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A note on the Hausdorff dimension of the singular set for minimizers of the Mumford–Shah energy


De Lellis, Camillo; Focardi, Matteo; Ruffini, Berardo (2014). A note on the Hausdorff dimension of the singular set for minimizers of the Mumford–Shah energy. Advances in Calculus of Variations, 7(4):539-545.

Abstract

We give a more elementary proof of a result by Ambrosio, Fusco and Hutchinson to estimate the Hausdorff dimension of the singular set of minimizers of the Mumford–Shah energy (see [Calc. Var. Partial Differential Equations 16 (2003), no. 2, 187–215, Theorem 5.6]). On the one hand, we follow the strategy of the above mentioned paper; but on the other hand our analysis greatly simplifies the argument since it relies on the compactness result proved by the first two authors in [J. Math. Pures Appl. 100 (2013), 391–409, Theorem 13] for sequences of local minimizers with vanishing gradient energy, and the regularity theory of minimal Caccioppoli partitions, rather than on the corresponding results for Almgren's area minimizing sets.

Abstract

We give a more elementary proof of a result by Ambrosio, Fusco and Hutchinson to estimate the Hausdorff dimension of the singular set of minimizers of the Mumford–Shah energy (see [Calc. Var. Partial Differential Equations 16 (2003), no. 2, 187–215, Theorem 5.6]). On the one hand, we follow the strategy of the above mentioned paper; but on the other hand our analysis greatly simplifies the argument since it relies on the compactness result proved by the first two authors in [J. Math. Pures Appl. 100 (2013), 391–409, Theorem 13] for sequences of local minimizers with vanishing gradient energy, and the regularity theory of minimal Caccioppoli partitions, rather than on the corresponding results for Almgren's area minimizing sets.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2014
Deposited On:14 Jan 2016 11:00
Last Modified:05 Apr 2016 19:16
Publisher:De Gruyter
ISSN:1864-8258
Publisher DOI:https://doi.org/10.1515/acv-2013-0107

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