Royden Decomposition for Harmonic Maps with Finite Total Energy

We prove the harmonic map version of the Royden decomposition in the sense that given any bounded C¹-map f with finite total energy on a complete Riemannian manifold into a Cartan-Hadamard manifold, there exists a unique bounded harmonic map with finite total energy from the manifold into the Cartan-Hadamard manifold taking the same boundary value at each harmonic boundary point as that of f.