Geodesic
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, Tome 15 (2009) no. 3, pp. 576-598

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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.

DOI : 10.1051/cocv:2008041
Classification : 49Q20, 49J45, 35B38, 35J60
Keywords: Mumford-Shah functional, Ambrosio-Tortorelli functional, gamma-convergence, critical points, brittle fracture 
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     author = {Francfort, Gilles A. and Le, Nam Q. and Serfaty, Sylvia},
     title = {Critical points of {Ambrosio-Tortorelli} converge to critical points of {Mumford-Shah} in the one-dimensional {Dirichlet} case},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {576--598},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {3},
     year = {2009},
     doi = {10.1051/cocv:2008041},
     mrnumber = {2542574},
     zbl = {1168.49041},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2008041/}
}
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Francfort, Gilles A.; Le, Nam Q.; 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, Tome 15 (2009) no. 3, pp. 576-598. doi: 10.1051/cocv:2008041

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