Timeline for Determine the location of the boundary where the heat changes fastest
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S May 16, 2021 at 17:02 | history | bounty ended | CommunityBot | ||
| S May 16, 2021 at 17:02 | history | notice removed | CommunityBot | ||
| S May 8, 2021 at 15:29 | history | bounty started | student | ||
| S May 8, 2021 at 15:29 | history | notice added | student | Draw attention | |
| May 5, 2021 at 13:18 | comment | added | Willie Wong | The fact that this holds for the ellipse is essentially due to the fact that in this case, the solution $u$ has constant Hessian. (Your ellipse can be written as the level set of some quadratic function. Up to a multiplicative scaling and a constant addition, that defining quadratic function is the solution to your PDE.) | |
| May 5, 2021 at 12:19 | comment | added | student | @MateuszKwaśnicki, thanks! I did not realize that the solution can be explicitly written down in this case. | |
| May 5, 2021 at 9:40 | comment | added | leo monsaingeon | and the symmetries of the ellipse can mislead the intuition. try experimenting with more generic domains, perhaps? | |
| May 5, 2021 at 9:33 | comment | added | Mateusz Kwaśnicki | For an ellipse, $u$ is just a quadratic function, no need for numerical experiments. | |
| May 5, 2021 at 9:29 | comment | added | student | @MateuszKwaśnicki, Numerical results on ellipses do imply that the conjecture is true, at least on this class of domains, so I'm guessing this is valid for general convex domains. | |
| May 5, 2021 at 8:35 | comment | added | Mateusz Kwaśnicki | Note that this is not a local problem, so I would not expect that the answer can be given in terms of local characteristics, such as curvature. To be specific: a tiny modification of the domain near an extremal point $p$ should not affect the answer much, but it can significantly change the curvature near $p$. | |
| May 5, 2021 at 8:24 | history | asked | student | CC BY-SA 4.0 |