{"title":"Topology of singular set of semiconcave function via Arnaud's theorem","authors":"Tianqi Shi, Wei Cheng, Jiahui Hong","doi":"10.3934/cpaa.2023053","DOIUrl":null,"url":null,"abstract":"We proved the (local) path-connectedness of certain subset of the singular set of semiconcave functions with linear modulus in general. In some sense this result is optimal. The proof is based on a theorem by Marie-Claude Arnaud (M.-C. Arnaud, \\textit{Pseudographs and the Lax-Oleinik semi-group: a geometric and dynamical interpretation}. Nonlinearity, \\textbf{24}(1): 71-78, 2011.). We also gave a new proof of the theorem in time-dependent case.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2022-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications on Pure and Applied Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/cpaa.2023053","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We proved the (local) path-connectedness of certain subset of the singular set of semiconcave functions with linear modulus in general. In some sense this result is optimal. The proof is based on a theorem by Marie-Claude Arnaud (M.-C. Arnaud, \textit{Pseudographs and the Lax-Oleinik semi-group: a geometric and dynamical interpretation}. Nonlinearity, \textbf{24}(1): 71-78, 2011.). We also gave a new proof of the theorem in time-dependent case.
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