Timeline for Variation on the Sobolev space $H^1_0$
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Nov 5, 2013 at 14:45 | vote | accept | timur | ||
| Nov 4, 2013 at 14:17 | comment | added | timur | @PietroMajer: That's a valid point. Thank you! | |
| Nov 4, 2013 at 8:13 | comment | added | Jean Van Schaftingen | I think that your definition should be the standard definition of $H^1_0 (\Omega)$: it is quite natural in variational problems to take functions verifying the boundary conditions and then complete the space for analytical reasons. Having a large set which is completed might allows to construct counterexamples without having to pass to the limit. On the other hand I think that pedagogically, the compact support assumption somehow an overkill. | |
| Nov 3, 2013 at 15:49 | comment | added | Pietro Majer | As to the pedagogical aspect, I'd not deviate from the standard definition, nor introduce new spaces, but I would certainly remark that an $H^1$ limit of functions with compact support in $\Omega$ may well have nonzero derivatives at the boundary. An example in one variable is enough to make it clear. I would rather insist on the beneficial aspects of the standard definition. | |
| Nov 3, 2013 at 13:14 | answer | added | Jean Van Schaftingen | timeline score: 3 | |
| Sep 30, 2013 at 19:59 | answer | added | Michael Renardy | timeline score: 3 | |
| Sep 30, 2013 at 17:30 | comment | added | Terry Tao | This doesn't directly answer the question, but I'd like point out that even a very simple domain such as the exterior of the unit ball in R^3 can have a very subtle Sobolev space theory, see the recent preprint arxiv.org/abs/1205.5784 , and in particular the discussion of the distinction between $H^{s,p}_0(\Omega)$ and $H^{s,p}_{00}(\Omega)$. | |
| Sep 30, 2013 at 13:29 | answer | added | paul garrett | timeline score: 2 | |
| Sep 29, 2013 at 15:30 | history | edited | timur | CC BY-SA 3.0 |
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| Sep 29, 2013 at 15:16 | answer | added | Michael Renardy | timeline score: 5 | |
| Sep 29, 2013 at 14:59 | history | edited | timur | CC BY-SA 3.0 |
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| Sep 29, 2013 at 14:32 | history | asked | timur | CC BY-SA 3.0 |