The Best Constant of Sobolev Inequality Corresponding to a Bending Problem of a Beam under Tension on an Elastic Foundation(Theory,Applied Integrable Systems,<Special Issue>Joint Symposium of JSIAM Activity Groups 2009)

Abstract

We consider 4 two-point boundary value problems for bending of a beam supported by uniformly distributed springs with spring constant q>0 on a fixed floor under tension p>0. The tension is relatively strong, that is (p/2)^2>q. We have treated clamped, Dirichlet, Neumann and free boundary conditions and found their Green functions. As an application, we have found the best constants of the corresponding Sobolev inequality, which are equal to the maximum of diagonal values of Green functions. Putting p=a^2+b^2, q=a^2b^2, a>b>0, we have found explicit forms of the best constants in terms of a, b.