We approximate, in the sense of $\Gamma $-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.

Keywords: 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://www.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://www.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://www.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, Volume 13 (2007) no. 1, pp. 135-162. doi : 10.1051/cocv:2007008. http://www.numdam.org/articles/10.1051/cocv:2007008/

[1] Free-discontinuity problems generated by singular perturbation. Proc. Roy. Soc. Edinburgh Sect. A 6 (1998) 1115-1129. | Zbl

, and ,[2] Free-discontinuity problems via functionals involving the ${L}^{1}$-norm of the gradient and their approximations. Interfaces and free boundaries 1 (1999) 17-37. | Zbl

, and ,[3] Free discontinuity problems generated by singular perturbation: the $n$-dimensional case. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 449-469. | Zbl

and ,[4] A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. B 3 (1989) 857-881. | Zbl

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

, and ,[6] Approximation of functionals depending on jumps by elliptic functionals via $\Gamma $-convergence. Comm. Pure Appl. Math. XLIII (1990) 999-1036. | Zbl

and ,[7] On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7) VI (1992) 105-123. | Zbl

and ,[8] Relaxation results for some free discontinuity problems. J. Reine Angew. Math. 458 (1995) 1-18. | Zbl

, and ,[9] Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85 (2000) 609-646. | Zbl

and ,[10] Approximation of free-discontinuity problems. Lect. Notes Math. 1694, Springer Verlag, Berlin (1998). | MR | Zbl

.[11] On the non-local approximation of free-discontinuity problems. Comm. Partial Differential Equations 23 (1998) 817-829. | Zbl

and ,[12] Non-local approximation of the Mumford-Shah functional. Calc. Var. 5 (1997) 293-322. | Zbl

and ,[13] Discrete approximation of the Mumford-Shah functional in dimension two. ESAIM: M2AN 33 (1999) 651-672. | Numdam | Zbl

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

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

and ,[18] Nonlocal approximation of nonisotropic free-discontinuity problems. SIAM J. Appl. Math. 59 (1999) 1507-1519. | Zbl

and ,[19] An Introduction to $\Gamma $-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] Non-local approximation of free-discontinuity functionals with linear growth: the one-dimensional case. Ann. Mat. Pura Appl. (to appear). | MR | Zbl

and ,[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] Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12-49. | Zbl

and ,[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

,*Cited by Sources: *