Calculus of Variations and Geometric Measure Theory

On the regularity of Mumford-Shah minimizers in dimension 3

Antoine Lemenant (Université de Lorraine (Nancy))

created by depascal on 27 Mar 2009

1 apr 2009


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 $$F(u,K):=\int{\Omega}
2 + \int{\Omega \backslash K}
\nabla u
2+H{N-1}(K)$$ 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).