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

Issue		ESAIM: COCV Volume 13, Number 4, October-December 2007


Page(s)		717 - 734
DOI		10.1051/cocv:2007032
Published online		20 July 2007

ESAIM: COCV, Vol. 13, N°4, pp. 717-734
DOI: 10.1051/cocv:2007032

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

Andrea Braides¹, Antonin Chambolle² and Margherita Solci³

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 December 12, 2005. Revised July 17, 2006. Published online July 20, 2007.)

Abstract

We consider, in an open subset $\Omega$ 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 is defined only on $\Omega\setminus E$ . We compute the lower semicontinuous envelope of such energies. This relaxation has to take into account the fact that in the limit, the "holes" 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, $\Gamma$ -convergence

© EDP Sciences, SMAI 2007

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