Timeline for Existence of an extension operator $E: W_0^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^d)$?
Current License: CC BY-SA 4.0
9 events
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| Apr 16 at 7:56 | history | edited | Hannes | CC BY-SA 4.0 |
incorporated comments
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| Apr 12 at 11:43 | comment | added | Hannes | Bof, I need to think about this for a second and possibly revert my answer to its previous state :-D It IS delicate.. | |
| Apr 12 at 11:42 | comment | added | Ayman Moussa | @Perelman, I think you're right, I just took for granted that $E$ arrives in $W^{s,p}(\mathbf{R}^d)$ because of the last sentence of your post. My answer only tells you that if $E$ is well-defined then it has to be bounded. | |
| Apr 12 at 10:46 | comment | added | Perelman | May I add to "...Zero extension is still a bounded linear operator for the given $W^{s,p}_0(\Omega)$ by Ayman's abstract answer...": Thats unfortunately not necessarily the case, since we do not know if $E$ is even a well defined map into $W^{s,p}(\mathbb{R}^d)$ (see (1) in my post). I did not realize that too in the first view. | |
| Apr 12 at 9:13 | history | edited | Hannes | CC BY-SA 4.0 |
deleted 96 characters in body
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| Apr 12 at 7:56 | comment | added | Jochen Glueck | This is great, thanks a lot for the reference! | |
| Apr 12 at 6:54 | comment | added | Hannes | @JochenGlueck Sure! (Else I would not have dared to say anything about "classical" :-)) How about Theorem 5.3.4 in Edmunds and Evans: Spectral Theory and Differential Operators? | |
| Apr 11 at 16:14 | comment | added | Jochen Glueck | Do you happen to know a reference for an embedding result that relates that weight $\operatorname{dist}^{-k}_{\partial \Omega}$ to $W_0^{k,p}(\Omega)$ (for $k=1$ or maybe even for larger $k$)? I tried to search for it, but did not find much. | |
| Apr 9 at 9:57 | history | answered | Hannes | CC BY-SA 4.0 |