On the boundary behavior of positive solutions of elliptic differential equations

Logunov, A.

doi:10.1090/spmj/1377

On the boundary behavior of positive solutions of elliptic differential equations
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by A. Logunov
Translated by: the author

St. Petersburg Math. J. 27 (2016), 87-102

DOI: https://doi.org/10.1090/spmj/1377

Published electronically: December 7, 2015

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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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