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

TL;DR

This paper investigates the Hausdorff dimension of elliptic measures associated with solutions to quasilinear elliptic PDEs in space, establishing $\sigma$-finiteness and providing examples with lower Hausdorff dimension.

Contribution

It generalizes previous results for the $p$-Laplacian and extends Wolff's theorem to higher dimensions and broader PDE classes.

Findings

Elliptic measure is concentrated on a set of $\sigma$-finite $(n-1)$-dimensional Hausdorff measure for $p>n$.

The same Hausdorff measure concentration result holds for $p=n$ under boundary assumptions.

Constructed examples show the Hausdorff dimension can be less than or equal to $n-1- ext{delta}$ for certain $p$ and domain conditions.

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 $\leq n-1-\delta$ for $p\geq n$ 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 $n=2$.