Timeline for Heating a long cylinder: steady states
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Jun 1 at 14:39 | comment | added | Beni Bogosel | @MateuszKwaśnicki: I don't think the Dirichlet Laplace eigenvalues need to be considered here. The volumic source term is zero. You need rather the Steklov eigenvalues: $-\Delta u=0$, $\partial_n u = \sigma_k u$ on the boundary? | |
| Oct 2, 2022 at 17:18 | vote | accept | Leo Moos | ||
| Sep 26, 2022 at 23:02 | comment | added | Mateusz Kwaśnicki | Yes, something of that kind, although I thought about pointwise bounds. I tried to give some details in an answer below. $L^2$ bounds might actually be much easier to establish, as the problem becomes essentially 1-D. | |
| Sep 26, 2022 at 22:59 | answer | added | Mateusz Kwaśnicki | timeline score: 6 | |
| Sep 26, 2022 at 21:46 | history | edited | Leo Moos | CC BY-SA 4.0 |
clarified question
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| Sep 26, 2022 at 20:30 | history | became hot network question | |||
| Sep 26, 2022 at 17:04 | answer | added | Alexandre Eremenko | timeline score: 5 | |
| Sep 26, 2022 at 16:07 | history | edited | Leo Moos | CC BY-SA 4.0 |
made a mistake in last argument
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| Sep 26, 2022 at 15:52 | history | edited | Leo Moos | CC BY-SA 4.0 |
added calculation in a comment
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| Sep 26, 2022 at 14:24 | comment | added | Leo Moos | @MateuszKwaśnicki Cool, that's very neat! So something like $\lvert u \rvert \leq C \lvert \mathrm{e}^{-Ct} \varphi \rvert$? Do you have any idea whether bounds like this can be obtained without resorting to Brownian motion? I'm just asking because I was hoping to use this setting to gain a better grasp of a similar, but non-linear problem. | |
| Sep 26, 2022 at 14:14 | comment | added | Mateusz Kwaśnicki | Much stronger: the density function of the exit time from $D$ (of the 2-D Brownian) motion decays exponentially fast, so the harmonic measure in the infinite case also decays exponentially fast. By periodization, the same is true for the reflecting boundary. That is, the "weight" associated with $\partial D \times \{x\}$ is roughly $\exp(-\sqrt{\lambda_1} |x|) / \sqrt{4 \lambda_1}$, where $\lambda_1$ is the smallest eigenvalue of $\Delta$ in $D$ with Dirichlet boundary conditions. | |
| Sep 26, 2022 at 13:40 | history | edited | Leo Moos | CC BY-SA 4.0 |
clarified question
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| Sep 26, 2022 at 13:21 | history | undeleted | Leo Moos | ||
| Sep 26, 2022 at 12:35 | history | deleted | Leo Moos | via Vote | |
| Sep 26, 2022 at 12:29 | history | asked | Leo Moos | CC BY-SA 4.0 |