Cahiers du CEREMADE

Unité Mixte de Recherche du C.N.R.S. N°7534
 
Abstract : G. Alberti, G. Bouchitté and G. Dal Maso recently found sufficient conditions for the minimizers of the (nonconvex) Mumford-Shah 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 Mumford-Shah case in dimension one, it can be made more precise and leads to the conclusion that for every minimizer, the calibration exists---although, still, in a very weak (asymptotical) sense.
 
 
CONVEX REPRESENTATION FOR LOWER SEMICONTINUOUS ENVELOPES OF FUNCTIONALS IN L1
CHAMBOLLE Antonin
2000-1
12-01-2000
 
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