Partial regularity and smooth topology-preserving approximations of rough domains

John M. Ball, Arghir Zarnescu

Research output: Contribution to journalArticle

5 Citations (Scopus)
14 Downloads (Pure)

Abstract

For a bounded domain Ω⊂Rm,m≥2, of class C0 , the properties are studied of fields of ‘good directions’, that is the directions with respect to which ∂Ω 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 ∂Ω , in terms of which a corresponding flow can be defined. Using this flow it is shown that Ω can be approximated from the inside and the outside by diffeomorphic domains of class C∞ . Whether or not the image of a general continuous field of good directions (pseudonormals) defined on ∂Ω is the whole of Sm−1 is shown to depend on the topology of Ω . These considerations are used to prove that if m=2,3 , or if Ω has nonzero Euler characteristic, there is a point P∈∂Ω in the neighbourhood of which ∂Ω is Lipschitz. The results provide new information even for more regular domains, with Lipschitz or smooth boundaries.
Original languageEnglish
Article number13
JournalCalculus of Variations and Partial Differential Equations
Volume56
Issue number1
Early online date12 Jan 2017
DOIs
Publication statusPublished - Feb 2017

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