4 problems

Let $\Omega\subset\mathbb{R}^d$ be a bounded Lipschitz domain. Let $\mu_n(\Omega)$ be the Neumann eigenvalues of the Laplacian, enumerated with multiplicity, and define … With … wh…

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$, let $V(\Omega)$ be its volume, let $\omega_n$ be the volume of the unit ball in $\mathbb{R}^n$, and set … For a positive integer…

Let $\Omega$ be a bounded domain in $\mathbb{S}^{1}\times\mathbb{R}$, and write $|\Omega|$ for its area. Pólya's small-area conjecture. There exists a positive number $A_{0}$ such…

Let $\Omega$ be the fixed domain considered in the proof, let $\lambda_j(\Omega)$ denote its $j$th Dirichlet eigenvalue, and let $Q=Q(k)$ be the index defined there so that the rel…