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

Abstract

We prove the maximum principle and the comparison prin- ciple 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 bound- ary of graphs. The maximum principle and the comparison principle are interesting topics in studying the behavior of solutions of some equations. In particular, the max- imum principle and the comparison principle enable us to control the behavior of solutions of a certain equation by the boundary data of the solutions. Such a study has been developed for several equations. For instance, Holopainen and Soardi (1) proved the maximum principle and the comparison principle of p-harmonic functions on finite subsets of graphs. Later, Kim and Chung (2) generalized the result of Holopainen and Soardi by extending the result into infinite subsets. However, those results are related to the real boundary, not the ideal boundary like infinite boundary. In fact, in the case that a given graph has no boundary, we need to lean on the ideal boundary in controlling the behavior of solutions at infinity of the graph. In line with the viewpoint, we rebuild the maximum principle and the comparison principle to control the whole behavior of p-harmonic functions on the graphs. In this paper, we newly suggest the maximum principle and the comparison principle of p-harmonic functions on a graph, in such a way that they are described in terms of the p-harmonic boundary as follows: