# MAXIMUM PRINCIPLE AND COMPARISON PRINCIPLE OF p-HARMONIC FUNCTIONS VIA p-HARMONIC BOUNDARY OF GRAPHS

• Published : 2012.11.30
• 34 7

#### Abstract

We prove the maximum principle and the comparison principle of $p$-harmonic functions via $p$-harmonic boundary of graphs. By applying the comparison principle, we also prove the solvability of the boundary value problem of $p$-harmonic functions via $p$-harmonic boundary of graphs.

#### Keywords

maximum principle;comparison principle;p-harmonic function;p-harmonic boundary;boundary value problem

#### References

1. I. Holopainen and P. M. Soardi, p-harmonic functions on graphs and manifolds, Manuscripta Math. 94 (1997), no. 1, 95-110. https://doi.org/10.1007/BF02677841
2. J.-H. Kim and S.-Y. Chung, Comparison principles for the p-Laplacian on nonlinear networks, J. Difference Equ. Appl. 16 (2010), no. 10, 1151-1163. https://doi.org/10.1080/10236190902787633
3. M. Yamasaki, Ideal boundary limit of discrete Dirichlet functions, Hiroshima Math. J. 16 (1986), no. 2, 353-360.

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)