Harmonic measure and quantitative connectivity: geometric characterization of the L^p-solvability of the Dirichlet problem. Part I

2017/12/11 by Steve Hofmann, José María Martell, Hofmann, Steve +1
Mathematics · #31B05 #35J25 #42B25 #42B37 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Graph theory and applications #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1712.03696

openalex publication_date 2017/12/11 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

Let Ω⊂ ℝⁿ⁺¹ be an open set, not necessarily connected, with an n-dimensional uniformly rectifiable boundary. We show that ∂Ω may be approximated in a "Big Pieces" sense by boundaries of chord-arc subdomains of Ω, and hence that harmonic measure for Ω is weak-A_∞ with respect to surface measure on ∂Ω, provided that Ω satisfies a certain weak version of a local John condition. Under the further assumption that Ω satisfies an interior Corkscrew condition, and combined with our previous work, and with recent work of Azzam, Mourgoglou and Tolsa, this yields a geometric characterization of domains whose harmonic measure is quantitatively absolutely continuous with respect to surface measure and hence a haracterization of the fact that the associated L^p-Dirichlet problem is solvable for some finite p.

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