ON THE LAPLACE-BELTRAMI OPERATOR IN STRATIFIED SETS COMPOSED OF PUNCTURED CIRCLES AND SEGMENTS

Kanguzhin B.E. Akanbay Y. Dosmagulova K.A.
25 March 2025 al-Farabi Kazakh State National University

KazNU Bulletin. Mathematics, Mechanics, Computer Science Series
2025 #125 Issue 1 29 - 41 pp.

This paper discusses the introduction of local coordinates on the circle S¹ and the analysis of various classes of functions defined on it. It is proved that every smooth function on the circle corresponds to a smooth 2π-periodic function on the real axis. The Laplace-Beltrami operator on S¹ is introduced using the apparatus of exterior differential forms and the Hodge operator. Its explicit expression in local coordinates is calculated, and it is shown that it can be reduced to the double differentiation operator. Then, the spectral analysis of the Laplace-Beltrami operator is performed, its eigenvalues and the corresponding eigenfunctions expressed in terms of the Chebyshev polynomials of the first and second kind are found. Well-solved problems for the Laplace-Beltrami operator on a punctured circle are written out. In the final paragraph of the article On the Laplace-Beltrami operator on stratified sets composed of punctured circles and segmentsthe eigenvalues and systems of eigenfunctions on one stratified set composed of two punctured circles and a finite interval are written out.

Laplace-Beltrami operator , one-dimensional punctured sphere , well-posed problems

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Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Al-Farabi Kazakh National University, Almaty, Kazakhstan
Ghent University, Ghent, Belgium

Institute of Mathematics and Mathematical Modeling
Al-Farabi Kazakh National University
Ghent University

10 лет помогаем публиковать статьи Международный издатель

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