Optimal lower bound for the first eigenvalue of the fourth order equation

In this paper we will find optimal lower bound for the first eigenvalue of the fourth order equation with integrable potentials when the L1L1 norm of potentials is known. We establish the minimization characterization for the first eigenvalue of the measure differential equation, which plays an important role in the extremal problem of ordinary differential equation. The conclusion of this paper will illustrate a new and very interesting phenomenon that the minimizing measures will no longer be located at the center of the interval when the norm is large enough.