σ-Finiteness of elliptic measures for quasilinear elliptic PDE in space

In this paper we study the Hausdorff dimension of a elliptic measure μf 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 ≤n−1−δ for p≥n for some δ which depends on various constants including p. The first result generalizes the authors previous work in [3] when the PDE is the p-Laplacian and the second result generalizes the well known theorem of Wolff in [24] when p=2 and n=2.