Math, Ann. 240, 231-250 (1979) Am l~ by Springer-Verlag1979 Uniqueness of Harmonic Mappings and of Solutions of Elliptic Equations on Riemannian Manifolds Willi J~iger Institut for Angewandte Mathematik, Im Neuenheimer Feld 294, D-6900 Heidelberg, Federal Republic of Germany Helmut Kaul* Fachbereich Mathematik der Universit~itTiibingen, Auf der Morgenstelle 10, D-7400Ttibingen, Federal Republic of Germany 1. Introduction and Statement of the Results In this paper we prove a maximum principle for a distance - like function of two harmonic maps ,fl,f2:MoN of Riemannian manifolds M, N, where M is compact and has a non-void boundary 0M. More general, instead of harmonic maps we consider solutions of an elliptic equation Af =b(df), (1.1) where b is a given tensor field with quadratic growth in the differential d f, satisfying a Lipschitz condition, and A denotes the generalized Laplace operator, naturally connected with the energy integral E(f) = ½ S ]dJ] 2d Volta M by the Euler-Lagrange equations. Our maximum principle includes an uniqueness theorem for the associated Dirichtet problem. More than that, it is also important for proving existence of solutions : It allows to reduce the solution of Dirichlet's problem for continuous boundary values dM~N to the case of smooth boundary values. Existence results for smooth boundary values have been obtained by Morrey [12], Hamilton [6], Hildebrandt et al. [8], and Kaul [11]. In a recent paper [ 15], v. Wahl proved the existence of smooth solutions of the Dirichlet problem for non-linear elliptic systems using a continuity method. In the main step he is combining estimates for linear equations and estimates as stated in our maximum principle. Using his arguments and the results of this paper one gets existence for Eq. (1.1) without using variational techniques. * The work of the secondly-named author was supported by the Sonderforschungsbereich Theoretische Mathematik at the Universityof Bonn 0025-5831/79/0240/0231/$04.00