12 January 2017
Calculus of Variations and Partial Differential Equations
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.
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