69 problems

Let $\Omega\subset\mathbb{R}^{N+1}$ be a bounded region whose boundary is homeomorphic to the unit sphere in $\mathbb{R}^{N+1}$. Williams conjecture. If $\partial\Omega$ is homeomo…

Fundamental gap conjecture. The fundamental gap is at least $3\pi^2/D^2$.

Let $\Omega\subset\mathbb{R}^n$ be a bounded convex domain, and let $u$ be a first Dirichlet eigenfunction of $\Omega$. Write $\operatorname{diam}(\Omega)$ for its diameter and…

Let $s\in(0,1)$, and consider the fractional Laplacian associated with a censored stable process on a convex domain. The best Hardy constant is the largest constant for which the c…

Let $\Omega\subset\subset D$ be convex domains with $C^2$ boundaries in $\mathbb{C}^n$, let $K$ be a convex compact subset of $\mathbb{C}^n$, and let $f$ be continuous on…

Let $D\subset\mathbb{C}^n$ be a bounded, convex domain, and let ${\mathcal Y}$ be a functional Hilbert space of holomorphic functions on $D$ in which the polynomials are contained…

Folklore conjecture. Every bounded convex domain that admits a finite volume quotient is biholomorphic to a bounded symmetric space.

Let $P$ be a convex polygonal domain, and let $\Omega$ be a simply-connected smoothly bounded domain. Two domains are Steklov isospectral if their Steklov spectra, counted with mul…

Uniqueness conjecture. If $\Omega$ is convex, then $\psi_\Omega$ has a unique critical point.

Let $\Omega$ be a bounded convex domain in the plane, let $\lambda_1$ be its first Dirichlet eigenvalue, and let $u_1$ be the corresponding $L^2$-normalized eigenfunction. Define ……

Let $\Omega$ be a bounded convex domain and let $V$ be a potential; consider the Schrödinger operator $-\Delta+V$ on $\Omega$ with Dirichlet boundary conditions. The fundamental ga…

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

Let $\Omega\subset\mathbb C^N$ be a bounded convex domain. Visibility–Denjoy–Wolff conjecture. Then $\Omega$ is visible if and only if it enjoys the Denjoy–Wolff property. The conj…

Let $\Omega\subset\mathbb C^N$ be a bounded convex domain whose boundary contains no non-trivial analytic discs. A sequential horosphere is a set of the form … where…

Let $\Omega\subset\mathbb C^N$ be a bounded convex domain, let $z_0\in\Omega$, and let $\mathcal G_{z_0}$ be the family of all geodesic segments and geodesic rays starting from…

Let $\Omega\subset\mathbb C^N$ be a bounded convex domain. It is complex visible if for every distinct $p,q\in\partial\Omega$ there exist a compact set $K\Subset\Omega$ and open ne…

The cluster-set conjecture. If $p,q\in\Gamma(\{\gamma_k\})$ and $p\ne q$, then either

Let $\Omega\subset \mathbb C^N$ be a bounded convex domain, with $N\geq 2$. A bounded convex domain is visible when it satisfies the visibility property for complex geodesics, and…

Let $\Omega\subset\mathbb R^n$ be a convex domain, and let $u$ be its ground state eigenfunction for the Dirichlet Laplacian, normalized in $L^\infty$. Ishige–Salani–Takatsu's conj…

Let $G$ be a noncompact real semisimple Lie group, let $P$ be a parabolic subgroup of $G$, and let $G/P$ be the corresponding flag manifold. A divisible convex domain is a convex d…

Henrot–Lucardesi–Philippin conjecture. Every convex planar domain satisfies

Let $\Omega$ be a smooth bounded convex domain, and let $\mu_p(\Omega)$ denote the best constant in the Hardy–Sobolev inequality considered in the paper. Non-attainability conjectu…

Let $\mathfrak{A}_{co}$ be the class of non-empty open convex subsets of $\mathbb{R}^m$, and consider the variational problem … For $m=2$, the thinning-rectangle conjecture. The su…

Jerison's monotonicity conjecture. The monotonicity property holds for centrally symmetric convex domains. The source notes that eigenfunctions can fail to be monotone in general.…

Let $\Omega\subset\mathbb{R}^2$ be a bounded, open, convex set, and let … where $\mu_1(\Omega)$ and $\sigma_1(\Omega)$ are respectively the first nonzero Neumann and Steklov eigenv…