We clarified the variational meaning of the special values $\zeta(2M)\
(M=1,2,3,\cdots)$ of Riemann zeta function $\zeta(s)$. These are essentially the
best constant of Sobolev inequality. In the background we consider
Dirichlet-Neumann boundary value problem for a differential operator
$(-1)^M(d/dx)^{2M}$. Its Green function is found and expressed in terms of the
well-known Bernoulli polynomial. The supremum of the diagonal value of Green
function is equal to the best constant for corresponding Sobolev inequality.
Discrete version of the corresponding Sobolev inequality is also presented.
Publié le : 2009-11-15
Classification:
Sobolev inequality,
best constant,
Green function,
reproducing kernel,
Bernoulli polynomial,
Riemann zeta function,
34B27,
46E35
@article{1257544215,
author = {Yamagishi, Hiroyuki},
title = {The best constant of Sobolev inequality corresponding to Dirichlet-Neumann
boundary value problem for $(-1)^M(d/dx)^{2M}$},
journal = {Hiroshima Math. J.},
volume = {39},
number = {1},
year = {2009},
pages = { 421-442},
language = {en},
url = {http://dml.mathdoc.fr/item/1257544215}
}
Yamagishi, Hiroyuki. The best constant of Sobolev inequality corresponding to Dirichlet-Neumann
boundary value problem for $(-1)^M(d/dx)^{2M}$. Hiroshima Math. J., Tome 39 (2009) no. 1, pp. 421-442. http://gdmltest.u-ga.fr/item/1257544215/