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

  • Andrea Braides (a1), Antonin Chambolle (a2) and Margherita Solci (a3)

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.


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

  • Andrea Braides (a1), Antonin Chambolle (a2) and Margherita Solci (a3)

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