Quantitative periodic homogenization for symmetric non-local stable-like operators

TL;DR

This paper establishes quantitative rates of convergence for homogenization of symmetric stable-like non-local operators with periodic coefficients, revealing how boundary decay influences the homogenization speed.

Contribution

It provides the first known quantitative homogenization results for stable-like operators in periodic environments, including explicit convergence rates depending on the stability index.

Findings

Convergence rate of order ε^{(2−α)/2} for α in (1,2)

Convergence rate of order ε^{α/2} for α in (0,1)

Boundary decay affects the homogenization rate

Abstract

Homogenization for non-local operators in periodic environments has been studied intensively. So far, these works are mainly devoted to the qualitative results, that is, to determine explicitly the operators in the limit. To the best of authors' knowledge, there is no result concerning the convergence rates of the homogenization for stable-like operators in periodic environments. In this paper, we establish a quantitative homogenization result for symmetric $\alpha$-stable-like operators on $\R^d$ with periodic coefficients. In particular, we show that the convergence rate for the solutions of associated Dirichlet problems on a bounded domain $D$ is of order $$ \varepsilon^{(2-\alpha)/2}\I_{\{\alpha\in (1,2)\}}+\varepsilon^{\alpha/2}\I_{\{\alpha\in (0,1)\}}+\varepsilon^{1/2}|\log \e|^2\I_{\{\alpha=1\}}, $$ while, when the solution to the equation in the limit is in $C^2_c(D)$, the convergence rate becomes $$ \varepsilon^{2-\alpha}\I_{\{\alpha\in (1,2)\}}+\varepsilon^{\alpha}\I_{\{\alpha\in (0,1)\}}+\varepsilon |\log \e|^2\I_{\{\alpha=1\}}. $$ This indicates that the boundary decay behaviors of the solution to the equation in the limit affects the convergence rate in the homogenization.