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
19 events
| when toggle format | what | by | license | comment | |
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| Apr 12 at 10:35 | history | edited | Perelman | CC BY-SA 4.0 |
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| Apr 9 at 10:12 | history | edited | Hannes | CC BY-SA 4.0 |
typo in Gagliardo norm in last formula
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| Apr 9 at 10:09 | history | edited | YCor |
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| Apr 9 at 9:57 | answer | added | Hannes | timeline score: 2 | |
| Apr 7 at 21:18 | vote | accept | Perelman | ||
| Apr 7 at 19:39 | answer | added | Ayman Moussa | timeline score: 2 | |
| Apr 7 at 19:18 | comment | added | Pietro Majer | ok I didn’t realise the norms are different, thankyou | |
| Apr 7 at 18:58 | comment | added | Perelman | @PietroMajer This would work if $W_0^{s,p}(\Omega)$ and $W^{s,p}(\mathbb{R}^d)$ have equivalent norms. I think we need to show that $\|Eu\|_{W^{s,p}(\mathbb{R}^d)}\leq C\|u\|_{W^{s,p}(\Omega)}$ | |
| Apr 7 at 18:42 | comment | added | Pietro Majer | A natural definition of $C^\infty_0(\Omega)$ is: all $C^\infty$ functions $u:\mathbb R^d\to\mathbb R$ with $\text{supp}(u)\subset \Omega$. So $W^{s,p}_0(\Omega)$ is increasing by inclusion wrto $\Omega$, and we don't even have to extend anything. | |
| Apr 7 at 18:19 | comment | added | Perelman | @PietroMajer yes | |
| Apr 7 at 18:03 | comment | added | Pietro Majer | So by the definition you are adopting $W^{s,p}(\mathbb R^d)=W^{s,p}_0(\mathbb R^d)$, and $W^{s,p}_0$ is defined above, right? | |
| Apr 7 at 17:27 | comment | added | Perelman | Thsi defintion holds for arbtrary open $\Omega \subseteq \mathbb{R}^d$ | |
| Apr 7 at 17:26 | history | edited | Perelman | CC BY-SA 4.0 |
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| Apr 7 at 17:24 | comment | added | Pietro Majer | You may add the definition of $W^{s,p}(\mathbb R^d)$ too, by completeness sake | |
| Apr 7 at 17:20 | history | edited | Perelman | CC BY-SA 4.0 |
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| Apr 7 at 17:11 | history | edited | Perelman | CC BY-SA 4.0 |
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| Apr 7 at 15:53 | answer | added | Liding Yao | timeline score: 1 | |
| S Apr 7 at 15:07 | review | First questions | |||
| Apr 7 at 19:45 | |||||
| S Apr 7 at 15:07 | history | asked | Perelman | CC BY-SA 4.0 |