Timeline for Upper bound for the first eigenvalue of the Laplacian on surfaces with boundary
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| S Apr 10, 2023 at 2:03 | history | bounty ended | CommunityBot | ||
| S Apr 10, 2023 at 2:03 | history | notice removed | CommunityBot | ||
| S Apr 2, 2023 at 0:34 | history | bounty started | Eduardo Longa | ||
| S Apr 2, 2023 at 0:34 | history | notice added | Eduardo Longa | Authoritative reference needed | |
| Jan 5, 2023 at 17:40 | comment | added | Neal | Intuition behind the cylinder counterexample in the linked question is: if "most" of the volume is "close to" the boundary, then $\lambda_1$ goes like $(\mbox{distance between boundary components})^{-2}$. So I suspect the idea can be adapted for any topological surface with boundary to produce a class of metrics with constant area and $\lambda_1$ growing without bound. | |
| Jan 5, 2023 at 0:23 | comment | added | Eduardo Longa | @L.F.Cavenaghi I don’t see how to relate the eigenvalues of circles and surfaces. I’m assuming Dirichlet boundary condition… | |
| Jan 5, 2023 at 0:05 | comment | added | L.F. Cavenaghi | some remarks: $\lambda_1$ is continuous at the $C^0$-topology in the space of metrics for $\Sigma$ fixed. Now since the classification of surfaces with boundary only depends on removing open discs from closed surfaces, I suppose that $\lambda_1$ can be parametrized by unions of circles, precisely, to understand $\lambda_1$ it suffices to understand the number $k$ of connected components on the boundary and the eigenvalue of each circle in the boundary. So I think that such a quantity is infinite for any surface with $k\geq 2$-boundary connected components. | |
| Jan 4, 2023 at 23:27 | history | asked | Eduardo Longa | CC BY-SA 4.0 |