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

¹ Dip. di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.
² CMAP, École Polytechnique, CNRS, 91128 Palaiseau, France; antonin.chambolle@polytechnique.fr
³ DAP, Università di Sassari, Palazzo Pou Salit, 07041 Alghero, Italy.

Received: 12 December 2005
Revised: 17 July 2006

Abstract


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.