Let be a bounded simply connected domain in the complex plane, . Let be a neighborhood of , let be fixed, and let be a positive weak solution to the Laplace equation in Assume that has zero boundary values on in the Sobolev sense and extend to by putting on Then there exists a positive finite Borel measure on with support contained in and such that
whenever If and if is the Green function for with pole at then the measure coincides with harmonic measure at , , 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 . In particular, we prove results, for , , reminiscent of the famous result of Makarov concerning the Hausdorff dimension of the support of harmonic measure in simply connected domains.
Soit une région bornée et simplement connexe dans le plan complexe . Soit un voisinage de . Pour , on considère une solution positive -harmonique faible de l’équation de Laplace dans Supposons que s’annule sur au sens de Sobolev et qu’elle s’étend dans avec en . Alors il existe une mesure positive finie de Borel dans avec support contenu dans telle que
pour tout Si et si est la fonction de Green pour avec pole , alors la mesure est la mesure harmonique au point , , 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 . En particulier, on obtient des résultats, pour , , qui rappèllent le fameux résultat de Makarov concernant la dimension de Hausdorff pour le support de la mesure harmonique des régions simplement connexes.
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
@article{AIF_2011__61_2_689_0, author = {Lewis, John L. and Nystr\"om, Kaj and Poggi-Corradini, Pietro}, title = {$p$ {Harmonic} {Measure} in {Simply} {Connected} {Domains}}, journal = {Annales de l'Institut Fourier}, pages = {689--715}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {61}, number = {2}, year = {2011}, doi = {10.5802/aif.2626}, zbl = {1241.35071}, mrnumber = {2895070}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2626/} }
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 SP - 689 EP - 715 VL - 61 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2626/ DO - 10.5802/aif.2626 LA - en ID - AIF_2011__61_2_689_0 ER -
%0 Journal Article %A Lewis, John L. %A Nyström, Kaj %A Poggi-Corradini, Pietro %T $p$ Harmonic Measure in Simply Connected Domains %J Annales de l'Institut Fourier %D 2011 %P 689-715 %V 61 %N 2 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2626/ %R 10.5802/aif.2626 %G en %F AIF_2011__61_2_689_0
Lewis, John L.; Nyström, Kaj; Poggi-Corradini, Pietro. $p$ Harmonic Measure in Simply Connected Domains. Annales de l'Institut Fourier, Volume 61 (2011) no. 2, pp. 689-715. doi : 10.5802/aif.2626. http://www.numdam.org/articles/10.5802/aif.2626/
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