1. The Robin Laplacian

Monotonicity of the spectral gap
The conjecture was originally formulated by R. Smits [MR1381604]. The following formulation can be found in a recent preprint by R. Laugesen [arXiv:1905.07658].Conjecture 1.1.
[R. Smits] For a bounded convex domain $\Omega \subset \mathbb{R}^n$, the spectral gap $\lambda_2(\Omega, \alpha)\lambda_1(\Omega, \alpha)$ is strictly increasing as a function of $\alpha>0$. In particular, the Neumann gap provides a lower bound for the Dirichlet gap: $$ \mu_1(\Omega)<\lambda_2(\Omega)\lambda_1(\Omega). $$ 
Bareket’s conjecture
In a paper by M. Bareket [MR430552], it was conjectured that the lowest Robin eigenvalue $\lambda_1(\Omega,\alpha)$ on a bounded domain $\Omega$ with a boundary parameter $\alpha<0$ is maximized by the ball among all sets of equal measure. \REMOVE{In the planar case,} P. Freitas and D. Krejcirik [MR3350222] disproved the general statement by considering the limit $\alpha\to \infty$ and comparing asymptotic expansion for $\lambda_1(B_r, \alpha)$ with that of $\lambda_1(B_{r_1}\setminus B_{r_2}, \alpha)$, where $B_r$ stands for the ball of radius $r >0$ centred at the origin. In the same paper, it was proved that when $\alpha$ is small enough the disk is the unique maximizer.Conjecture 1.2.
[D. Krejcirik] Let $\lambda_1(\Omega, \alpha)$ be the lowest Robin eigenvalue on a bounded domain $\Omega$ with a negative boundary parameter $\alpha$. In two dimensions, the disk maximizes $\lambda_1(\Omega, \alpha)$ among all simply connected domains of the same area.
 In three and higher dimensions, the ball maximizes $\lambda_1(\Omega, \alpha)$ among all convex domains of the same volume.
\ADD{reference to my paper with Antunes} 
Discrete Bareket’s conjecture
In the same spirit as Bareket’s conjecture one can ask what polygon maximizes $\lambda_1(\Omega, \alpha)$, $\alpha<0$, among all polygons of the same area and having no more than $N$ sides. Similarly, one can ask the corresponding question when $\alpha>0$ in which case the problem is to minimize the eigenvalue. The following conjecture concerns the case $N = 3$.Conjecture 1.3.
[V. Lotoreichik, Z. Lu] Among all triangles of a given area $\lambda_1(\Omega, \alpha)$ is: minimized by the equilateral triangle when $\alpha>0$.
 maximized by the equilateral triangle when $\alpha<0$.

Negative Robin eigenvalues on a locally deformed halfspace
Consider the Robin Laplacian with a negative boundary parameter $\alpha$ on a locally perturbed halfspace (its boundary is a hyperplane away from a bounded region). The essential spectrum of this operator is expected to be $[\alpha^2, \infty)$. In the strong coupling limit $\alpha\to \infty$, existence of negative eigenvalues below the essential spectrum reduces to showing that the maximum for the mean curvature of the boundary is positive [MR3626320].Problem 1.4.
[V. Lotoreichik] Prove or disprove that there are always negative eigenvalues below $\alpha^2$.
Cite this as: AimPL: Shape optimization with surface interactions, available at http://aimpl.org/shapesurface.