Tokyo Journal of Mathematics

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

Yorimasa OSHIME and Kohtaro WATANABE

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1313074446

Digital Object Identifier
doi:10.3836/tjm/1313074446

Mathematical Reviews number (MathSciNet)
MR2866638

Zentralblatt MATH identifier
1237.34028

Subjects
Primary: 34B27: Green functions
Secondary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

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


Export citation

References

  • Y. Oshime, Y. Kametaka and H. Yamagishi, The best constant of $L^p$ Sobolev inequality corresponding to the periodic boundary value problem for $(d/dx)^{4m}$, Sci. Math. Jpn., 68 (2008), 313–321.
  • Y. Oshime, On the best constant for $L^{p}$ Sobolev inequality, Sci. Math. Jpn., 68 (2008), 333–344.
  • Y. Kametaka, H. Yamagishi, K. Watanabe, A. Nagai and K. Takemura, The best constant of Sobolev inequality corresponding to Dirichlet boundary value problem for $(-1)^M(d/dx)^{2M}$, Sci. Math. Jpn., 68 (2008), 299–311.
  • K. Watanabe, Y. Kametaka, A. Nagai, H. Yamagishi and K. Takemura, Symmetrization of Functions and the Best Constant of 1-DIM $L^{p}$ Sobolev Inequality, J. Ineq. and Appl., 2009 (2009), 874631.
  • S. Saitoh, Integral transforms, reproducing kernels and their applications, Addison Wesley Longman, U.K., 1997.
  • W. Richardson, Steepest decent and the least $c$ for Sobolev's inequality, Bull. London Math. Soc., 18 (1986) 478–484.
  • G. A. Kalyabin, Sharp Constant in Inequalities for Intermediate Derivatives (the Gabushin Case),Functional Analysis and it's Appl., 38 (2004), 184–191.
  • Y. Kametaka, K. Watanabe, A. Nagai and S. Pyatkov, The best constant of Sobolev inequality in an $n$ dimensional Euclidean space, Sci. Math. Jpn., 61 (2004) 295–303.
  • Y. Kametaka, H.Yamagishi, K. Watanabe, A. Nagai and K. Takemura, Riemann zeta function, Bernoulli polynomials and the best constant of Sobolev inequality, Sci. Math. Jpn., 65 (2007), 333–359.
  • A. Nagai, K. Takemura, Y. Kametaka, K. Watanabe and H. Yamagishi, Green function for boundary value problem of $2M$-th order linear ordinary equations with free boundary condition, Far East J. Appl. Math., 26 (2007), 393–406.
  • K. Watanabe, Y. Kametaka, A. Nagai, K. Takemura and H. Yamagishi, The best constant of Sobolev inequality on a bounded interval, J. Math. Anal. Appl., 340 (2008), 699–706.
  • H. Yamagishi, Y. Kametaka, A. Nagai, K. Watanabe and K. Takemura, Riemann zeta function and the best constants of five series of Sobolev inequalities, RIMS Kôkyûroku Bessatsu, to appear. \endthebibliography,