Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Dirichlet problem for harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature

Author: Bent Fuglede
Journal: Trans. Amer. Math. Soc. 357 (2005), 757-792
MSC (2000): Primary 58E20, 49N60; Secondary 58A35
Published electronically: May 10, 2004
MathSciNet review: 2095630
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This is a continuation of the Cambridge Tract ``Harmonic maps between Riemannian polyhedra'', by J. Eells and the present author. The variational solution to the Dirichlet problem for harmonic maps with countinuous boundary data is shown to be continuous up to the boundary, and thereby uniquely determined. The domain space is a compact admissible Riemannian polyhedron with boundary, while the target can be, for example, a simply connected complete geodesic space of nonpositive Alexandrov curvature; alternatively, the target may have upper bounded curvature provided that the maps have a suitably small range. Essentially in the former setting it is further shown that a harmonic map pulls convex functions in the target back to subharmonic functions in the domain.

References [Enhancements On Off] (What's this?)

  • [A1] Alexandrov, A.D., A theorem on triangles in a metric space and some of its applications, (In Russian), Trudy Math. Inst. Steklov 38 (1951), 5-23. MR 14:198a
  • [A2] Alexandrov, A.D., Über eine Verallgemeinerung der Riemannschen Geometrie, Schr. Forschungsinst. Math. Berlin 1 (1957), 33-84. MR 19:304h
  • [AB1] Alexander, S.B., and R.L. Bishop, The Hadamard-Cartan theorem in locally convex metric spaces, L'Enseignement Math. 36 (1990), 309-320. MR 92c:53044
  • [AB2] Alexander, S.B., and R.L. Bishop, Comparison theorems for curves of bounded geodesic curvature in metric spaces of curvature bounded above, Diff. Geom. Appl. 6 (1996), 67-86. MR 97a:53051
  • [De] Deny, J., Méthodes hilbertiennes en théorie du potentiel, Potential Theory, C.I.M.E, Stresa 1969, Ed. Cremonese, Roma, 1970, pp. 121-201. MR 44:1833
  • [EF] Eells, J., and B. Fuglede, Harmonic maps between Riemannian polyhedra, Cambridge Tracts in Mathematics No. 142, Cambridge University Press, 2001. MR 2002h:58017
  • [EP] Eells, J., and J.C. Polking, Removable singularities of harmonic maps, Indiana Univ. Math. J. 33 (1984), 859-871. MR 86e:58018
  • [F1] Fuglede, B., Finely Harmonic Functions, Lecture Notes in Math., No. 289, Springer, Berlin, 1972. MR 56:8883
  • [F2] Fuglede, B., Fonctions BLD et fonctions finement surharmoniques, Sém. Théorie du Potentiel Paris, No. 6, Lecture Notes in Math. No. 906, Springer, Berlin, 1982, pp. 126-157. MR 84b:31009
  • [F3] Fuglede, B., Hölder continuity of harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature, Calc. Var. 16 (2003), 375-403.
  • [F4] Fuglede, B., Finite energy maps from Riemannian polyhedra to metric spaces, Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 2, 433-458.
  • [F5] Fuglede, B., Dirichlet problems for harmonic maps from regular domains, Manuscript.
  • [GH] Giaquinta, M. and S. Hildebrandt, A priori estimates for harmonic mappings, J. Reine Angew. Math. 336 (1982), 124-164. MR 84b:58035
  • [Her] Hervé, R.-M., Un principe du maximum pour les sous-solutions locales d'une équation uniformément elliptique de la forme $Lu=-\sum _{i}\partial /\partial x_{i}\bigl (\sum _{j}a_{ij}\partial u/\partial x_{j}\bigr )=0$, Ann. Inst. Fourier (Grenoble) 14 (2) (1964), 493-508. MR 30:5040
  • [HKW] Hildebrandt, S., H. Kaul, and K.O. Widman, An existence theorem for harmonic mappings of Riemannian manifolds, Acta Math. 138 (1977), 1-16. MR 55:6478
  • [Ish] Ishihara, T., A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ. 19 (1979), 215-229. MR 80k:58045
  • [JK] Jäger, W., and H. Kaul, Uniqueness and stability of harmonic maps and their Jacobi fields, Manuscripta Math. 28 (1979), 269-291. MR 80j:58030
  • [Jo1] Jost, J., Equilibrium maps between metric spaces, Calc. Var. 2 (1994), 173-204. MR 98a:58049
  • [Jo2] Jost, J., Generalized Dirichlet forms and harmonic maps, Calc. Var. P.D.E. 5 (1997), 1-19. MR 98f:31014
  • [KM] Kilpeläinen, T., and J. Malý, Supersolutions to degenerate elliptic equations on quasi open sets, Comm. P.D.E. 17 (1992), 371-405. MR 93g:31022
  • [KS] Korevaar, N.J., and R.M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), 561-659. Reprinted as Chapter X in Lectures on harmonic maps, by R. Schoen and S.T. Yau, Conf. Proc. and Lecture Notes in Geometry and Topology, Vol. II (1997), 204-310. MR 95b:58043
  • [La] Landkof, N.S., Foundations of Modern Potential Theory, Springer, Berlin, 1972. MR 50:2520
  • [R] Reshetnyak, Y.G., Inextensible [=Non-expansive] mappings in a space of curvature no greater than $K$, Siberian Math. J. 9 (1968), 683-689. MR 39:6235
  • [Sbg] Schoenberg, I.J., Remarks to Maurice Fréchet's article: Sur la définition axiomatique d'une classe d'espaces distancés vectoriellement applicables sur l'espace de Hilbert, Ann. Math. 36 (1935), 724-732.
  • [Se1] Serbinowski, T., Boundary regularity of harmonic maps to nonpositively curved metric spaces, Comm. Anal. Geom. 2 (1994), 1-15. MR 95k:58050
  • [Se2] Serbinowski, T., Harmonic maps into metric spaces with curvature bounded above, Thesis, Univ. Utah, 1995.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 58E20, 49N60, 58A35

Retrieve articles in all journals with MSC (2000): 58E20, 49N60, 58A35

Additional Information

Bent Fuglede
Affiliation: Department of Mathematics, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark

Keywords: Harmonic map, Dirichlet problem, Riemannian manifold, Riemannian polyhedron, geodesic space, Alexandrov curvature
Received by editor(s): February 18, 2003
Received by editor(s) in revised form: September 2, 2003
Published electronically: May 10, 2004
Dedicated: In memory of Professor Heinz Bauer
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society