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'Regularity' of Energy Minimizer maps between Riemannian polyhedra

In the past decades, there has been a wide range of activity in the study of the existence of of energy minimizers in various homotopy classes of maps between smooth riemannian manifolds, See [11], [15], [16], [18], [19], [20] and the references therein. In [11], Eells and Sampson obtained a fundamental theorem on the existence of harmonic maps in each free homotopy class of maps with target manifolds of non-positive sectional curvature which was generalized to the case of target manifolds without focal points by Xin [23]. In sharp contrast to the smooth case, very little is known for the case of singular spaces. In [17], Korevaar and Schoen expanded the theory of harmonic maps between smooth riemannian manifolds to the case of maps between certain singular spaces. For instance, admissible riemannian polyhedra are prototypes of the relevant singular spaces because these being both geodesic, harmonic (in the sense of Brelot cf. [10] ch 2) and Dirichlet spaces and provide a wealth of examples as well. Let us mention here some examples of riemannian polyhedra (cf. [10] ch 8):