Minimization of the first positive eigenvalue for the beam equation with indefinite weight

Abstract

In this paper, we obtain the sharp estimate of the first positive eigenvalue for the beam equation $y^{\left(4\right)}\left(t\right)=-\lambda m\left(t\right)y\left(t\right)$with Lidstone boundary condition, where weight function $m$ is allowed to change sign. We first establish a variational characterization for the first positive eigenvalue of the measure differential equation (MDE) $d{y}^{\left(3\right)}\left(t\right)=-\lambda y\left(t\right)d\mu \left(t\right),$and solve the corresponding minimization problem of the first positive eigenvalue for the MDE, where $\mu$ is a suitable measure. Then by finding the relationship between minimization problem for the first positive eigenvalue of ordinary differential equation (ODE) and that of MDE, we obtain the explicit sharp lower bound of the first positive eigenvalue for the indefinite beam equation.