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

We consider, in an open subset Ω
ofRN, energies depending on the perimeter of a subsetEС Ω
(or some equivalent surface integral) and on a function u which is defined only onE. 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 \Emay collapse into a
discontinuity ofu, 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.