The unique continuation property for a nonlinear equation on trees

TL;DR

This paper investigates the unique continuation property for solutions of a nonlinear game p-Laplacian equation on trees, identifying subsets where zero solutions imply trivial solutions everywhere.

Contribution

It characterizes subsets of trees with the unique continuation property for a nonlinear p-Laplacian, extending understanding of solution behavior on such structures.

Findings

Identifies conditions for subsets to have the unique continuation property.

Provides a characterization of these subsets within the tree structure.

Enhances understanding of nonlinear equations on discrete structures.

Abstract

In this paper we study the game $p-$Laplacian on a tree, that is, $$ u(x)=\frac{\alpha}2\left\{\max_{y\in \S(x)}u(y) + \min_{y\in \S(x)}u(y)\right\} + \frac{\beta}{m}\sum_{y\in \S(x)} u(y), $$ here $x$ is a vertex of the tree and $S(x)$ is the set of successors of $x$. We study the family of the subsets of the tree that enjoy the unique continuation property, that is, subsets $U$ such that $u\mid_U=0$ implies $u \equiv 0$.