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Journal of Convex Analysis 25 (2018), No. 4, 1197--1221
Copyright Heldermann Verlag 2018

A Convex Decomposition Formula for the Mumford-Shah Functional in Dimension One

Marcello Carioni
Max Planck Institute for Mathematics in the Science, Inselstrasse 22, 04103 Leipzig, Germany
and: Institut für Mathematik, Universität Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany

We study the convex lift of Mumford-Shah type functionals in the space of rectifiable currents and we prove a convex decomposition formula in dimension one, for finite linear combinations of SBV graphs. We use this result to prove the equivalence between the minimum problems for the Mumford-Shah functional and the lifted one and, as a consequence, we obtain a weak existence result for calibrations in one dimension.

Keywords: Mumford-Shah functional, convex lift, rectifiable currents, calibrations.

MSC: 49K99, 49Q20, 39B62

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