Mathematics – Analysis of PDEs
Scientific paper
2007-02-07
Mathematics
Analysis of PDEs
6 pages
Scientific paper
Let $x_0\in\Omega\subset\Bbb{R}^n$, $n\ge 2$, be a domain and let $m\ge 2$. We will prove that a solution $u$ of the polyharmonic equation $\Delta^mu=0$ in $\Omega\setminus\{x_0\}$ has a removable singularity at $x_0$ if and only if $|\Delta^ku(x)|=o(|x-x_0|^{2-n})\quad\forall k=0,1,2,...,m-1$ as $|x-x_0|\to 0$ for $n\ge 3$ and $=o(\log (|x-x_0|^{-1}))\quad\forall k=0,1,2,...,m-1$ 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 $n=2$.
No associations
LandOfFree
Removable singularity of the polyharmonic equation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Removable singularity of the polyharmonic equation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Removable singularity of the polyharmonic equation will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-218940