Korol’kova E.S., Korol’kov S.A. Boundary problems for harmonic functions on unbounded open sets of riemannian manifolds
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http://dx.doi.org/10.15688/jvolsu1.2013.1.4
Korol’kova Elеna Sеrgееvna
Engineer OOO “Gazprom transgaz Volgograd”
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Ul. Raboche-Krest’yanskaya, 58, 400074 Volgograd, Russian Federation
Korol’kov Sеrgеy Alеksееvich
Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Mathematical Analysis and Function Theory Volgograd State University
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Prospekt Universitetskij, 100, 400062 Volgograd, Russian Federation
Abstract. We study harmonic functions on unbounded open set of Riemannian manifold and establish some existence and uniqueness results.
Let M be a smooth connected noncompact Riemannian manifold without boundary and Ω be a simply connected unbounded open set of M with C1- smooth boundary ∂Ω. Let 
be a smooth exhaustion of M i.e. sequence of precompact open subsets of M with C1-smooth boundaries ∂Bk such that 
for all k. In what follows we assume 
sets Bk ∩Ω are simply connected, ∂Bk and ∂Ω are transversal for all k.
Two continuous in M (in Ω, in ∂Ω, resp.) functions f1 and f2 are called equivalent in M (in Ω, in ∂Ω, resp.) 
if for some smooth exhaustion of M the following relation holds: 
| f1 - f2 | = 0
It isn’t hard to prove that ‘~’ is actually an equivalence relation and does not depend on the choice of a smooth exhaustion of M.
A continuous function f in Ω (in M, resp.) is called admissible in Ω (on M, resp.) if there is an harmonic function u in Ω (in M, resp.) such that 
Let B be an compact (with C1-smooth boundary) in M and 
be the solutions of the following Dirichlet problems:
By the maximum principle, the sequence is point-wise increasing and converges to an harmonic in M ∖ B function 
It is easy to see that 
The function νm is called the capacity potential of the compact B relatively to M.
We say that M is strong if 
It’s easy to verify that notion of strong manifold does not depend on choose the compact B.
We have the following result.
Theorem. Let M be a strong manifold, Ω ⊂ M be an unbounded open set with C1-smooth boundary ∂Ω, f be an admissible continuous in Ω function and φ — continuous in ∂Ω function such that 
Then there exists unique function u in Ω such that
Also we establish some existence and uniqueness results and prove solvability of the Dirichlet problem with continuous boundary data on a spherically symmetric manifolds with noncompact boundary.
Key words: boundary problems, harmonic functions, Riemannian manifolds, Dirichlet problem, model manifolds, manifolds with end.
Boundary problems for harmonic functions on unbounded open sets of riemannian manifolds by Korol’kova E.S., Korol’kov S.A. is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №1 (18) 2013 pp. 45-58
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