## 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 $$\boldsymbol{A}$$.

### MSC:

 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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### References:

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