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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the remainder term of the Berezin inequality on a convex domain

Author: Simon Larson
Journal: Proc. Amer. Math. Soc. 145 (2017), 2167-2181
MSC (2010): Primary 35P15; Secondary 47A75
Published electronically: November 18, 2016
MathSciNet review: 3611329
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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 $\sigma \geq 3/2$ 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 $\Omega \subset \mathbb {R}^n$ 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 $\lambda _k$, which for a certain range of $k$ improves the Li–Yau inequality for convex domains. However, for convex domains one can by using different methods obtain even stronger lower bounds for $\lambda _k$.

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Simon Larson
Affiliation: Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden

Keywords: Dirichlet-Laplace operator, semi-classical estimates, Berezin–Li–Yau inequality
Received by editor(s): December 15, 2015
Received by editor(s) in revised form: December 31, 2015, July 13, 2016, and July 15, 2016
Published electronically: November 18, 2016
Communicated by: Michael Hitrik
Article copyright: © Copyright 2016 American Mathematical Society