{"title":"Rectifiability of the free boundary for varifolds","authors":"L. De Masi","doi":"10.1512/iumj.2021.70.9401","DOIUrl":null,"url":null,"abstract":"Abstract. We establish a partial rectifiability result for the free boundary of a k-varifold V . Namely, we first refine a theorem of Grüter and Jost by showing that the first variation of a general varifold with free boundary is a Radon measure. Next we show that if the mean curvature H of V is in L for some p ∈ [1, k], then the set of points where the k-density of V does not exist or is infinite has Hausdorff dimension at most k − p. We use this result to prove, under suitable assumptions, that the part of the first variation of V with positive and finite (k− 1)-density is (k− 1)-rectifiable.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1512/iumj.2021.70.9401","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 7
Abstract
Abstract. We establish a partial rectifiability result for the free boundary of a k-varifold V . Namely, we first refine a theorem of Grüter and Jost by showing that the first variation of a general varifold with free boundary is a Radon measure. Next we show that if the mean curvature H of V is in L for some p ∈ [1, k], then the set of points where the k-density of V does not exist or is infinite has Hausdorff dimension at most k − p. We use this result to prove, under suitable assumptions, that the part of the first variation of V with positive and finite (k− 1)-density is (k− 1)-rectifiable.
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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