Timeline for Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $
Current License: CC BY-SA 3.0
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Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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Jan 11, 2016 at 14:07 | review | Close votes | |||
Jan 12, 2016 at 0:14 | |||||
Jan 11, 2016 at 13:51 | comment | added | Deane Yang | Thomas's remark, along with the definitions of the two spaces you're trying to decide between, immediately gives the answer. This is more appropriate for math.stackexchange.com | |
Jan 11, 2016 at 10:30 | comment | added | Thomas Richard | Maybe you already know that, but by integration by parts your norm is equal to $\left(\int_\Omega|D^2u|^2dx\right)^{1/2}$ since $\langle \nabla u,\nabla\Delta u\rangle=-|D^2u|^2+\tfrac{1}{2}\Delta|\nabla u|^2$ where $|D^2u|^2$ is the Frobenius norm of the Hessian. | |
Jan 11, 2016 at 3:56 | comment | added | Nate Eldredge | I don't see how it can possibly be $W^{2,2}_0(\Omega)$. For instance, the completion certainly can't contain any function whose Laplacian isn't radially symmetric. | |
Jan 11, 2016 at 0:54 | history | asked | Hheepp | CC BY-SA 3.0 |