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

¹ 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: 19 September 2007
Revised: 29 February 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.