COMMUNICATIONS IN ANALYSIS AND GEOMETRY Volume 1, Number 3, 327-346, 1993 EVOLUTION OF HARMONIC MAPS WITH DIRICHLET BOUNDARY CONDITIONS YUNMEI CHEN* AND FANG HUA LIN** INTRODUCTION 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 with- out boundary. We denote M U dM by M. Since (AT, h) can be isometrically embedded into an Euclidean space M fc , for some k > n, we may view TV as a submanifold of R fe . In local coordinates on M, the energy of a map u : M —> N ^-> R^ is given by M here and here after (g a(3 ) — (^a/?) -1 ,^ = det(5 a/ g),l <(*.,&< m and a sum- mation convention is employed. The Euler-Lagrange equation associated with the functional (0.1) is (0.2) Au = A(u)(du, du) , where A denotes the Laplace-Beltram operator on M and A{u) is the second fundamental form of A^ in R^ at u. *The research is partially supported by NSF grant DMS#9123532. **The research is partially supported by NSF grant DMS# 9149555.