8 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…

Friedman's conjecture. A Faber–Krahn type inequality should hold for regular trees with given volume.

Let $\triangle$ be a triangle, with area $|\triangle|$, perimeter $|\partial\triangle|$, and first Dirichlet eigenvalue $\lambda_1(\triangle)$. Antunes–Freitas conjecture. There ex…

Let $u=gamma$ be the Gaussian measure, let $L=-Deltagamma$ be the Ornstein–Uhlenbeck operator, and let $d(Omega)$ be the Gaussian Fraenkel asymmetry. The quantitative Gaussian Fab…

Let $\Omega \subset \mathbb{R}^2$ be a $C^\infty$ simply connected domain, and let $\mu^\Omega(E)$ be the first eigenvalue of the auxiliary operator $H_E^\Omega$. Let $\mathbb{D}$…

Bhattacharya–Weitsman–Nadirashvili conjecture. There exists a dimensional constant $C>0$ such that

Bhattacharya–Weitsman–Nadirashvili conjecture. There exists a dimensional constant $\sigma>0$ such that

Let $1<p<n$. Consider the three assertions for a domain $\Omega\subset\mathbb R^n$: (i) the $p$-Faber–Krahn inequality, (ii) the corresponding isocapacitary inequality, and (iii) t…