Absolute Continuity of the Harmonic Measure on Low Dimensional Rectifiable Sets

Abstract

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 $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> is an open domain in $$\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> with mild topological conditions, we can even characterize the $$n-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> dimensional uniformly rectifiability of the boundary $$\partial \Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> by the $$A_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> -absolute continuity of the harmonic measure on $$\partial \Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> with respect to the surface measure. In low dimension ( $$d&lt;n-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ), David and Mayboroda tackled one direction of the above characterization, i.e. proved that if $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> is a d -dimensional uniformly rectifiable set, then the harmonic measure (associated to an suitable degenerate elliptic operator) on $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> is $$A_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> -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.

Locations

• Journal of Geometric Analysis - View - PDF