Harmonic mappings into manifolds with boundary

In this paper we deal with harmonic maps from a compact Riemannian manifold into a manifold with boundary. In this case, a weak harmonic map in de definition a solution to a differential inclusion. In the first part of the paper we investigate the general properties of weak harmonic maps, which can be seen a solutions to a system of elliptic differential equations. In the second part we concentrate our attention on the heat flow method for harmonic maps. The result we achieve in this context extends a result by Chen and Struwe .