The Best Constant of $L^p$ Sobolev Inequality Including $j$-th Derivative Corresponding to Periodic and Neumann Boundary Value Problem for $(-1)^M(d/dx)^{2M}$

Abstract

In this paper, we study the best constant of $L^p$ Sobolev inequality including $j$-th derivative: \begin{align*} \sup_{0\leq y\leq 1}\big\vert u^{(j)}(y)\big\vert \leq C \Bigg(\int_0^1 \big|\,u^{(M)}(x)\,\big|^p dx \Bigg)^{1/p}\,, \end{align*} where $u$ is an element of Sobolev space with periodic or Neumann boundary condition. The best constant can be expressed by $L^q$ norm of Bernoulli polynomial. In [1, 4], the best constant of the above inequality was obtained for the case of $1<p<\infty$ and $j=0$. This paper extends the results of [1, 4] to $j=1,2,3,\ldots,M-1$.