Minkowski-type Estimates on the Quantitative Strata of the Generalized Critical set of Green's functions for Two-Sided NTA Domains arising from a Free-Boundary Problem for Harmonic Measure

Abstract

In this work, we prove three things. The main results are two different results on Minkowski-type estimates on the quantitative strata of the generalized critical set of Green's functions of 2-Sided NTA domains arising from a free-boundary problem for harmonic measure. The first uses simpler techniques and obtains weaker results. The second employs much more complicated machinery and obtains a much stronger result which completely subsumes the results of the first approach. The third result contained in this work is the construction of two families of rectifiable sets which fail to be uniformly rectifiable as dramatically as possible which still retaining nice topological and measure theoretic properties. This third result is independent of the first two, and represents joint work with Max Goering.