Seminar

One hundred years ago F. and M. Riesz established that harmonic measure is absolutely continuous with respect to arc length measure, for any simply connected domain in the complex plane with a rectifi- able boundary. Roughly speaking, under some background topological assumptions, some “regularity” or “flatness” of the boundary implies that harmonic measure behaves well with respect to the arc length measure. In this talk we will present a quantitative converse in higher dimensions which is also valid for a class of variable coefficient operators: if the domain satisfies quantitative topological conditions and harmonic/elliptic measure behaves well with respect to surface measure in some scale-invariant way, then the boundary is quantitatively “regular” or “flat”. More precisely, let Ω ⊂ R n+1 , n ≥ 2, be an open set with Ahlfors-David regular boundary and assume that Ω satis- fies the Harnack Chain condition plus an interior (but not exterior) Corkscrew condition (these are quantitative or scale-invariant versions of path-connectedness and openness respectively). Let L be a real second order divergence form elliptic operator such that the first order derivatives of the coefficients satisfy a Carleson measure condition. Under these background hypotheses, we obtain that if the associated elliptic measures for L and L > are absolute continuous with respect to surface measure on ∂Ω, with scale invariant higher integrability of the Poisson kernels, then Ω has exterior corkscrews. Hence, Ω is indeed an NTA domain and, since ∂Ω is Ahlfors-David regular, it follows that ∂Ω is quantitatively rectifiable.

Joint work with S. Hofmann and T. Toro.