# Non-local approximation of free-discontinuity problems with linear growth

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 13, Issue: 1, page 135-162
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topLussardi, Luca, and Vitali, Enrico. "Non-local approximation of free-discontinuity problems with linear growth." ESAIM: Control, Optimisation and Calculus of Variations 13.1 (2007): 135-162. <http://eudml.org/doc/249997>.

@article{Lussardi2007,

abstract = {
We approximate, in the sense of
Γ-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.
},

author = {Lussardi, Luca, Vitali, Enrico},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Variational approximation; free discontinuities.; variational approximation; free discontinuities},

language = {eng},

month = {2},

number = {1},

pages = {135-162},

publisher = {EDP Sciences},

title = {Non-local approximation of free-discontinuity problems with linear growth},

url = {http://eudml.org/doc/249997},

volume = {13},

year = {2007},

}

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

DA - 2007/2//

PB - EDP Sciences

VL - 13

IS - 1

SP - 135

EP - 162

AB -
We approximate, in the sense of
Γ-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.

LA - eng

KW - Variational approximation; free discontinuities.; variational approximation; free discontinuities

UR - http://eudml.org/doc/249997

ER -

## References

top- R. Alicandro, A. Braides and M.S. Gelli, Free-discontinuity problems generated by singular perturbation. Proc. Roy. Soc. Edinburgh Sect. A6 (1998) 1115–1129.
- R. Alicandro, A. Braides and J. Shah, Free-discontinuity problems via functionals involving the L1-norm of the gradient and their approximations. Interfaces and free boundaries1 (1999) 17–37.
- R. Alicandro and M.S. Gelli, Free discontinuity problems generated by singular perturbation: the n-dimensional case. Proc. Roy. Soc. Edinburgh Sect. A130 (2000) 449–469.
- L. Ambrosio, A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. B3 (1989) 857–881.
- L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000).
- L. Ambrosio and V.M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $\Gamma $-convergence. Comm. Pure Appl. Math.XLIII (1990) 999–1036.
- L. Ambrosio and V.M. Tortorelli, On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7)VI (1992) 105–123.
- G. Bouchitté, A. Braides and G. Buttazzo, Relaxation results for some free discontinuity problems. J. Reine Angew. Math.458 (1995) 1–18.
- B. Bourdin and A. Chambolle, Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math.85 (2000) 609–646.
- A. Braides. Approximation of free-discontinuity problems. Lect. Notes Math.1694, Springer Verlag, Berlin (1998).
- A. Braides and A. Garroni, On the non-local approximation of free-discontinuity problems. Comm. Partial Differential Equations23 (1998) 817–829.
- A. Braides and G. Dal Maso, Non-local approximation of the Mumford-Shah functional. Calc. Var.5 (1997) 293–322.
- A. Chambolle and G. Dal Maso, Discrete approximation of the Mumford-Shah functional in dimension two. ESAIM: M2AN33 (1999) 651–672.
- G. Cortesani, Sequence of non-local functionals which approximate free-discontinuity problems. Arch. Rational Mech. Anal.144 (1998) 357–402.
- G. Cortesani, A finite element approximation of an image segmentation problem. Math. Models Methods Appl. Sci.9 (1999) 243–259.
- G. Cortesani and R. Toader, Finite element approximation of non-isotropic free-discontinuity problems. Numer. Funct. Anal. Optim.18 (1997) 921–940.
- G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies. Nonlinear Anal.38 (1999) 585–604.
- G. Cortesani and R. Toader, Nonlocal approximation of nonisotropic free-discontinuity problems. SIAM J. Appl. Math.59 (1999) 1507–1519.
- G. Dal Maso, An Introduction to$\Gamma $-Convergence. Birkhäuser, Boston (1993).
- 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.
- 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).
- M. Morini, Sequences of singularly perturbed functionals generating free-discontinuity problems. SIAM J. Math. Anal.35 (2003) 759–805.
- 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.
- 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.
- J. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion, in IEEE conference on computer vision and pattern recognition (1996).
- J. Shah, Uses of elliptic approximations in computer vision. In R. Serapioni and F. Tomarelli, editors, Progress in Nonlinear Differential Equations and Their Applications25 (1996).
- L. Simon, Lectures on Geometric Measure Theory. Centre for Mathematical Analysis, Australian National University (1984).

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.