Harmonic maps on domains with piecewise Lipschitz continuous metrics

For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from $(\mathrm{\Omega },g)$ to a compact Riemannian manifold $(N,h)\subset {\mathbb{R}}^k$ without boundary. We generalize the notion of stationary harmonic map and prove the partial regularity. We also discuss the global Lipschitz and piecewise $C^{1,\alpha }$-regularity of harmonic maps from $(\mathrm{\Omega },g)$ manifolds that support convex distance functions.