Γ-limits of convolution functionals
ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 2, pp. 486-515.

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.

DOI: 10.1051/cocv/2012018
Classification: 49Q20, 49J45, 49M30
Keywords: free discontinuities, Γ-convergence, anisotropy
@article{COCV_2013__19_2_486_0,
     author = {Lussardi, Luca and Magni, Annibale},
     title = {$\Gamma $-limits of convolution functionals},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {486--515},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {2},
     year = {2013},
     doi = {10.1051/cocv/2012018},
     zbl = {1263.49010},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2012018/}
}
TY  - JOUR
AU  - Lussardi, Luca
AU  - Magni, Annibale
TI  - $\Gamma $-limits of convolution functionals
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2013
SP  - 486
EP  - 515
VL  - 19
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2012018/
DO  - 10.1051/cocv/2012018
LA  - en
ID  - COCV_2013__19_2_486_0
ER  - 
%0 Journal Article
%A Lussardi, Luca
%A Magni, Annibale
%T $\Gamma $-limits of convolution functionals
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2013
%P 486-515
%V 19
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2012018/
%R 10.1051/cocv/2012018
%G en
%F COCV_2013__19_2_486_0
Lussardi, Luca; Magni, Annibale. $\Gamma $-limits of convolution functionals. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 2, pp. 486-515. doi : 10.1051/cocv/2012018. http://www.numdam.org/articles/10.1051/cocv/2012018/

[1] R. Alicandro and M. S. Gelli, Free discontinuity problems generated by singular perturbation : the n-dimensional case. Proc. R. Soc. Edinb. Sect. A 130 (2000) 449-469. | MR | Zbl

[2] R. Alicandro, A. Braides, and M.S. Gelli, Free-discontinuity problems generated by singular perturbation. Proc. R. Soc. Edinburgh Sect. A 6 (1998) 1115-1129. | MR | Zbl

[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. | MR | Zbl

[4] L. Ambrosio and V.M. Tortorelli, On the approximation of free discontinuity problems. Boll. Unione Mat. Ital. B (7) VI (1992) 105-123. | MR | Zbl

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

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

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

[8] A. Braides, Approximation of free-discontinuity problems. Lect. Notes Math. 1694 (1998). | MR | Zbl

[9] A. Braides, Γ-convergence for beginners. Oxford University Press (2002). | MR | Zbl

[10] A. Braides and G. Dal Maso, Non-local approximation of the Mumford-Shah functional. Calc. Var. 5 (1997) 293-322. | MR | Zbl

[11] A. Braides and A. Garroni, On the non-local approximation of free-discontinuity problems. Commut. Partial Differ. Equ. 23 (1998) 817-829. | MR | Zbl

[12] A. Chambolle and G. Dal Maso, Discrete approximation of the Mumford-Shah functional in dimension two. ESAIM : M2AN 33 (1999) 651-672. | Numdam | MR | Zbl

[13] G. Cortesani, Sequence of non-local functionals which approximate free-discontinuity problems. Arch. R. Mech. Anal. 144 (1998) 357-402. | MR | Zbl

[14] G. Cortesani, A finite element approximation of an image segmentation problem. Math. Models Methods Appl. Sci. 9 (1999) 243-259. | MR | Zbl

[15] G. Cortesani and R. Toader, Finite element approximation of non-isotropic free-discontinuity problems. Numer. Funct. Anal. Optim. 18 (1997) 921-940. | MR | Zbl

[16] G. Cortesani and R. Toader, Nonlocal approximation of nonisotropic free-discontinuity problems. SIAM J. Appl. Math. 59 (1999) 1507-1519. | MR | Zbl

[17] G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies. Nonlinear Anal. 38 (1999) 585-604. | MR | Zbl

[18] G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser, Boston (1993). | Zbl

[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. | MR | Zbl

[20] L. Lussardi, An approximation result for free discontinuity functionals by means of non-local energies. Math. Methods Appl. Sci. 31 (2008) 2133-2146. | MR

[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. | MR | Zbl

[22] L. Lussardi and E. Vitali, Non local approximation of free-discontinuity problems with linear growth. ESAIM : COCV 13 (2007) 135-162. | Numdam | MR | Zbl

[23] M. Morini, Sequences of singularly perturbed functionals generating free-discontinuity problems. SIAM J. Math. Anal. 35 (2003) 759-805. | MR | Zbl

[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. | MR | Zbl

[25] M. Negri, A non-local approximation of free discontinuity problems in SBV and SBD. Calc. Var. 25 (2006) 33-62. | MR | Zbl

Cited by Sources: