Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.14122 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915198784241664 |
|---|---|
| author | Steinerberger, Stefan Venkatraman, Raghavendra |
| author_facet | Steinerberger, Stefan Venkatraman, Raghavendra |
| contents | We consider Laplacian eigenfunctions on a domain $Ω\subset \mathbb{R}^d$. Under Neumann boundary conditions, the first eigenfunction is constant and the others have mean value 0. The situation is different for Dirichlet boundary conditions: on `generic' domains, one would expect that every eigenfunction has nonzero mean value. The other extreme is the ball in $\mathbb{R}^d$, where among the first $n$ eigenfunctions only $\sim n^{1/d}$ have a mean value different from zero. We prove that this rate is sharp in \textit{any} smooth domain, up to a logarithmic factor: in any smooth domain~$Ω$, among the first $n$ Dirichlet eigenfunctions at least $ (\log{n})^{-1/2} \cdot n^{1/d} $ have a nonzero mean. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_14122 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dirichlet eigenfunctions with nonzero mean value Steinerberger, Stefan Venkatraman, Raghavendra Analysis of PDEs Spectral Theory We consider Laplacian eigenfunctions on a domain $Ω\subset \mathbb{R}^d$. Under Neumann boundary conditions, the first eigenfunction is constant and the others have mean value 0. The situation is different for Dirichlet boundary conditions: on `generic' domains, one would expect that every eigenfunction has nonzero mean value. The other extreme is the ball in $\mathbb{R}^d$, where among the first $n$ eigenfunctions only $\sim n^{1/d}$ have a mean value different from zero. We prove that this rate is sharp in \textit{any} smooth domain, up to a logarithmic factor: in any smooth domain~$Ω$, among the first $n$ Dirichlet eigenfunctions at least $ (\log{n})^{-1/2} \cdot n^{1/d} $ have a nonzero mean. |
| title | Dirichlet eigenfunctions with nonzero mean value |
| topic | Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2312.14122 |