23 problems

Let $\Omega$ be a bounded domain, let $\sigma>0$ be the Robin boundary parameter, and let $\mu_1(\sigma)\leq\mu_2(\sigma)\leq\cdots$ denote the Robin Laplacian eigenvalues. Laugese…

Let $\Omega\subset\mathbb{R}^N$ be a bounded connected domain, let $\sigma>0$, and let $\mu_1\leq\mu_2\leq\cdots$ be the Robin Laplacian eigenvalues defined by … where $\mathbf n$…

Let $\Omega\subset\mathbb R^d$ be a bounded domain, and let $\{\lambda_k^1\}_{k\in\mathbb N}$ and $\{\mu_k^1\}_{k\in\mathbb N}$ denote respectively the Dirichlet and Neumann spectr…

Let $Q$ be either of the two isospectral mixed Dirichlet-Neumann problems on a quarter-sphere shown in the source, and let $Q_{a,2}$ be the axisymmetric problem with the Dirichlet…

The genus-2 extremal metric conjecture. There exists a metric on a surface of genus $2$ that attains the upper bound

Lin's conjecture. There exists a unitary matrix $U\in M_n$ such that

Exponential separation conjecture. There exist two absolute positive constants $C_*$ and $\alpha_*$ such that

Let $\triangle$ be a triangle, with area $|\triangle|$, perimeter $|\partial\triangle|$, and first Dirichlet eigenvalue $\lambda_1(\triangle)$. Siudeja's conjecture. For any triang…

Laugesen–Siudeja's conjecture. The functional $\mathcal{F}(\triangle)$ is minimized uniquely by the equilateral triangle, and its minimum value is $7\sqrt{3}\pi^2/12$. This is a sh…

Cox et al.'s conjecture. There exist constants $C_1,C_2>0$, possibly depending only on $n$, such that

Let $M$ be a closed $n$-dimensional manifold with Riemannian metric $g$ in a conformal class $C$, and let $\lambda_1(M,g),\ldots,\lambda_{m+1}(M,g)$ be the first $m+1$ nonzero Lapl…

Let $L$ be an L-shaped domain with canonically embedded edges $e_1,\ldots,e_6$, and let $\lambda_1^D(L)$ denote the first mixed Dirichlet-Neumann eigenvalue with Dirichlet boundary…

Rectangular eigenvalue-gap conjecture. We conjecture that

Let $\{\lambda_k\}_{k\ge 1}$ and $\{\mu_k\}_{k\ge 1}$ be the eigenvalues of ${\mathsf H}_{\rm D}$ and ${\mathsf H}_{\rm N}$, respectively, enumerated in non-decreasing order and re…

Let $n\geq 5$, and let $\Theta_n(K)$ denote the quantity defined in the paper for the eigenvalue estimates on the hemisphere, with $K\geq 1$. Universal-bound conjecture. For every…

Let $n$ be the dimension of the hemisphere, let $\lambda_k(\mathbb{S}^{n}_{+})$ denote its Dirichlet eigenvalues, and let $k_n$ be the threshold from the upper bound … where…

Let $M^n$ be an $n$-dimensional compact minimal submanifold of the unit sphere $S^{n+p}(1)$, and let $\overline{\Lambda}_j$ denote the eigenvalues of the closed Laplace–Beltrami pr…

Let $\Omega$ be a bounded domain with piecewise smooth boundary in the $n$-dimensional Euclidean space $\mathbb{R}^{n}$. Let $\Lambda_j$ be the $j$-th Dirichlet eigenvalue of the L…

Let $\Omega$ be a bounded domain in the $n$-dimensional Euclidean space $\mathbb{R}^{n}$. Let $\Lambda_i$ denote the $i$-th Dirichlet eigenvalue of the Laplace operator, and let…

Let $\Omega$ be a convex domain. Let $c_{m,t}$ and $c_{m,n}$ denote the best constants in the tangential and normal Maxwell inequalities, respectively, and let $\lambda_1$ be the f…

Let $\Omega$ be a domain, let $\beta>0$, and let $\Omega^*$ be a ball with the same volume as $\Omega$. Denote by $\lambda_1(\Omega,\beta)$ and $\lambda_2(\Omega,\beta)$ the first…

Let $\Omega\subset\mathbb{R}^d$ be a domain with volume $|\Omega|$, let $\lambda_j$ be the eigenvalues of its Dirichlet Laplacian, and let $F$ and $G$ be related by the transforms…

Let $M$ be a submanifold as in the generalized Reilly inequality, with dimension $n$, eigenvalue index $k$, potential $q$, and generalized Reilly constant … For $q=0$, the constant…