Neumann domains of planar analytic eigenfunctions
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909344298172416 |
|---|---|
| author | Anoop, T. V. Bobkov, Vladimir Ghosh, Mrityunjoy |
| author_facet | Anoop, T. V. Bobkov, Vladimir Ghosh, Mrityunjoy |
| contents | Along with the partition of a planar bounded domain $Ω$ by the nodal set of a fixed eigenfunction of the Laplace operator in $Ω$, one can consider another natural partition of $Ω$ by, roughly speaking, gradient flow lines of a special type (separatrices) of this eigenfunction. Elements of such partition are called Neumann domains and their boundaries are Neumann lines. When the eigenfunction is a Morse function, this partition corresponds to the Morse--Smale complex and its fundamental properties have been systematically investigated by Band & Fajman (2016). Although, in the case of general position, eigenfunctions are always of the Morse type, particular eigenfunctions can possess degenerate critical points. In the present work, we propose a way to characterize Neumann domains and lines of an arbitrary eigenfunction. Instead of requiring the nondegeneracy of critical points of the eigenfunction, its real analyticity is principally used. The analyticity allows for the presence of degenerate critical points but significantly limits their possible diversity. Even so, the eigenfunction can possess curves of critical points, which have to belong naturally to the Neumann lines set, as well as critical points of a saddle-node type. We overview all possible types of degenerate critical points in the eigenfunction's critical set and provide a numerically based evidence that each of them can be observed for particular eigenfunctions. Alongside with [Band & Fajman, 2016], our approach is inspired by a little-known note of Weinberger that appeared back in 1963, where a part of the Neumann line set, under the name of "effectless cut", was explicitly introduced and studied for the first eigenfunctions in domains with nontrivial topology. In addition, we provide an asymptotic counting of Neumann domains for a disk and rectangles in analogy with the Pleijel constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07811 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Neumann domains of planar analytic eigenfunctions Anoop, T. V. Bobkov, Vladimir Ghosh, Mrityunjoy Analysis of PDEs Dynamical Systems Spectral Theory 35P05, 58K05, 26E05, 35P15 Along with the partition of a planar bounded domain $Ω$ by the nodal set of a fixed eigenfunction of the Laplace operator in $Ω$, one can consider another natural partition of $Ω$ by, roughly speaking, gradient flow lines of a special type (separatrices) of this eigenfunction. Elements of such partition are called Neumann domains and their boundaries are Neumann lines. When the eigenfunction is a Morse function, this partition corresponds to the Morse--Smale complex and its fundamental properties have been systematically investigated by Band & Fajman (2016). Although, in the case of general position, eigenfunctions are always of the Morse type, particular eigenfunctions can possess degenerate critical points. In the present work, we propose a way to characterize Neumann domains and lines of an arbitrary eigenfunction. Instead of requiring the nondegeneracy of critical points of the eigenfunction, its real analyticity is principally used. The analyticity allows for the presence of degenerate critical points but significantly limits their possible diversity. Even so, the eigenfunction can possess curves of critical points, which have to belong naturally to the Neumann lines set, as well as critical points of a saddle-node type. We overview all possible types of degenerate critical points in the eigenfunction's critical set and provide a numerically based evidence that each of them can be observed for particular eigenfunctions. Alongside with [Band & Fajman, 2016], our approach is inspired by a little-known note of Weinberger that appeared back in 1963, where a part of the Neumann line set, under the name of "effectless cut", was explicitly introduced and studied for the first eigenfunctions in domains with nontrivial topology. In addition, we provide an asymptotic counting of Neumann domains for a disk and rectangles in analogy with the Pleijel constant. |
| title | Neumann domains of planar analytic eigenfunctions |
| topic | Analysis of PDEs Dynamical Systems Spectral Theory 35P05, 58K05, 26E05, 35P15 |
| url | https://arxiv.org/abs/2410.07811 |