Minimization of the lowest positive Neumann-Dirichlet eigenvalue for general indefinite Sturm-Liouville problems

The aim of this paper is to obtain the sharp estimate for the lowest positive eigenvalue ${\lambda }_0^{ND+}$ for the general Sturm–Liouville problem$y^{″}=q\left(t\right)y+\lambda m\left(t\right)y,$ with the Neumann-Dirichlet boundary conditions, where q is a nonnegative potential and another potential m admits to change sign. First, we will study the optimal lower bound for the smallest positive eigenvalue in the measure differential equations to make our results more applicable. Second, based on the relationship between the minimization problem of the smallest positive eigenvalue for the ODE and the one for the MDE, we find the explicit optimal lower bound of the smallest positive eigenvalue for the general Sturm–Liouville equation.