Asymptotic analysis of the first eigenvalue for the Sturm-Liouville problems with coupled boundary conditions

Abstract

The present paper is concerned with the asymptotic behavior of the first eigenvalue on the jump set of the Sturm-Liouville eigenvalue problems with coupled boundary conditions. By transforming the original problem into two problems with separated boundary conditions, we obtain the asymptotic formula of the first eigenvalue and the first normalized eigenfunction in terms of the coefficients in the equations. Particularly, we prove that the unique zero of the first eigenfunction tends to the zero of the first eigenfunction of the corresponding Laplace problem. The results indicate that the asymptotic behavior is only related to the values of the coefficients in the neighborhood of the endpoints.