The best constant of the m-th Sobolev inequality corresponding to the clamped boundary value problem for (−1)^M (d/dx)^2M (m = 0, 1, 2)

Abstract

The m-th Sobolev inequality is that the square of the maximum of the derivative of the m-th order is estimated from the above by the constant multiple of the Sobolev norm. In the background, we have the ordinary differential equation of the 2M-th order simple type with the clamped boundary condition. The solution of this equation is given by the Green function. The best constant of the m-th Sobolev inequality is also expressed by using the Green function.