$p$ Harmonic Measure in Simply Connected Domains

Lewis, John L.; Nyström, Kaj; Poggi-Corradini, Pietro

Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 689-715.

Soit $\Omega$ une région bornée et simplement connexe dans le plan complexe $ℂ$. Soit $N$ un voisinage de $\partial \Omega$. Pour $1<p<\infty$, on considère une solution positive $p$-harmonique faible $\widehat{u}$ de l’équation de $p$ Laplace dans $\Omega \cap N.$ Supposons que $\widehat{u}$ s’annule sur $\partial \Omega$ au sens de Sobolev et qu’elle s’étend dans $N\setminus \Omega$ avec $\widehat{u}\equiv 0$ en $N\setminus \Omega$. Alors il existe une mesure positive finie de Borel $\widehat{\mu }$ dans $ℂ$ avec support contenu dans $\partial \Omega$ telle que

$$\begin{array}{c}\hfill \int |\nabla \widehat{u}{|}^{p-2}\langle \nabla \widehat{u},\nabla \phi \rangle dA=-\int \phi d\widehat{\mu }\end{array}$$

pour tout $\phi \in C_0^{\infty }\left(N\right).$ Si $p=2$ et si $\widehat{u}$ est la fonction de Green pour $\Omega$ avec pole $x\in \Omega \setminus \overline{N}$, alors la mesure $\widehat{\mu }$ est la mesure harmonique au point $x$, $\omega ={\omega }^x$, pour l’équation de Laplace. Dans ce travail on continue l’ étude commencée par le premier auteur, en établissant des nouveaux résultats, pour les régions simplement connexe, concernant la dimension de Hausdorff du support de la mesure $\widehat{\mu }$. En particulier, on obtient des résultats, pour $1<p<\infty$, $p\ne 2$, qui rappèllent le fameux résultat de Makarov concernant la dimension de Hausdorff pour le support de la mesure harmonique des régions simplement connexes.

Let $\Omega$ be a bounded simply connected domain in the complex plane, $ℂ$. Let $N$ be a neighborhood of $\partial \Omega$, let $p$ be fixed, $1<p<\infty ,$ and let $\widehat{u}$ be a positive weak solution to the $p$ Laplace equation in $\Omega \cap N.$ Assume that $\widehat{u}$ has zero boundary values on $\partial \Omega$ in the Sobolev sense and extend $\widehat{u}$ to $N\setminus \Omega$ by putting $\widehat{u}\equiv 0$ on $N\setminus \Omega .$ Then there exists a positive finite Borel measure $\widehat{\mu }$ on $ℂ$ with support contained in $\partial \Omega$ and such that

$$\begin{array}{c}\hfill \int |\nabla \widehat{u}{|}^{p-2}\langle \nabla \widehat{u},\nabla \phi \rangle dA=-\int \phi d\widehat{\mu }\end{array}$$

whenever $\phi \in C_0^{\infty }\left(N\right).$ If $p=2$ and if $\widehat{u}$ is the Green function for $\Omega$ with pole at $x\in \Omega \setminus \overline{N}$ then the measure $\widehat{\mu }$ 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 $\widehat{\mu }$. 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.

Classification : 35J25, 35J70
Keywords: Harmonic function, harmonic measure, $p$ harmonic measure, $p$ harmonic function, simply connected domain, Hausdorff measure, Hausdorff dimension
Mot clés : fonction harmonique, mesure harmonique, mesure $p$ harmonique, fonction $p$ harmonique, région simplement connexe, mesure de Hausdorff, dimension de Hausdorff

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