# Γ-limits of convolution functionals

ESAIM: Control, Optimisation and Calculus of Variations (2013)

- Volume: 19, Issue: 2, page 486-515
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topLussardi, Luca, and Magni, Annibale. "Γ-limits of convolution functionals." ESAIM: Control, Optimisation and Calculus of Variations 19.2 (2013): 486-515. <http://eudml.org/doc/272771>.

@article{Lussardi2013,

abstract = {We compute the Γ-limit of a sequence of non-local integral functionals depending on a regularization of the gradient term by means of a convolution kernel. In particular, as Γ-limit, we obtain free discontinuity functionals with linear growth and with anisotropic surface energy density.},

author = {Lussardi, Luca, Magni, Annibale},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {free discontinuities; Γ-convergence; anisotropy; -convergence},

language = {eng},

number = {2},

pages = {486-515},

publisher = {EDP-Sciences},

title = {Γ-limits of convolution functionals},

url = {http://eudml.org/doc/272771},

volume = {19},

year = {2013},

}

TY - JOUR

AU - Lussardi, Luca

AU - Magni, Annibale

TI - Γ-limits of convolution functionals

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2013

PB - EDP-Sciences

VL - 19

IS - 2

SP - 486

EP - 515

AB - We compute the Γ-limit of a sequence of non-local integral functionals depending on a regularization of the gradient term by means of a convolution kernel. In particular, as Γ-limit, we obtain free discontinuity functionals with linear growth and with anisotropic surface energy density.

LA - eng

KW - free discontinuities; Γ-convergence; anisotropy; -convergence

UR - http://eudml.org/doc/272771

ER -

## References

top- [1] R. Alicandro and M. S. Gelli, Free discontinuity problems generated by singular perturbation : the n-dimensional case. Proc. R. Soc. Edinb. Sect. A130 (2000) 449–469. Zbl0978.49014MR1769236
- [2] R. Alicandro, A. Braides, and M.S. Gelli, Free-discontinuity problems generated by singular perturbation. Proc. R. Soc. Edinburgh Sect. A6 (1998) 1115–1129. Zbl0920.49007MR1664085
- [3] L. Ambrosio and V.M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Commut. Pure Appl. Math. XLIII (1990) 999–1036. Zbl0722.49020MR1075076
- [4] L. Ambrosio and V.M. Tortorelli, On the approximation of free discontinuity problems. Boll. Unione Mat. Ital. B (7) VI (1992) 105–123. Zbl0776.49029MR1164940
- [5] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). Zbl0957.49001MR1857292
- [6] G. Bouchitté, A. Braides and G. Buttazzo, Relaxation results for some free discontinuity problems. J. Reine Angew. Math.458 (1995) 1–18. Zbl0817.49015MR1310950
- [7] B. Bourdin and A. Chambolle, Implementation of an adaptive finite-element approximation of the Mumford–Shah functional. Numer. Math.85 (2000) 609–646. Zbl0961.65062MR1771782
- [8] A. Braides, Approximation of free-discontinuity problems. Lect. Notes Math. 1694 (1998). Zbl0909.49001MR1651773
- [9] A. Braides, Γ-convergence for beginners. Oxford University Press (2002). Zbl1198.49001MR1968440
- [10] A. Braides and G. Dal Maso, Non-local approximation of the Mumford–Shah functional. Calc. Var.5 (1997) 293–322. Zbl0873.49009MR1450713
- [11] A. Braides and A. Garroni, On the non-local approximation of free-discontinuity problems. Commut. Partial Differ. Equ.23 (1998) 817–829. Zbl0907.49009MR1632744
- [12] A. Chambolle and G. Dal Maso, Discrete approximation of the Mumford–Shah functional in dimension two. ESAIM : M2AN 33 (1999) 651–672. Zbl0943.49011MR1726478
- [13] G. Cortesani, Sequence of non-local functionals which approximate free-discontinuity problems. Arch. R. Mech. Anal.144 (1998) 357–402. Zbl0926.49007MR1656480
- [14] G. Cortesani, A finite element approximation of an image segmentation problem. Math. Models Methods Appl. Sci.9 (1999) 243–259. Zbl0937.65072MR1674564
- [15] G. Cortesani and R. Toader, Finite element approximation of non-isotropic free-discontinuity problems. Numer. Funct. Anal. Optim.18 (1997) 921–940. Zbl0903.49002MR1485987
- [16] G. Cortesani and R. Toader, Nonlocal approximation of nonisotropic free-discontinuity problems. SIAM J. Appl. Math.59 (1999) 1507–1519. Zbl0944.49012MR1692635
- [17] G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies. Nonlinear Anal.38 (1999) 585–604. Zbl0939.49024MR1709990
- [18] G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser, Boston (1993). Zbl0816.49001
- [19] E. De Giorgi, Free discontinuity problems in calculus of variations, in Frontiers in pure and applied mathematics, edited by R. Dautray. A collection of papers dedicated to Jacques-Louis Lions on the occasion of his sixtieth birthday, Paris 1988. North-Holland Publishing Co., Amsterdam (1991) 55–62. Zbl0758.49002MR1110593
- [20] L. Lussardi, An approximation result for free discontinuity functionals by means of non-local energies. Math. Methods Appl. Sci.31 (2008) 2133–2146. Zbl05373390MR2467182
- [21] L. Lussardi and E. Vitali, Non local approximation of free-discontinuity functionals with linear growth : the one dimensional case. Ann. Mat. Pura Appl.186 (2007) 722–744. Zbl1136.49030MR2317787
- [22] L. Lussardi and E. Vitali, Non local approximation of free-discontinuity problems with linear growth. ESAIM : COCV 13 (2007) 135–162. Zbl1136.49029MR2282106
- [23] M. Morini, Sequences of singularly perturbed functionals generating free-discontinuity problems. SIAM J. Math. Anal.35 (2003) 759–805. Zbl1058.49021MR2048406
- [24] M. Negri, The anisotropy introduced by the mesh in the finite element approximation of the Mumford–Shah functional. Numer. Funct. Anal. Optim.20 (1999) 957–982. Zbl0953.49024MR1728172
- [25] M. Negri, A non-local approximation of free discontinuity problems in SBV and SBD. Calc. Var.25 (2006) 33–62. Zbl1087.65010MR2183854