Estimates for Nonlinear Harmonic Measures on Trees

Abstract

In this paper we give some estimates for nonlinear harmonic measures on trees. In particular, we estimate in terms of the size of a set $D$ the value at the origin of the solution to $u(x)=F((x,0),\dots ,(x,m-1))$ for every $x\in {\mathbb{T}}_m,$ a directed tree with $m$ branches with initial datum $f+{\chi }_D$. Here $F$ is an averaging operator on ${\mathbb{R}}^m$, $x$ is a vertex of a directed tree ${\mathbb{T}}_m$ with regular $m$-branching and $(x,i)$ denotes a successor of that vertex for $0\le i\le m-1$.

19 pages, 1 figure

References in corpus (2)

• Asymptotic statistical characterizations of p-harmonic functions of two variables
• The unique continuation property for a nonlinear equation on trees

Cited by in corpus (3)

• Dirichlet-to-Neumann maps on Trees
• Systems involving mean value formulas on trees
• The two membranes problem in a regular tree