10 problems

Bögli–Kennedy–Lang conjecture. If $\operatorname{Re}\alpha \to -\infty$, then there exists an infinite family of analytic branches of absolutely divergent eigenvalues such that, if…

Length-scaled spectral-ratio conjecture. For every $\alpha\geq-2\pi$, the quantity

Ball spectral-ratio conjecture. For $\alpha\geq-1/R$ with $\alpha\neq0$, the ratio

Andrews–Clutterbuck–Hauer conjecture. The Robin spectral gap is minimized by the degenerate box, identified with the line segment $(0,D)$:

Freitas–Laugesen conjecture. The ball maximizes

Rayleigh–Bossel conjecture. For each $\alpha\in\mathbb R$, the scale-invariant quantity

Second-eigenvalue concavity conjecture. The function $\lambda_2(\Omega;\alpha)$ is concave in $\alpha>0$. Concavity is known for domains with a suitable symmetry, including balls a…

Spectral-ratio monotonicity conjecture. The map

Fix positive numbers $A$ and $\alpha$. For a Lipschitz domain $\Omega\subset\mathbb{R}^2$ of area $A$, let $\lambda_j(\Omega,\alpha)$ denote its $j$th Robin eigenvalue. Let the can…

Fix a dimension $d\geq 2$ and an integer $k\geq 3$. Let $\Omega\subset\mathbb{R}^d$ be a sufficiently smooth domain of volume $1$, and let $\mathcal{B}_k$ denote the disjoint union…