Boundary Structure and Size in Terms of Interior and Exterior Harmonic Measures in Higher Dimensions

In this work we introduce the use of powerful tools from geometric measure theory (GMT) to study problems related to the size and structure of sets of mutual absolute continuity for the harmonic measure ω of a domain Ω = Ω ⊂ R and the harmonic measure ω− of Ω−, Ω− = int(Ω) in dimension n ≥ 3. These tools come mainly from Preiss’ work (see [19]), in which he proved that if the m-density of a Radon measure μ in R exists and is positive and finite, for μ-almost every point of R, then μ is m-rectifiable; see [18] for all the relevant definitions. These techniques are combined with the blow-up analysis developed by Kenig-Toro [14], the properties of harmonic functions on non-tangentially accessible (NTA) domains [11] and the monotonicity formula of Alt-Caffarelli-Friedman [1] to obtain analogs for n ≥ 3 of some well-known results when n = 2. Let us first briefly describe some of the 2-dimensional results. Thus, let Ω ⊂