Variational Problems with Obstacles and Harmonic Maps

Let (M, g) be an oriented Riemannian manifold of finite dimension and let C be a closed subset of the Euclidean space R k. In this paper we illustrate some results on harmonic maps u: M →C which were obtained in collaboration with Chen Yun-Mei. Our first purpose was to find a generalization of the usual definition of a harmonic map between two Riemannian manifolds in order to consider less regular target spaces. Our second aim was to extend a result by Chen and Struwe about the heat flow of harmonic mappings into manifolds with boundary.