Goncharov Yu.V., Losеv A.G., Svеtlov A.V. Harmonic functions on cones of model manifolds

 
 
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

AbstractThe 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
3 2014 2 pic1
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.

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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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