$p$ Harmonic Measure in Simply Connected Domains

Lewis, John L., Nyström, Kaj, and Poggi-Corradini, Pietro. "$p$ Harmonic Measure in Simply Connected Domains." Annales de l’institut Fourier 61.2 (2011): 689-715. <http://eudml.org/doc/219746>.

@article{Lewis2011,
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 &lt; p &lt; \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 &lt; p &lt; \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.},
affiliation = {University of Kentucky Department of Mathematics Lexington, KY 40506-0027 (USA); Umeå University Department of Mathematics 90187 Umeå (Sweden); Kansas State University Cardwell Hall Department of Mathematics Manhattan, KS 66506 (USA)},
author = {Lewis, John L., Nyström, Kaj, Poggi-Corradini, Pietro},
journal = {Annales de l’institut Fourier},
keywords = {Harmonic function; harmonic measure; $p$ harmonic measure; $p$ harmonic function; simply connected domain; Hausdorff measure; Hausdorff dimension; -harmonic measure},
language = {eng},
number = {2},
pages = {689-715},
publisher = {Association des Annales de l’institut Fourier},
title = {$p$ Harmonic Measure in Simply Connected Domains},
url = {http://eudml.org/doc/219746},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Lewis, John L.
AU - Nyström, Kaj
AU - Poggi-Corradini, Pietro
TI - $p$ Harmonic Measure in Simply Connected Domains
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 2
SP - 689
EP - 715
AB - 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 &lt; p &lt; \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 &lt; p &lt; \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.
LA - eng
KW - Harmonic function; harmonic measure; $p$ harmonic measure; $p$ harmonic function; simply connected domain; Hausdorff measure; Hausdorff dimension; -harmonic measure
UR - http://eudml.org/doc/219746
ER -