Bounds and optimization of the minimum eigenvalue for a vibrating system

We consider the problem of the oscillation of a string fixed at one end with a mass connected to a spring at the other end. The problem of minimizing the first eigenvalue of the system subject to a fixed total mass constraint is investigated. We discuss both a Sturm-Liouville and a Stieltjes integral formulation of the boundary value problem. For small spring constant, the minimum eigenvalue for both formulations is obtained by concentrating all the mass at the end with the spring. For large spring constants, the Stieltjes eigenvalue is minimized by a point mass at an interior point. We also formulate the problem with an $\alpha$-norm constraint on the density $\rho$ in which case the optimal eigenpair satisfies a nonlinear boundary value problem. Numerical evidence suggests that this case tends to the point-mass case at the end as $\alpha\rightarrow 1+.$