Boundary Value Problems for harmonic functions on domains in Sierpinski gaskets

TL;DR

This paper develops explicit Poisson integral formulas for harmonic functions on specific domains within Sierpinski gaskets with Cantor set boundaries, addressing open problems in fractal boundary value theory.

Contribution

It introduces explicit formulas and characterizations for harmonic functions on fractal domains in Sierpinski gaskets, advancing boundary value problem solutions in fractal analysis.

Findings

Explicit Poisson integral formulas derived for fractal domains.

Characterization of finite energy harmonic functions on these domains.

Energy estimates relating boundary data to harmonic functions.

Abstract

We study boundary value problems for harmonic functions on certain domains in the level-$l$ Sierpinski gaskets $\mathcal{SG}_l$($l\geq 2$) whose boundaries are Cantor sets. We give explicit analogues of the Poisson integral formula to recover harmonic functions from their boundary values. Three types of domains, the left half domain of $\mathcal{SG}_l$ and the upper and lower domains generated by horizontal cuts of $\mathcal{SG}_l$ are considered at present. We characterize harmonic functions of finite energy and obtain their energy estimates in terms of their boundary values. This paper settles several open problems raised in previous work.