A relaxation result for energies defined on pairs set-function and applications
Dip. di Matematica,
Università di Roma “Tor Vergata”,
via della Ricerca Scientifica,
2 CMAP, École Polytechnique, CNRS, 91128 Palaiseau, France; email@example.com
3 DAP, Università di Sassari, Palazzo Pou Salit, 07041 Alghero, Italy.
Revised: 17 July 2006
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.
Mathematics Subject Classification: 49J45 / 49Q20 / 49Q10
Key words: Relaxation / free discontinuity problems / Γ-convergence
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