Math. Z. 147, 225-236 (1976) Mathematische Zeitschrift 9 by Springer-Verlag 1976 Dirichlet's Boundary Value Problem for Harmonic Mappings of Riemannian Manifolds Stefan Hildebrandt ~, Hehnut Kaul 1 Kjell-Ove Widman 2. Mathematisches Institut der Universitiit Bonn, Wegelerstr. 10, D-5300 Bonn, Germany 2 Department of Mathematics, Link6ping University, S-58183 Link6ping, Sweden Dedicated to Ernst H61der on his 75th birthday, April 2,1976 1. Introduction Let f be a compact, connected, n-dimensional Riemannian manifold of class C #, with non-void boundary X, and interior O. In terms of local coordinates x= (x 1..... x") on f, the line element on Y' will be denoted by da 2 =7~b(x)dx~ dx b. Furthermore, let J/g be a complete Riemannian manifold without boundary, of dimension N~2, and of class C 4. The metric on J/l will be denoted by (3, tl)~a, Jr ~ ]r~ = (~, ~)~t, for ~, ~/e T~, and the corresponding line element in local coordi- nates u = (u 1..... u N) b y ds 2 = gik(U) dui du k. With every mapping UeCI(CLJ/I) we can associate an invariant eeC~ IR), defined by e = ev= glk(U)7~ D ~ui D ~ u k and called the energy density of U. Here u = u(x) is a representation of U in local coordinates x and u on 5~ and rig, respectively. The energy E(U) of a map UeCI(f2, JC[) is defined by E(U)= ~ ed~" X where ~ stands for the n-dimensional Lebesgue measure on K r induced by the metric. A mapping U~C2((2, ,///g) is said to be harmonic, if it satisfies the Euler * This work was partially carried out under the auspices of the Sonderforschungsbereich Theore- tische Mathematik at the University of Bonn. The two firstly-named authors are also grateful for the hospitality of I Link6ping University. Here and in the sequel, repeated Greek indices c~,fi, ... are to be summed from 1 to n, Latin indices i, k,... from 1 to N.