We consider, in an open subset Omega of R-N, energies depending on the perimeter of a subset E is an element of Omega ( or some equivalent surface integral) and on a function u which is defined only on Omega\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.

Braides, A., Chambolle, A., & Solci, M. (2007). A relaxation result for energies defined on pairs set-function and applications. ESAIM. COCV, 13(4), 717-734 [10.1051/cocv:2007032].

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

BRAIDES, ANDREA;
2007

Abstract

We consider, in an open subset Omega of R-N, energies depending on the perimeter of a subset E is an element of Omega ( or some equivalent surface integral) and on a function u which is defined only on Omega\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.
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/05 - Analisi Matematica
English
Con Impact Factor ISI
relaxation; free discontinuity problems; Gamma-convergence
Braides, A., Chambolle, A., & Solci, M. (2007). A relaxation result for energies defined on pairs set-function and applications. ESAIM. COCV, 13(4), 717-734 [10.1051/cocv:2007032].
Braides, A; Chambolle, A; Solci, M
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/28854
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