Well-posed problems for the Laplace-Beltrami operator on a stratified set consisting of punctured circles and segments
Kanguzhin B.E. Mustafina M.O. Kaiyrbek O.A.
30 September 2025E.A. Buketov Karaganda University Publish house
Bulletin of the Karaganda University. Mathematics Series
2025#2025Issue 3150 - 163 pp.
The Laplace-Beltrami operator is studied on a stratified set consisting of two punctured circles and an interval. A complete description of all well-posed boundary value problems for the Laplace-Beltrami operator on such a set is given. In the second part of the paper, a class of self-adjoint well-posed problems for the Laplace-Beltrami operator on the specified stratified set is identified. The obtained results can be considered as a generalization of known results on geometric graphs. In particular, the stratified set under consideration can be interpreted as graphs with loops. Studies on the spectral asymptotics of Sturm-Liouville operators on plane curves homotopic to a finite interval are also closely related to the present results paper. Since the punctured circle is diffeomorphic to a finite interval, the spectral methods applied to differential operators on a finite interval can be modified to study the spectral properties of differential operators on the punctured circle. The main results of this paper are obtained by modifications of methods that were previously used in the study of the asymptotic behavior of the eigenvalues of the Sturm-Liouville operator on a finite interval.
differential operators , eigenvalue , graph , inhomogeneous equation , inhomogeneous system of equations , Laplace-Beltrami operator , local coordinate , punctured circle , unique solution
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East Kazakhstan University named after Sarsen Amanzholov, 30th Guards Division street, Ust-Kamenogorsk, 070000, Kazakhstan
Al-Farabi Kazakh National University, 71 Al-Farabi Avenue, Almaty, 050040, Kazakhstan
Institute of Mathematics and Mathematical Modeling
East Kazakhstan University named after Sarsen Amanzholov
Al-Farabi Kazakh National University
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