Timeline for Are all positive eigenfunctions principal eigenfunctions?
Current License: CC BY-SA 4.0
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| Aug 14, 2023 at 21:06 | history | edited | Connor Mooney | CC BY-SA 4.0 |
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| Aug 14, 2023 at 21:05 | comment | added | Connor Mooney | @Jochen: True. To be precise I've added that we consider Dirichlet boundary conditions, $L = \Delta$, and smooth domains. As you point out, the argument works in more general scenarios provided the eigenfunctions are in $C^2(\Omega) \cap C^1(\overline{\Omega})$. | |
| Aug 14, 2023 at 21:01 | history | edited | Connor Mooney | CC BY-SA 4.0 |
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| Aug 14, 2023 at 16:53 | comment | added | Jochen Glueck | +1 A small remark: I think one does not only need smoothness of $\partial \Omega$ but also elliptic boundary regularity to get that $u$ and $u_0$ are $C^1$ up to the boundary (which is needed to apply Hopf's lemma and also to get that $u_0$ is dominated by a multiple of $u$, which is part of the first step in your argument). This might somewhat restrict the boundary conditions one can allow for (but I'm not sure for precisely which boundary conditions elliptic boundary conditions still holds or goes wrong). | |
| Aug 14, 2023 at 15:13 | history | edited | Connor Mooney | CC BY-SA 4.0 |
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| Aug 14, 2023 at 10:23 | comment | added | Connor Mooney | @Giorgio: thanks for your comment. Yes, I've clarified in my answer that the eigenvalue associated to $u_0$ satisfies $\lambda_0 \leq \lambda$, hence $L(u-u_0) = -\lambda u + \lambda_0 u_0 \leq 0$. | |
| Aug 14, 2023 at 10:21 | history | edited | Connor Mooney | CC BY-SA 4.0 |
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| Aug 14, 2023 at 8:14 | comment | added | Giorgio Metafune | Does your argument work if $u, u_0$ are eigenfunctions with respect to different eigenvalues? | |
| Aug 14, 2023 at 2:57 | history | edited | Connor Mooney | CC BY-SA 4.0 |
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| Aug 13, 2023 at 23:59 | history | answered | Connor Mooney | CC BY-SA 4.0 |