Timeline for Solving $\Delta \text{tr}(h) - \mathrm{div}(\mathrm{div}(h)) + \text{tr}(h) = f$ on $S^2$
Current License: CC BY-SA 4.0
6 events
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| Dec 15, 2021 at 16:19 | vote | accept | Laithy | ||
| Dec 14, 2021 at 20:51 | comment | added | Willie Wong | Actually: thinking a bit more: if you have a trace-free symmetric two tensor, then the double divergence only contains those spherical harmonics with $\ell \geq 2$. Since the eigenvalue $2$ corresponds to the $\ell = 1$ spherical harmonic, the obstruction you saw still persists: that if $f$ has a spherical harmonic component with $\ell = 1$ the equation has no solution. | |
| Dec 14, 2021 at 18:00 | comment | added | Willie Wong | Since you are working on the round sphere, you can actually do this by spherical harmonics. See e.g. the set-up in Czimek's APDE paper link.springer.com/article/10.1007/s40818-017-0039-3#Sec5 starting around equation (2.7). | |
| Dec 14, 2021 at 16:35 | answer | added | MyShepherd | timeline score: 4 | |
| Dec 14, 2021 at 6:41 | history | edited | YCor | CC BY-SA 4.0 |
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| Dec 14, 2021 at 5:49 | history | asked | Laithy | CC BY-SA 4.0 |