19 problems

Let a planar domain have fixed area, and let its first Dirichlet eigenvalue of the Laplacian be denoted by $\lambda_1$. Rayleigh's conjecture. Among planar domains of fixed area, t…

Let $\Omega\subset\mathbb R^N$ be a bounded domain, let $B\subset\mathbb R^N$ be a Euclidean ball, and set $A_N=\lambda_2(B)/\lambda_1(B)=j_{\nu+1,1}^2/j_{\nu,1}^2$, where…

For each integer $N\geq 3$, let $P_N$ be a regular $N$-gon whose area is fixed, and let $\lambda_1(P_N)$ denote its first Dirichlet eigenvalue. Antunes–Freitas conjecture. The sequ…

Let $\Omega\subset\mathbb{R}^2$ be an open set of finite measure, and let $\lambda_3(\Omega)$ denote the third strictly positive Dirichlet eigenvalue of the Laplacian on $\Omega$,…

Let $\Omega\subset\mathbb{R}^n$ be a regular bounded domain, and let $\lambda_1^D(\Omega)<\lambda_2^D(\Omega)\leq\cdots$ be its Dirichlet eigenvalues. Let $\mathbb{B}^n$ be an $n$-…

Dirichlet monotonicity with volume scaling. The function

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with smooth boundary, and let $\lambda_3(\Omega)$ denote its third Dirichlet eigenvalue. The planar third-eigenvalue conjecture.…

Pólya–Szegő conjecture. The unique solution is the regular polygon with $n$ sides and area $\pi$.

Let $\Omega\subset\mathbb{R}^2$ be a domain, and let $u_2$ be a second eigenfunction of the Dirichlet Laplacian on $\Omega$. A closed nodal line is a closed connected component of…

A polygon is an $n$-gon of fixed area. Pólya's conjecture. Among $n$-gons of fixed area, the regular one minimizes the first Dirichlet eigenvalue. The conjecture is used as a hypot…

Let $d$ be the dimension and let the $(d+1)^\mathrm{th}$ Dirichlet eigenvalue be considered among domains in dimension $d$. Bareket's conjecture. The ball optimises this eigenvalue…

Let $c>0$. For an open or quasi-open set $\Omega\subset\mathbb R^2$ with $|\Omega|=c$, let $\lambda_3(\Omega)$ denote the third eigenvalue of the Dirichlet Laplacian on $\Omega$. D…

Let $T$ be a triangular domain, and let $\lambda_1(T)\leq\lambda_2(T)$ denote its first two Dirichlet eigenvalues. Antunes–Freitas triangular spectral gap conjecture. The differenc…

Let $\Omega$ be a bounded plane domain with diameter $D$, let $\lambda_2$ be its second Dirichlet eigenvalue, and let $j_{1,1}$ denote the first positive zero of the Bessel functio…

Let $\Omega$ be a bounded plane domain of diameter $D$, and let $0<\lambda_1\leq\lambda_2\leq\cdots$ be the Dirichlet eigenvalues of its Laplacian. Disk eigenvalue-sum conjecture.…

Let $\lambda_1<\lambda_2$ be the first two Dirichlet eigenvalues of a triangle, and let $D$ denote its diameter. Dirichlet gap conjecture. The quantity … is minimal for the equilat…

Let $\Omega$ be an open set in $\mathbb{R}^m$, with $m=2,3,\ldots$, and let $\lambda_k(\Omega)$ denote the $k$-th eigenvalue of the Dirichlet Laplacian on $\Omega$. Consider … More…

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain, and let $\lambda_1(\Omega)\leq\lambda_2(\Omega)\leq\dots$ denote its Dirichlet eigenvalues. Among planar domains with fixed are…

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain, and let $\lambda_{1}(\Omega)$ denote its first Dirichlet eigenvalue. Let $F(\Omega)$ be the geometric quantity appearing in T…