Harmonic Polynomials and Free Boundary Regularity for Harmonic Measure from Two Sides
Matthew Badger · 2011
Harmonic Polynomials and Free Boundary Regularity for Harmonic Measure from Two Sides Matthew Badger Chair of the Supervisory Committee: Professor Tatiana Toro Mathematics We use tools from geometric measure theory to catalog fine behavior of harmonic measure on a class of two-sided domains Ω ⊂ R in n-dimensional Euclidean space, with n ≥ 3. Assume the interior Ω = Ω and exterior Ω− = R Ω of Ω are NTA domains, equipped with harmonic measures ω and ω−, respectively. We prove that if ω and ω− are mutually absolutely continuous and the logarithm of their Radon-Nikodym derivative dω−/dω+ has vanishing mean oscillation, then the boundary ∂Ω can be written as a finite disjoint union of sets Γk (1 ≤ k ≤ d) with the following properties. For each Q ∈ Γk, every blow-up of ∂Ω centered atQ is the zero set of a homogeneous harmonic polynomial of degree k which separates space into two connected components; the set Γ1 of “flat points” is relatively open and locally Reifenberg flat with vanishing constant; and the setΓ2∪· · ·∪Γd of “singularities” has harmonic measure zero.