24 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…

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…

Square-optimality conjecture. The square is a global minimiser under both the area and perimeter constraints.

Let $B\geq 0$ be the strength of a constant magnetic field. Let $\Omega^*(B)$ be a minimizing domain, and let $h(\Omega^*(B))$ and $R(\Omega^*(B))$ denote its height and radius. Le…

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…

Oja's linear-regret conjecture. There exists such a sequence $\boldsymbol{A}_1,\dots,\boldsymbol{A}_T$, chosen obliviously and independently of Oja's initial random vector, for whi…

Let $\Omega$ be a smooth domain, and let $P_{\delta}$ denote the concentration points associated with the optimizers as $\delta\to0$. Maximal mean-curvature concentration conjectur…

Let $B_R$ be a disk in dimension $n=2$, and let $m_0$ denote the stationarity-breaking threshold for the ball shape. The functional under consideration is denoted by…

Let $D\subset\mathbb{R}^n$ be an optimization domain for the Neumann problem for the Laplace operator, and let $\varphi$ be a scalar phase-field with interpolation coefficient $a$…

Quantitative stability conjecture. There exists $\eta>0$ such that, for every $V\in\mathcal M(\Omega)$,

Equality and disconnectedness conjecture. The solutions of these two problems are equal and are disconnected.

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…

Qualitative similarity conjecture. The optimal weight $\widehat p$ is qualitatively very similar to $p^*$.

Hexagonal-pattern conjecture. For large $N$ and strong point interactions, the optimal pattern should be approximately hexagonal; this is not expected to hold in the weak-coupling…

Ground-state optimization conjecture. The inequality

Let $\beta>0$ and $0<\delta<\dfrac{\beta|\Omega|}{\beta+1}$. For a measurable set $D\subset\Omega$ with $|D|=\delta$, define … and let … Here $D^*$ denotes a minimizer of this opti…

Truncated Rellich-function maximizer conjecture. In this case, a maximizer is given by

Nonattainment conjecture. For almost every value of the constraint parameter $L$, this problem has no solution. This concerns the existence of an optimal measurable boundary subset…

Nonexistence conjecture. The global minimizer of the decay rate without frequency constraints does not exist:

Let $M>0$, and consider the lattice-point problem associated with the line $f(x)=M-x$, with maximizing parameter $s>0$ and parameter $r\to\infty$. An asymptotic maximizer would be…

Let $b_1$ and $b_2$ be constants satisfying $0\le b_1<b_2<+\infty$. For each admissible frequency parameter $\alpha$, let $\beta(\alpha)$ denote the minimal decay rate among the co…

Minimal-energy equality conjecture. The three inequalities in Proposition 1 are equalities.

Let a ball contain two conducting materials with conductivities $\beta$ and $\beta$ and with a fixed prescribed proportion. Consider the first eigenvalue of the associated Dirichle…

Let $\Gamma$ be an $N$-armed star graph with sector angles $\beta_1,\ldots,\beta_N$ satisfying the geometric constraints for a star, and let the ground-state energy of the correspo…