The dimension of planar elliptic measures arising from Lipschitz matrices in Reifenberg flat domains

TL;DR

This paper proves that for planar Reifenberg flat domains with small constants, the elliptic measure associated with Lipschitz elliptic operators has Hausdorff dimension at most 1, extending previous harmonic measure results.

Contribution

It establishes a dimension bound for elliptic measures in Reifenberg flat domains with Lipschitz coefficients, generalizing Wolff's harmonic measure results.

Findings

Elliptic measure has Hausdorff dimension at most 1 in specified domains.

Existence of a boundary subset with full elliptic measure and finite one-dimensional Hausdorff measure.

Extension of Wolff's harmonic measure results to more general elliptic operators.

Abstract

In this paper we show that, given a planar Reifenberg flat domain with small constant and a divergence form operator associated to a real (not necessarily symmetric) uniformly elliptic matrix with Lipschitz coefficients, the Hausdorff dimension of its elliptic measure is at most 1. More precisely, we prove that there exists a subset of the boundary with full elliptic measure and with $\sigma$-finite one-dimensional Hausdorff measure. For Reifenberg flat domains, this result extends a previous work of Thomas H. Wolff for the harmonic measure.