ESAIM: Control, Optimisation and Calculus of Variations

Research Article

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

Braides, Andreaa1, Chambolle, Antonina2 and Solci, Margheritaa3

a1 Dip. di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.

a2 CMAP, École Polytechnique, CNRS, 91128 Palaiseau, France;

a3 DAP, Università di Sassari, Palazzo Pou Salit, 07041 Alghero, Italy.


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.

(Received December 12 2005)

(Revised July 17 2006)

(Online publication July 20 2007)

Key Words:

  • Relaxation;
  • free discontinuity problems;
  • Γ-convergence

Mathematics Subject Classification:

  • 49J45;
  • 49Q20;
  • 49Q10