20 problems

Let $\Omega$ range over bounded domains in $\mathbb{R}^n$ with fixed volume, and let $\beta<0$ be the Robin parameter. Bareket's conjecture. The ball maximizes the first Robin eige…

Let $\Omega\subset\mathbb{R}^2$ be a domain of fixed area, and let $\operatorname{l}_{1,\alpha}(\Omega)$ denote its first Robin eigenvalue for $\alpha<0$. For sufficiently small…

Let $\Omega$ be a domain in $\mathbb{R}^n$ with prescribed volume, let $\beta<0$ be the Robin parameter, and let $\operatorname{l}_{1}(\Omega,\beta)$ denote the first Robin eigenva…

Let $\mathcal{O}mega\tenrm ext$ be the exterior of a bounded non-convex domain $\mathcal{O}mega$ in the hyperbolic plane, and define the critical boundary parameter by … Here…

Let the Robin Laplacian be defined on a bounded domain in the hyperbolic plane, with negative boundary parameter, and let the domain class consist of simply-connected planar domain…

Let $n=2$, let $B_R$ be a disk of radius $R$, and let $\beta$ be a fixed perimeter or area constraint. For a bounded simply connected open set $\Omega\subset\mathbb{R}^2$, write…

Let $B_R$ be a ball and let $\lambda_1^\alpha(\Omega^\text{ext})$ denote the first Robin eigenvalue of the exterior domain associated with a nearly spherical perturbation…

Second Robin eigenvalue conjecture. If $\mathcal B$ is a disk, then

Suppose $F:\mathbb D\to\Omega$ is a conformal map of the unit disk onto a Jordan–Lipschitz domain $\Omega$, and let $L(\Omega)$ denote the perimeter of $\Omega$. For $\alpha>0$, th…

Let $\Omega$ be a convex bounded domain in $\mathbb R^n$, with volume $V$ and surface area $S$, and let $\mathbb B$ be the unit ball. For…

Let $\Omega$ be a convex bounded planar domain, let $L(\Omega)$ denote its perimeter, and let $\lambda_1(\Omega;\alpha/L(\Omega))$ be its first Robin eigenvalue with perimeter-scal…

Let $Omegasubsetmathbb{R}^d$, $dgeq 2$, be a domain satisfying suitable regularity assumptions, and let $R^Omegaalpha$ be the Laplacian in $L^2(Omega)$ with Robin parameter…

One-dimensional sharp lower-bound conjecture. The sharp lower bound for given $\alpha$ and $D$ should correspond to the gap for the corresponding one-dimensional problem. Consequen…

Let $\mu(c)$ denote the parameter arising in the explicit solution of the fundamental Robin eigenvalue problem for a rectangle or box, as defined in the paper. Complete monotonicit…

Let $\Omega\subset\mathbb{R}^d$ be a domain with prescribed surface area, and let $\lambda_1^\alpha(\Omega)$ be the first eigenvalue of the Robin problem with $\alpha<0$. Surface-a…

Let $\Omega\subset\mathbb{R}^d$ be a convex domain of fixed volume, and let $\lambda_1^\alpha(\Omega)$ be the first eigenvalue of the Robin problem with $\alpha<0$. Convex-domain R…

Let $\Omega\subset\mathbb{R}^d$ be a domain of fixed volume, and let $\lambda_1^\alpha(\Omega)$ be the first eigenvalue of the Robin problem … where $\alpha<0$. Ball-to-shell trans…

Let $\Omega$ be a domain of fixed volume $|\Omega|$, let $\alpha>0$ be the Robin boundary parameter, and let $\lambda_n^\alpha(\Omega)$ denote its $n^\mathrm{th}$ Robin eigenvalue.…

Fix a volume and let $\alpha\leq0$. Consider the first Robin eigenvalue $\lambda_1^\alpha(\Omega)$ among domains of that volume. A domain is reflection-symmetric when it is invaria…

Reverse Bossel–Daners conjecture. One has