The best constant of three kinds of discrete Sobolev inequalities on regular polyhedron. (English) Zbl 1288.46028

Summary: We consider three kinds of discrete Sobolev inequalities corresponding to a graph Laplacian \(\boldsymbol{A}\) on regular \(M\)-hedrons for \(M=4,6,8,12,20\). The discrete heat kernel \(\boldsymbol{H}(t)=\exp(-t\boldsymbol{A})\), the Green matrix \(\boldsymbol{G}(a)=(\boldsymbol{A}+a\boldsymbol{I})^{-1}\) and the 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 \(\boldsymbol{A}\).


46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables
35R02 PDEs on graphs and networks (ramified or polygonal spaces)
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[1] F. R. K. Chung and S.-T. Yau, Eigenvalues of graphs and Sobolev inequalities, Combin. Probab. Comput. 4 (1995), 11-25. · Zbl 0843.05073
[2] Y. Kametaka, K. Watanabe, H. Yamagishi, A. Nagai and K. Takemura, The Best Constant of Discrete Sobolev Inequality on Regular Polyhedron, Transactions of the Japan Society for Industrial and Applied Mathematics 21 (2011), 289-308 [in Japanese]. · Zbl 1232.34041
[3] A. Nagai, Y. Kametaka, H. Yamagishi, K. Takemura and K. Watanabe, Discrete Bernoulli polynomials and the best constant of discrete Sobolev inequality, Funkcial. Ekvac. 51 (2008), 307-327. · Zbl 1158.46026
[4] H. Yamagishi, A. Nagai, K. Watanabe, K. Takemura and Y. Kametaka, The best constant of discrete Sobolev inequality corresponding to a bending problem of a string, Kumamoto J. Math. 25 (2012), 1-15. · Zbl 1261.46026
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