Issue |
ESAIM: COCV
Volume 20, Number 1, January-March 2014
|
|
---|---|---|
Page(s) | 190 - 221 | |
DOI | https://doi.org/10.1051/cocv/2013061 | |
Published online | 27 January 2014 |
Variational approximation of a functional of Mumford–Shah type in codimension higher than one
Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126, Pisa, Italy
f.ghiraldin@sns.it
Received:
24
January
2013
Revised:
5
April
2013
In this paper we consider a new kind of Mumford–Shah functional E(u, Ω) for maps u : ℝm → ℝn with m ≥ n. The most important novelty is that the energy features a singular set Su of codimension greater than one, defined through the theory of distributional jacobians. After recalling the basic definitions and some well established results, we prove an approximation property for the energy E(u, Ω) via Γ −convergence, in the same spirit of the work by Ambrosio and Tortorelli [L. Ambrosio and V.M. Tortorelli, Commun. Pure Appl. Math. 43 (1990) 999–1036].
Résumé
Dans cet article on considère une nouvelle fonctionnelle du type de Mumford–Shah E(u, Ω) pour des applications u : ℝm → ℝn avec m ≥ n. La nouveauté principale est que l’énergie présente un ensemble singulier Su de codimension supérieure à un, défini par la théorie des déterminant au sense de distributions. Après avoir rappelé les définitions de base et certains résultats classiques, nous prouvons une propriété d’approximation pour l’énergie E(u, Ω) par Γ-convergence, dans le même esprit de Ambrosio et Tortorelli [L. Ambrosio and V.M. Tortorelli, Commun. Pure Appl. Math. 43 (1990) 999–1036].
Mathematics Subject Classification: 49Q20 / 49J45 / 49Q15
Key words: Jacobian / Γ-convergence / higher codimension / Mumford–Shah / Ginzburg–Landau / phase transition
© EDP Sciences, SMAI, 2014
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