A characterization of cut locus for tion id-i-eq1"> tion-source format-t-e-x" xmlns:search="http://marklogic.com/appservices/search">\(C^1\) hypersurfaces

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• 作者：Tatsuya Miura
• 关键词：Distance function ; Eikonal equation ; Singularity ; Ridge ; Cut locus ; Radius of curvature
• 刊名：NoDEA : Nonlinear Differential Equations and Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：23
• 期：6
• 全文大小：636 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1420-9004
• 卷排序：23

文摘

Let \(\Omega \) be an open set in \(\mathbb {R}^n\) with \(C^1\)-boundary and \(\Sigma \) be the skeleton of \(\Omega \), which consists of points where the distance function to \(\partial \Omega \) is not differentiable. This paper characterizes the cut locus (ridge) \(\overline{\Sigma }\), which is the closure of the skeleton, by introducing a generalized radius of curvature and its lower semicontinuous envelope. As an application we give a sufficient condition for vanishing of the Lebesgue measure of \(\overline{\Sigma }\).