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Partial regularity and smooth topology-preserving approximations of rough domains

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Abstract

For a bounded domain \(\Omega \subset {\mathbb R}^m, m\ge 2,\) 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 \(S^{m-1}\) 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.

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Notes

  1. i.e. for each point \(P\in \partial \Omega \) there exists a neighbourhood U(P) in \({\mathbb R}^m\) and a non-zero vector \(b(P)\in {\mathbb R}^m\) such that \(x+tb\in \Omega , \text{ for } \text{ all } x\in \overline{\Omega }\cap U(P), 0<t<1\).

  2. if \(m\ge 4\) under the possibly unnecessary assumption that the Euler characteristic of \(\Omega \) is nonzero.

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Acknowledgements

The research of both authors was partly supported by EPSRC grants EP/E010288/1 and by the EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1). The research of JMB was also supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 291053 and by a Royal Society Wolfson Research Merit Award. The research of AZ was partially supported by the Basque Government through the BERC 2014-2017 program; and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323. We are especially grateful to Marc Lackenby for the suggestion of an example forming the basis of the second part of Proposition 6.1 as well as for other advice, to Vladimir Šverák, whose perceptive questions after a seminar of JMB at the University of Minnesota on an earlier version of the paper led to radical improvements, and to Rob Kirby for long discussions concerning smoothing of manifolds that are incorporated in Remark 5.4. We are also grateful to Moe Hirsch for his interest and various references, and to Nigel Hitchin for useful comments. Finally we are grateful to anonymous referees for pointing out to us the connection with the theory of smoothing of manifolds, for a simplification of our original example in the second part of Proposition 6.1, and for mentioning various relevant references.

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Authors and Affiliations

  1. Oxford Centre for Nonlinear PDE, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    John M. Ball

  2. IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013, Bilbao, Bizkaia, Spain

    Arghir Zarnescu

  3. BCAM, Basque Center for Applied Mathematics, Mazarredo 14, E48009, Bilbao, Basque Country, Spain

    Arghir Zarnescu

  4. “Simion Stoilow” Institute of the Romanian Academy, 21 Calea Griviţei Street, 010702, Bucharest, Romania

    Arghir Zarnescu

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  1. John M. Ball
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Correspondence to John M. Ball.

Additional information

Communicated by L. Ambrosio.

In memoriam J. Bryce McLeod.

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Ball, J.M., Zarnescu, A. Partial regularity and smooth topology-preserving approximations of rough domains. Calc. Var. 56, 13 (2017). https://doi.org/10.1007/s00526-016-1092-6

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