The best constant of Sobolev inequality corresponding to clamped-free boundary value problem for (-1)M(d/dx)2M

Abstract

Green function of the clamped-free boundary value problem for (-1)^M(d/dx)^2M on the interval (-1,1) is obtained. Its Green function is a reproducing kernel for a suitable set of Hilbert space and an inner product. By using the fact, the best constant of Sobolev inequality corresponding to this boundary value problem is obtained as a function of M. The best constant is the maximal value of the diagonal value G(y,y) of Green function G(x,y).