9 problems

Let the optimal domain be a domain in $\mathbb{R}^3$ minimizing the first eigenvalue of the curl operator under the relevant volume constraint. A spheromak is a torus whose central…

Transition-threshold conjecture. For odd $k$, the threshold $b_k$ is given by the upper bound

Let $\xi(D)=\lambda_2(D)/\lambda_1(D)$ be the fundamental ratio of the first two Dirichlet–Laplacian eigenvalues. Consider the moduli space of triangles after quotienting by the sy…

Let $P_k\subset\mathbb{R}^2$ be an open $k$-gon, and let $P_k^*\subset\mathbb{R}^d$ be an open regular $k$-gon. For $k\geq 3$, define the fundamental ratio by…

Let $\Omega$ be the domain, let $d$ be the distance to $\partial\Omega$, and let $\overline{\lambda}(\tau)$ be the maximal principal eigenvalue obtained by replacing the infimum in…

Let $\Omega$ be the domain, let $d$ be the distance to $\partial\Omega$, and let $v_\tau$ denote the vector field minimizing the principal eigenvalue under the drift constraint of…

Let $\Omega\in\mathcal{O}$ be a general domain, let $d$ denote the distance to $\partial\Omega$, and let $v_\tau$ be the minimizing vector fields associated with the drift optimiza…

Let … denote the two optimization problems defined earlier in the paper for the relevant flat-torus eigenvalue spectrum. Agreement conjecture. The solutions to these two optimizati…

Let $\Lambda^t_k(\gamma)$ denote the supremum of the normalized $k$-th Laplace–Beltrami eigenvalue over closed orientable Riemannian surfaces of genus $\gamma$, with $k\geq 1$. Let…