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Existence and non-existence of minimal graphic and p-harmonic functions

Part of: Partial differential equations on manifolds; differential operators Global differential geometry Other generalizations

Published online by Cambridge University Press:  25 January 2019

Jean-Baptiste Casteras ,
Esko Heinonen  and
Ilkka Holopainen
Jean-Baptiste Casteras
Affiliation:
Departement de Mathematique, Universite libre de Bruxelles, CP 214, Boulevard du Triomphe, B-1050 Bruxelles, Belgium (jeanbaptiste.casteras@gmail.com)
Esko Heinonen
Affiliation:
Department of Mathematics and Statistics, P.O.B. 68 (Gustaf Hällströmin katu 2b), 00014 University of Helsinki, Finland (esko.heinonen@helsinki.fi; ilkka.holopainen@helsinki.fi)
Ilkka Holopainen
Affiliation:
Department of Mathematics and Statistics, P.O.B. 68 (Gustaf Hällströmin katu 2b), 00014 University of Helsinki, Finland (esko.heinonen@helsinki.fi; ilkka.holopainen@helsinki.fi)
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Abstract

We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold M with only one end if M has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constant minimal graphic and p-harmonic functions on rotationally symmetric Cartan-Hadamard manifolds under optimal assumptions on the sectional curvatures.

Keywords

Mean curvature equationp-Laplace equationDirichlet problemHadamard manifold

MSC classification

Primary: 58J32: Boundary value problems on manifolds 31C45: Other generalizations (nonlinear potential theory, etc.)
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 341 - 366
Copyright
Copyright © Royal Society of Edinburgh 2019

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