Title:  On the dimension of a certain measure arising from a quasilinear elliptic Partial Differential Equation

Abstract:  In the first part of this dissertation we will discuss the Hausdorff dimension of a measure, $\mu$, associated to a positive weak solution to a certain quasilinear elliptic PDE in simply connected domain in the plane. Our work generalizes the work of Lewis and coauthors when the PDE is p-Laplace equation and also for $p=2$ the well known theorem of Makarov regarding the Hausdorff dimension of harmonic measure relative to a point in the domain.

In the second part of the talk we will present a recent result in the study of the Hausdorff dimension of $\mu$ in space associated to a positive weak solution to the same PDE in an open subset and vanishing on a portion of the boundary of that open set. We show that this measure is concentrated on a set of $\sigma-$finite  $n-1$ dimensional Hausdorff measure for $p>n$ and the same result holds for $p=n$ with an assumption on the boundary. We also construct an example of a domain in space for which corresponding measure has Hausdorff dimension strictly less than $n-1$ for $p\geq n$.