Timeline for Does the function with the αth-weak partial derivative has the βth-weak partial derivative with β≤α?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Mar 17, 2020 at 18:00 | comment | added | user131781 | Of course (plus 5). | |
| Mar 17, 2020 at 17:58 | comment | added | Michael Renardy | @user131781: And it is possible that $u_2$ is a locally integrable function, but $u_2'$ is not. | |
| Mar 17, 2020 at 17:52 | comment | added | user131781 | There seems to be some misunderstanding here—we are not talking about regularity in the classical sense. $u$, as a locally integrable function, is a distribution and so has (distributional) derivatives of all orders. The point of the given definition is not to specify when a derivative exists but when this derivative is not just a distribution but even a (specific) locally integrable function, namely $v$. If $u_{xy}=0$, then $u$ is a sum $u_1(x)+u_2(y)$ (under suitable conditions on the domain) where the two summands are distributions of a single variable.. | |
| Mar 17, 2020 at 17:32 | comment | added | Wentao Hu | @MichaelRenardy Thank you very much! | |
| Mar 17, 2020 at 16:56 | comment | added | Michael Renardy | It is not true. Consider functions $u(x,y)$ which are independent of $x$. Then $u_{xy}=0$. But this does not tell you anything about the regularity of $u_y$. | |
| Mar 17, 2020 at 16:40 | review | First posts | |||
| Mar 17, 2020 at 17:19 | |||||
| Mar 17, 2020 at 16:35 | history | asked | Wentao Hu | CC BY-SA 4.0 |