Calculus of Variations and Geometric Measure Theory
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C. De Lellis - M. Focardi - B. Ruffini

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

created by focardi on 01 Jul 2013
modified on 26 Feb 2015


Published Paper

Inserted: 1 jul 2013
Last Updated: 26 feb 2015

Journal: Adv. Calc. Var.
Volume: 7
Number: 4
Pages: 539-545
Year: 2014
Doi: 10.1515/ acv-2013-0107


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 2, 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 4, 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.

Keywords: Local minimizer, Mumford-Shah energy, singular set


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