*Published Paper*

**Inserted:** 12 dec 2005

**Last Updated:** 2 feb 2008

**Journal:** ESAIM: COCV

**Volume:** 13

**Pages:** 717-734

**Year:** 2007

**Abstract:**

We consider, in an open subset $\Omega$ of $\re^N$, energies depending on the perimeter of a subset $E\subset\Omega$ (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'' $\Omega\setminus E$ 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.

**Keywords:**
relaxation, Free-discontinuity problems, Image processing, Ambrosio-Tortorelli

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