a1 Dip. di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.
a2 CMAP, École Polytechnique, CNRS, 91128 Palaiseau, France; email@example.com
a3 DAP, Università di Sassari, Palazzo Pou Salit, 07041 Alghero, Italy.
We consider, in an open subset Ω of , energies depending on the perimeter of a subset (or some equivalent surface integral) and on a function u which is defined only on . 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.
(Received December 12 2005)
(Revised July 17 2006)
(Online publication July 20 2007)
Mathematics Subject Classification: