Partial regularity and smooth topology-preserving approximations of rough domains

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.