Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

marcello.carioni@uni-graz.at

Keywords

  • Mumford-Shah functional
  • convex lift
  • rectifiable currents
  • calibrations

Mathematics Subject Classification

49K99, 49Q20, 39B62

Suggested citation

M. Carioni. “A Convex Decomposition Formula for the Mumford-Shah Functional in Dimension One.” Journal of Convex Analysis 25 (2018), No. 4, 1197–1221.

Copyright Heldermann Verlag 2018