Timeline for Laplace equation on the disk with Robin boundary condition
Current License: CC BY-SA 4.0
9 events
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| Apr 9, 2021 at 8:47 | history | edited | username | CC BY-SA 4.0 |
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| Apr 9, 2021 at 8:44 | comment | added | username | @JacobLu added a paragraph about regularity | |
| Apr 9, 2021 at 7:55 | history | edited | username | CC BY-SA 4.0 |
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| Apr 9, 2021 at 4:13 | vote | accept | Jacob Lu | ||
| Apr 9, 2021 at 4:11 | comment | added | Jacob Lu | Many thanks for the detailed answer! This is very helpful! I understand that in general, the regularity of eigenfunctions should be similar to $b$. But can one rigorously rule out the possibility that the eigenfunction $f$ is smooth while $b$ is piecewisely-constant (discontinuous)? | |
| Apr 8, 2021 at 8:53 | history | edited | username | CC BY-SA 4.0 |
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| Apr 6, 2021 at 18:54 | comment | added | username | @JacobLu Yes, they are as smooth as is allowed by the domain and b. If the domain is very smooth, then they will have the smoothness inherited from b (the maximum smoothness b allows). For example, if $b$ is bounded but not differentiable, then $u\in W^{1-1/p,p}(\partial\Omega)$ for any $p$. | |
| Apr 6, 2021 at 4:26 | comment | added | Jacob Lu | Many thanks for the answer! I am not very familiar with the spectrum property of the Robin problem. Do you mean that for general $b$ satisfying some integrability, there always exist eigenvalues? Is it possible for these eigenfunctions to be smooth? | |
| Apr 3, 2021 at 9:46 | history | answered | username | CC BY-SA 4.0 |