Well-posed problems for the Laplace-Beltrami operator on a stratified set consisting of punctured circles and segments

Abstract

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.