$\sigma$-finiteness of elliptic measures for quasilinear elliptic PDE in space

Abstract

In this paper we study the Hausdorff dimension of a elliptic measure ${\mu }_f$ in space associated to a positive weak solution to a certain quasilinear elliptic 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 the corresponding measure has Hausdorff dimension $\le n-1-\delta$ for $p\ge n$ for some $\delta$ which depends on various constants including $p$. The first result generalizes the authors previous work when the PDE is the $p-$Laplacian and the second result generalizes the well known theorem of Wolff when $p=2$ and $n=2$.

arXiv comment35 pages, 3 figures, shortened title and some minor changes