Evolution of harmonic maps with Dirichlet boundary conditions

In this paper we shall study a left over problem concerning the heat flow of harmonic maps on manifolds with boundary. Let (M, g) be a compact smooth m-dimensional Reimannian manifold with nonempty smooth boundary <9M, and let (iV, h) be a compact smooth n-dimensional Reimannian manifold without boundary. We denote M U dM by M. Since (AT, h) can be isometrically embedded into an Euclidean space M, for some k > n, we may view TV as a submanifold of R. In local coordinates on M, the energy of a map u : M —> N ^-> R^ is given by