# On the dimension of a certain measure in the plane

@article{Akman2013OnTD,
title={On the dimension of a certain measure in the plane},
author={Murat Akman},
journal={arXiv: Analysis of PDEs},
year={2013}
}
• M. Akman
• Published 24 January 2013
• Mathematics
• arXiv: Analysis of PDEs
We study the Hausdorff dimension of a measure related to a positive weak solution of a certain partial differential equation in a simply connected domain in the plane. Our work generalizes work of Lewis and coauthors when the measure is $p$ harmonic and also for $p=2$, the well known theorem of Makarov regarding the Hausdorff dimension of harmonic measure relative to a point in a simply connected domain.
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## References

SHOWING 1-10 OF 23 REFERENCES
On the dimension of p-harmonic measure
• Mathematics
• 2005
In this paper we study the dimension of a measure associated with a positive p-harmonic function which vanishes on the boundary of a certain domain.
p Harmonic Measure in Simply Connected Domains Revisited
@Let be a bounded simply connected domain in the complex plane, C. Let N be a neighborhood of @, let p be xed, 1 < p <1; and let ^ be a positive weak solution to the p Laplace equation in \N: Assume
p Harmonic Measure in Simply Connected Domains
— Let Ω be a bounded simply connected domain in the complex plane, C. Let N be a neighborhood of ∂Ω, let p be fixed, 1 < p < ∞, and let û be a positive weak solution to the p Laplace equation in Ω ∩
Quasiconformal geometry of monotone mappings
This paper concerns a class of monotone mappings, in a Hilbert space, that can be viewed as a nonlinear version of the class of positive invertible operators. Such mappings are proved to be open,
Hausdorff dimension of harmonic measures in the plane
• Mathematics
• 1988
for all continuous u: aQ--*R, where /~ is the Perron solution of the Dirichlet problem with boundary values u. We must assume here that E has positive capacity, but not that 92 is regular for the
Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Pms-48)
• Mathematics
• 2009
This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis.
Uniform limits of certain A-harmonic functions with applications to quasiregular mappings
• Mathematics
• 1991
Let u1, 1t2t...ru* be nonconstant uniform limits (on compact subsets) of ,4 harmonic functions in {c : lrl < n} C R' where .4 satisfies certain elliptic structure conditions.
Uniformly fat sets
In this paper we study closed sets E which are "locally uniformly fat" with respect to a certain nonlinear Riesz capacity. We show that E is actually "locally uniformly fat" with respect to a weaker
Nonlinear Potential Theory of Degenerate Elliptic Equations
• Mathematics
• 1993
Introduction. 1: Weighted Sobolev spaces. 2: Capacity. 3: Supersolutions and the obstacle problem. 4: Refined Sobolev spaces. 5: Variational integrals. 6: A-harmonic functions. 7: A superharmonic
On the logarithm of the minimizing integrand for certain variational problems in two dimensions
• Mathematics
• 2012
AbstractLet f be a smooth convex homogeneous function of degreep, 1 < p < ∞, on $${\mathbb{C} \setminus \{0\}.}$$ We show that if u is a minimizer for the functional whose integrand is {f(\nabla v