Potential theory on Sierpiński carpets with applications to uniformization
This research is motivated by the study of the geometry of fractal sets and is focused on uniformization problems: transformation of sets to canonical sets, using maps that preserve the geometry in some sense. More specifically, the main question addressed is the uniformization of planar Sierpiński carpets by square Sierpiński carpets, using methods of potential theory on carpets.
We first develop a potential theory and study harmonic functions on planar Sierpiński carpets. We introduce a discrete notion of Sobolev spaces on Sierpiński carpets and use this to define harmonic functions. Our approach differs from the classical approach of potential theory in metric spaces because it takes the ambient space that contains the carpet into account. We prove basic properties such as the existence and uniqueness of the solution to the Dirichlet problem, Liouville's theorem, Harnack's inequality, strong maximum principle, and equicontinuity of harmonic functions.
Then we utilize this notion of harmonic functions to prove a uniformization result for Sierpiński carpets. Namely, it is proved that every planar Sierpiński carpet whose peripheral disks are uniformly fat, uniform quasiballs can be mapped to a square Sierpiński carpet with a map that preserves carpet modulus. If the assumptions on the peripheral circles are strengthened to uniformly relatively separated, uniform quasicircles, then the map is a quasisymmetry. The real part of the uniformizing map is the solution of a certain Dirichlet-type problem. Then a harmonic conjugate of that map is constructed using the methods developed by Rajala.