On the Boundary Behavior of Positive Solutions of Elliptic Differential Equations

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 $u$. Consider a point $x\in \partial B_1$ and let $n(x)$ denote the unit normal vector at $x$. Let $\alpha$ be a number in $(-1,n-1]$ and $A \in [0,+\infty) $. We prove 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 $\alpha=0$ it follows from the theorems by Rudin and Loomis which claim that a positive harmonic function has a limit along the normal iff the boundary measure has the derivative at the corresponding point of the boundary. For $\alpha=n-1$ it concerns about the point mass of $\mu$ at $x$ and it follows from the Beurling minimal principle. For the general case of $\alpha \in (-1,n-1)$ we prove it with the help of the Wiener Tauberian theorem in a similar way to Rudin's approach. Unfortunately this approach works for a ball or a half-space only but not for a general kind of domain. In dimension $2$ one can use conformal mappings and generalise the statement above to sufficiently smooth domains, in dimension $n\geq 3$ we showed that this generalisation is possible for $\alpha\in [0,n-1]$ due to harmonic measure estimates. The last method leads to an extension of the theorems by Loomis, Ramey and Ullrich on non-tangential limits of harmonic functions to positive solutions of elliptic differential equations with Holder continuous coefficients.