A Relaxation Result for Energies Dened on Pairs Set-function and Applications a Relaxation Result for Energies Dened 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 de ned 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.