8 problems

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

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…

Let $\Omega$ be a bounded domain in Euclidean space $\mathbb{R}^{n}$, and let $\lambda_i$ be the $i$-th Dirichlet eigenvalue of the Laplacian on $\Omega$. Let $\mathbb{B}^{n}$ be a…

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…