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

Abstract

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


How to cite

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Braides, 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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