Document 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 Solci31 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
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,
-convergence
© EDP Sciences, SMAI 2007