A relaxation result for energies defined on pairs set-function and applications
ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 4, p. 717-734

We consider, in an open subset ω of N , energies depending on the perimeter of a subset Eω (or some equivalent surface integral) and on a function u which is defined only on ωE. We compute the lower semicontinuous envelope of such energies. This relaxation has to take into account the fact that in the limit, the “holes” E may collapse into a discontinuity of u, whose surface will be counted twice in the relaxed energy. We discuss some situations where such energies appear, and give, as an application, a new proof of convergence for an extension of Ambrosio-Tortorelli’s approximation to the Mumford-Shah functional.

DOI : https://doi.org/10.1051/cocv:2007032
Classification:  49J45,  49Q20,  49Q10
Keywords: relaxation, free discontinuity problems, Γ-convergence
@article{COCV_2007__13_4_717_0,
     author = {Braides, Andrea and Chambolle, Antonin and Solci, Margherita},
     title = {A relaxation result for energies defined on pairs set-function and applications},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {13},
     number = {4},
     year = {2007},
     pages = {717-734},
     doi = {10.1051/cocv:2007032},
     zbl = {pre05212111},
     mrnumber = {2351400},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2007__13_4_717_0}
}
Braides, Andrea; Chambolle, Antonin; Solci, Margherita. A relaxation result for energies defined on pairs set-function and applications. ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 4, pp. 717-734. doi : 10.1051/cocv:2007032. http://www.numdam.org/item/COCV_2007__13_4_717_0/

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