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

Published online by Cambridge University Press:  27 January 2014

Francesco Ghiraldin
Francesco Ghiraldin*
Affiliation:
Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126, Pisa, Italy. f.ghiraldin@sns.it
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Abstract

In this paper we consider a new kind of Mumford–Shah functionalE(u, Ω) for mapsu : ℝm → ℝnwith m ≥ n. The most important novelty is that theenergy features a singular set Su ofcodimension greater than one, defined through the theory of distributional jacobians.After recalling the basic definitions and some well established results, we prove anapproximation property for the energy E(u, Ω) viaΓ −convergence, in the same spirit of the work by Ambrosio andTortorelli [L. Ambrosio and V.M. Tortorelli, Commun. Pure Appl. Math.43 (1990) 999–1036].

Keywords

JacobianΓ-convergencehigher codimensionMumford–ShahGinzburg–Landauphase transition

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Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 1 , January 2014 , pp. 190 - 221
Copyright
© EDP Sciences, SMAI, 2014

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