### 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 |
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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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## Profiles

## John Ball

- School of Mathematical & Computer Sciences - Professor
- School of Mathematical & Computer Sciences, Mathematics - Professor

Person: Academic (Research & Teaching)