BGD domains in p.c.f. self-similar sets I: boundary value problems for harmonic functions

TL;DR

This paper investigates boundary value problems for harmonic functions on p.c.f. self-similar sets, introducing flux transfer matrices to describe hitting probabilities and providing energy estimates based on boundary values.

Contribution

It introduces flux transfer matrices for boundary problems on p.c.f. sets with graph-directed boundaries, linking harmonic functions to boundary measures.

Findings

Existence of finite flux transfer matrices for boundary hitting probabilities.

Representation of harmonic functions via boundary integrals.

Two-sided energy estimates in terms of boundary data.

Abstract

We study the boundary value problems for harmonic functions on open connected subsets of post-critically finite (p.c.f.) self-similar sets, on which the Laplacian is defined through a strongly recurrent self-similar local regular Dirichlet form. For a p.c.f. self-similar set $K$, we prove that for any open connected subset $\Omega\subset K$ whose "geometric" boundary is a graph-directed self-similar set, there exists a finite number of matrices called $\textit{flux transfer matrices}$ whose products generate the hitting probability from a point in $\Omega$ to the "resistance" boundary $\partial \Omega$. The harmonic functions on $\Omega$ can be expressed by integrating functions on $\partial \Omega$ against the probability measures. Furthermore, we obtain a two-sided estimate of the energy of a harmonic function in terms of its values on $\partial \Omega$.