The first Robin eigenvalue with negative boundary parameter

• We consider the Robin Laplacian with a negative boundary parameter.
• Bareket conjectured in 1977 that the ball maximises the first eigenvalue.
• We derive the asymptotics for spherical shells in the large negative regime.
• This provides a counterexample, showing the conjecture cannot hold in general.
• We show that the conjecture holds for 2D domains with a small boundary parameter.

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.