29 problems

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, let $\mu_1(\Omega)\leq\mu_2(\Omega)\leq\cdots$ denote its first nontrivial Neumann eigenvalues of the Ornstein–Uhlenbeck operat…

Let $\a0\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary. Write $0=\mu_0(\Omega)<\mu_1(\Omega)\leq\mu_2(\Omega)\leq\cdots$ for its Neumann eigenvalues, let…

Let a chain domain consist of $k$ planar domains connected in a line, and consider its Neumann eigenfunctions ordered by increasing eigenvalue. Courant sharpness conjecture. The fi…

Let $M^n_\kappa$ be the simply connected $n$-dimensional Riemannian manifold of constant sectional curvature $\kappa\in\{-1,1\}$, and let $\Omega\subset M^n_\kappa$ be a bounded co…

Let $P$ be a path graph with $n$ vertices, and let $\lambda_{n-2}(P)$ denote its largest Neumann eigenvalue. Path-graph Neumann eigenvalue conjecture. The largest Neumann eigenvalu…

Ba{n}uelos and Burdzy's conjecture. If $\mu_1(\Omega)$ has multiplicity $2$, then

Let $T\subseteq M_{\kappa}$ be a non-equilateral geodesic triangle of constant curvature, where $\kappa\in\mathbb{R}$, and consider its second Neumann eigenvalue. Simplicity conjec…

Let $\Omega\subseteq\mathbb R^k$ be a convex set, let $V$ be a convex function on $\Omega$, and define the weighted Neumann operator … Let the first nonconstant eigenfunction of…

Let $\Omega\subseteq\mathbb R^d$ be a convex bounded domain, and let $\phi_\Omega$ be its first nontrivial Neumann eigenfunction. Hot Spots conjecture. The function $\phi_\Omega$ a…

Neumann monotonicity with volume scaling. The function

Higher-dimensional Neumann monotonicity conjecture. The function

Let $\mathbb{M}^{n}(\kappa)$ be an $n$-dimensional complete simply connected Riemannian manifold of constant sectional curvature $\kappa\in\{1,-1\}$, and let $\Omega$ be a bounded…

Let $\a0\Omega$ be a planar convex domain, let $P(\Omega)$ denote its perimeter, and let $\mu_1(\Omega)$ denote its first nonzero Neumann eigenvalue. Laugesen–Polterovich–Siudeja's…

Let $\alpha_{k,d}$ denote the quantity defined earlier in the paper as the infimum of the relevant ratio of Neumann eigenvalues over pairs of convex domains in dimension $d$. Asymp…

Let $\Omega$ be a convex domain in $\mathbb{R}^2$, let $L(\Omega)$ denote its perimeter, and let $\mu_2(\Omega)$ be its first nonzero Neumann eigenvalue. Laugesen's conjecture. … E…

Let $\Omega$ be a bounded convex domain in $\mathbb{R}^n$ with piecewise smooth boundary. Denote by $\lambda_k(\Omega)$ the $k$th positive Neumann eigenvalue of the Laplacian on…

Perimeter conjecture. The inequality $P(\Omega)^2\mu_1(\Omega)\leq 16\pi^2$ holds for every plane convex domain $\Omega$. Equality is attained for a square and an equilateral trian…

Let $\Omega$ be a symmetric domain in $\mathbb{R}^2$ with a line of symmetry. Let $\mu_2$ be the second Neumann Laplacian eigenvalue, and suppose that all modes associated with…

Let $\Omega$ be a convex domain in $\mathbb{R}^2$ such that $\kappa_1=\mu_2$. Here $\kappa_1$ denotes the first eigenvalue of the variational problem defined in the paper, $\mu_2$…

Let $\Omega\subset\mathbb{R}^2$ be a planar convex domain, let $P(\Omega)$ denote its perimeter, and let $\mu_1(\Omega)$ denote the second Neumann eigenvalue of the Laplacian. Laug…

Maximal normalized Neumann eigenvalue conjecture. The maximal value of $M_2(\Omega)$ over convex domains is attained both by the equilateral triangle $E$ and by the square $S$.

Consider the shape-optimization problem for the third and fourth interior Neumann eigenvalues of a planar domain. The candidate maximizing shapes are described by modified equipote…

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$, with Neumann-Laplacian eigenvalues … Let $\Omega^{\star}$ denote the ball in $\mathbb{R}^N$ having the same measure as $\Omega$.…

For $b>0$, let $R_b$ be the rectangle whose Neumann eigenvalues are $\mu_{p,q}=\frac{\pi^2p^2}{b^2}+\pi^2q^2b^2$ for $(p,q)\in\mathbb Z_{\geq 0}^2$, ordered as…

Let $\Omega$ be a bounded planar domain of area $A$, and let $0=\mu_1<\mu_2\leq\mu_3\leq\cdots$ be its Neumann Laplace eigenvalues. For $t>0$, define the area-normalized Neumann he…