Minimization of first eigenvalue of fourth order measure differential equation

What is the smallest axial compressive load to cause the beam buckle when the total bounded elastic force is given? Mathematically,this corresponds to a minimization problem for the first eigenvalue of the fourth order measure differential equation subjects to Dirichletboundary condition. In this paper, we establish a variational characterization for the first eigenvalue of thefourth order measure differential equation; we then give a complete solution to a minimization problem on the first eigenvalue; so thatwe can answer the above physical problem in a mathematical way.