Variational analysis of a perturbed free boundary problem

We consider maps u:X→Y of Riemannian manifolds which are locally energy minimizing subject to a constraint of the form Im(u) where M denotes a smooth bounded domain contained in a regular ball B of the target manifold. In general local minima are regular up to a set of vanishing Hausdorff measure, here we show for star-shaped obstacles M singular points can be exluded.