Timeline for Elliptic equations and Fredholms alternative in the non-compact case
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 22, 2023 at 16:41 | comment | added | Overflowian | @G.Blaickner Well in the setting of manifolds with cylindrical ends, under certain conditions, if the source is L^2-orthogonal to the kernel of the formal adjoint of the operator then there is a solution which decays exponentially over the ends but this doesn't mean that the solution will be compactly supported. | |
| Jul 21, 2023 at 15:00 | comment | added | G. Blaickner | @Overflowian actually, my question is motivated from this post. Rbega2 claimed that a necessaty condition for compactly supported solutions is that the source is orthogonal to the kernel of L. I wanted to know if it is also sufficient. Since in this case my Poisson equation can be solved, as $\delta\omega$ is clearly orthogonal to harmonic L^2 functions, since harmonic L^2 functions are closed. | |
| Jul 21, 2023 at 14:54 | comment | added | Overflowian | @G.Blaickner in view of your last comment then this might be relevant: mathoverflow.net/questions/442169/poisson-equation-on-manifolds | |
| Jul 21, 2023 at 14:45 | comment | added | G. Blaickner | More precisly, the example I am interested most, is whether there are compactly supported smooth solutions of the equation $\Delta_g f=\delta\omega$ for some compactly supported smooth 1 form omega. | |
| Jul 21, 2023 at 14:44 | comment | added | G. Blaickner | Thanks, I will check it out! Also, I just realized, my precise formulation also makes not a lot of sense since $\mathrm{ker}(L\vert_{c})$ is probably empty for many operators by unique continuation | |
| Jul 21, 2023 at 14:34 | comment | added | Liviu Nicolaescu | This is tricky. There are few results in the noncompact case since noncompactness can manifest itself in so many ways. An old result that covers a special case of non compactness is due to Lockhard and McOwen. numdam.org/item/ASNSP_1985_4_12_3_409_0 | |
| Jul 21, 2023 at 14:00 | history | asked | G. Blaickner | CC BY-SA 4.0 |