John L. Lewis; Kaj Nyström; Pietro Poggi-Corradini
$p$ Harmonic Measure in Simply Connected Domains
(Mesure $p$ harmonique dans les régions simplement connexes)
Annales de l'institut Fourier, 61 no. 2 (2011), p. 689-715, doi: 10.5802/aif.2626
Article PDF | Reviews MR 2895070 | Zbl 1241.35071
Class. Math.: 35J25, 35J70
Keywords: Harmonic function, harmonic measure, $p$ harmonic measure, $p$ harmonic function, simply connected domain, Hausdorff measure, Hausdorff dimension
Résumé - Abstract
Let $ \Omega $ be a bounded simply connected domain in the complex plane, $ \mathbb{C} $. Let $ N $ be a neighborhood of $ \partial \Omega $, let $ p $ be fixed, $ 1 < p < \infty , $ and let $ \hat{u} $ be a positive weak solution to the $ p $ Laplace equation in $ \Omega \cap N. $ Assume that $ \hat{u} $ has zero boundary values on $ \partial \Omega $ in the Sobolev sense and extend $ \hat{u} $ to $ N \setminus \Omega $ by putting $ \hat{u} \equiv 0 $ on $ N \setminus \Omega . $ Then there exists a positive finite Borel measure $ \hat{\mu }$ on $ \mathbb{C} $ with support contained in $ \partial \Omega $ and such that
\begin{eqnarray*} \int | \nabla \hat{u} |^{p - 2} \, \langle \nabla \hat{u} , \nabla \phi \rangle \, dA = - \int \phi \, d \hat{\mu }\end{eqnarray*}whenever $ \phi \in C_0^\infty ( N ). $ If $ p = 2$ and if $\hat{u}$ is the Green function for $ \Omega $ with pole at $x\in \Omega \setminus \bar{N}$ then the measure $\hat{\mu }$ coincides with harmonic measure at $x$, $\omega =\omega ^x$, associated to the Laplace equation. In this paper we continue the studies initiated by the first author by establishing new results, in simply connected domains, concerning the Hausdorff dimension of the support of the measure $\hat{\mu }$. In particular, we prove results, for $ 1 < p < \infty $, $p\ne 2$, reminiscent of the famous result of Makarov concerning the Hausdorff dimension of the support of harmonic measure in simply connected domains.
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