Sturm–Liouville Problems Involving Distribution Weights and an Application to Optimal Problems

This paper is concerned with Sturm–Liouville problems (SLPs) with distribution weights and sets up the min–max principle and Lyapunov-type inequality for such problems. As an application, the paper solves the following optimization problems: If the first eigenvalue of a string vibration problem is known, what is the minimal total mass and by which distribution of weight is it attained; if both the first eigenvalue and the total mass are known, what is the corresponding results on the string mass? The vibration problem leads to a SLP with the spectral parameter in both the equation and the boundary conditions. Our main method is to incorporate this problem into the framework of classical SLPs with weights in an appropriate space by transforming it into the one with distribution weight, which provides a different idea for the investigation of the SLPs with spectral parameter in boundary condition.