Regularity of minimizers of a Ginzburg-Landau type energy with metric cone target space

In this paper, we consider an energy of the type Dirichlet energy plus potential term, for a map with values into a metric cone. We investigate the regularity of minimizers of such functionals, and prove that they are always locally Hölder continuous. We establish that Lipschitz continuity is achieved in some cases where the target space has non-positive curvature, and show examples for which the minimizers are not Lipschitz continuous.