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Asymptotic Behaviour of Cuboids Optimising Laplacian Eigenvalues
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0002-0057-8211
2017 (English)In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 89, no 4, p. 607-629Article in journal (Refereed) Published
Abstract [en]

We prove that in dimension n≥ 2 , within the collection of unit-measure cuboids in Rn (i.e. domains of the form ∏i=1n(0,an)), any sequence of minimising domains RkD for the Dirichlet eigenvalues λk converges to the unit cube as k→ ∞. Correspondingly we also prove that any sequence of maximising domains RkN for the Neumann eigenvalues μk within the same collection of domains converges to the unit cube as k→ ∞. For n= 2 this result was obtained by Antunes and Freitas in the case of Dirichlet eigenvalues and van den Berg, Bucur and Gittins for the Neumann eigenvalues. The Dirichlet case for n= 3 was recently treated by van den Berg and Gittins. In addition we obtain stability results for the optimal eigenvalues as k→ ∞. We also obtain corresponding shape optimisation results for the Riesz means of eigenvalues in the same collection of cuboids. For the Dirichlet case this allows us to address the shape optimisation of the average of the first k eigenvalues.

Place, publisher, year, edition, pages
Springer Basel , 2017. Vol. 89, no 4, p. 607-629
Keywords [en]
Asymptotics, Cuboids, Eigenvalues, Laplacian, Spectral optimisation
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:kth:diva-227117DOI: 10.1007/s00020-017-2407-5Scopus ID: 2-s2.0-85032668674OAI: oai:DiVA.org:kth-227117DiVA, id: diva2:1204560
Funder
Swedish Research Council
Note

QC 20180508

Available from: 2018-05-08 Created: 2018-05-08 Last updated: 2018-05-08Bibliographically approved

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Larson, Simon

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