Timeline for Under which conditions does this PDE have unique solutions
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jun 17, 2022 at 14:17 | answer | added | Robert Bryant | timeline score: 3 | |
| Jun 17, 2022 at 9:30 | comment | added | mlainz | @WillieWong this is an interesting suggestion, but I think that it restricts too much the classes of solution that I can consider. Locally this is equivalent than considering that the rotational vanishes. The obvious generalization would be to set $δu = g$.. | |
| Jun 16, 2022 at 20:10 | answer | added | MyShepherd | timeline score: 4 | |
| Jun 16, 2022 at 19:36 | comment | added | Willie Wong | The other way to use the Hodge idea is to fix $u$ to purely a gradient. Is that possible in your case? The scalar equation $\Delta u = f(x, \nabla u)$ certainly has been previously studied. | |
| Jun 16, 2022 at 18:54 | history | edited | mlainz | CC BY-SA 4.0 |
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| Jun 16, 2022 at 18:50 | comment | added | mlainz | I also though about using the Hodge decomposition and fix the codifferential of $u$ and try to use boundary conditions to determine the harmonic part, but I do not think that this approach is possible if you allow $g$ to depend on $u$ | |
| Jun 16, 2022 at 18:45 | comment | added | mlainz | Also, you are right that $f$ is not a top form, but a fiber bundle morphism from $E \to M$ to $\Lambda^k(M) \to M$, so $f(\cdot, u(\cdot))$ is the actual volume form. I only look for locally defined solution, so the manifold is completely irrelevant. | |
| Jun 16, 2022 at 18:37 | comment | added | mlainz | @WillieWong You are right about the EDIT, I will remove it. | |
| Jun 16, 2022 at 17:08 | comment | added | Willie Wong | The first displayed equation, contrary to asserted, is not linear with constant coefficients. | |
| Jun 16, 2022 at 16:18 | history | edited | mlainz | CC BY-SA 4.0 |
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| S Jun 16, 2022 at 15:34 | review | First questions | |||
| Jun 16, 2022 at 17:48 | |||||
| S Jun 16, 2022 at 15:34 | history | asked | mlainz | CC BY-SA 4.0 |