

For a family of systems of linear elasticity with rapidly oscillating periodic coef?cients, we establishsharp boundary estimates with either Dirichlet or Neumann conditions, uniform down to the microscopicscale, without smoothness assumptions on the coef?cients. 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 L pestimates 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.

Original languageEnglish
Pages (from-to)653-694
Number of pages42
JournalAnalysis and PDE
Volume10
Issue number3
DOIs
StatePublished - 2017

Keywords

  • Convergence rates
  • Homogenization
  • Lipschitz estimates
  • Rellich estimates
  • Systems of elasticity

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

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