Homogenization for locally periodic elliptic problems on a domain

TL;DR

This paper establishes the rates at which solutions to locally periodic elliptic problems converge to their homogenized limits, depending on the regularity of the effective operator, with applications to various boundary value problems.

Contribution

It provides explicit convergence rates for the resolvent operators of elliptic problems with locally periodic coefficients, extending homogenization theory to less regular settings.

Findings

Convergence rates depend on the regularity of the effective operator.

Rates are $ ext{O}( ext{} extstylerac{1}{p} ext{)}$ for the gradient approximation.

Results apply to Dirichlet, Neumann, and mixed boundary conditions.

Abstract

Let $\Omega$ be a Lipschitz domain in $\mathbb R^d$, and let $\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$ be a strongly elliptic operator on $\Omega$. We suppose that $\varepsilon$ is small and the function $A$ is Lipschitz in the first variable and periodic in the second, so the coefficients of $\mathcal A^\varepsilon$ are locally periodic and rapidly oscillate. Given $\mu$ in the resolvent set, we are interested in finding the rates of approximations, as $\varepsilon\to0$, for $(\mathcal A^\varepsilon-\mu)^{-1}$ and $\nabla(\mathcal A^\varepsilon-\mu)^{-1}$ in the operator topology on $L_p$ for suitable $p$. It is well-known that the rates depend on regularity of the effective operator $\mathcal A^0$. We prove that if $(\mathcal A^0-\mu)^{-1}$ and its adjoint are bounded from $L_p(\Omega)^n$ to the Lipschitz--Besov space $\Lambda_p^{1+s}(\Omega)^n$ with $s\in(0,1]$, then the rates are, respectively, $\varepsilon^s$ and $\varepsilon^{s/p}$. The results are applied to the Dirichlet, Neumann and mixed Dirichlet--Neumann problems for strongly elliptic operators with uniformly bounded and $\operatorname{VMO}$ coefficients.