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On pseudospheres that are quasispheres.

John L. Lewis; Andrew Vogel — 2001

Revista Matemática Iberoamericana

We construct bounded domains D not equal to a ball in n ≥ 3 dimensional Euclidean space, R, for which ∂D is homeomorphic to a sphere under a quasiconformal mapping of R and such that n - 1 dimensional Hausdorff measure equals harmonic measure on ∂D.

On pseudospheres.

John L. Lewis; Andrew Vogel — 1991

Revista Matemática Iberoamericana

On the dimension of $p$ -harmonic measure in space

John L. Lewis; Kaj Nyström; Andrew Vogel — 2013

Journal of the European Mathematical Society

Let $Ω \subset ℝ^{n}, n \geq 3$ , and let $p, 1 < p < \infty, p \neq 2$ , be given. In this paper we study the dimension of $p$ -harmonic measures that arise from non-negative solutions to the $p$ -Laplace equation, vanishing on a portion of $\partial Ω$ , in the setting of $δ$ -Reifenberg flat domains. We prove, for $p \geq n$ , that there exists $\tilde{δ} = \tilde{δ} (p, n) > 0$ small such that if $Ω$ is a $δ$ -Reifenberg flat domain with $δ < \tilde{δ}$ , then $p$ -harmonic measure is concentrated on a set of $σ$ -finite $H^{n - 1}$ -measure. We prove, for $p \geq n$ , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of $p$ -harmonic measure...

Symmetry theorems and uniform rectifiability.

Lewis, John L.; Vogel, Andrew L. — 2007

Boundary Value Problems [electronic only]

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