8 problems

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

Courant-sharpness conjecture. If $\lambda_{0,q,h}(S)$ is negative, then no associated eigenfunction is Courant-sharp.

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 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 $\Omega\subset\mathbb{R}^d$ 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…

Reverse Bossel–Daners conjecture. One has