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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the remainder term of the Berezin inequality on a convex domain
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by Simon Larson PDF
Proc. Amer. Math. Soc. 145 (2017), 2167-2181 Request permission

Abstract:

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 $\lambda _k$.

References
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Additional Information
  • Simon Larson
  • Affiliation: Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
  • Email: simla@math.kth.se
  • 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
  • © Copyright 2016 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 2167-2181
  • MSC (2010): Primary 35P15; Secondary 47A75
  • DOI: https://doi.org/10.1090/proc/13386
  • MathSciNet review: 3611329