ESAIM: COCV
DOI: 10.1051/cocv:2008041

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

Gilles A. Francfort¹, Nam Q. Le² and Sylvia Serfaty<sup>3

1</sup>  LPMTM, Université Paris 13, Av. J.B. Clément, 93430 Villetaneuse, France. francfor@galilee.univ-paris13.fr
²  Courant Institute of Mathematical Sciences, 251 Mercer St, New York, NY 10012, USA. quangle@cims.nyu.edu
³  Courant Institute of Mathematical Sciences, 251 Mercer St, New York, NY 10012, USA. serfaty@cims.nyu.edu

Received September 19, 2007. Revised February 29, 2008. Published online June 24, 2008.

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.

Mathematics Subject Classification. 49Q20, 49J45, 35B38, 35J60

Key words: Mumford-Shah functional, Ambrosio-Tortorelli functional, Gamma-convergence, critical points, brittle fracture


© EDP Sciences, SMAI 2008