Timeline for On a core for Neumann Laplacians
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 16, 2021 at 11:56 | comment | added | sharpe | @GiorgioMetafune Thank you for your comment. I also think your idea is available for $L^2$-Neumann Laplacians. | |
| Oct 16, 2021 at 11:52 | comment | added | Giorgio Metafune | Sorry, the second even modified is too small. For example in 1d consider $D^2$ with Neumann b.c at $x=0$. The domain consists of all $C^2$ functions $u$ with $u'(0)=0$ and if $u''(0) \neq 0$ such a $u$ cannot be approximated by functions which are locally constant near 0. However this works in Sobolev spaces. | |
| Oct 16, 2021 at 11:42 | comment | added | Giorgio Metafune | Yes, sure. I have been too vague. In the half plane $\{y>0\}$ I would say $u(x,y)=u(x,0)$ for $y \leq \delta$, for a certain $\delta>0$. In a regular domain one should reformulate using a tubolar neighborhood and requiring that $u$ does not depend on the distance from the boundary, for small distance. However I am not sure if it is true, | |
| Oct 16, 2021 at 11:32 | comment | added | sharpe | @GiorgioMetafune Thank you for your kind answer. I was not aware that Schauder estimates are useful. Could you tell me the definition of "near the boundary"? | |
| Oct 16, 2021 at 11:21 | comment | added | Giorgio Metafune | To see that the first is a core you can use Schauder estimates. If $u_n-\Delta u_n=f_n$ with Neumann b.c. and $f_n \in C^1$, $f_n \to u-\Delta u$ in the sup norm, then $u_n \in C^2$ and $(u_n) \to u$ in the graph norm. The second is too small, I guess. I would substitute with functions for which only the normal derivative is zero near the boundary. | |
| Oct 16, 2021 at 10:59 | history | asked | sharpe | CC BY-SA 4.0 |