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The first Robin eigenvalue with negative boundary parameter |
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| Authors: | Pedro Freitas,David Krejčiří k |
| Affiliation: | 1. Department of Mathematics, Faculty of Human Kinetics & Group of Mathematical Physics, Universidade de Lisboa, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, P-1649-003 Lisboa, Portugal;2. Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, 250 68 ?e?, Czech Republic |
| Abstract: | We give a counterexample to the long standing conjecture that the ball maximises the first eigenvalue of the Robin eigenvalue problem with negative parameter among domains of the same volume. Furthermore, we show that the conjecture holds in two dimensions provided that the boundary parameter is small. This is the first known example within the class of isoperimetric spectral problems for the first eigenvalue of the Laplacian where the ball is not an optimiser. |
| Keywords: | Eigenvalue optimisation Laplacian Robin boundary condition Negative parameter Asymptotics Disproval of Bareket's conjecture |
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