On the dimension of a certain Borel measure in the plane

In this paper we study the Hausdorff dimension of a measure μ related to a positive weak solution, u, of a certain partial differential equation in ω ∩ N where ω ⊂ C is a bounded simply connected domain and N is a neighborhood of ∂ω . u has continuous boundary value 0 on ∂ω and is a weak solution to 2 ∑ i,j=1 ∂ / ∂ xi(fηiηj(∇u(z))uxj(z))=0 in ω ∩ N. Also f(η), η ∈ C is homogeneous of degree p and ∇f is δ-monotone on C for some δ > 0. Put u ≡ 0 in N \ ω . Then μ is the unique positive finite Borel measure with support on ∂ω satisfying ∫c 〈∇f(∇u(z)),∇φ(z)〉dA = -∫∂ωφ(z)dμ for every φ ∈ C∞0(N) . Our work generalizes work of Lewis and coauthors when the above PDE is the p Laplacian (i.e., f(η) = |η|p) and also for p = 2, the well known theorem of Makarov regarding the Hausdorff dimension of harmonic measure relative to a point in ω.