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Γ-limits of convolutionfunctionals

Published online by Cambridge University Press:  23 January 2013

Luca Lussardi  and
Annibale Magni
Luca Lussardi
Affiliation:
Dipartimento di Matematica e Fisica “N. Tartaglia”, Università Cattolica del Sacro Cuore, via dei Musei 41, 25121 Brescia, Italy. [email protected]
Annibale Magni
Affiliation:
Mathematisches Institut Abt. für Reine Mathematik, Albert-Ludwigs Universität Freiburg, Eckerstrasse 1, 79104 Freiburg im Breisgau, Germany; [email protected]
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Abstract

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

Keywords

Free discontinuitiesΓ-convergenceanisotropy
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 2 , April 2013 , pp. 486 - 515
Copyright
© EDP Sciences, SMAI, 2013

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References

Alicandro, R. and Gelli, M. S., Free discontinuity problems generated by singular perturbation : the n-dimensional case. Proc. R. Soc. Edinb. Sect. A 130 (2000) 449–469. Google Scholar
Alicandro, R., Braides, A., and Gelli, M.S., Free-discontinuity problems generated by singular perturbation. Proc. R. Soc. Edinburgh Sect. A 6 (1998) 1115–1129. Google Scholar
Ambrosio, L. and Tortorelli, V.M., Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Commut. Pure Appl. Math. XLIII (1990) 999–1036. Google Scholar
Ambrosio, L. and Tortorelli, V.M., On the approximation of free discontinuity problems. Boll. Unione Mat. Ital. B (7) VI (1992) 105–123. Google Scholar
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000).
Bouchitté, G., Braides, A. and Buttazzo, G., Relaxation results for some free discontinuity problems. J. Reine Angew. Math. 458 (1995) 1–18. Google Scholar
Bourdin, B. and Chambolle, A., Implementation of an adaptive finite-element approximation of the Mumford–Shah functional. Numer. Math. 85 (2000) 609–646. Google Scholar
Braides, A., Approximation of free-discontinuity problems. Lect. Notes Math. 1694 (1998). Google Scholar
A. Braides, Γ -convergence for beginners. Oxford University Press (2002).
Braides, A. and Dal Maso, G., Non-local approximation of the Mumford–Shah functional. Calc. Var. 5 (1997) 293–322. Google Scholar
Braides, A. and Garroni, A., On the non-local approximation of free-discontinuity problems. Commut. Partial Differ. Equ. 23 (1998) 817–829. Google Scholar
Chambolle, A. and Dal Maso, G., Discrete approximation of the Mumford–Shah functional in dimension two. ESAIM : M2AN 33 (1999) 651–672. Google Scholar
Cortesani, G., Sequence of non-local functionals which approximate free-discontinuity problems. Arch. R. Mech. Anal. 144 (1998) 357–402. Google Scholar
Cortesani, G., A finite element approximation of an image segmentation problem. Math. Models Methods Appl. Sci. 9 (1999) 243–259. Google Scholar
Cortesani, G. and Toader, R., Finite element approximation of non-isotropic free-discontinuity problems. Numer. Funct. Anal. Optim. 18 (1997) 921–940. Google Scholar
Cortesani, G. and Toader, R., Nonlocal approximation of nonisotropic free-discontinuity problems. SIAM J. Appl. Math. 59 (1999) 1507–1519. Google Scholar
Cortesani, G. and Toader, R., A density result in SBV with respect to non-isotropic energies. Nonlinear Anal. 38 (1999) 585–604. Google Scholar
G. Dal Maso, An Introduction to Γ -Convergence. Birkhäuser, Boston (1993).
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.
Lussardi, L., An approximation result for free discontinuity functionals by means of non-local energies. Math. Methods Appl. Sci. 31 (2008) 2133–2146. Google Scholar
Lussardi, L. and Vitali, E., Non local approximation of free-discontinuity functionals with linear growth : the one dimensional case. Ann. Mat. Pura Appl. 186 (2007) 722–744. Google Scholar
Lussardi, L. and Vitali, E., Non local approximation of free-discontinuity problems with linear growth. ESAIM : COCV 13 (2007) 135–162. Google Scholar
Morini, M., Sequences of singularly perturbed functionals generating free-discontinuity problems. SIAM J. Math. Anal. 35 (2003) 759–805. Google Scholar
Negri, M., The anisotropy introduced by the mesh in the finite element approximation of the Mumford–Shah functional. Numer. Funct. Anal. Optim. 20 (1999) 957–982. Google Scholar
Negri, M., A non-local approximation of free discontinuity problems in SBV and SBD. Calc. Var. 25 (2006) 33–62. Google Scholar