Illinois Journal of Mathematics

The spectrum of the $p$-Laplacian and $p$-harmonic morphisms on graphs

Hiroshi Takeuchi

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For a real number $p$ with $1<p<\infty$ we consider the spectrum of the $p$-Laplacian on graphs, $p$-harmonic morphisms between two graphs, and estimates for the solutions of $p$-Laplace equations on graphs. More precisely, we prove a Cheeger type inequality and a Brooks type inequality for infinite graphs. We also define $p$-harmonic morphisms and horizontally conformal maps between two graphs and prove that these two concepts are equivalent. Finally, we give some estimates for the solutions of $p$-Laplace equations, which coincide with Green kernels in the case $p=2$.

Article information

Illinois J. Math., Volume 47, Number 3 (2003), 939-955.

First available in Project Euclid: 13 November 2009

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Zentralblatt MATH identifier

Primary: 31C20: Discrete potential theory and numerical methods
Secondary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 31C05: Harmonic, subharmonic, superharmonic functions 52B60: Isoperimetric problems for polytopes


Takeuchi, Hiroshi. The spectrum of the $p$-Laplacian and $p$-harmonic morphisms on graphs. Illinois J. Math. 47 (2003), no. 3, 939--955. doi:10.1215/ijm/1258138202.

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