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We consider, in an open subset Ω of ${\mathbb R}^N$ 
, energies depending on the perimeter of a subset $E\subset\Omega$ 
(or some equivalent surface integral) and on a function u which isdefined only on $\Omega\setminus E$ 
. We compute the lower semicontinuous envelopeof such energies. This relaxation has to take intoaccount the fact that in the limit, the “holes” E maycollapse into a discontinuity of u, whose surface will be countedtwice in the relaxed energy. We discuss some situations where suchenergies appear, and give, as an application, a new proofof convergence for an extensionof Ambrosio-Tortorelli's approximation to the Mumford-Shah functional.
