Free boundary regularity for harmonic measures and Poisson kernels

One of the basic aims of this paper is to study the relationship between the geometry of \hypersurface like" subsets of Euclidean space and the properties of the measures they support. In this context we show that certain doubling properties of a measure determine the geometry of its support. A Radon measure is said to be doubling with constant C if C times the measure of the ball of radius r centered on the support is greater than the measure of the ball of radius 2r and the same center. We prove that if the doubling constant of a measure on R n+1 is close to the doubling constant of the ndimensional Lebesgue measure then its support is well approximated by ndimensional a‐ne spaces, provided that the support is relatively ∞at to start with. Primarily we consider sets which are boundaries of domains in R n+1 . The n-dimensional Hausdorfi measure may not be deflned on the boundary of a domain in R n+1 . Thus we turn our attention to the harmonic measure which is well behaved under minor assumptions (see Section 3). We obtain a new characterization of locally ∞at domains in terms of the doubling properties of their harmonic measure (see Section 3). Along these lines we investigate how the \weak" regularity of the Poisson kernel of a domain determines the geometry of its boundary. Sections 5 and 6 pursue this goal, as in Alt and Cafiarelli’s work (see [AC], [C1], [C2]), and also Jerison’s [J]. In both cases the goal is to prove that, under the appropriate technical conditions at \∞at points" of the boundary, the oscillation of the Poisson kernel controls the oscillation of the unit normal vector. The difierence between our work and the work in [AC] is that we measure the oscillation in an integral sense (BMO estimates)