Uniform Calderón-Zygmund estimates in multiscale elliptic homogenization

2024/05/24 by Niu, Weisheng, Zhuge, Jinping · 1 citation
#35B27 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2405.15149

Abstract

This paper is concerned with the elliptic equation -div (A_ε ∇ u_ε) = div f in a bounded C¹ domain, where A_ε takes a form of A_ε(x) = A(x/ε₁, x/ε₂,⋯, x/ε_n), with A(y₁,y₂,⋯,y_n) being 1-periodic in each y_i. We prove the uniform Calderón-Zygmund estimate, namely, the uniform L^p boundedness of the linear map f↦ ∇ u_ε for any p∈ (1,∞) with a constant independent of small parameters (ε₁,ε₂,⋯, ε_n) ∈ (0,1]ⁿ. Our result includes the uniform Calderón-Zygmund estimate in quasiperiodic elliptic homogenization (even without the Diophantine condition), which was previously unknown. The proof novelly combines the Dirichlet's theorem on the simultaneous Diophantine approximation from number theory, a technique of reperiodization, reiterated periodic homogenization and a large-scale real-variable argument. Using the idea of reperiodization, we also obtain some large-scale or mesoscopic-scale Lipschitz estimates.

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• Optimal convergence rates in multiscale elliptic homogenization

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