Timeline for Pde system problem
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 25, 2017 at 10:24 | comment | added | Robert Bryant | @exxxit8: Yes, up to isometry. | |
| Nov 25, 2017 at 8:35 | comment | added | MathDG | @RobertBryant, and these are the only nontrivial metrics (for $c=0$ and $\neq 0$), that the surface can admits, is it right? | |
| Nov 24, 2017 at 21:16 | comment | added | MathDG | @RobertBryant, then is enough to say that the surface is smooth for consider both metrics not singular.Thank you for your help | |
| Nov 24, 2017 at 21:10 | comment | added | Robert Bryant | @exxxit8: Yes, that is correct. | |
| Nov 24, 2017 at 20:58 | comment | added | MathDG | @RobertBryant, then is the same for $c=0$, in both cases ($c=0$ and $c \neq 0$ ) the metric is singular in $r=0$, but if I consider only smooth surfaces, the metrics (in both cases) are not singular because $r=0$ will be not contained in the surfaces..is it right? | |
| Nov 24, 2017 at 20:44 | comment | added | Robert Bryant | @exxxit8: It is not singular except at $r=0$, but then the metric is singular there, so you could certainly say that the solution is not singular on a smooth surface. | |
| Nov 24, 2017 at 18:00 | comment | added | MathDG | @RobertBryant, the solution with $c \neq 0$ is not singular, I am right? | |
| Nov 15, 2017 at 21:34 | comment | added | MathDG | @Robert Bryant, it is exactly this, I did not think that we are in a simply-connected domain where $u$ is nonzero, then there isn't reason to ask if there is a boundary...thank you very much! | |
| Nov 15, 2017 at 21:14 | comment | added | Robert Bryant | @AlexanderPigazzini: I'm not sure exactly what you are asking. A non-zero solution $u$ cannot go to zero in finite distance, even if $c$ is not zero. The solution as I've described it would be valid on any simply-connected domain on which $u$ is nonzero, so I'm not sure what the boundary, if one exists, would have to do with anything (assuming that the metric and solution are sufficiently differentiable on the boundary). Does that answer your question? | |
| Nov 15, 2017 at 20:45 | comment | added | MathDG | @Robert Bryant, sorry if I intrude in the discussion, but the last solution ($c \neq 0$) is valid for surfaces with boundary ($\partial S \neq0$) or is the same also for surfaces without boundary ($\partial S=0$)? | |
| Nov 15, 2017 at 14:23 | comment | added | Robert Bryant | @exxxit8: Yes. The analysis is essentially the same: Set $u=-hf$, and the first equation becomes $\Delta u + u^2 + cu = 0$. On the open set where $u\not=0$, it is positive, so set $u = 4/r^2$, and we find that $\omega_1 = \mathrm{d}r$ and $\omega_2 = r^5\mathrm{e}^{(c/4)r^2}\mathrm{d}\theta$ for some angular coordinate $\theta$. The metric $$g=\mathrm{d}r^2 +r^{10}\mathrm{e}^{(c/2)r^2}\mathrm{d}\theta ^2$$ is therefore not flat, so there are no nontrivial solutions on the flat plane, but there is a nontrivial solution $f = -4/(hr^2)$ with $g$ as background metric. | |
| Nov 15, 2017 at 13:36 | comment | added | MathDG | you are right! I read fast. And for $c$ not zero, Do you expect something like that? | |
| Nov 15, 2017 at 13:18 | comment | added | Robert Bryant | @exxxit8: You have not read my analysis carefully. I showed that, when $h$ is nonzero, then, on the open set where $f$ is nonzero, it has a particular form (as described above). Obviously, $f=0$ is a solution, but that can be discarded as trivial. | |
| Nov 15, 2017 at 13:11 | comment | added | MathDG | you say there is only one solution (ie $ f $ constant and $ h = 0 $) because if $ h $ is not zero, $ f $ is a not smooth function..is this so? instead in the case of Igor Khavkine (which is the case where the metric is flat) the solution for $h$ not zero is $f=0$. | |
| Nov 15, 2017 at 12:00 | comment | added | MathDG | thank you very much for your answer, then if I understand for the case with $c$ not zero, the answer is the same, i.e. $f$ constant and $h=c=0$ ? | |
| Nov 15, 2017 at 11:53 | vote | accept | MathDG | ||
| Nov 15, 2017 at 10:44 | history | edited | Robert Bryant | CC BY-SA 3.0 |
added 121 characters in body
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| Nov 15, 2017 at 10:36 | history | answered | Robert Bryant | CC BY-SA 3.0 |