SUR LA REGULARITE DES MINIMISEURS DE MUMFORD-SHAH EN DIMENSION 3 ET SUPERIEURE

Abstract : In this thesis we study some aspects about the regularity of a minimizer for the Mumford-Shah functional. The story takes place mostly in dimension 3 but some results are still true in higher dimension. The first part is about global minimizers in RN. We prove that if (u;K) is a global minimizer and if K is a cone smooth enough, then u (modulo constants) is a homogenous function of degree 1/2 in R^N\K. Thank to this we can link the existence of a global minimizer and the spectrum of the spherical laplacian in S^N-1\K. A consequence is that there is no global minimizer with K an angular sector in R3. In the second part we prove a regularity theorem near a minimal cone of type P, Y and T. We show that if K is close enough (in distance) to a cone of type P, Y or T in a certain ball, then K is the C^1,alpha image of a P, Y or T in a smaller ball. This is a generalisation of a theorem of L. Ambrosio, N. Fusco and D. Pallara [AFP07]. The technics employed are not specific to dimension 3 and it should be used to prove some results in any dimension for a Mumford-Shah minimizer whenever some regularity result about almost minimal sets would exist.
Document type :
Theses
Complete list of metadatas

https://tel.archives-ouvertes.fr/tel-00288822
Contributor : Antoine Lemenant <>
Submitted on : Wednesday, June 18, 2008 - 4:24:33 PM
Last modification on : Thursday, January 11, 2018 - 6:12:18 AM
Long-term archiving on : Friday, May 28, 2010 - 10:37:35 PM

Identifiers

  • HAL Id : tel-00288822, version 1

Collections

Citation

Antoine Lemenant. SUR LA REGULARITE DES MINIMISEURS DE MUMFORD-SHAH EN DIMENSION 3 ET SUPERIEURE. Mathématiques [math]. Université Paris Sud - Paris XI, 2008. Français. ⟨tel-00288822⟩

Share

Metrics

Record views

220

Files downloads

831