Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 187-209

ON THE DIMENSION OF A CERTAIN MEASURE IN THE PLANE

Murat Akman

University of Kentucky, Mathematics Department
Lexington, Kentucky, 40506, U.S.A.; murat.akman 'at' uky.edu

Abstract. 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 \cap N where \Omega \subset C is a bounded simply connected domain and N is a neighborhood of \partial\Omega. u has continuous boundary value 0 on \partial\Omega and is a weak solution to

\sum_{i,j=1}^{2}\frac{\partial}{\partial x_{i}}(f_{\eta_{i}\eta_{j}} (\nabla u(z)) u_{x_j}}(z)) = 0 in \Omega \cap N.

Also f(\eta), \eta \in C is homogeneous of degree p and \nabla f is \delta-monotone on C for some \delta > 0. Put u \equiv 0 in N \setminus \Omega. Then \mu is the unique positive finite Borel measure with support on \partial\Omega satisfying

\int_C <\nabla f(\nabla u(z)), \nabla\phi(z)> dA = -\int_{\partial\Omega} \phi(z) d\mu

for every \phi\in C₀^\infty(N). Our work generalizes work of Lewis and coauthors when the above PDE is the p Laplacian (i.e., f(\eta) = |\eta|^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.

2010 Mathematics Subject Classification: Primary 35J25, 37F35.

Key words: Hausdorff dimension, dimension of a measure, p-harmonic measure.

Reference to this article: M. Akman: On the dimension of a certain measure in the plane. Ann. Acad. Sci. Fenn. Math. 39 (2014), 187-209.

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doi:10.5186/aasfm.2014.3923