The Best Constant of Sobolev Inequality Corresponding to a Bending Problem of a Beam under Tension on an Elastic Foundation II

Abstract

Abstract. We consider 2-point boundary value problem (BVP) for 4-th order ordinary differential equation with Periodic, Dirichlet, Dirichlet-Neumann and Neumann boundary conditions. A solution of BVP represents bending of a beam on an elastic foundation under a tension. BVP has an unique solution which is described using Green function. As its application, we obtain the best constant of the corresponding Sobolev inequality is equal to the maximum of diagonal value of Green function. The engineering meaning of Sobolev inequality is that the maximum displacement of bending can be estimated by the constant multiple of its potential energy.