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

We consider, in an open subset Ω of R , 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 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” Ω \ 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.