Partial regularity and smooth topology-preserving approximations of rough domains

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 ${\mathbb{S}}^{m-1}$ 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.

Final version appeared in Calc. Var PDE 56, Issue 1, 2017

References in corpus (1)

• The Dirichlet boundary problem for second order parabolic operators satisfying Carleson condition

Cited by in corpus (2)

• Continuum limit of total variation on point clouds
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