On the dimension of a certain measure in the plane

TL;DR

This paper investigates the Hausdorff dimension of a measure associated with solutions to specific PDEs in planar domains, extending classical results for harmonic and p-harmonic measures.

Contribution

It generalizes existing theorems on harmonic measure dimension to broader classes of PDE-related measures in the plane.

Findings

Extended Makarov's theorem to new PDE measures

Provided bounds on Hausdorff dimension for these measures

Connected measure properties to PDE solutions in planar domains

Abstract

We study the Hausdorff dimension of a measure related to a positive weak solution of a certain partial differential equation in a simply connected domain in the plane. Our work generalizes work of Lewis and coauthors when the measure is $p$ harmonic and also for $p=2$, the well known theorem of Makarov regarding the Hausdorff dimension of harmonic measure relative to a point in a simply connected domain.