# 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

## Access Full Article

top## Abstract

top## How to cite

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

top- G. Alberti and A. DeSimone, Wetting of rough surfaces: a homogenization approach. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.461 (2005) 79–97.
- L. Ambrosio and A. Braides, Functionals defined on partitions of sets of finite perimeter, I: integral representation and Γ-convergence. J. Math. Pures. Appl.69 (1990) 285–305.
- L. Ambrosio and A. Braides, Functionals defined on partitions of sets of finite perimeter, II: semicontinuity, relaxation and homogenization. J. Math. Pures. Appl.69 (1990) 307–333.
- L. Ambrosio and V.M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math.43 (1990) 999–1036.
- L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000).
- G. Bellettini, M. Paolini and S. Venturini, Some results on surface measures in calculus of variations. Ann. Mat. Pura Appl.170 (1996) 329–357.
- E. Bonnetier and A. Chambolle, Computing the equilibrium configuration of epitaxially strained crystalline films. SIAM J. Appl. Math.62 (2002) 1093–1121.
- G. Bouchitté and P. Seppecher, Cahn and Hilliard fluid on an oscillating boundary. Motion by mean curvature and related topics (Trento, 1992), de Gruyter, Berlin (1994) 23–42.
- G. Bouchitté, A. Braides and G. Buttazzo, Relaxation results for some free discontinuity problems. J. Reine Angew. Math.458 (1995) 1–18.
- B. Bourdin and A. Chambolle, Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math.85 (2000) 609–646.
- A. Braides, Approximation of Free-Discontinuity Problems. Lect. Notes Math.1694, Springer, Berlin (1998).
- A. Braides, Γ -convergence for Beginners. Oxford University Press, Oxford (2002).
- A. Braides, A handbook of Γ>-convergence, in Handbook of Differential Equations. Stationary Partial Differential Equations, Vol. 3, M. Chipot and P. Quittner Eds., Elsevier (2006).
- A. Braides and V. Chiadò Piat, Integral representation results for functionals defined in $SBV(\Omega ;\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}{\mathrm{R}}^{\mathrm{m}})$. J. Math. Pures Appl.75 (1996) 595–626.
- A. Braides and R. March, Approximation by Γ-convergence of a curvature-depending functional in Visual Reconstruction. Comm. Pure Appl. Math.58 (2006) 71–121.
- A. Braides and M. Solci, A remark on the approximation of free-discontinuity problems. Manuscript (2003).
- A. Braides, A. Defranceschi and E. Vitali, Homogenization of free discontinuity problems. Arch. Rational Mech. Anal.135 (1996) 297–356.
- B. Buffoni, Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation. Arch. Ration. Mech. Anal.173 (2004) 25–68.
- A. Chambolle and M. Solci, Interaction of a bulk and a surface energy with a geometrical constraint. SIAM J. Math. Anal.39 (2007) 77–102.
- A. Chambolle, E. Séré and C.Zanini, Progressive water-waves: a global variational approach. (In preparation).
- E. Giusti, Minimal surfaces and functions of bounded variation. Birkhäuser Verlag, Basel (1984).
- J.M. Morel and S. Solimini, Variational Methods in Image Segmentation. Progr. Nonlinear Differ. Equ. Appl.14, Birkhäuser, Basel (1995).
- D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math.42 (1989) 577–685.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.