Tangent measures and absolute continuity of harmonic measure

2015/07/03 by Azzam, Jonas, Mourgoglou, Mihalis · 1 citation
#28A33 #28A75 #28A78 #31A15 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1507.00926

Abstract

We show that for uniform domains Ω⊆ ℝ^d+1 whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to α-dimensional Hausdorff measure unless α≤ d. We employ a lemma that shows that at almost every nondegenerate point, we may find a tangent measure of harmonic measure whose support is the boundary of yet another uniform domain whose harmonic measure resembles the tangent measure.

Cited by

• On the dimension drop for harmonic measure on uniformly non-flat Ahlfors-David regular boundaries

Related

• Harmonic measure is absolutely continuous with respect to the Hausdorff measure on all low-dimensional uniformly rectifiable sets
• Mutual absolute continuity of interior and exterior harmonic measure\n implies rectifiability
• On a two-phase problem for harmonic measure in general domains
• Uniform domains with rectifiable boundaries and harmonic measure
• Absolute continuity of harmonic measure for domains with lower regular boundaries