W1,p estimates for elliptic homogenization problems in nonsmooth domains

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Let Lε = -div(A(x/ε)∇), ε > 0 be a family of second order elliptic operators with real, symmetric coefficients on a bounded Lipschitz domain Ω in ℝn, subject to the Dirichlet boundary condition. Assuming that A(x) is periodic and belongs to VMO, we show that there exists δ > 0 independent of ε such that Riesz transforms ∇(Lε)-1/2 are uniformly bounded on LP(Ω), where 1 < p < 3 + δ if n ≥ 3, and 1 < p < 4 + δ if n = 2. The ranges of p's are sharp. In the case of C 1 domains, we establish the uniform LP boundedness of ∇(Lε)-1/2 for 1 < p < ∞ and n ≥ 2. As a consequence, we obtain the uniform W1,p estimates for the elliptic homogenization problem Lεuε = div f in Ω, uε = 0 on ∂Ω.

Original languageEnglish
Pages (from-to)2283-2298
Number of pages16
JournalIndiana University Mathematics Journal
Issue number5
StatePublished - 2008


  • Homogenization
  • Lipschitz domain
  • Riesz transform

ASJC Scopus subject areas

  • Mathematics (all)


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