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, Volume 15 (2009) no. 3, p. 576-598

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 : https://doi.org/10.1051/cocv:2008041
Classification:  49Q20,  49J45,  35B38,  35J60
Keywords: Mumford-Shah functional, Ambrosio-Tortorelli functional, gamma-convergence, critical points, brittle fracture
@article{COCV_2009__15_3_576_0,
     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},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {3},
     year = {2009},
     pages = {576-598},
     doi = {10.1051/cocv:2008041},
     zbl = {1168.49041},
     mrnumber = {2542574},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2009__15_3_576_0}
}
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, Volume 15 (2009) no. 3, pp. 576-598. doi : 10.1051/cocv:2008041. http://www.numdam.org/item/COCV_2009__15_3_576_0/

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