{"title":"距离函数的边界正则性和埃克纳方程","authors":"Nikolai Nikolov, Pascal J. Thomas","doi":"arxiv-2409.01774","DOIUrl":null,"url":null,"abstract":"We study the gain in regularity of the distance to the boundary of a domain\nin $\\R^m$. In particular, we show that if the signed distance function happens\nto be merely differentiable in a neighborhood of a boundary point, it and the\nboundary have to be $\\mathcal C^{1,1}$ regular. Conversely, we study the\nregularity of the distance function under regularity hypotheses of the\nboundary. Along the way, we point out that any solution to the eikonal\nequation, differentiable everywhere in a domain of the Euclidean space, admits\na gradient which is locally Lipschitz.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"1 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Boundary regularity for the distance functions, and the eikonal equation\",\"authors\":\"Nikolai Nikolov, Pascal J. Thomas\",\"doi\":\"arxiv-2409.01774\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the gain in regularity of the distance to the boundary of a domain\\nin $\\\\R^m$. In particular, we show that if the signed distance function happens\\nto be merely differentiable in a neighborhood of a boundary point, it and the\\nboundary have to be $\\\\mathcal C^{1,1}$ regular. Conversely, we study the\\nregularity of the distance function under regularity hypotheses of the\\nboundary. Along the way, we point out that any solution to the eikonal\\nequation, differentiable everywhere in a domain of the Euclidean space, admits\\na gradient which is locally Lipschitz.\",\"PeriodicalId\":501142,\"journal\":{\"name\":\"arXiv - MATH - Complex Variables\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Complex Variables\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.01774\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.01774","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Boundary regularity for the distance functions, and the eikonal equation
We study the gain in regularity of the distance to the boundary of a domain
in $\R^m$. In particular, we show that if the signed distance function happens
to be merely differentiable in a neighborhood of a boundary point, it and the
boundary have to be $\mathcal C^{1,1}$ regular. Conversely, we study the
regularity of the distance function under regularity hypotheses of the
boundary. Along the way, we point out that any solution to the eikonal
equation, differentiable everywhere in a domain of the Euclidean space, admits
a gradient which is locally Lipschitz.