Inserted: 27 may 2014
Last Updated: 27 may 2014
Journal: Bruno Pini Mathematical Analysis Seminar
We review some issues about the regularity theory of local minimizers of the Mumford \& Shah energy in the 2-dimensional case. In particular, we stress upon some recent results obtained in collaboration with Camillo De Lellis (Universitaet ̈Zuerich). On one hand, we deal with basic regularity, more precisely we survey on an elementary proof of the equivalence between the weak and strong formulation of the problem established in Ref. 16. On the other hand, we discuss fine regularity properties by outlining an higher integrability result for the density of the volume part proved in Ref. 17. The latter, in turn, implies an estimate on the Hausdorff dimension of the singular set of minimizers according to the results in Ref. 2 (see also Ref. 18).
Keywords: Local minimizers, Density lower bound, higher integrability of the approximate gradient, regularity of the singular set., Mumford & Shah variational model