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Unité
Mixte de Recherche du C.N.R.S. N°7534 

Abstract : G. Alberti, G. BouchittÃ© and G. Dal Maso recently found sufficient conditions for the minimizers of the (nonconvex) MumfordShah functional. Their method consists in an extension of the calibration method (that is used for the characterization of minimal surfaces), adapted to this functional. The existence of a calibration, given a minimizer of the functional, remains an open problem. We introduce a general framework for the study of this problem. We first observe that, roughly, the minimization of any functional of a scalar function can be achieved by minimizing a convex functional, in higher dimension. Although this principle is in general too vague, in some situations, including the MumfordShah case in dimension one, it can be made more precise and leads to the conclusion that for every minimizer, the calibration existsalthough, still, in a very weak (asymptotical) sense. 





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