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Non-local approximation of free-discontinuity problems with linear growth. (English) Zbl 1136.49029

Summary: 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.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49J45 Methods involving semicontinuity and convergence; relaxation
49M30 Other numerical methods in calculus of variations (MSC2010)
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References:

[1] R. Alicandro , A. Braides and M.S. Gelli , Free-discontinuity problems generated by singular perturbation . Proc. Roy. Soc. Edinburgh Sect. A 6 ( 1998 ) 1115 - 1129 . Zbl 0920.49007 · Zbl 0920.49007 · doi:10.1017/S0308210500027256
[2] R. Alicandro , A. Braides and J. Shah , 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 0947.49011 · Zbl 0947.49011 · doi:10.4171/IFB/2
[3] R. Alicandro and M.S. Gelli , Free discontinuity problems generated by singular perturbation: the \(n\)-dimensional case . Proc. Roy. Soc. Edinburgh Sect. A 130 ( 2000 ) 449 - 469 . Zbl 0978.49014 · Zbl 0978.49014
[4] L. Ambrosio , A compactness theorem for a new class of functions of bounded variation . Boll. Un. Mat. Ital. B 3 ( 1989 ) 857 - 881 . Zbl 0767.49001 · Zbl 0767.49001
[5] L. Ambrosio , N. Fusco and D. Pallara , Functions of Bounded Variation and Free Discontinuity Problems . Oxford University Press ( 2000 ). MR 1857292 | Zbl 0957.49001 · Zbl 0957.49001
[6] L. Ambrosio and V.M. Tortorelli , Approximation of functionals depending on jumps by elliptic functionals via \(\Gamma \)-convergence . Comm. Pure Appl. Math. XLIII ( 1990 ) 999 - 1036 . Zbl 0722.49020 · Zbl 0722.49020 · doi:10.1002/cpa.3160430805
[7] L. Ambrosio and V.M. Tortorelli , On the approximation of free discontinuity problems . Boll. Un. Mat. Ital. B (7) VI ( 1992 ) 105 - 123 . Zbl 0776.49029 · Zbl 0776.49029
[8] G. Bouchitté , A. Braides and G. Buttazzo , Relaxation results for some free discontinuity problems . J. Reine Angew. Math. 458 ( 1995 ) 1 - 18 . Zbl 0817.49015 · Zbl 0817.49015 · doi:10.1515/crll.1995.458.1
[9] B. Bourdin and A. Chambolle , Implementation of an adaptive finite-element approximation of the Mumford-Shah functional . Numer. Math. 85 ( 2000 ) 609 - 646 . Zbl 0961.65062 · Zbl 0961.65062 · doi:10.1007/s002110000099
[10] A. Braides . Approximation of free-discontinuity problems . Lect. Notes Math. 1694, Springer Verlag, Berlin ( 1998 ). MR 1651773 | Zbl 0909.49001 · Zbl 0909.49001 · doi:10.1007/BFb0097344
[11] A. Braides and A. Garroni , On the non-local approximation of free-discontinuity problems . Comm. Partial Differential Equations 23 ( 1998 ) 817 - 829 . Zbl 0907.49009 · Zbl 0907.49009 · doi:10.1080/03605309808821367
[12] A. Braides and G. Dal Maso , Non-local approximation of the Mumford-Shah functional . Calc. Var. 5 ( 1997 ) 293 - 322 . Zbl 0873.49009 · Zbl 0873.49009 · doi:10.1007/s005260050068
[13] A. Chambolle and G. Dal Maso , Discrete approximation of the Mumford-Shah functional in dimension two . ESAIM: M2AN 33 ( 1999 ) 651 - 672 . Numdam | Zbl 0943.49011 · Zbl 0943.49011 · doi:10.1051/m2an:1999156
[14] G. Cortesani , Sequence of non-local functionals which approximate free-discontinuity problems . Arch. Rational Mech. Anal. 144 ( 1998 ) 357 - 402 . Zbl 0926.49007 · Zbl 0926.49007 · doi:10.1007/s002050050121
[15] G. Cortesani , A finite element approximation of an image segmentation problem . Math. Models Methods Appl. Sci. 9 ( 1999 ) 243 - 259 . Zbl 0937.65072 · Zbl 0937.65072 · doi:10.1142/S0218202599000130
[16] G. Cortesani and R. Toader , Finite element approximation of non-isotropic free-discontinuity problems . Numer. Funct. Anal. Optim. 18 ( 1997 ) 921 - 940 . Zbl 0903.49002 · Zbl 0903.49002 · doi:10.1080/01630569708816801
[17] G. Cortesani and R. Toader , A density result in SBV with respect to non-isotropic energies . Nonlinear Anal. 38 ( 1999 ) 585 - 604 . Zbl 0939.49024 · Zbl 0939.49024 · doi:10.1016/S0362-546X(98)00132-1
[18] G. Cortesani and R. Toader , Nonlocal approximation of nonisotropic free-discontinuity problems . SIAM J. Appl. Math. 59 ( 1999 ) 1507 - 1519 . Zbl 0944.49012 · Zbl 0944.49012 · doi:10.1137/S0036139997327691
[19] G. Dal Maso , An Introduction to \(\Gamma \)-Convergence . Birkhäuser, Boston ( 1993 ). MR 1201152 | Zbl 0816.49001 · Zbl 0816.49001
[20] E. De Giorgi . 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 0758.49002 · Zbl 0758.49002
[21] L. Lussardi and E. Vitali , Non-local approximation of free-discontinuity functionals with linear growth: the one-dimensional case . Ann. Mat. Pura Appl. (to appear). MR 2317787 | Zbl 1136.49030 · Zbl 1136.49030 · doi:10.1007/s10231-006-0028-8
[22] M. Morini , Sequences of singularly perturbed functionals generating free-discontinuity problems . SIAM J. Math. Anal. 35 ( 2003 ) 759 - 805 . Zbl 1058.49021 · Zbl 1058.49021 · doi:10.1137/S0036141001395388
[23] 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 . Zbl 0953.49024 · Zbl 0953.49024 · doi:10.1080/01630569908816934
[24] S. Osher and J.A. Sethian , Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations . J. Comput. Phys. 79 ( 1988 ) 12 - 49 . Zbl 0659.65132 · Zbl 0659.65132 · doi:10.1016/0021-9991(88)90002-2
[25] J. Shah , A common framework for curve evolution, segmentation and anisotropic diffusion , in IEEE conference on computer vision and pattern recognition ( 1996 ).
[26] J. Shah , 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 1414486 | Zbl 0871.65120 · Zbl 0871.65120
[27] L. Simon , Lectures on Geometric Measure Theory . Centre for Mathematical Analysis, Australian National University ( 1984 ). MR 756417 | Zbl 0546.49019 · Zbl 0546.49019
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