$σ$-finiteness of elliptic measures for quasilinear elliptic PDE in space

In this paper we study the Hausdorff dimension of a elliptic measure $μ_{f}$ 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 $σ-$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 $\leq n-1-δ$ for $p\geq n$ for some $δ$ 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 $n=2$.

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