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. 
Hausdorff dimension and $\sigma$ finiteness of $p-$harmonic measures in space when $p\geq n$
In this paper we study a p harmonic measure, associated with a positive p harmonic function \hat{u} defined in an open set O, subset of R^n, and vanishing on a portion \Gamma of boundary of O. If p>n
On the Dimension of a Certain Measure Arising from a Quasilinear Elliptic Partial Differential Equation
OF DISSERTATION ON THE DIMENSION OF A CERTAIN MEASURE ARISING FROM A QUASILINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATION We study the Hausdor↵ dimension of a certain Borel measure associated to a
Failure of Fatou type theorems for solutions to PDE of $p$-Laplace type in domains with flat boundaries
Let R denote Euclidean n space and given k a positive integer let Λk ⊂ R, 1 ≤ k < n− 1, n ≥ 3, be a k-dimensional plane with 0 ∈ Λk. If p > n− k, we first study the Martin boundary problem for
The Brunn--Minkowski inequality and a Minkowski problem for 𝒜-harmonic Green's function
Abstract In this article we study two classical problems in convex geometry associated to 𝒜{\mathcal{A}}-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modelled on the p-Laplace
The Brunn-Minkowski inequality and a Minkowski problem for $\mathcal{A}$-harmonic Green's function
In this article we study two classical problems in convex geometry associated to $\mathcal{A}$-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace equation. Let
On a Bernoulli-type overdetermined free boundary problem
In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in \cite{HS1} to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs
On a Theorem of Wolff Revisited
We study $p$-harmonic functions, $ 1 0, - \infty < x < \infty \} $ and $B( 0, 1 ) = \{ z : |z| < 1 \}$. We first show for fixed $ p$, $1 < p\neq 2 < \infty$, and for all large integers $N\geq N_0$
The Brunn-Minkowski Inequality and A Minkowski Problem for Nonlinear Capacity
TLDR
This article studies two classical potential-theoretic problems in convex geometry and an inequality of Brunn-Minkowski type for a nonlinear capacity in Laplace equation and its solutions in an open set.

References

SHOWING 1-10 OF 23 REFERENCES
On the dimension of p-harmonic measure
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
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)
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
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
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
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
...
1
2
3
...