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$ p$ harmonic measure in simply connected domains revisited


Author: John L. Lewis
Journal: Trans. Amer. Math. Soc. 367 (2015), 1543-1583
MSC (2010): Primary 35J25, 35J70
DOI: https://doi.org/10.1090/S0002-9947-2014-05979-5
Published electronically: November 6, 2014
MathSciNet review: 3286492
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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

$\displaystyle \int \vert \nabla \hat u \vert^{p - 2} \, \langle \nabla \hat u , \nabla \phi \rangle \, dA = - \int \phi \, d \hat \mu$      

whenever $ \phi \in C_0^\infty ( N ). $ In this paper we continue our studies by establishing endpoint type results for the Hausdorff dimension of this measure in simply connected domains. Our results are similar to the well known result of Makarov concerning harmonic measure in simply connected domains.

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Additional Information

John L. Lewis
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: johnl@uky.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-05979-5
Received by editor(s): September 18, 2012
Published electronically: November 6, 2014
Additional Notes: The author was partially supported by DMS-0900291 and the Institut Mittag Leffler (Djursholm, Sweden). He thanks the staff at the Institut for their gracious hospitality
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.