We approximate, in the sense of -convergence, free-discontinuity functionals with linear growth in the gradient by a sequence of non-local integral functionals depending on the average of the gradients on small balls. The result extends to higher dimension what we already proved in the one-dimensional case.
Mots-clés : variational approximation, free discontinuities
@article{COCV_2007__13_1_135_0,
author = {Lussardi, Luca and Vitali, Enrico},
title = {Non-local approximation of free-discontinuity problems with linear growth},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {135--162},
publisher = {EDP-Sciences},
volume = {13},
number = {1},
year = {2007},
doi = {10.1051/cocv:2007008},
mrnumber = {2282106},
zbl = {1136.49029},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv:2007008/}
}
TY - JOUR AU - Lussardi, Luca AU - Vitali, Enrico TI - Non-local approximation of free-discontinuity problems with linear growth JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2007 SP - 135 EP - 162 VL - 13 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2007008/ DO - 10.1051/cocv:2007008 LA - en ID - COCV_2007__13_1_135_0 ER -
%0 Journal Article %A Lussardi, Luca %A Vitali, Enrico %T Non-local approximation of free-discontinuity problems with linear growth %J ESAIM: Control, Optimisation and Calculus of Variations %D 2007 %P 135-162 %V 13 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2007008/ %R 10.1051/cocv:2007008 %G en %F COCV_2007__13_1_135_0
Lussardi, Luca; Vitali, Enrico. Non-local approximation of free-discontinuity problems with linear growth. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 1, pp. 135-162. doi : 10.1051/cocv:2007008. http://archive.numdam.org/articles/10.1051/cocv:2007008/
[1] , and, Free-discontinuity problems generated by singular perturbation. Proc. Roy. Soc. Edinburgh Sect. A 6 (1998) 1115-1129. | Zbl
[2] , and, Free-discontinuity problems via functionals involving the -norm of the gradient and their approximations. Interfaces and free boundaries 1 (1999) 17-37. | Zbl
[3] and, Free discontinuity problems generated by singular perturbation: the -dimensional case. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 449-469. | Zbl
[4] , A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. B 3 (1989) 857-881. | Zbl
[5] , and, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). | MR | Zbl
[6] and, Approximation of functionals depending on jumps by elliptic functionals via -convergence. Comm. Pure Appl. Math. XLIII (1990) 999-1036. | Zbl
[7] and, On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7) VI (1992) 105-123. | Zbl
[8] , and, Relaxation results for some free discontinuity problems. J. Reine Angew. Math. 458 (1995) 1-18. | Zbl
[9] and, Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85 (2000) 609-646. | Zbl
[10] . Approximation of free-discontinuity problems. Lect. Notes Math. 1694, Springer Verlag, Berlin (1998). | MR | Zbl
[11] and, On the non-local approximation of free-discontinuity problems. Comm. Partial Differential Equations 23 (1998) 817-829. | Zbl
[12] and, Non-local approximation of the Mumford-Shah functional. Calc. Var. 5 (1997) 293-322. | Zbl
[13] and, Discrete approximation of the Mumford-Shah functional in dimension two. ESAIM: M2AN 33 (1999) 651-672. | Numdam | Zbl
[14] , Sequence of non-local functionals which approximate free-discontinuity problems. Arch. Rational Mech. Anal. 144 (1998) 357-402. | Zbl
[15] , A finite element approximation of an image segmentation problem. Math. Models Methods Appl. Sci. 9 (1999) 243-259. | Zbl
[16] and, Finite element approximation of non-isotropic free-discontinuity problems. Numer. Funct. Anal. Optim. 18 (1997) 921-940. | Zbl
[17] and, A density result in SBV with respect to non-isotropic energies. Nonlinear Anal. 38 (1999) 585-604. | Zbl
[18] and, Nonlocal approximation of nonisotropic free-discontinuity problems. SIAM J. Appl. Math. 59 (1999) 1507-1519. | Zbl
[19] , An Introduction to -Convergence. Birkhäuser, Boston (1993). | MR | Zbl
[20] . Free discontinuity problems in calculus of variations, in Frontiers in pure and applied mathematics. A collection of papers dedicated to Jacques-Louis Lions on the occasion of his sixtieth birthday. June 6-10, Paris 1988, Robert Dautray, Ed., Amsterdam, North-Holland Publishing Co. (1991) 55-62. | Zbl
[21] and, Non-local approximation of free-discontinuity functionals with linear growth: the one-dimensional case. Ann. Mat. Pura Appl. (to appear). | MR | Zbl
[22] , Sequences of singularly perturbed functionals generating free-discontinuity problems. SIAM J. Math. Anal. 35 (2003) 759-805. | Zbl
[23] , The anisotropy introduced by the mesh in the finite element approximation of the Mumford-Shah functional. Numer. Funct. Anal. Optim. 20 (1999) 957-982. | Zbl
[24] and, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12-49. | Zbl
[25] , A common framework for curve evolution, segmentation and anisotropic diffusion, in IEEE conference on computer vision and pattern recognition (1996).
[26] , Uses of elliptic approximations in computer vision. In R. Serapioni and F. Tomarelli, editors, Progress in Nonlinear Differential Equations and Their Applications 25 (1996). | MR | Zbl
[27] , Lectures on Geometric Measure Theory. Centre for Mathematical Analysis, Australian National University (1984). | MR | Zbl
Cité par Sources :
