[CvGmt News] Seminari Calcolo delle Variazioni

depascal at dm.unipi.it depascal at dm.unipi.it
Fri Mar 27 11:24:28 CET 2009

Seminari di Calcolo delle Variazioni

Mercoledi' 1 Aprile, Sala riunioni Dipartimento di Matematica
ORE 16.30

Dr. Antoine Lemenant (Centro di Ricerca Matematica Ennio De Giorgi)

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).
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You find this news in the cvgmt preprint server: http://cvgmt.sns.it/news/20090401a/