Timeline for Smooth Sobolev extension from $W^{1,p}(U)$ to $W^{1,p} (\mathbb{R}^n) $

6 events

when toggle format	what		by	license	comment
Apr 22, 2012 at 0:09	answer	added	Piero D'Ancona		timeline score: 6
Apr 21, 2012 at 22:40	comment	added	Analysis Now		For my case, $n = 2$.
Apr 21, 2012 at 22:37	comment	added	Analysis Now		@ Prof. Deane Yang : Thank you. If we assume $ p > n $, then is there any condition on $u$ that would make sure that the smooth Sobolev function $u$ on $ U $ can be extended to $Eu \in C^{\infty}(\mathbb{R}^n) \cap W^{1,p}(\mathbb{R}^n)$. Is Holder continuity of $u$ or that $ u $ is Lipchitz enough for this ? Actually, I am looking at this problem as a part of another problem, and there I have $ p > 2, u $ is Lipchitz upto the boundary, because its derivatives are bounded in the interior.
Apr 21, 2012 at 21:58	comment	added	Deane Yang		If $p > n$, then this won't work, because $W^{1,p} \subset C^{\alpha}$ for some $\alpha > 0$. So if $u$ is not Holder continuous on $\overline{U}$, it cannot be extended in $W^{1,p}$ beyond $\overline{U}$.
Apr 21, 2012 at 21:19	answer	added	Pietro Majer		timeline score: 4
Apr 21, 2012 at 20:58	history	asked	Analysis Now	CC BY-SA 3.0