Sharp estimates of lowest positive Neumann eigenvalue for general indefinite Sturm-Liouville problems

Abstract

Given two measures μ,ν and their total variations, we study the minimization of Neumann eigenvalues for measure differential equationdy•=y(t)dμ(t)+λydν(t). By solving the infinitely dimensional minimization problem of the lowest positive Neumann eigenvalue for the measure differential equation, we obtain the optimal lower bound of the lowest positive Neumann eigenvalue for the Sturm-Liouville problemy″=q(t)y+λm(t)y, where q(t) is a nonnegative potential function and the other potential function m(t) admits to change sign.