Xin Chen, Zhen-Qing Chen, Takashi Kumagai, Jian Wang
{"title":"Quantitative periodic homogenization for symmetric non-local stable-like operators","authors":"Xin Chen, Zhen-Qing Chen, Takashi Kumagai, Jian Wang","doi":"arxiv-2409.08120","DOIUrl":null,"url":null,"abstract":"Homogenization for non-local operators in periodic environments has been\nstudied intensively. So far, these works are mainly devoted to the qualitative\nresults, that is, to determine explicitly the operators in the limit. To the\nbest of authors' knowledge, there is no result concerning the convergence rates\nof the homogenization for stable-like operators in periodic environments. In\nthis paper, we establish a quantitative homogenization result for symmetric\n$\\alpha$-stable-like operators on $\\R^d$ with periodic coefficients. In\nparticular, we show that the convergence rate for the solutions of associated\nDirichlet problems on a bounded domain $D$ is of order $$\n\\varepsilon^{(2-\\alpha)/2}\\I_{\\{\\alpha\\in\n(1,2)\\}}+\\varepsilon^{\\alpha/2}\\I_{\\{\\alpha\\in (0,1)\\}}+\\varepsilon^{1/2}|\\log\n\\e|^2\\I_{\\{\\alpha=1\\}}, $$ while, when the solution to the equation in the\nlimit is in $C^2_c(D)$, the convergence rate becomes $$ \\varepsilon^{2-\\alpha}\\I_{\\{\\alpha\\in\n(1,2)\\}}+\\varepsilon^{\\alpha}\\I_{\\{\\alpha\\in (0,1)\\}}+\\varepsilon |\\log\n\\e|^2\\I_{\\{\\alpha=1\\}}. $$ This indicates that the boundary decay behaviors of\nthe solution to the equation in the limit affects the convergence rate in the\nhomogenization.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"1 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.08120","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.