CRM: Centro De Giorgi
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ERC School on Free Discontinuity Problems*

course: Fine regularity results for Mumford-Shah minimizers: higer integrability of gradient and estimates on the Hausdorff dimension of the singular set

speaker: Matteo Focardi (Universita' di Firenze)

abstract: Since its very introduction in the late 80's, the Mumford-Shah variational model for image segmentation has been a source of many challenges. It has inspired several beautiful ideas and the development of new theories that have proven to be effective in different fields.

Despite all these efforts we are still faraway from giving a solution to the Mumford-Shah conjecture.

In this course we will review the state of the art of the theory in this respect: starting from an overview on the basic regularity properties to the more recent contributions. In particular, we will focus the attention on the higher integrability property enjoyed by the gradients of the minimizers, and how this fact is related to the mentioned conjecture.

Tentative plan of the hours:

1. The Mumford-Shah model: weak and strong formulation, recallings on SBV theory

2. The Mumford-Shah conjecture: known results and the regularity theory by Ambrosio-Fusco-Pallara

3. An estimate on the Hausdorff dimension of the singular set and its link with the Mumford-Shah conjecture

4. An estimate on the Hausdorff dimension of the set of triple junctions: a simplified proof via Caccioppoli partitions

5. Higher integrability of the gradient I

6. Higher intregrability of the gradient II

Mon 7 Jul, 14:30 - 16:00, Aula Dini
Tue 8 Jul, 11:00 - 12:30, Aula Dini
Wed 9 Jul, 9:00 - 10:30, Aula Dini
Thu 10 Jul, 11:00 - 12:30, Aula Dini
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