A uniqueness theorem for superharmonic functions in $\textbf {R}^{n}$

Abstract:

Let $s(x)$ be a nonnegative superharmonic function defined on the $n$-ball ${B^n}(y;r)$ in ${{\mathbf {R}}^n}, n \geqslant 3$. If $s(x)$ tends to zero "too rapidly" as $x$ tends to a single point $\xi$ on the boundary of ${B^n}(y;r)$, then we prove that $s \equiv 0$. The same result can then be extended to domains $D \subseteq {{\mathbf {R}}^n}$, whose boundary $\partial D$ is locally ${C^1}$ at $\xi \in \partial D$. These results generalize some earlier work of the author and Ü. Kuran for $n = 2$.