Removable singularity of the polyharmonic equation

Abstract

Let $x_0\in \Omega \subset {\mathbb{R}}^n$, $n\ge 2$, be a domain and let $m\ge 2$. We will prove that a solution $u$ of the polyharmonic equation ${\Delta }^{m}u=0$ in $\Omega \setminus \{x_0\}$ has a removable singularity at $x_0$ if and only if $|{\Delta }^{k}u(x)|=o(|x-x_0{|}^{2-n})\forall k=0,1,2,...,m-1$ as $|x-x_0|\to 0$ for $n\ge 3$ and $=o(\log (|x-x_0{|}^{-1}))\forall k=0,1,2,...,m-1$ as $|x-x_0|\to 0$ for $n=2$. For $m\ge 2$ we will also prove that $u$ has a removable singularity at $x_0$ if $|u(x)|=o(|x-x_0{|}^{2m-n})$ as $|x-x_0|\to 0$ for $n\ge 3$ and $|u(x)|=o(|x-x_0{|}^{2m-2}\log (|x-x_0{|}^{-1}))$ as $|x-x_0|\to 0$ for $n=2$.

6 pages