On the remainder term of the Berezin inequality on a convex domain
We study the Dirichlet eigenvalues of the Laplacian on a convex domain in $\mathbb{R}^n$, with $n\geq 2$. In particular, we generalize and improve upper bounds for the Riesz means of order $σ\geq 3/2$ established in an article by Geisinger, Laptev and Weidl. This is achieved by refining estimates fo...
Main Author: | |
---|---|
Format: | Text |
Language: | unknown |
Published: |
arXiv
2015
|
Subjects: | |
Online Access: | https://dx.doi.org/10.48550/arxiv.1509.06705 https://arxiv.org/abs/1509.06705 |
id |
ftdatacite:10.48550/arxiv.1509.06705 |
---|---|
record_format |
openpolar |
spelling |
ftdatacite:10.48550/arxiv.1509.06705 2023-05-15T17:07:17+02:00 On the remainder term of the Berezin inequality on a convex domain Larson, Simon 2015 https://dx.doi.org/10.48550/arxiv.1509.06705 https://arxiv.org/abs/1509.06705 unknown arXiv https://dx.doi.org/10.1090/proc/13386 arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Spectral Theory math.SP Mathematical Physics math-ph FOS Mathematics FOS Physical sciences 35P15 Primary, 47A75 Secondary article-journal Article ScholarlyArticle Text 2015 ftdatacite https://doi.org/10.48550/arxiv.1509.06705 https://doi.org/10.1090/proc/13386 2022-04-01T12:05:04Z We study the Dirichlet eigenvalues of the Laplacian on a convex domain in $\mathbb{R}^n$, with $n\geq 2$. In particular, we generalize and improve upper bounds for the Riesz means of order $σ\geq 3/2$ established in an article by Geisinger, Laptev and Weidl. This is achieved by refining estimates for a negative second term in the Berezin inequality. The obtained remainder term reflects the correct order of growth in the semi-classical limit and depends only on the measure of the boundary of the domain. We emphasize that such an improvement is for general $Ω\subset\mathbb{R}^n$ not possible and was previously known to hold only for planar convex domains satisfying certain geometric conditions. As a corollary we obtain lower bounds for the individual eigenvalues $λ_k$, which for a certain range of $k$ improves the Li--Yau inequality for convex domains. However, for convex domains one can use different methods to obtain even stronger such lower bounds. : Revised and accepted version. 13 pages Text laptev DataCite Metadata Store (German National Library of Science and Technology) |
institution |
Open Polar |
collection |
DataCite Metadata Store (German National Library of Science and Technology) |
op_collection_id |
ftdatacite |
language |
unknown |
topic |
Spectral Theory math.SP Mathematical Physics math-ph FOS Mathematics FOS Physical sciences 35P15 Primary, 47A75 Secondary |
spellingShingle |
Spectral Theory math.SP Mathematical Physics math-ph FOS Mathematics FOS Physical sciences 35P15 Primary, 47A75 Secondary Larson, Simon On the remainder term of the Berezin inequality on a convex domain |
topic_facet |
Spectral Theory math.SP Mathematical Physics math-ph FOS Mathematics FOS Physical sciences 35P15 Primary, 47A75 Secondary |
description |
We study the Dirichlet eigenvalues of the Laplacian on a convex domain in $\mathbb{R}^n$, with $n\geq 2$. In particular, we generalize and improve upper bounds for the Riesz means of order $σ\geq 3/2$ established in an article by Geisinger, Laptev and Weidl. This is achieved by refining estimates for a negative second term in the Berezin inequality. The obtained remainder term reflects the correct order of growth in the semi-classical limit and depends only on the measure of the boundary of the domain. We emphasize that such an improvement is for general $Ω\subset\mathbb{R}^n$ not possible and was previously known to hold only for planar convex domains satisfying certain geometric conditions. As a corollary we obtain lower bounds for the individual eigenvalues $λ_k$, which for a certain range of $k$ improves the Li--Yau inequality for convex domains. However, for convex domains one can use different methods to obtain even stronger such lower bounds. : Revised and accepted version. 13 pages |
format |
Text |
author |
Larson, Simon |
author_facet |
Larson, Simon |
author_sort |
Larson, Simon |
title |
On the remainder term of the Berezin inequality on a convex domain |
title_short |
On the remainder term of the Berezin inequality on a convex domain |
title_full |
On the remainder term of the Berezin inequality on a convex domain |
title_fullStr |
On the remainder term of the Berezin inequality on a convex domain |
title_full_unstemmed |
On the remainder term of the Berezin inequality on a convex domain |
title_sort |
on the remainder term of the berezin inequality on a convex domain |
publisher |
arXiv |
publishDate |
2015 |
url |
https://dx.doi.org/10.48550/arxiv.1509.06705 https://arxiv.org/abs/1509.06705 |
genre |
laptev |
genre_facet |
laptev |
op_relation |
https://dx.doi.org/10.1090/proc/13386 |
op_rights |
arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
op_doi |
https://doi.org/10.48550/arxiv.1509.06705 https://doi.org/10.1090/proc/13386 |
_version_ |
1766062659461971968 |