Optimisation and monotonicity of the second Robin eigenvalue on a planar exterior domain
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915136276529152 |
|---|---|
| author | Krejcirik, David Lotoreichik, Vladimir |
| author_facet | Krejcirik, David Lotoreichik, Vladimir |
| contents | We consider the Laplace operator in the exterior of a compact set in the plane, subject to Robin boundary conditions. If the boundary coupling is sufficiently negative, there are at least two discrete eigenvalues below the essential spectrum. We state a general conjecture that the second eigenvalue is maximised by the exterior of a disk under isochoric or isoperimetric constraints. We prove an isoelastic version of the conjecture for the exterior of convex domains. Finally, we establish a monotonicity result for the second eigenvalue under the condition that the compact set is strictly star-shaped and centrally symmetric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_14286 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimisation and monotonicity of the second Robin eigenvalue on a planar exterior domain Krejcirik, David Lotoreichik, Vladimir Optimization and Control Mathematical Physics Analysis of PDEs Spectral Theory We consider the Laplace operator in the exterior of a compact set in the plane, subject to Robin boundary conditions. If the boundary coupling is sufficiently negative, there are at least two discrete eigenvalues below the essential spectrum. We state a general conjecture that the second eigenvalue is maximised by the exterior of a disk under isochoric or isoperimetric constraints. We prove an isoelastic version of the conjecture for the exterior of convex domains. Finally, we establish a monotonicity result for the second eigenvalue under the condition that the compact set is strictly star-shaped and centrally symmetric. |
| title | Optimisation and monotonicity of the second Robin eigenvalue on a planar exterior domain |
| topic | Optimization and Control Mathematical Physics Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2307.14286 |