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

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.

arXiv commentThe reverse Hölder inequality in the proof of (5.11) has been replaced by standard estimates in NTA domains. Minor typos corrected. References updated. Accepted for publication in Analysis and Mathematical Physics

References in corpus (4)

• The Green function estimates for strongly elliptic systems of second order
• Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains
• Tangent measures of elliptic harmonic measure and applications
• $L^2$-boundedness of gradients of single layer potentials and uniform rectifiability