Harmonic measure and quantitative connectivity: geometric characterization of the L(p) solvability of the Dirichlet problem
Abstract
It is wellknown that quantitative, scale invariant absolute continuity (more precisely, the weak$A_\infty$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $ \Omega\subset \mathbb{R}^{n+1}$ with AhlforsDavid regular boundary, is equivalent to the solvability of the Dirichlet problem in $\Omega$, with data in $L^p(\partial\Omega)$ for some $p<\infty$. In this paper, we give a geometric characterization of the weak$A_\infty$ property, of harmonic measure, and hence of solvability of the $L^p$ Dirichlet problem for some finite $p$. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with AhlforsDavid regularity of the boundary) that are natural, and in a certain sense optimal: we provide counterexamples in the absence of either of them (or even one of the two, upper or lower, AhlforsDavid bounds); moreover, the examples show that the upper and lower AhlforsDavid bounds are each quantitatively sharp.
 Publication:

Inventiones Mathematicae
 Pub Date:
 December 2020
 DOI:
 10.1007/s00222020009845
 arXiv:
 arXiv:1907.07102
 Bibcode:
 2020InMat.222..881A
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Analysis of PDEs;
 31B05;
 35J25;
 42B25;
 42B37
 EPrint:
 This paper is a combination of arXiv:1712.03696 and arXiv:1803.07975 To appear in Invent. Math