# 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} }

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.

## 10 Citations

Hausdorff dimension and $\sigma$ finiteness of $p-$harmonic measures in space when $p\geq n$

- Mathematics
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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…

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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

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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

- MathematicsAdvances in Calculus of Variations
- 2019

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

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- 2018

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

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- 2019

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

- Mathematics
- 2020

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

- Computer ScienceMemoirs of the American Mathematical Society
- 2022

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.

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