Rectifiability and Harmonic Measure

Tuesday, April 5, 2016 - 4:00pm
111 MSB
Simon Bortz (Advisor: Prof. Steve Hofmann)

Abstract: This dissertation is concerned with the interplay of the geometry of the boundary of a given domain (or open set) and its harmonic measure. First we prove structure theorems for uniformly rectifiable sets, which in turn yield information about harmonic measure. Then we turn our attention to sets that are merely rectifiable and which satisfy some additional hypotheses; we again prove structure results, which give that surface measure is absolutely continuous with respect to harmonic measure. Finally, we look in the other direction; we show that (under certain hypotheses) that weak regularity of the Poisson kernel on both sides of a domain yields the same regularity for the unit outer normal to the domain.