Timeline for Elliptic operator becomes Fredholm
Current License: CC BY-SA 3.0
12 events
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| Oct 20, 2017 at 2:30 | comment | added | Bombyx mori | I think it is comes from the usual PDE situation where you impose two boundary conditions, say. Neumann on one end, Dirichlet on the other. Then using Fourier transform and some elementary tools like separation of variables, you can solve the PDE uniquely. This obviously does not totally carry through to the PsiDO case, but the idea is the same. | |
| Oct 20, 2017 at 1:36 | comment | added | DLIN | @Bombyxmori twice cylindrical end means "$\mathbb R\times Y$", the manifold $X$ has a cylindrical end $\mathbb R^+\times Y$. I do not understand why the author consider the Laplacian on whole tube "$\mathbb R\times Y$" rather than the half $\mathbb R^+\times Y$. | |
| Oct 18, 2017 at 15:11 | comment | added | Bombyx mori | @WillieWong: Yes! Thanks for the correction. I think at least the set-up of the problem, if you place it as $\Delta g=0$ and $g=f$ on the boundary, with $g\in W^{p,w}_{2}$ being the weighted Sobolev space is understandable to non-specialist. To me elliptic operators is just a natural extension of the generalized Laplacian. The gist of the original proof in the paper he linked seems to be relying on some perturbation argument in analytic Fredholm theory, which is standard for people work in the field. But I did not have time to check it in detail. | |
| Oct 18, 2017 at 15:06 | history | edited | Bombyx mori | CC BY-SA 3.0 |
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| Oct 18, 2017 at 15:00 | comment | added | Willie Wong | not being an expert in the field so I am not sure, but do you mean "Calderon projector" instead of what you wrote? | |
| Oct 18, 2017 at 13:20 | comment | added | Bombyx mori | I do not know what you mean by "twice cylinderal end". Actually I found the proof in the paper to be quite readable, so I am a bit confused. | |
| Oct 18, 2017 at 5:16 | comment | added | DLIN | Just this party, I do not understand why it is enough to say a operator $L$ on $X$ is Fredholm, if we can say that $L$ on the "twice cylindrical end" is Fredholm. | |
| Oct 18, 2017 at 2:02 | history | edited | Bombyx mori | CC BY-SA 3.0 |
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| Oct 18, 2017 at 1:50 | comment | added | Bombyx mori | @DLIN: Maybe I am not getting your idea correctly. Do you mean your $L^{p,w}_{2}$ is just the space in page 410 of the paper? Can you elaborate your difficulty in following the author's proof in the paper you linked? | |
| Oct 18, 2017 at 1:35 | comment | added | DLIN | $L^{p,w}_2$ is the $W^p_{i,w}$ is the paper you cited. | |
| Oct 17, 2017 at 15:23 | history | edited | Bombyx mori | CC BY-SA 3.0 |
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| Oct 17, 2017 at 15:18 | history | answered | Bombyx mori | CC BY-SA 3.0 |