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

Published online by Cambridge University Press:  20 July 2007

Andrea Braides ,
Antonin Chambolle  and
Margherita Solci
Andrea Braides
Affiliation:
Dip. di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.
Antonin Chambolle
Affiliation:
CMAP, École Polytechnique, CNRS, 91128 Palaiseau, France; antonin.chambolle@polytechnique.fr
Margherita Solci
Affiliation:
DAP, Università di Sassari, Palazzo Pou Salit, 07041 Alghero, Italy.
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Abstract


We consider, in an open subset Ω 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.


Keywords

Relaxationfree discontinuity problemsΓ-convergence
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 4 , October 2007 , pp. 717 - 734
Copyright
© EDP Sciences, SMAI, 2007

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