A relaxation result for energies defined on pairs set-function

We consider, in an open subset, energies depending on the perimeter of a subset E (or some equivalent surface integral) and on a function u which is defined only on the complement. 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.