Timeline for Square Integrable Harmonic Functions in an Infinite Strip
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 8, 2015 at 13:45 | comment | added | Ali | It means solving $ \Delta_g u = 0$ with u being a tempered distribution. | |
| Jul 8, 2015 at 6:20 | comment | added | Alexandre Eremenko | what are the harmonic functions "in the distributional sense"?? | |
| Jul 7, 2015 at 23:30 | answer | added | Robert Israel | timeline score: 3 | |
| Jul 7, 2015 at 22:16 | comment | added | Christian Remling | @Ali: I'm not sure I understand your objections; maybe my comment was slightly misleading. I'm convolving with the Poisson kernel for upper half space. This gives me $\sim r^{-3}$ decay in the unrestricted variables; see here: en.wikipedia.org/wiki/Poisson_kernel#On_the_upper_half-space | |
| Jul 7, 2015 at 21:46 | comment | added | Ali | I suspect that it might not be difficult to construct specific counter examples but I want to understand a bit better what's going wrong ... Specifically is this because of the euclidean metric and if so can one characterize different metrics for which L^2 harmonic functions in (S,g) would be the trivial function | |
| Jul 7, 2015 at 21:43 | comment | added | Ali | Would you clarify why you think that would give an L^2 function? Poisson kernel is not square integrable as it is like 1/r | |
| Jul 7, 2015 at 21:32 | comment | added | Christian Remling | It's easy to build counterexamples that are actually harmonic in a half-space, by convolving the Poisson kernel with a compactly supported function. | |
| Jul 7, 2015 at 20:40 | comment | added | Joonas Ilmavirta | One notable difference to the case of the whole space is that there are bounded harmonic functions, like $\sinh(x_1)\sin(x_2)$. | |
| Jul 7, 2015 at 20:10 | history | asked | Ali | CC BY-SA 3.0 |