Convex Representation for Lower Semicontinuous Envelopes of Functionals in L1
Chambolle, Antonin (2001), Convex Representation for Lower Semicontinuous Envelopes of Functionals in L1, Journal of Convex Analysis, 8, 1, p. 149-170
TypeArticle accepté pour publication ou publié
Journal nameJournal of Convex Analysis
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Abstract (EN)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.
Subjects / Keywordsminimizer; calibration; Mumford-Shah functional
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