A relaxation result for energies defined on pairs set-function and applications

Braides, Andrea; Chambolle, Antonin; Solci, Margherita

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 717-734.

We consider, in an open subset $\omega$ of ${ℝ}^N$, 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.

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