  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.
