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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Positive harmonic functions vanishing on the boundary for the Laplacian in unbounded horn-shaped domains


Authors: Dimitry Ioffe and Ross Pinsky
Journal: Trans. Amer. Math. Soc. 342 (1994), 773-791
MSC: Primary 60J50; Secondary 31C35
DOI: https://doi.org/10.1090/S0002-9947-1994-1211410-9
MathSciNet review: 1211410
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Abstract: Denote points $ \bar x \in {R^{d + 1}}$, $ d \geq 2$, by $ \bar x = (\rho ,\theta ,z)$, where $ \rho > 0$, $ \theta \in {S^{d - 1}}$, and $ z \in R$. Let $ a:[0,\infty ) \to (0,\infty )$ be a nondecreasing $ {C^2}$-function and define the "horn-shaped" domain $ \Omega = \{ \bar x = (\rho ,\theta ,z):\vert z\vert < a(\rho )\} $ and its unit "cylinder" $ D = \{ \bar x = (\rho ,\theta ,z) \in \Omega :\rho < 1\} $. Under appropriate regularity conditions on a, we prove the following theorem: (i) If $ {\smallint ^\infty }a(\rho )/{\rho ^2}d\rho = \infty $, then the Martin boundary at infinity for $ \frac{1}{2}\Delta $ in $ \Omega $ is a single point, (ii) If $ {\smallint ^\infty }a(\rho )/{\rho ^2}d\rho < \infty $, then the Martin boundary at infinity for $ \frac{1}{2}\Delta $ in $ \Omega $ is homeomorphic to $ {S^{d - 1}}$. More specifically, a sequence $ \{ ({\rho _n},{\theta _n},{z_n})\} _{n = 1}^\infty \subset \Omega $ satisfying $ {\lim _{n \to \infty }}{\rho _n} = \infty $ is a Martin sequence if and only if $ {\lim _{n \to \infty }}{\theta _n}$ exists on $ {S^{d - 1}}$. From (i), it follows that the cone of positive harmonic functions in $ \Omega $ vanishing continuously on $ \partial \Omega $ is one-dimensional. From (ii), it follows easily that the cone of positive harmonic functions on $ \Omega $ vanishing continuously on $ \partial \Omega $ is generated by a collection of minimal elements which is homeomorphic to $ {S^{d - 1}}$.

In particular, the above result solves a problem stated by Kesten, who asked what the Martin boundary is for $ \frac{1}{2}\Delta $ in $ \Omega $ in the case $ a(\rho ) = 1 + {\rho ^\gamma }$, $ 0 < \gamma < 1$. Our method of proof involves an analysis as $ \rho \to \infty $ of the exit distribution on $ \partial D$ for Brownian motion starting from $ (\rho ,\theta ,z) \in \Omega $ and conditioned to hit D before exiting $ \Omega $.


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DOI: https://doi.org/10.1090/S0002-9947-1994-1211410-9
Article copyright: © Copyright 1994 American Mathematical Society

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