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December 2014 The Best Constant of $L^p$ Sobolev Inequality Including $j$-th Derivative Corresponding to Periodic and Neumann Boundary Value Problem for $(-1)^M(d/dx)^{2M}$
Kohtaro WATANABE, Hiroyuki YAMAGISHI
Tokyo J. Math. 37(2): 485-501 (December 2014). DOI: 10.3836/tjm/1422452804

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 $j=1,2,3,\ldots,M-1$.

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Kohtaro WATANABE. Hiroyuki YAMAGISHI. "The Best Constant of $L^p$ Sobolev Inequality Including $j$-th Derivative Corresponding to Periodic and Neumann Boundary Value Problem for $(-1)^M(d/dx)^{2M}$." Tokyo J. Math. 37 (2) 485 - 501, December 2014. https://doi.org/10.3836/tjm/1422452804

Information

Published: December 2014
First available in Project Euclid: 28 January 2015

zbMATH: 1296.62018
MathSciNet: MR3304692
Digital Object Identifier: 10.3836/tjm/1422452804

Subjects:
Primary: 34B27
Secondary: 46E35

Rights: Copyright © 2014 Publication Committee for the Tokyo Journal of Mathematics

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Vol.37 • No. 2 • December 2014
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