{"title":"Approximation of divergence-free vector fields vanishing on rough planar sets","authors":"Giacomo Del Nin, Bian Wu","doi":"arxiv-2409.09880","DOIUrl":null,"url":null,"abstract":"Given any divergence-free vector field of Sobolev class $W^{m,p}_0(\\Omega)$\nin bounded open subset $\\Omega \\subset \\mathbb{R}^2$, we are interested in\napproximating it in $W^{m,p}$ with divergence-free smooth vector fields\ncompactly supported in $\\Omega$. We show that this approximation property holds\nin the following cases. For $p>2$, this holds given that $\\partial \\Omega$ has\nzero Lebesgue measure (a weaker but more technical condition is sufficient);\nFor $p \\leq 2$, this holds if $\\Omega^c$ can be decomposed into finitely many\ndisjoint closed set, each of which is connected or $d$-Ahlfors regular for some\n$d\\in[0,2]$. This has links to the uniqueness of weak solutions to the Stokes\nequation in $\\Omega$. For H\\\"older spaces, we prove this property in general\nbounded domains.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"19 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Analysis of PDEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09880","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
Given any divergence-free vector field of Sobolev class $W^{m,p}_0(\Omega)$
in bounded open subset $\Omega \subset \mathbb{R}^2$, we are interested in
approximating it in $W^{m,p}$ with divergence-free smooth vector fields
compactly supported in $\Omega$. We show that this approximation property holds
in the following cases. For $p>2$, this holds given that $\partial \Omega$ has
zero Lebesgue measure (a weaker but more technical condition is sufficient);
For $p \leq 2$, this holds if $\Omega^c$ can be decomposed into finitely many
disjoint closed set, each of which is connected or $d$-Ahlfors regular for some
$d\in[0,2]$. This has links to the uniqueness of weak solutions to the Stokes
equation in $\Omega$. For H\"older spaces, we prove this property in general
bounded domains.