Timeline for Singularity of L^1-solutions to elliptic PDEs on the puntured ball
Current License: CC BY-SA 4.0
14 events
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| Oct 16, 2020 at 19:12 | history | edited | T. Le | CC BY-SA 4.0 |
I've added a new update which confirms the answer to the original question.
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| Oct 15, 2020 at 23:31 | comment | added | T. Le | @GiorgioMetafune I believe you're right! I was afraid that $Dv$ not being zero everywhere except at the origin may be a problem but yes, it's a distribution of order zero away from the origin so the argument still works. | |
| Oct 15, 2020 at 22:05 | comment | added | Giorgio Metafune | Why does not your argument apply to $v=\phi u$, where $\phi$ is a cut-off equal 1 near the origin? In such a case $Dv$ is no longer $0$, out of the origin, but a distribution of order 0 (assuming smoothness of $u$ out of the origin) and your argument should work. Maybe I am missing something. | |
| Oct 15, 2020 at 20:47 | comment | added | T. Le | I've updated with some details. | |
| Oct 15, 2020 at 20:46 | history | edited | T. Le | CC BY-SA 4.0 |
added 572 characters in body
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| Oct 15, 2020 at 20:35 | comment | added | T. Le | Sorry, I was not very clear. Since $D(u)=0$ everywhere except at the origin, $D(u)$ is a distribution supported as the origin, hence it is $L(\delta)$, where $L$ is a differential operator with constant coefficients and $\delta$ is the Dirac distribution at the origin. The order of $L$ is at most $N$ since the order of $D$ is $N$. | |
| Oct 15, 2020 at 16:12 | comment | added | Giorgio Metafune | Could you give some more detail? How do you deduce the estimate for the Fourier transform of $Du$? And what is $L(\delta)$? Thank you | |
| Oct 15, 2020 at 2:30 | history | edited | T. Le | CC BY-SA 4.0 |
provide an update to answering the question
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| Sep 29, 2020 at 21:36 | comment | added | Giorgio Metafune | Thank you very much the answers. I see the arguments but I cannot conclude for more general operators, too. | |
| Sep 29, 2020 at 18:41 | comment | added | Daniele Tampieri | @GiorgioMetafune I mean the two sided estimate of the Green's function: I should have made my comment clearer. And I do not know how to answer the question according to the idea of T. Le. | |
| Sep 29, 2020 at 17:07 | comment | added | T. Le | @GiorgioMetafune Here is what I had in mind. If $\Delta(u)$ has order 2, then $u$ must be a combination of a harmonic function (smooth at the origin), first partial derivatives, and second partial derivatives of the fundamental solution $F$. But since second partial derivatives of $F$ do not belong to $L^1$, $u$ cannot be in $L^1$ unless the coefficients of all second derivatives are zero. | |
| Sep 29, 2020 at 16:50 | comment | added | Giorgio Metafune | @Daniele Tampieri You mean the result for the two side estimates of the Green function, I guess. And then how to conclude that $D(u)$ has order less than 2? I have not even understood how @ T. Le concludes for the Laplacian. | |
| Sep 29, 2020 at 12:33 | comment | added | Daniele Tampieri | For general elliptic divergence-form partial differential operators with bounded measurable coefficients, the result is a consequence of a general theorem proved by Littman, Weinberger and Stampacchia for the Green's functions of these operators, for general domains and for the ball: you can find all the relevant references in the linked Q&A. | |
| Sep 28, 2020 at 23:30 | history | asked | T. Le | CC BY-SA 4.0 |