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In this paper we consider a new kind of Mumford-Shah functional E(u, Ω )f or maps u : R m → R n 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). Resume. Dans cet article on considere une nouvelle fonctionnelle du type de Mumford-Shah E(u, Ω) pour des applications u : R m → R n avec m ≥ n. La nouveautep rincipale est que l'´ presente un ensemble singulier Su de codimension superieureun, defini par la theorie des determinant au sense de distributions. Apres avoir rappelel es dde base et certains resultats classiques, nous prouvons une propriete d'approximation pour l'´energie E(u, Ω )p arΓ-convergence, dans le meme 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.

Variational approximation of a functional of Mumford–Shah type in codimension higher than one