Boundary Value Problems for a Family of Domains in the Sierpinski Gasket

TL;DR

This paper investigates harmonic functions on specific domains within the Sierpinski gasket, providing characterizations, boundary behavior analysis, and explicit Green's function constructions for these fractal domains.

Contribution

It introduces a new framework for understanding harmonic functions and boundary value problems on fractal domains in the Sierpinski gasket, including extension operators and Green's functions.

Findings

Characterization of harmonic functions via boundary values

Explicit Green's function construction for certain domains

Conditions for existence of extension operators

Abstract

For a family of domains in the Sierpinski gasket, we study harmonic functions of finite energy, characterizing them in terms of their boundary values, and study their normal derivatives on the boundary. We characterize those domains for which there is an extension operator for functions of finite energy. We give an explicit construction of the Green's function for these domains.