52 problems

Let $\Omega$ be a domain, let $A$ be the prescribed parameter, and let $D=D_{\alpha,A}$ be an optimal configuration. Say that $D$ has the same symmetries as $\Omega$ when every sym…

Consider an annulus of thickness $\tau$, and write $\delta=A/|\Omega|$ for the prescribed area fraction. Let $D=D_{\alpha,A}$ be an optimal configuration, and let symmetry breaking…

Let $\Omega_h$ be a dumbbell domain, and let $D=D_{\alpha,A}$ be an optimal configuration. Dumbbell symmetry conjecture. For every $\alpha>0$, there is $\rho_1(\alpha,h)>0$ such th…

Let $\Omega_h$ be a dumbbell domain, let $D=D_{\alpha,A}$ be an optimal configuration, and let $\psi_{\Omega}$ be the function whose maxima determine the relevant localization regi…

Let $D$ be an optimal configuration in a planar domain, and let $\partial D$ denote its free boundary. Two-dimensional free-boundary regularity conjecture. In dimension two, the fr…

Let $\Omega$ be a convex domain, and let $D=D_{\alpha,A}$ be an optimal configuration in $\Omega$. Write $D^c$ for its complement and let $\overline{\alpha}_\Omega(A)$ denote the t…

Let $B$ denote the ball in $\mathbb{R}^3$, and let $\lambda_1(\Omega)$ be the first Maxwell eigenvalue of a domain $\Omega$. Consider domains subject to either a fixed-volume constrai…

Let $B$ denote the ball in $\mathbb{R}^3$, and let the first three Maxwell eigenvalues of a domain $\Omega$ be $\lambda_1(\Omega),\lambda_2(\Omega),\lambda_3(\Omega)$. For $j=1,2,3$, let…

Neural-network complexity conjecture. The complexity of the architecture, measured by the number of layers and its width, is controlled by $4\|\tau_{\Omega}\|_{\infty}4$ and…

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain, let $\nu$ denote the unit outward normal on $\partial\Omega$, and let $\beta>0$. Consider the overdetermined system ……

Let $\Omega \subset \mathbb R^2$ be a bounded simply-connected planar domain with area $|\Omega|=1$, and let $n\in\mathbb N$. For a homogeneous magnetic field of strength $B\geq 0$…

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with $C^2$ boundary, and for $a>0$ let $\mathcal{R}_a$ be the $\overline\partial$-Robin Laplacian on $L^2(\Omega)$, with first e…

Assume that $m\geq0$. For a bounded domain $\Omega\subset\mathbb{R}^2$ with $C^2$ boundary, let $\lambda_\Omega(\theta)$ denote the first nonnegative eigenvalue of the quantum dot…

Equilateral triangle torsional-rigidity conjecture. The equilateral triangle of unit width minimizes the torsional rigidity among shapes having unit minimal width.

Let $B\geq 0$ be the strength of a constant magnetic field, and let $\Omega^*(B)$ be a domain minimizing the first eigenvalue among the admissible domains of fixed volume. The magn…

Let $k\geq 2$, let $A$ be a $\lambda_k$-minimizer, and let $u_k$ be the corresponding eigenfunction. Assume that $\lambda_k$ is simple. Ball-support conjecture. For every copy…

Let $\Omega$ be a bounded smooth convex domain in $\mathbb{R}^n$. Let $X(t,x)$ be the flow defined by … where $\nu$ is the unit outer normal, $u(t)$ is the torsion function on the…

Conjecture on polygonal extremals. Except for the ball, domains on the upper boundary of $\mathcal{E}^C$ are polygonal; regular polygons lie on that upper boundary; and there exist…

Let $n\geq2$, let $B$ be a constant magnetic field, and let $\lambda_n(\Omega,B)$ denote the $n$th eigenvalue of the magnetic Dirichlet Laplacian on a bounded open domain…

Let $B\geq0$ be a constant magnetic field, let $\Omega\subset\mathbb{R}^2$ be a bounded open domain with $|\Omega|=1$, and let $\lambda_2(\Omega,B)$ be the second eigenvalue of the…

Pólya functional bounds conjecture. For every bounded convex planar domain $D$,

Let $\Omega\subset\mathbb{R}^2$ be a bounded, open, convex set, and let … where $\mu_1(\Omega)$ and $\sigma_1(\Omega)$ are respectively the first nonzero Neumann and Steklov eigenv…

Let $\Omega_{a,b}\subset\mathbb{R}^2$ be the right triangle with vertices $O=(0,0)$, $A=(a,0)$ and $B=(0,b)$, where $a,b>0$, and write $\lambda_1(a,b)=\lambda_1(\Omega_{a,b})$. The…

Let $\Omega$ range over planar pentagons with a fixed area, and let $T(\Omega)$ denote their Torsional rigidity. Saint-Venant's conjecture. Among all pentagons of given area, the r…

Let $n\geq 2$ and $0<p<n$. Let $K\subset\mathbb{R}^n$ be a compact set with an irreducible group of isometries, let $M$ be a volume-preserving invertible linear map, and write…