Abstract
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.
Original language  English 

Article number  13 
Journal  Calculus of Variations and Partial Differential Equations 
Volume  56 
Issue number  1 
Early online date  12 Jan 2017 
DOIs  
Publication status  Published  Feb 2017 
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John Ball
 School of Mathematical & Computer Sciences  Professor
 School of Mathematical & Computer Sciences, Mathematics  Professor
Person: Academic (Research & Teaching)