Timeline for Is $\int_M\Delta u = 0$ if $u$ is not $C^2$ on a set of measure zero?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| May 16, 2020 at 13:20 | vote | accept | L.F. Cavenaghi | ||
| May 16, 2020 at 3:52 | answer | added | Alexandre Eremenko | timeline score: 4 | |
| May 16, 2020 at 1:57 | comment | added | Deane Yang | And the Laplacian of $u$ is defined using weak derivatives as a distribution over the full domain. | |
| May 16, 2020 at 1:10 | comment | added | Deane Yang | If $u$ is well-defined as a distribution on $M$, then the answer is yes, if the integral is properly interpreted, namely. as the evaluation of the distribution on the constant function $1$. | |
| May 16, 2020 at 0:15 | comment | added | Nate Eldredge | Oh, I didn't see the added assumption of dimension 2. | |
| May 16, 2020 at 0:14 | comment | added | L.F. Cavenaghi | @NateEldredge I don't see how your counter example works in dimension greater or equal $2$ since on those it does hold that the integral of the Laplacian of $C^2$ function vanishes. | |
| May 16, 2020 at 0:13 | history | edited | L.F. Cavenaghi | CC BY-SA 4.0 |
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| May 16, 2020 at 0:09 | comment | added | Nate Eldredge | Now I think my first comment is a counterexample again, with $S$ equal to a single point. | |
| May 16, 2020 at 0:08 | history | edited | L.F. Cavenaghi | CC BY-SA 4.0 |
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| May 16, 2020 at 0:02 | history | edited | L.F. Cavenaghi | CC BY-SA 4.0 |
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| May 16, 2020 at 0:02 | comment | added | Nate Eldredge | If you mean instead something like "almost everywhere equal to a $C^2$ function $v$" then the answer is certainly yes, in the sense of weak derivatives, because then $\Delta u = \Delta v$ almost everywhere. | |
| May 16, 2020 at 0:00 | comment | added | Nate Eldredge | You need to be clear what "$C^2$ at almost every point" means. For example if $M$ is $S^1$ realized as $[0,1]$ with endpoints identified, the function $u(x) = x^2$ is $C^2$ except at one point and clearly doesn't satisfy the desired conclusion. | |
| May 15, 2020 at 23:46 | history | asked | L.F. Cavenaghi | CC BY-SA 4.0 |