Abstract

We prove that a wide class of singularly perturbed functionals generates as $\Gamma$-limit a functional related to a free-discontinuity problem. Several applications of the result are shown.

MSC codes

  1. 49K10

Keywords

  1. free-discontinuity problems
  2. singular perturbations
  3. $\Gamma$-convergence

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
Giovanni Alberti, Guy Bouchitté, Pierre Seppecher, Phase transition with the line‐tension effect, Arch. Rational Mech. Anal., 144 (1998), 1–46
2.
Roberto Alicandro, Andrea Braides, Maria Gelli, Free‐discontinuity problems generated by singular perturbation, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1115–1129
3.
R. Alicandro, M. Gelli, Free discontinuity problems generated by singular perturbation: the n‐dimensional case, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 449–469
4.
L. Ambrosio, A compactness theorem for a new class of functions of bounded variation, Boll. Un. Mat. Ital. B (7), 3 (1989), 857–881
5.
L. Ambrosio, Existence theory for a new class of variational problems, Arch. Rational Mech. Anal., 111 (1990), 291–322
6.
Luigi Ambrosio, Nicola Fusco, Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, 2000xviii+434
7.
Luigi Ambrosio, Vincenzo Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via Γ‐convergence, Comm. Pure Appl. Math., 43 (1990), 999–1036
8.
Guy Bouchitté, Andrea Braides, Giuseppe Buttazzo, Relaxation results for some free discontinuity problems, J. Reine Angew. Math., 458 (1995), 1–18
9.
G. Bouchitté, C. Dubs, P. Seppecher, Regular approximation of free‐discontinuity problems, Math. Models Methods Appl. Sci., 10 (2000), 1073–1097
10.
Andrea Braides, Approximation of free‐discontinuity problems, Lecture Notes in Mathematics, Vol. 1694, Springer‐Verlag, 1998xii+149
11.
Andrea Braides, Γ‐convergence for beginners, Oxford Lecture Series in Mathematics and its Applications, Vol. 22, Oxford University Press, 2002xii+218
12.
Andrea Braides, Alessandra Coscia, The interaction between bulk energy and surface energy in multiple integrals, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 737–756
13.
A. Braides, G. Dal Maso, Non‐local approximation of the Mumford‐Shah functional, Calc. Var. Partial Differential Equations, 5 (1997), 293–322
14.
Andrea Braides, Gianni Dal Maso, Adriana Garroni, Variational formulation of softening phenomena in fracture mechanics: the one‐dimensional case, Arch. Ration. Mech. Anal., 146 (1999), 23–58
15.
Andrea Braides, Maria Gelli, Limits of discrete systems with long‐range interactions, J. Convex Anal., 9 (2002), 363–399, Special issue on optimization (Montpellier, 2000)
16.
Francine Catté, Pierre‐Louis Lions, Jean‐Michel Morel, Tomeu Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29 (1992), 182–193
17.
Guido Cortesani, Sequences of non‐local functionals which approximate free‐discontinuity problems, Arch. Rational Mech. Anal., 144 (1998), 357–402
18.
Guido Cortesani, Rodica Toader, A density result in SBV with respect to non‐isotropic energies, Nonlinear Anal., 38 (1999), 585–604
19.
Gianni Dal Maso, An introduction to Γ‐convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston Inc., 1993xiv+340
20.
Ennio De Giorgi, Luigi Ambrosio, New functionals in the calculus of variations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 82 (1988), 199–210
21.
Massimo Gobbino, Finite difference approximation of the Mumford‐Shah functional, Comm. Pure Appl. Math., 51 (1998), 197–228
22.
Massimo Gobbino, Maria Mora, Finite‐difference approximation of free‐discontinuity problems, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 567–595
23.
Stefan Müller, Singular perturbations as a selection criterion for periodic minimizing sequences, Calc. Var. Partial Differential Equations, 1 (1993), 169–204
24.
David Mumford, Jayant Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577–685
25.
L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115–162
26.
P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern and Mach. Intell., 12 (1990), pp. 629–639.

Information & Authors

Information

Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 759 - 805
ISSN (online): 1095-7154

History

Published online: 1 August 2006

MSC codes

  1. 49K10

Keywords

  1. free-discontinuity problems
  2. singular perturbations
  3. $\Gamma$-convergence

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.