Partial regularity and smooth topology-preserving approximations of rough domains

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• 作者：John M. Ball ; Arghir Zarnescu
• 关键词：Mathematics Subject Classification41A29 ; 54C50 ; 34C40
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：56
• 期：1
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-0835
• 卷排序：56

文摘

For a bounded domain \(\Omega \subset {\mathbb R}^m, m\ge 2,\) of class \(C^0\), the properties are studied of fields of ‘good directions’, that is the directions with respect to which \(\partial \Omega \) can be locally represented as the graph of a continuous function. For any such domain there is a canonical smooth field of good directions defined in a suitable neighbourhood of \(\partial \Omega \), in terms of which a corresponding flow can be defined. Using this flow it is shown that \(\Omega \) can be approximated from the inside and the outside by diffeomorphic domains of class \(C^\infty \). Whether or not the image of a general continuous field of good directions (pseudonormals) defined on \(\partial \Omega \) is the whole of \(S^{m-1}\) is shown to depend on the topology of \(\Omega \). These considerations are used to prove that if \(m=2,3\), or if \(\Omega \) has nonzero Euler characteristic, there is a point \(P\in \partial \Omega \) in the neighbourhood of which \(\partial \Omega \) is Lipschitz. The results provide new information even for more regular domains, with Lipschitz or smooth boundaries.