We discuss regularity questions for harmonic maps from a n-dimen-sional Riemannian polyhedral complex X to a non-positively curved metric space. The main theorems assert, assuming Lipschitz regularity of the metric on the domain complex, that such maps are locally Holder continuous with explicit bounds of the Holder constant and exponent on the energy of the map and the geome-try of the domain and locally Lipschitz continuous away from the (n — 2)- skeleton of the complex. Moreover, if x is a point on the k-skeleton (k ≤ n — 2) we give explicit dependence of the Holder exponent at a point near x on the combinatorial and geometric information of the link of x in X and the link of the k-dimensional skeleton in X at x.
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