ON THE DIMENSION OF A CERTAIN MEASURE IN THE PLANE

In this paper we study the Hausdorff dimension of a measure mu related to a positive weak solution, u, of a certain partial differential equation in Omega boolean AND N where Omega subset of C is a bounded simply connected domain and N is a neighborhood of partial derivative Omega u has continuous boundary value 0 on partial derivative Omega and is a weak solution to Sigma(2)(i,j=1)partial derivative/partial derivative x(i) (f(eta z eta j) (del u(z)) u(xj) (z)) = 0 in Omega boolean AND N. Also f(eta), eta is an element of C is homogeneous of degree p and del f is delta-monotone on C for some delta > 0. Put u 0 in N \ Omega Then mu is the unique positive finite Borel measure with support on partial derivative Omega satisfying integral(c) <del f (del u(z)), del phi(z))> dA = -integral(partial derivative Omega) phi(z) d mu for every phi is an element of (C0N)-N-infinity(). Our work generalizes work of Lewis and coauthors when the above PDE is the p Laplacian (i.e., f(eta) = vertical bar eta vertical bar(p)) and also for p = 2, the well known theorem of Makarov regarding the Hausdorff dimension of harmonic measure relative to a point in Omega.