# 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.

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.

## Citations

1 citation in Web of Science®
3 citations in Scopus®