The best constant of Sobolev inequality corresponding to Dirichlet-Neumann boundary value problem for $(-1)^M(d/dx)^{2M}$

Abstract

We clarified the variational meaning of the special values $\zeta(2M)\ (M=1,2,3,\cdots)$ of Riemann zeta function $\zeta(s)$. These are essentially the best constant of Sobolev inequality. In the background we consider Dirichlet-Neumann boundary value problem for a differential operator $(-1)^M(d/dx)^{2M}$. Its Green function is found and expressed in terms of the well-known Bernoulli polynomial. The supremum of the diagonal value of Green function is equal to the best constant for corresponding Sobolev inequality. Discrete version of the corresponding Sobolev inequality is also presented.