Journal of Convex Analysis 08 (2001), No. 1, 149--170
Copyright Heldermann Verlag 2001
Convex Representation for Lower Semicontinuous Envelopes of Functionals in L1
Antonin Chambolle
Ceremade - CNRS, Université de Paris-Dauphine, 75775 Paris 16, France
G. Alberti, G. Bouchitte and G. Dal Maso [The calibration method for the Mumford-Shah functional, C. R. Acad. Sci. Paris 329, Serie I (1999) 249--254] 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 in this paper 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.
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