Partial regularity and smooth topology-preserving approximations of rough domains

Author: 

Ball, J
Zarnescu, A

Publication Date: 

12 January 2017

Journal: 

Calculus of Variations and Partial Differential Equations

Last Updated: 

2020-11-19T09:51:41.697+00:00

Issue: 

13

Volume: 

56

DOI: 

10.1007/s00526-016-1092-6

page: 

1-32

abstract: 

For a bounded domain Ω⊂Rm,m≥2,Ω⊂Rm,m≥2, of class C0C0, the properties are studied of fields of ‘good directions’, that is the directions with respect to which ∂Ω∂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∂Ω, in terms of which a corresponding flow can be defined. Using this flow it is shown that ΩΩ can be approximated from the inside and the outside by diffeomorphic domains of class C∞C∞. Whether or not the image of a general continuous field of good directions (pseudonormals) defined on ∂Ω∂Ω is the whole of Sm−1Sm−1 is shown to depend on the topology of ΩΩ. These considerations are used to prove that if m=2,3m=2,3, or if ΩΩ has nonzero Euler characteristic, there is a point P∈∂ΩP∈∂Ω in the neighbourhood of which ∂Ω∂Ω is Lipschitz. The results provide new information even for more regular domains, with Lipschitz or smooth boundaries.

Symplectic id: 

668628

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article