Extremal problems of the density for vibrating string equations with applications to gap and ratio of eigenvalues (Q2301726)

In this paper, the authors obtain the infimum of the densities for vibrating string equations \[ -y''=\lambda w y,\, y=y(x),\, x\in (0,1) \] together with Dirichlet-type boundary conditions \(y(0)=y(1)=0\) in terms of the gap and ratio of the first two eigenvalues. Here, the density \(w\) is a nonnegative integrable function on \([0, 1]\). As a main result of this investigation the authors prove a generalized version of the Lyapunov inequality involving the first two eigenvalues. Furthermore, they find some new estimates of the gap and ratio for the first and second eigenvalues of the above-mentioned boundary-value problem.