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

Published online by Cambridge University Press:  24 June 2008

Gilles A. Francfort ,
Nam Q. Le  and
Sylvia Serfaty
Gilles A. Francfort
Affiliation:
LPMTM, Université Paris 13, Av. J.B. Clément, 93430 Villetaneuse, France. [email protected]
Nam Q. Le
Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer St, New York, NY 10012, USA. [email protected]
Sylvia Serfaty
Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer St, New York, NY 10012, USA. [email protected]
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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.

Keywords

Mumford-Shah functionalAmbrosio-Tortorelli functionalGamma-convergencecritical pointsbrittle fracture
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 3 , July 2009 , pp. 576 - 598
Copyright
© EDP Sciences, SMAI, 2008

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