Timeline for On the weak derivative of $|u|^{(p-2)/2}u$
Current License: CC BY-SA 4.0
20 events
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| Apr 14 at 15:30 | history | edited | Perelman | CC BY-SA 4.0 |
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| Apr 6 at 16:44 | comment | added | Perelman | @TerryTao By the way, regarding your post on mathoverflow.net/questions/23478/…, if this case hasn't been proposed as an answer already, I think this might also be a nice example of a false belief in mathematics, that we can always apply the chain rule for weak derivatives. But this is by no means intended to cause offense. I really appreciate your answer. | |
| Apr 6 at 16:41 | comment | added | Perelman | @TerryTao Thank you for the book recommendation. I've thought about the hint for a while, and I might still overlook something, but as far as I can see, $F$ is already $C^1$. Its derivative is $\frac{2}{p^\prime}|x|^{\frac{2}{p^\prime}-1}$ $(p^\prime<2)$. However, the derivative is unbounded, so we don't have access to the chain rule for weak derivatives as far as I know; $F^\prime$ needs to be in $L^\infty$ for that. Unfortunately, $F^\prime_\varepsilon$ also isn't bounded. But we could cut off the derivative and consider the limit case. But I wasnt succesfull at that yet. | |
| Apr 1 at 14:39 | comment | added | Terry Tao | Actually, on closer inspection, the fractional chain rule is not needed as one is working here with a full derivative rather than a fractional one. One way to proceed is to replace $F(x)$ by the regularization $F_\varepsilon(x) = (\varepsilon^2+x^2)^{\frac{1}{p'}-\frac{1}{2}} x$ and then take weak limits as $\varepsilon \to 0$. By the way, for a modern introduction to paradifferential calculus (including the fractional chain rule), I recommend Taylor's "Tools for PDE". | |
| Apr 1 at 14:00 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Additions + minor math Jaxing
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| Apr 1 at 5:03 | comment | added | Terry Tao | I think the proof provided by Raviart is incomplete at this step, but Lemma 1.1 is easy to establish in any event: write $|v|^{p-2} v$ as $F( |v|^{\frac{p-2}{2}} v)$, where $F(x) := |x|^{\frac{2}{p'}-1}x$, and apply the fractional chain rule (which perhaps was not fully available in Raviart's time). | |
| Mar 31 at 14:18 | history | edited | Perelman | CC BY-SA 4.0 |
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| Mar 31 at 14:11 | vote | accept | Perelman | ||
| Mar 30 at 20:17 | history | edited | Perelman | CC BY-SA 4.0 |
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| Mar 30 at 18:27 | answer | added | Ayman Moussa | timeline score: 3 | |
| Mar 30 at 16:12 | comment | added | Perelman | @JingeonAn-Lacroix do you have an idea how to show for a smooth function approaching $u$, that its weak derivatives also converge in $L^1_{loc}$ at least | |
| Mar 30 at 16:11 | history | undeleted | Perelman | ||
| Mar 30 at 15:36 | history | deleted | Perelman | via Vote | |
| Mar 30 at 15:33 | comment | added | Perelman | I cant quite follow, how does that imply that for $u$ being some function that its weak derivative exists? | |
| Mar 30 at 15:04 | comment | added | Jingeon An-Lacroix | For example, you can mollify $u$ and get the wanted identity, then show the identity is preserved when mollification sent to identity? Of course we have to show some estimates along the way? | |
| Mar 30 at 14:49 | comment | added | Perelman | Where do you know that $u$ has a weak derivative or $|u|^{(p-2)/2}$? | |
| Mar 30 at 14:46 | comment | added | Jingeon An-Lacroix | What difficulty are you having? Just applying the product rule (with being careful with conditions) is not enough? | |
| Mar 30 at 14:43 | history | edited | Perelman | CC BY-SA 4.0 |
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| S Mar 30 at 13:16 | review | First questions | |||
| Mar 30 at 14:30 | |||||
| S Mar 30 at 13:16 | history | asked | Perelman | CC BY-SA 4.0 |