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June 2013 The Best Constant of Three Kinds of Discrete Sobolev Inequalities on Regular Polyhedron
Yoshinori KAMETAKA, Atsushi NAGAI, Kazuo TAKEMURA, Kohtaro WATANABE, Hiroyuki YAMAGISHI
Tokyo J. Math. 36(1): 253-268 (June 2013). DOI: 10.3836/tjm/1374497523

Abstract

We consider three kinds of discrete Sobolev inequalities corresponding to a graph Laplacian $\boldsymbol{A}$ on regular $M$-hedron for $M=4,6,8,12,20$. Discrete heat kernel $\boldsymbol{H}(t)=\exp(-t\boldsymbol{A})$, Green matrix $\boldsymbol{G}(a)=(\boldsymbol{A}+a\boldsymbol{I})^{-1}$ and pseudo Green matrix $\boldsymbol{G}_*$ are obtained and investigated in a detailed manner. The best constants of the inequalities are given by means of eigenvalues of $\boldsymbol{A}$.

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Yoshinori KAMETAKA. Atsushi NAGAI. Kazuo TAKEMURA. Kohtaro WATANABE. Hiroyuki YAMAGISHI. "The Best Constant of Three Kinds of Discrete Sobolev Inequalities on Regular Polyhedron." Tokyo J. Math. 36 (1) 253 - 268, June 2013. https://doi.org/10.3836/tjm/1374497523

Information

Published: June 2013
First available in Project Euclid: 22 July 2013

zbMATH: 1288.46028
MathSciNet: MR3112387
Digital Object Identifier: 10.3836/tjm/1374497523

Subjects:
Primary: 46E39
Secondary: 35K08

Rights: Copyright © 2013 Publication Committee for the Tokyo Journal of Mathematics

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Vol.36 • No. 1 • June 2013
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