Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford–Shah functional |
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| Authors: | Luigi Ambrosio Nicola Fusco John E. Hutchinson |
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| Affiliation: | (1) Scuola Normale Superiore, Piazza dei Cavalieri, 56100 Pisa, Italy (e-mail: luigi@ambrosio.sns.it) , IT;(2) Dipartimento di Matematica, Via Cintia, 80126 Napoli, Italy (e-mail: n.fusco@unina.it) , IT;(3) School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia (e-mail: John.Hutchinson@anu.edu.au) , AU |
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| Abstract: | The paper is concerned with the higher regularity properties of the minimizers of the Mumford–Shah functional. It is shown that, near to singular points where the scaled Dirichlet integral tends to 0, the discontinuity set is close to an Almgren area minimizing set. As a byproduct, the set of singular points of this type has Hausdorff dimension at most N-2, N being the dimension of the ambient space. Assuming higher integrability of the gradient this leads to an optimal estimate of the Hausdorff dimension of the full singular set. Received: 5 July 2001 / Accepted: 29 November 2001 / Published online: 23 May 2002 |
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| Keywords: | Mathematics Subject Classification (2000): 49J45 49Q20 |
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