Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 39, 2014, 187-209

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*)> d*A*
= -\int_{\partial\Omega} \phi(*z*) d\mu

for every \phi\in *C*_{0}^{\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.

doi:10.5186/aasfm.2014.3923

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