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Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles


Authors: D. Buoso and P. Freitas
Journal: Proc. Amer. Math. Soc. 148 (2020), 1109-1120
MSC (2010): Primary 35J30; Secondary 35P15, 49R50, 74K20
DOI: https://doi.org/10.1090/proc/14792
Published electronically: October 18, 2019
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Abstract: We study the behaviour of extremal eigenvalues of the Dirichlet biharmonic operator over rectangles with a given fixed area. We begin by proving that the principal eigenvalue is minimal for a rectangle for which the ratio between the longest and the shortest side lengths does not exceed $ 1.066459$. We then consider the sequence formed by the minimal $ k$th eigenvalue and show that the corresponding sequence of minimising rectangles converges to the square as $ k$ goes to infinity.


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Additional Information

D. Buoso
Affiliation: École Polytechnique Fédéral de Lausanne, EPFL SB MATH, SCI-SB-JS, MA B3 514 (Bâtiment MA), Station 8, CH-1015 Lausanne, Switzerland
Email: davide.buoso@epfl.ch

P. Freitas
Affiliation: Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, P-1049-001 Lisboa, Portugal; Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, P-1749-016 Lisboa, Portugal
Email: psfreitas@fc.ul.pt

DOI: https://doi.org/10.1090/proc/14792
Keywords: Biharmonic operator, shape optimisation, rectangles, eigenvalues, isoperimetric inequality
Received by editor(s): January 3, 2019
Received by editor(s) in revised form: June 19, 2019
Published electronically: October 18, 2019
Additional Notes: This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portugal) through project Extremal spectral quantities and related problems (PTDC/MAT-CAL/4334/2014)
Most of the research in this paper was carried out while the first author held a post-doctoral position at the University of Lisbon within the scope of this project. The first author is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2019 American Mathematical Society