## Korol’kova E.S., Korol’kov S.A. Boundary problems for harmonic functions on unbounded open sets of riemannian manifolds

http://dx.doi.org/10.15688/jvolsu1.2013.1.4

Korol’kova Elеna Sеrgееvna

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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 C1smooth 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: | f- f| = 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 MB 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.  E. S. Korol’kova, S. A. Korol’kov URL: https://mp.jvolsu.com/index.php/en/component/attachments/download/160 766 Downloads