Extremal norm for potentials of Sturm-Liouville eigenvalue problems with separated boundary conditions

Abstract

For the n-th eigenvalue of a Sturm-Liouville eigenvalue problem with separated boundary conditions, we express the infimum of the $L^1[0,1]$ norm of potentials, in terms of a parameter $\lambda$ and the boundary conditions. Also we indicate where the infimum can be attained. As an application, we obtain the extremum of the n-th eigenvalue of a problem for potentials on a sphere in $L^1[0,1]$.

Keywords

• Sturm-Liouville problem
• eigenvalue
• inverse problem
• Lyapunov-type inequality
• extremal problem