Homogenization for locally periodic elliptic problems on a domain

2020/06/10 by Nikita N. Senik, Senik, Nikita N. · 1 citation
Computer Science · Engineering · Mathematics · #35B27 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2006.05856

openalex publication_date 2020/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ω be a Lipschitz domain in \mathbb R^d, and let \mathcal A^ε=-divA(x,x/ε)∇ be a strongly elliptic operator on Ω. We suppose that ε is small and the function A is Lipschitz in the first variable and periodic in the second, so the coefficients of \mathcal A^ε are locally periodic and rapidly oscillate. Given μ in the resolvent set, we are interested in finding the rates of approximations, as ε→0, for (\mathcal A^ε-μ)^-1 and ∇(\mathcal A^ε-μ)^-1 in the operator topology on L_p for suitable p. It is well-known that the rates depend on regularity of the effective operator \mathcal A⁰. We prove that if (\mathcal A⁰-μ)^-1 and its adjoint are bounded from L_p(Ω)ⁿ to the Lipschitz--Besov space Λ_p^1+s(Ω)ⁿ with s∈(0,1], then the rates are, respectively, ε^s and ε^s/p. The results are applied to the Dirichlet, Neumann and mixed Dirichlet--Neumann problems for strongly elliptic operators with uniformly bounded and VMO coefficients.

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