Partial regularity and smooth topology-preserving approximations of rough domains |
| |
Authors: | Email author" target="_blank">John?M?BallEmail author Arghir?Zarnescu |
| |
Institution: | 1.Oxford Centre for Nonlinear PDE, Mathematical Institute,University of Oxford,Oxford,UK;2.IKERBASQUE, Basque Foundation for Science,Bilbao,Spain;3.BCAM, Basque Center for Applied Mathematics,Bilbao,Spain;4.“Simion Stoilow” Institute of the Romanian Academy,Bucharest,Romania |
| |
Abstract: | 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. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|