The Laplacian with Robin boundary conditions is known to satisfy a Faber-Krahn-type inequality for positive values of the boundary parameter. As this parameter becomes negative after crossing the Neumann case, the inequality reverses and a maximiser for the first eigenvalue is now expected to exist among domains with a given volume. This was conjectured to be the ball by Bareket in 1977, but in 2015, together with David Krejčiřík, we proved that for large negative values of the parameter this could not be the case [FK15]. TO the best of our knowledge, this was the first known example where the ball is not the extremal domain for the first eigenvalue of the Laplacian. In that paper we also proved that in the planar case, and for sufficiently small values of the parameter, the disk remains the maximiser. For fixed perimeter, we proved that the disk is the maximiser independently of the (negative) value of the bondary parameter [AFK17].
In [FL21] we showed that in any dimension and for a specific interval of the boundary parameter
that includes the range where the second eigenvalue remains positive, the ball is also the maximiser of the second
eigenvalue among domains with a given volume..
For positive values of the boundary parameter we considered the case of rectangles and unions of rectangles, showing that there
are regions where the union of k equal squares will minimise the kth eigenvalue, among domains of fixed area [FK21]. As
conjectured in [AFK13], a similar result is expected to happen also for general domains, with k equal balls minimising
the kth eigenvalue among domains with a given volume (Krahn's dream).
[AFK13] Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian. ESAIM: Control, Optimisation and Calculus of Variations 19 (2013), 438–459 (with Pedro R. S. Antunes and James B. Kennedy).
[AFK17] Bounds and extremal domains for Robin eigenvalues with negative boundary parameter. Adv. Calc. Var. 10 (2017), 357–379 (with Pedro R. S. Antunes and David Krejčiřík).
[FK15] The first Robin eigenvalue with negative boundary parameter. Adv. Math. 280 (2015), 322–339 (with David Krejčiřík).
[FK21] Extremal domains and Pólya-type inequalities for the Robin Laplacian on rectangles and unions of rectangles, Int. Math. Res. Not. IMRN (2021), no. 18, 13730–13782 (with James B. Kennedy).
[FL20] From Steklov to Neumann and beyond, via Robin: the Szegö way. Canadian J. Math. 72 (2020), 1024–1043 (with Richard S. Laugesen).
[FL21] From Neumann to Steklov and beyond, via Robin: the Weinberger way. Amer. J. Math. 143 (2021), 969–994 (with Richard S. Laugesen).