
## 1. The Robin Laplacian

1. ### 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: $$\mu_1(\Omega)<\lambda_2(\Omega)-\lambda_1(\Omega).$$
It was checked by Smits that the conjecture is true for the disk and also for rectangular boxes in all dimensions. For the special case of rectangular boxes, monotonicity of the gap holds for all $-\infty<\alpha<\infty$, see Theorem 2.1 in [arXiv:1905.07658]. However, it is known that the monotonicity fails in general when $\alpha < 0$ becomes sufficiently large by absolute value.
• ### 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 $\alpha$.
1. In two dimensions, the disk maximizes $\lambda_1(\Omega, \alpha)$ among all simply connected domains of the same area.
2. 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}
Counterparts of this conjecture under fixed area (perimeter) of the boundary are already settled in two dimensions [MR3707083] for general domains and also in three and higher dimensions [arXiv:1810.06108], [arXiv:1906.11141] in the class of convex domains.
• ### 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:
1. minimized by the equilateral triangle when $\alpha>0$.
2. maximized by the equilateral triangle when $\alpha<0$.
• ### Negative Robin eigenvalues on a locally deformed half-space

Consider the Robin Laplacian with a negative boundary parameter $\alpha$ on a locally perturbed half-space (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.