Partial regularity and smooth topology-preserving approximations of rough domains

Abstract

For a bounded domain $\Omega\subset\mathbb{R}^m, m\geq 2,$ 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.