Absolute Continuity of the Harmonic Measure on Low Dimensional Rectifiable Sets

Abstract

AbstractIn the past decades, we learnt that uniform rectifiability is often a right candidate to go past Lipschitz boundaries in boundary value problems. If $$\Omega $$ Ω is an open domain in $$\mathbb {R}^n$$ R n with mild topological conditions, we can even characterize the $$n-1$$ n - 1 dimensional uniformly rectifiability of the boundary $$\partial \Omega $$ ∂ Ω by the $$A_\infty $$ A ∞ -absolute continuity of the harmonic measure on $$\partial \Omega $$ ∂ Ω with respect to the surface measure. In low dimension ($$d<n-1$$ d < n - 1 ), David and Mayboroda tackled one direction of the above characterization, i.e. proved that if $$\Gamma $$ Γ is a d-dimensional uniformly rectifiable set, then the harmonic measure (associated to an suitable degenerate elliptic operator) on $$\Gamma $$ Γ is $$A_\infty $$ A ∞ -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.