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

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We consider, in an open subset ω of ℝ N , energies depending on the perimeter of a subset E⊂ω (or some equivalent surface integral) and on a function u which is defined only on ω∖E. 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 : 10.1051/cocv:2007032
Classification : 49J45, 49Q20, 49Q10
Keywords: relaxation, free discontinuity problems, $\Gamma $-convergence 
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     title = {A relaxation result for energies defined on pairs set-function and applications},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {717--734},
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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, Tome 13 (2007) no. 4, pp. 717-734. doi: 10.1051/cocv:2007032

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