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

We consider, in an open subset ω of N , energies depending on the perimeter of a subset Eω (or some equivalent surface integral) and on a function u which is defined only on ωE. 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 : https://doi.org/10.1051/cocv:2007032
Classification:  49J45,  49Q20,  49Q10
Keywords: relaxation, free discontinuity problems, Γ-convergence
     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},
     publisher = {EDP-Sciences},
     volume = {13},
     number = {4},
     year = {2007},
     pages = {717-734},
     doi = {10.1051/cocv:2007032},
     zbl = {pre05212111},
     mrnumber = {2351400},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 13 (2007) no. 4, pp. 717-734. doi : 10.1051/cocv:2007032. http://www.numdam.org/item/COCV_2007__13_4_717_0/

[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. | Zbl pre05206138

[2] L. Ambrosio and A. Braides, Functionals defined on partitions of sets of finite perimeter, I 69 (1990) 285-305. | Zbl 0676.49028

[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 0676.49029

[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 0722.49020

[5] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000). | MR 1857292 | Zbl 0957.49001

[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 0890.49020

[7] E. Bonnetier and A. Chambolle, Computing the equilibrium configuration of epitaxially strained crystalline films. SIAM J. Appl. Math. 62 (2002) 1093-1121. | Zbl 1001.49017

[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 0807.76081

[9] G. Bouchitté, A. Braides and G. Buttazzo, Relaxation results for some free discontinuity problems. J. Reine Angew. Math. 458 (1995) 1-18. | Zbl 0817.49015

[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 0961.65062

[11] A. Braides, Approximation of Free-Discontinuity Problems. Lect. Notes Math. 1694, Springer, Berlin (1998). | MR 1651773 | Zbl 0909.49001

[12] A. Braides, Γ-convergence for Beginners. Oxford University Press, Oxford (2002). | MR 1968440 | Zbl pre01865939

[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;IR m ). J. Math. Pures Appl. 75 (1996) 595-626. | Zbl 0880.49010

[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 1098.49012

[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 0924.35015

[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 1110.76308

[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. | Zbl pre05240407

[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 775682 | Zbl 0545.49018

[22] J.M. Morel and S. Solimini, Variational Methods in Image Segmentation. Progr. Nonlinear Differ. Equ. Appl. 14, Birkhäuser, Basel (1995). | MR 1321598

[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 0691.49036