Electron. J. Differential Equations, Vol. 2017 (2017), No. 99, pp. 1-11.

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

Hongjie Guo, Jiangang Qi

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 $L^1[0,1]$.

Submitted January 7, 2017. Published April 11, 2017.
Math Subject Classifications: 34B20, 34L99, 34L05.
Key Words: Sturm-Liouville problem; eigenvalue; inverse problem; Lyapunov-type inequality; extremal problem.

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Hongjie Guo
Department of Mathematics
Shandong University at Weihai
Weihai 264209, China
email: ghj790629@163.com
Jiangang Qi
Department of Mathematics
Shandong University at Weihai
Weihai 264209, China
email: qijiangang@sdu.edu.cn

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