1 apr 2009

**Abstract.**

Mercoledi' 1 Aprile, Sala riunioni Dipartimento di Matematica

ORE 16.30

Title: On the regularity of Mumford-Shah minimizers in dimension 3.

Abstract :
In 1989, D. Mumford and J. Shah proposed to define
$$F(u,K):=\int_{{\Omega}u}-g^{2} + \int_{{\Omega} \backslash K}

\nabla u^{2+H}^{{N}-1}(K)$$
and to get a segmentation of the image $g$ they minimizes $F$ over all the admissible pairs $(u,K)$ where $K$ is a closed set of codimension 1 and $u$ is regular out of $K$. The regularity of the segmentation $K$ has been highly investigate during the last 20 years and the conjecture of D. Mumford and J. Shah about the regularity in $R^2$ is still not completely proved. In this talk I will present a new regularity result in dimension 3 that comes from a work in my Thesis directed by Guy David. The aim of the talk is to explain the link between Mumford-Shah minimizers and the Theorem of Jean Taylor
(1976) about almost minimal sets of soap bubble type in $R^3$. In particular, this result contains also a new proof of the Theorem of L. Ambrosio, N. Fusco and D. Pallara (1997).