Timeline for Dirichlet-type condition on Riemannian manifold
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 28, 2021 at 17:13 | comment | added | Mattis Bakken | Thanks so much! All you said is very helpful. | |
| Oct 28, 2021 at 16:40 | comment | added | Leo Moos | The answer should indeed be 'yes', at least if $S$ has codimension one. The result should follow from Cauchy-Kovalevskaya theory. Unfortunately I don't know the best reference, but I think Evans covers some of this. Whether there are some issues if $S$ is not embedded or $\dim S \leq \dim M - 2$ I do not know. | |
| Oct 28, 2021 at 16:36 | vote | accept | Mattis Bakken | ||
| Oct 28, 2021 at 16:25 | comment | added | Mattis Bakken | That's interesting. In my setup, I can actually assume that $u : S \to \mathbf{R}$ is analytic. So you say the answer to my question is 'yes' if $S$ and $u$ are analytic? | |
| Oct 28, 2021 at 16:21 | comment | added | Leo Moos | I think the issue would basically be the same: you can't extend non-analytic 'initial data'. That being said---and perhaps you're already aware of this---if the given function $u: S \to \mathbf{R}$ is analytic, then some form of Cauchy-Kovalevskaya should give you the desired harmonic extension. | |
| Oct 28, 2021 at 16:15 | comment | added | Mattis Bakken | (+1) Thanks for your answer; that's very interesting. In the examples that I had in mind, the submanifold $S$ was actually real-analytic (in fact holomorphic). Do you think there can still be an issue? | |
| Oct 28, 2021 at 16:08 | history | edited | Leo Moos | CC BY-SA 4.0 |
clarified answer
|
| Oct 28, 2021 at 15:45 | history | answered | Leo Moos | CC BY-SA 4.0 |