Uniform Calderón-Zygmund estimates in multiscale elliptic homogenization

Abstract

This paper is concerned with the elliptic equation $-\text{div}(A_{\epsilon }\nabla u_{\epsilon })=\text{div}f$ in a bounded $C^1$ domain, where $A_{\epsilon }$ takes a form of $A_{\epsilon }(x)=A(x/{\epsilon }_1,x/{\epsilon }_2,\cdots ,x/{\epsilon }_n)$, with $A(y_1,y_2,\cdots ,y_n)$ being 1-periodic in each $y_i$. We prove the uniform Calderón-Zygmund estimate, namely, the uniform $L^p$ boundedness of the linear map $f\mapsto \nabla u_{\epsilon }$ for any $p\in (1,\infty )$ with a constant independent of small parameters $({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n)\in (0,1{]}^n$. 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.

26 pages