Regularity issues for local minimizers of the Mumford & Shah energy in 2d

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 (Universitat 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 [16]. On the other hand, we discuss ne regularity properties by outlining an higher integrability result for the density of the volume part proved in [17]. The latter, in turn, implies an estimate on the Hausdor dimension of the singular set of minimizers according to the results in [2] (see also [18]).