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 Lipschitz boundaries in boundary value problems. If $$\Omega $$ ? an open domain $$\mathbb {R}^n$$ R n with mild topological conditions, can even characterize $$n-1$$ - 1 dimensional uniformly of $$\partial \Omega ? by $$A_\infty A ? -absolute continuity harmonic measure on respect surface measure. low dimension ( $$d<n-1$$ d < ), David and Mayboroda tackled one direction above characterization, i.e. proved if $$\Gamma ? d -dimensional rectifiable set, then (associated suitable degenerate elliptic operator) -absolutely continuous Hausdorff present article, use completely new approach give alternative significantly shorter proof Mayboroda’s result.