Goncharov Yu.V., Losеv A.G., Svеtlov A.V. Harmonic functions on cones of model manifolds
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Goncharov Yuriy Vladimirovich
Student, Institute of Mathematics and IT,
Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
Losеv Alеxandеr Gеorgiеvich
Doctor of Physical and Mathematical Sciences, Director, Institute of Mathematics and IT,
Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
Svеtlov Andrеy Vladimirovich
Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Mathematical Analysis and Function Theory,
Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
Abstract. The paper deals with harmonic functions on cones of model manifolds. M is called a cone of model manifold, if M = B ∪ D, where B is a non-empty precompact set and D is isometric to the product [r0,+∞) × Ω (r0> 0, Ω is a compact Riemannian manifold with non-empty smooth boundary) with the metric
ds2= dr2+ g2(r)dθ2.
Here g(r) is a positive smooth on [r0,+∞) function, and dθ is a metric on Ω. Note if Ω is a compact Riemannian manifold with no boundary, we have just a definition of model manifold.
Let’s H0(M) = {u : Δu = 0,u|∂M= 0},
and
where r0= const > 0, n = dim M.
The main results of the paper are following.
Theorem 1. Let’s manifold M has J = ∞. Then any bounded function u ∈ H0(M) is equal to zero identically.
Theorem 2. Let’s manifold M has J = ∞. Then for cone of positive harmonic functions from class H0(M) the dimension is equal to 1.
Key words: Laplace — Beltrami equation, Liouville type theorems, model Riemannian manifolds, cones of model manifolds, dimension of solutions’ space.
Harmonic Functions on Cones of Model Manifolds by Goncharov Yu.V., Losеv A.G., Svеtlov A.V. is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №3 (22) 2014 pp. 13-22
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