We consider, in an open subset $\omega $ 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.

Classification: 49J45, 49Q20, 49Q10

Keywords: relaxation, free discontinuity problems, $\Gamma $-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}, 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] Wetting of rough surfaces: a homogenization approach. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005) 79-97. | Zbl pre05206138

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

and ,[3] 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

and ,[4] Approximation of functionals depending on jumps by elliptic functionals via $\Gamma $-convergence. Comm. Pure Appl. Math. 43 (1990) 999-1036. | Zbl 0722.49020

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

, and ,[6] Some results on surface measures in calculus of variations. Ann. Mat. Pura Appl. 170 (1996) 329-357. | Zbl 0890.49020

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

and ,[8] 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

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

, and ,[10] Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85 (2000) 609-646. | Zbl 0961.65062

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

,[12] $\Gamma $-convergence for Beginners. Oxford University Press, Oxford (2002). | MR 1968440 | Zbl pre01865939

,[13] A handbook of $\Gamma $-convergence, in Handbook of Differential Equations. Stationary Partial Differential Equations, Vol. 3, M. Chipot and P. Quittner Eds., Elsevier (2006).

,[14] 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. | Zbl 0880.49010

and ,[15] Approximation by $\Gamma $-convergence of a curvature-depending functional in Visual Reconstruction. Comm. Pure Appl. Math. 58 (2006) 71-121. | Zbl 1098.49012

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

and ,[17] Homogenization of free discontinuity problems. Arch. Rational Mech. Anal. 135 (1996) 297-356. | Zbl 0924.35015

, and ,[18] 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] Interaction of a bulk and a surface energy with a geometrical constraint. SIAM J. Math. Anal. 39 (2007) 77-102. | Zbl pre05240407

and ,[20] Progressive water-waves: a global variational approach. (In preparation).

, and ,[21] Minimal surfaces and functions of bounded variation. Birkhäuser Verlag, Basel (1984). | MR 775682 | Zbl 0545.49018

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

and ,[23] Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577-685. | Zbl 0691.49036

and ,