ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - July 2009 - Volume 15, Issue 03  

Research Article

Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case

Francfort, Gilles A.a1, Le, Nam Q.a2 and Serfaty, Sylviaa3

a1 LPMTM, Université Paris 13, Av. J.B. Clément, 93430 Villetaneuse, France. francfor@galilee.univ-paris13.fr

a2 Courant Institute of Mathematical Sciences, 251 Mercer St, New York, NY 10012, USA. quangle@cims.nyu.edu

a3 Courant Institute of Mathematical Sciences, 251 Mercer St, New York, NY 10012, USA. serfaty@cims.nyu.edu

Abstract

Critical points of a variant of the Ambrosio-Tortorelli functional, for which non-zero Dirichlet boundary conditions replace the fidelity term, are investigated. They are shown to converge to particular critical points of the corresponding variant of the Mumford-Shah functional; those exhibit many symmetries. That Dirichlet variant is the natural functional when addressing a problem of brittle fracture in an elastic material.

(Received September 19 2007)

(Revised February 29 2008)

(Online publication June 24 2008)

Key Words:

  • Mumford-Shah functional;
  • Ambrosio-Tortorelli functional;
  • Gamma-convergence;
  • critical points;
  • brittle fracture

Mathematics Subject Classification:

  • 49Q20;
  • 49J45;
  • 35B38;
  • 35J60