Resumen de Extremal Problems of the Density for Vibrating String Equations with Applications to Gap and Ratio of Eigenvalues

Jiangang Qi, Jing Li, Bing Xie

• The present paper obtains the infimum of the densities for vibrating string equations in terms of the gap and ratio of the first two eigenvalues. The main result in this paper can be viewed as a new version of the Lyapunov inequality involving the first two eigenvalues. Some new estimates of the gap and ratio are obtained. Furthermore, the classical Lyapunov inequality can be deduced by the main conclusion.