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Transactions of the American Mathematical Society
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
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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.


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Additional Information

Bent Fuglede
Affiliation: Department of Mathematics, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
Email: fuglede@math.ku.dk

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03498-1
PII: S 0002-9947(04)03498-1
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