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

Engineer OOO “Gazprom transgaz Volgograd”
This email address is being protected from spambots. You need JavaScript enabled to view it.
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
This email address is being protected from spambots. You need JavaScript enabled to view it.
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.

Creative Commons License
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

Attachments:
Download this file (4_Korolkova_Korolkov.pdf) E. S. Korol’kova, S. A. Korol’kov
URL: https://mp.jvolsu.com/index.php/en/component/attachments/download/160
766 Downloads