21 problems

Let $\Omega$ be a bounded domain, let $\sigma>0$ be the Robin boundary parameter, and let $\mu_1(\sigma)\leq\mu_2(\sigma)\leq\cdots$ denote the Robin Laplacian eigenvalues. Laugese…

Let $\Omega\subset\mathbb{R}^N$ be a bounded connected domain, let $\sigma>0$, and let $\mu_1\leq\mu_2\leq\cdots$ be the Robin Laplacian eigenvalues defined by … where $\mathbf n$…

Let $\Omega\subset\mathbb{R}^d$ be a domain of volume $|\Omega|$, let $B_R$ be a ball with $|B_R|=|\Omega|$, and let $\lambda_1^\alpha(\Omega)$ denote the lowest eigenvalue of the…

Let $\Omega\subset\mathbb{R}^n$, with $n\geq 3$, be a domain with fixed surface area of the boundary, and consider the lowest eigenvalue of the Laplacian on $\Omega$ with a negativ…

Let $\Omega\subset\mathbb{R}^2$ be a simply connected planar domain with fixed area, and consider the lowest eigenvalue of the Laplacian on $\Omega$ with a negative Robin boundary…

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…

Let $\Omega\subset\mathbb R^2$ be a bounded simply-connected domain of fixed area, and let $\lambda_1^\alpha(\Omega)$ denote the first eigenvalue of the Robin Laplacian with parame…

Let ${\mathcal M}\subset\mathbb R^2$ be a simply-connected bounded domain of fixed area, with Robin parameter $\beta<0$, and let the lowest Robin eigenvalue be defined by … where…

Let ${\mathcal M}\subset\mathbb R^d$ be a smooth bounded domain of fixed volume, and let the Robin parameter satisfy $\beta<0$. The lowest Robin eigenvalue is defined by … where…

Let $\Omega$ be a general smooth domain in dimension $d\geq 2$, and let $\alpha\in\mathbb{C}$ be the complex Robin parameter for the Robin Laplacian on $\Omega$. Consider the eigen…

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…

Let $\Omega$ be a domain, let $\beta>0$, and let $\Omega^*$ be a ball with the same volume as $\Omega$. Denote by $\lambda_1(\Omega,\beta)$ and $\lambda_2(\Omega,\beta)$ the first…

Optimal-ball conjecture. For each $n$, there exists a positive value $\alpha_n$ such that, for all $0<\alpha<\alpha_n$, the $n^{\rm th}$ eigenvalue is minimised by $n$ identical ba…