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

Gilles A. Francfort; Nam Q. Le; Sylvia Serfaty

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 15, Issue: 3, page 576-598
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Francfort, Gilles A., Le, Nam Q., and Serfaty, Sylvia. "Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case." ESAIM: Control, Optimisation and Calculus of Variations 15.3 (2008): 576-598. <http://eudml.org/doc/90928>.

@article{Francfort2008,
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. },
author = {Francfort, Gilles A., Le, Nam Q., Serfaty, Sylvia},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Mumford-Shah functional; Ambrosio-Tortorelli functional; Gamma-convergence; critical points; brittle fracture; -convergence},
language = {eng},
month = {6},
number = {3},
pages = {576-598},
publisher = {EDP Sciences},
title = {Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case},
url = {http://eudml.org/doc/90928},
volume = {15},
year = {2008},
}

TY - JOUR
AU - Francfort, Gilles A.
AU - Le, Nam Q.
AU - Serfaty, Sylvia
TI - Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/6//
PB - EDP Sciences
VL - 15
IS - 3
SP - 576
EP - 598
AB - 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.
LA - eng
KW - Mumford-Shah functional; Ambrosio-Tortorelli functional; Gamma-convergence; critical points; brittle fracture; -convergence
UR - http://eudml.org/doc/90928
ER -

References

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  14. P.J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics107. Springer-Verlag, New York (1986).  Zbl0588.22001
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