Partial regularity and smooth topology-preserving approximations of rough domains

TL;DR

This paper investigates the properties of 'good directions' for rough domains in Euclidean space, showing how to approximate such domains smoothly and exploring the topological conditions affecting these directions.

Contribution

It introduces a canonical smooth field of good directions for $C^0$ domains and proves approximation by smooth diffeomorphic domains, linking topology to the behavior of pseudonormals.

Findings

Existence of a canonical smooth field of good directions near the boundary.

Approximation of rough domains by smooth diffeomorphic domains from inside and outside.

Topological conditions determine whether pseudonormals cover the entire sphere.

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.