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

Andrea Braides; Antonin Chambolle; Margherita Solci

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 13, Issue: 4, page 717-734
- ISSN: 1292-8119

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topBraides, Andrea, Chambolle, Antonin, and Solci, Margherita. "A relaxation result for energies defined on pairs set-function and applications." ESAIM: Control, Optimisation and Calculus of Variations 13.4 (2007): 717-734. <http://eudml.org/doc/250010>.

@article{Braides2007,

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.
},

author = {Braides, Andrea, Chambolle, Antonin, Solci, Margherita},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Relaxation; free discontinuity problems; Γ-convergence; relaxation; -convergence},

language = {eng},

month = {7},

number = {4},

pages = {717-734},

publisher = {EDP Sciences},

title = {A relaxation result for energies defined on pairs set-function and applications},

url = {http://eudml.org/doc/250010},

volume = {13},

year = {2007},

}

TY - JOUR

AU - Braides, Andrea

AU - Chambolle, Antonin

AU - Solci, Margherita

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2007/7//

PB - EDP Sciences

VL - 13

IS - 4

SP - 717

EP - 734

AB -
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.

LA - eng

KW - Relaxation; free discontinuity problems; Γ-convergence; relaxation; -convergence

UR - http://eudml.org/doc/250010

ER -

## References

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