Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles
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- by D. Buoso and P. Freitas PDF
- Proc. Amer. Math. Soc. 148 (2020), 1109-1120 Request permission
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.References
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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
- MR Author ID: 1050257
- 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
- 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
- © Copyright 2019 American Mathematical Society
- 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
- MathSciNet review: 4055938