The first Robin eigenvalue with negative boundary parameter

0446668 - ÚJF 2016 RIV US eng J - Journal Article
Freitas, P. - Krejčiřík, David
The first Robin eigenvalue with negative boundary parameter.
Advances in Mathematics. Roč. 280, AUG (2015), s. 322-339. ISSN 0001-8708. E-ISSN 1090-2082
R&D Projects: GA ČR(CZ) GA14-06818S
Institutional support: RVO:61389005
Keywords : Eigenvalue optimisation * Laplacian * Robin boundary condition * negative parameter * asymptotics * disproval of Bareket's conjecture
Subject RIV: BE - Theoretical Physics
Impact factor: 1.405, year: 2015

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.
Permanent Link: http://hdl.handle.net/11104/0248681