## Tokyo Journal of Mathematics

### The Best Constant of $L^p$ Sobolev Inequality Corresponding to Dirichlet Boundary Value Problem II

#### 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.

#### Article information

Source
Tokyo J. of Math. Volume 34, Number 1 (2011), 115-133.

Dates
First available in Project Euclid: 11 August 2011

https://projecteuclid.org/euclid.tjm/1313074446

Digital Object Identifier
doi:10.3836/tjm/1313074446

Mathematical Reviews number (MathSciNet)
MR2866638

Zentralblatt MATH identifier
1237.34028

#### Citation

OSHIME, Yorimasa; WATANABE, Kohtaro. The Best Constant of $L^p$ Sobolev Inequality Corresponding to Dirichlet Boundary Value Problem II. Tokyo J. of Math. 34 (2011), no. 1, 115--133. doi:10.3836/tjm/1313074446. https://projecteuclid.org/euclid.tjm/1313074446

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