We consider, in an open subset $\omega $ of ${\mathbb{R}}^{N}$, energies depending on the perimeter of a subset $E\subset \omega $ (or some equivalent surface integral) and on a function $u$ which is defined only on $\omega \setminus E$. 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.

Keywords: relaxation, free discontinuity problems, $\Gamma $-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}, pages = {717--734}, publisher = {EDP-Sciences}, volume = {13}, number = {4}, year = {2007}, doi = {10.1051/cocv:2007032}, mrnumber = {2351400}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2007032/} }

TY - JOUR AU - Braides, Andrea AU - Chambolle, Antonin AU - Solci, Margherita TI - A relaxation result for energies defined on pairs set-function and applications JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2007 SP - 717 EP - 734 VL - 13 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007032/ DO - 10.1051/cocv:2007032 LA - en ID - COCV_2007__13_4_717_0 ER -

%0 Journal Article %A Braides, Andrea %A Chambolle, Antonin %A Solci, Margherita %T A relaxation result for energies defined on pairs set-function and applications %J ESAIM: Control, Optimisation and Calculus of Variations %D 2007 %P 717-734 %V 13 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007032/ %R 10.1051/cocv:2007032 %G en %F 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/articles/10.1051/cocv:2007032/

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