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

Abstract

The best constant of $L^p$ Sobolev inequality for a function with Neumann boundary condition is obtained. The best constant is expressed by $L^q$ norm of $M$-th order Bernoulli polynomial. For $L^p$ Sobolev inequality, the equality holds for a function which is written by Green function with Neumann boundary value problem for $(-1)^M(d/dx)^{2M}$.