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June 2011 The Best Constant of $L^p$ Sobolev Inequality Corresponding to Dirichlet Boundary Value Problem II
Yorimasa OSHIME, Kohtaro WATANABE
Tokyo J. Math. 34(1): 115-133 (June 2011). DOI: 10.3836/tjm/1313074446

Abstract

Let $M = 2m$ $(m = 1,2,\ldots)$. In [1] the best constant of $L^p$ Sobolev inequality $$ \sup_{-1\leq x\leq 1}\vert u(x)\vert\leq C\Biggl( \int_{-1}^{1}\vert u^{(M)}(x)\vert^{p}dx\Biggr)^{1/p} $$ was obtained for $u$ satisfying $u, u^{(M)} \in L^{p}(-1,1)$ and $u^{(2i)}(\pm 1) = 0$ $(0\leq i\leq [(M-1)/2])$. On the other hand, for the case $M$ is odd, up to now, only the case $M =1$ was treated for technical difficulty; see [2]. This paper treats the case $M =3$ with different two approach, one is based on the property of the function associated with certain Green function and another is on the property of function space. For the latter approach, symmetrizations of functions play an important role.

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Yorimasa OSHIME. Kohtaro WATANABE. "The Best Constant of $L^p$ Sobolev Inequality Corresponding to Dirichlet Boundary Value Problem II." Tokyo J. Math. 34 (1) 115 - 133, June 2011. https://doi.org/10.3836/tjm/1313074446

Information

Published: June 2011
First available in Project Euclid: 11 August 2011

zbMATH: 1237.34028
MathSciNet: MR2866638
Digital Object Identifier: 10.3836/tjm/1313074446

Subjects:
Primary: 34B27
Secondary: 46E35

Rights: Copyright © 2011 Publication Committee for the Tokyo Journal of Mathematics

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Vol.34 • No. 1 • June 2011
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