A relaxation result for energies defined on pairs set-function and applications
ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 717-734.

We consider, in an open subset ω of ℝN, 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.

DOI : 10.1051/cocv:2007032
Classification : 49J45, 49Q20, 49Q10
Mots-clés : relaxation, free discontinuity problems, Γ-convergence 
@article{COCV_2007__13_4_717_0,
     author = {Braides, Andrea and Chambolle, Antonin and Solci, Margherita},
     title = {A relaxation result for energies defined on pairs set-function and applications},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {717--734},
     publisher = {EDP-Sciences},
     volume = {13},
     number = {4},
     year = {2007},
     doi = {10.1051/cocv:2007032},
     mrnumber = {2351400},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2007032/}
}
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
PY  - 2007
SP  - 717
EP  - 734
VL  - 13
IS  - 4
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/cocv:2007032/
DO  - 10.1051/cocv:2007032
LA  - en
ID  - COCV_2007__13_4_717_0
ER  - 
%0 Journal Article
%A Braides, Andrea
%A Chambolle, Antonin
%A Solci, Margherita
%T A relaxation result for energies defined on pairs set-function and applications
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2007
%P 717-734
%V 13
%N 4
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/cocv:2007032/
%R 10.1051/cocv:2007032
%G en
%F COCV_2007__13_4_717_0
Braides, Andrea; Chambolle, Antonin; Solci, Margherita. A relaxation result for energies defined on pairs set-function and applications. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 717-734. doi : 10.1051/cocv:2007032. https://www.numdam.org/articles/10.1051/cocv:2007032/

[1] 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.

[2] L. Ambrosio and A. Braides, Functionals defined on partitions of sets of finite perimeter, I 69 (1990) 285-305. | Zbl

[3] 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. | Zbl

[4] 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. | Zbl

[5] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000). | MR | Zbl

[6] G. Bellettini, M. Paolini and S. Venturini, Some results on surface measures in calculus of variations. Ann. Mat. Pura Appl. 170 (1996) 329-357. | Zbl

[7] E. Bonnetier and A. Chambolle, Computing the equilibrium configuration of epitaxially strained crystalline films. SIAM J. Appl. Math. 62 (2002) 1093-1121. | Zbl

[8] 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. | Zbl

[9] G. Bouchitté, A. Braides and G. Buttazzo, Relaxation results for some free discontinuity problems. J. Reine Angew. Math. 458 (1995) 1-18. | Zbl

[10] B. Bourdin and A. Chambolle, Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85 (2000) 609-646. | Zbl

[11] A. Braides, Approximation of Free-Discontinuity Problems. Lect. Notes Math. 1694, Springer, Berlin (1998). | MR | Zbl

[12] A. Braides, Γ-convergence for Beginners. Oxford University Press, Oxford (2002). | MR

[13] 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).

[14] A. Braides and V. Chiadò Piat, Integral representation results for functionals defined in SBV(Ω;IRm). J. Math. Pures Appl. 75 (1996) 595-626. | Zbl

[15] 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. | Zbl

[16] A. Braides and M. Solci, A remark on the approximation of free-discontinuity problems. Manuscript (2003).

[17] A. Braides, A. Defranceschi and E. Vitali, Homogenization of free discontinuity problems. Arch. Rational Mech. Anal. 135 (1996) 297-356. | Zbl

[18] B. Buffoni, Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation. Arch. Ration. Mech. Anal. 173 (2004) 25-68. | Zbl

[19] 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.

[20] A. Chambolle, E. Séré and C. Zanini, Progressive water-waves: a global variational approach. (In preparation).

[21] E. Giusti, Minimal surfaces and functions of bounded variation. Birkhäuser Verlag, Basel (1984). | MR | Zbl

[22] J.M. Morel and S. Solimini, Variational Methods in Image Segmentation. Progr. Nonlinear Differ. Equ. Appl. 14, Birkhäuser, Basel (1995). | MR

[23] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577-685. | Zbl

Cité par Sources :