Absolute Continuity of the Harmonic Measure on Low Dimensional Rectifiable Sets

Abstract

In the past decades, we learnt that uniform rectifiability is often a right candidate to go past Lipschitz boundaries in boundary value problems. If Omega is an open domain in mathbb {R}^n with mild topological conditions, we can even characterize the n-1 dimensional uniformly rectifiability of the boundary partial Omega by the A_infty -absolute continuity of the harmonic measure on partial Omega with respect to the surface measure. In low dimension (d<n-1), David and Mayboroda tackled one direction of the above characterization, i.e. proved that if Gamma is a d-dimensional uniformly rectifiable set, then the harmonic measure (associated to an suitable degenerate elliptic operator) on Gamma is A_infty -absolutely continuous with respect to the d-dimensional Hausdorff measure. In the present article, we use a completely new approach to give an alternative and significantly shorter proof of David and Mayboroda’s result.