{"title":"从 Sobolev 到 BV 的扩展域的零体积边界","authors":"Tapio Rajala, Zheng Zhu","doi":"10.1007/s13163-024-00485-6","DOIUrl":null,"url":null,"abstract":"<p>In this note, we prove that the boundary of a <span>\\((W^{1, p}, BV)\\)</span>-extension domain is of volume zero under the assumption that the domain <span>\\({\\Omega }\\)</span> is 1-fat at almost every <span>\\(x\\in \\partial {\\Omega }\\)</span>. Especially, the boundary of any planar <span>\\((W^{1, p}, BV)\\)</span>-extension domain is of volume zero.</p>","PeriodicalId":501429,"journal":{"name":"Revista Matemática Complutense","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Zero volume boundary for extension domains from Sobolev to BV\",\"authors\":\"Tapio Rajala, Zheng Zhu\",\"doi\":\"10.1007/s13163-024-00485-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this note, we prove that the boundary of a <span>\\\\((W^{1, p}, BV)\\\\)</span>-extension domain is of volume zero under the assumption that the domain <span>\\\\({\\\\Omega }\\\\)</span> is 1-fat at almost every <span>\\\\(x\\\\in \\\\partial {\\\\Omega }\\\\)</span>. Especially, the boundary of any planar <span>\\\\((W^{1, p}, BV)\\\\)</span>-extension domain is of volume zero.</p>\",\"PeriodicalId\":501429,\"journal\":{\"name\":\"Revista Matemática Complutense\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-02-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Revista Matemática Complutense\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s13163-024-00485-6\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Revista Matemática Complutense","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s13163-024-00485-6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Zero volume boundary for extension domains from Sobolev to BV
In this note, we prove that the boundary of a \((W^{1, p}, BV)\)-extension domain is of volume zero under the assumption that the domain \({\Omega }\) is 1-fat at almost every \(x\in \partial {\Omega }\). Especially, the boundary of any planar \((W^{1, p}, BV)\)-extension domain is of volume zero.
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