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On the boundary behavior of positive solutions of elliptic differential equations

Author: A. Logunov
Translated by: the author
Original publication: Algebra i Analiz, tom 27 (2015), nomer 1.
Journal: St. Petersburg Math. J. 27 (2016), 87-102
MSC (2010): Primary 35J67, 31B25, 35J08, 31B05
Published electronically: December 7, 2015
MathSciNet review: 3443267
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Abstract: Let $ u$ be a positive harmonic function in the unit ball $ B_1 \subset \mathbb{R}^n$, and let $ \mu $ be the boundary measure of $ u$. For a point $ x\in \partial B_1$, let $ n(x)$ denote the unit inner normal at $ x$. Let $ \alpha $ be a number in $ (-1,n-1]$, and let $ A \in [0,+\infty ) $. In the paper, it is proved that $ u(x+n(x)t)t^{\alpha } \to A$ as $ t \to +0$ if and only if $ \frac {\mu ({B_r(x)})}{r^{n-1}} r^{\alpha } \to C_\alpha A$ as $ r\to +0$, where $ {C_\alpha = \frac {\pi ^{n/2}}{\Gamma (\frac {n-\alpha +1}{2})\Gamma (\frac {\alpha +1}{2})}}$. For $ \alpha =0$, this follows from the theorems by Rudin and Loomis that claim that a positive harmonic function has a limit along the normal if and only if the boundary measure has the derivative at the corresponding point of the boundary. For $ \alpha =n-1$, this is related to the size of the point mass of $ \mu $ at $ x$ and in this case the claim follows from the Beurling minimum principle. For the general case of $ \alpha \in (-1,n-1)$, the proof employs the Wiener Tauberian theorem in a way similar to Rudin's approach. In dimension $ 2$, conformal mappings can be used to generalize the statement to sufficiently smooth domains; in dimension $ n\geq 3$ it is shown that this generalization is possible for $ \alpha \in [0,n-1]$ due to harmonic measure estimates. A similar method leads to an extension of results by Loomis, Ramey, and Ullrich on nontangential limits of harmonic functions to positive solutions of elliptic differential equations with Hölder continuous coefficients.

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Additional Information

A. Logunov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, 14th Line 29B, Vasilyevsky Island, St. Petersburg 199178, Russia

Keywords: Harmonic functions, Tauberian theorems
Received by editor(s): September 21, 2014
Published electronically: December 7, 2015
Additional Notes: Supported by the Russian Science Foundation (project no. 14.21-00035).
Article copyright: © Copyright 2015 American Mathematical Society