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

Abstract

This paper discusses the introduction of local coordinates on the circle $S^{1}$ 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 $2\pi$ -periodic function on the real axis. The Laplace-Beltrami operator on $S^{1}$ 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 segments" the eigenvalues and systems of eigenfunctions on one stratified set composed of two punctured circles and a finite interval are written out.