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 $μ$ 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 $α$ be a number in $(-1,n-1]$ and $A \in [0,+\infty) $. We prove that $u(x+n(x)t)t^α \to A$ as $t \to +0$ if and only if $\frac{μ({B_r(x)})}{r^{n-1}} r^α \to C_αA$ as $r\to+0$, where ${C_α= \frac{π^{n/2}}{Γ(\frac{n-α+1}{2})Γ(\frac{α+1}{2})}}$. For $α=0$ 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 $α=n-1$ it concerns about the point mass of $μ$ at $x$ and it follows from the Beurling minimal principle. For the general case of $α\in (-1,n-1)$ 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 $α\in [0,n-1]$ 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.