We consider, in an open subset Ω ofRN, energies depending on the perimeter of a subsetEС Ω (or some equivalent surface integral) and on a function u which is defined only onE. We compute the lower semicontinuous envelope of such energies. This relaxation has to take into account the fact that in the limit, the “holes” Ω \Emay collapse into a discontinuity ofu, 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.
A Relaxation result for energies defined on pairs set-function and applications / Solci, Margherita; Braides, Andrea; Chambolle, Antonin. - 13:4(2007), pp. 717-734. [10.1051/cocv:2007032]
A Relaxation result for energies defined on pairs set-function and applications
Solci, Margherita;
2007-01-01
Abstract
We consider, in an open subset Ω ofRN, energies depending on the perimeter of a subsetEС Ω (or some equivalent surface integral) and on a function u which is defined only onE. We compute the lower semicontinuous envelope of such energies. This relaxation has to take into account the fact that in the limit, the “holes” Ω \Emay collapse into a discontinuity ofu, 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.File | Dimensione | Formato | |
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