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ON THE REMAINDER TERM OF THE BEREZIN INEQUALITY ON A CONVEX DOMAIN
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-0057-8211
2017 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 145, no 5, p. 2167-2181Article in journal (Refereed) Published
Abstract [en]

We study the Dirichlet eigenvalues of the Laplacian on a convex domain in R-n, with n >= 2. In particular, we generalize and improve upper bounds for the Riesz means of order sigma >= 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 of 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 use different methods to obtain even stronger lower bounds for lambda(k)

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2017. Vol. 145, no 5, p. 2167-2181
Keywords [en]
Dirichlet-Laplace operator, semi-classical estimates, Berezin-Li-Yau inequality
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:kth:diva-205425DOI: 10.1090/proc/13386ISI: 000395809900031Scopus ID: 2-s2.0-85013627273OAI: oai:DiVA.org:kth-205425DiVA, id: diva2:1097234
Note

QC 20170522

Available from: 2017-05-22 Created: 2017-05-22 Last updated: 2017-05-22Bibliographically approved

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Larson, Simon

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