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Advances in Calculus of Variations

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Volume 10, Issue 4


Bounds and extremal domains for Robin eigenvalues with negative boundary parameter

Pedro R. S. Antunes
  • Group of Mathematical Physics, Faculdade de Ciências da Universidade de Lisboa, Campo Grande,Edifício C6 1749-016 Lisboa, Portugal
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/ Pedro Freitas
  • Department of Mathematics, Faculty of Human Kinetics & Group of Mathematical Physics, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Edifício C6 1749-016 Lisboa, Portugal
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/ David Krejčiřík
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  • Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, 25068 Řež, Czech Republic
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Published Online: 2016-08-02 | DOI: https://doi.org/10.1515/acv-2015-0045


We present some new bounds for the first Robin eigenvalue with a negative boundary parameter. These include the constant volume problem, where the bounds are based on the shrinking coordinate method, and a proof that in the fixed perimeter case the disk maximises the first eigenvalue for all values of the parameter. This is in contrast with what happens in the constant area problem, where the disk is the maximiser only for small values of the boundary parameter. We also present sharp upper and lower bounds for the first eigenvalue of the ball and spherical shells. These results are complemented by the numerical optimisation of the first four and two eigenvalues in two and three dimensions, respectively, and an evaluation of the quality of the upper bounds obtained. We also study the bifurcations from the ball as the boundary parameter becomes large (negative).

Keywords: Eigenvalue optimisation; Robin Laplacian; negative boundary parameter; Bareket’s conjecture

MSC 2010: 58J50; 35P15


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About the article

Received: 2015-11-09

Accepted: 2016-06-21

Published Online: 2016-08-02

Published in Print: 2017-10-01

Funding Source: Fundação para a Ciência e a Tecnologia

Award identifier / Grant number: PTDC/MAT-CAL/4334/2014

Award identifier / Grant number: IF/00177/2013

This research was partially supported by FCT (Portugal) through project PTDC/MAT-CAL/4334/ 2014. The research of the first author was partially supported by FCT, Portugal, through the program “Investigador FCT” with reference IF/00177/2013. The research of the third author was supported by the project RVO61389005 and the GACR grant No. 14-06818S.

Citation Information: Advances in Calculus of Variations, Volume 10, Issue 4, Pages 357–379, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2015-0045.

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