32 problems

Let $D\subset\mathbb{R}^2$ be a simply connected domain, and understand equality of domains modulo sets of measure zero. The Pompeiu property is the property characterized in the s…

Let $\Omega$ be a planar domain of area $|\Omega|$, let $\Lambda_N^{(2)}(\Omega)$ denote the optimal sum of the first Dirichlet eigenvalues over $N$-partitions of $\Omega$, and let…

Let $\Gamma$ be a planar domain and consider partitions of $\Gamma$ into $m$ parts minimizing the optimal partition energy. Hexagonal asymptotics conjecture. For large $m$, the opt…

Let a Jordan domain have finitely many corners with angle parameters $\alpha_j$, and let the truncated Grunsky determinant be regularized by subtracting its logarithmic divergence.…

Single-sign-switch conjecture. The function $\iota$ switches sign once on the positive part of $I$.

Boundary-condition conjecture. Similar results to Theorems and should hold for these alternative boundary conditions, in particular yielding corresponding asymptotics and limiting…

Classification conjecture for Serrin bands. Any Serrin band in $\mathbb{R}^2$ is, up to similarity, one of the periodic bands $\Omega_\tau$, with $\tau\in(0,1]$.

Let $D\subset\mathbb{R}^2$ be a bounded, simply connected Lipschitz domain. A domain has the Pompeiu/Schiffer property if it is not among the exceptional domains for the associated…

Let $D$ be a planar regular region with finitely many boundary components that are analytic Jordan curves. Let $c_{\beta}(z)$ denote the logarithmic capacity and let $B(z)$ denote…

Converse visibility conjecture. The converse of this sufficient condition is also true: every visibility domain satisfies the corresponding boundary condition, namely that there ex…

A reformulation of the MLC conjecture. The complement of the Mandelbrot set in $\mathbb{C}$ satisfies the visibility property outside a totally disconnected subset of the boundary.

Triangle fail-point conjecture. The fail point lies on the longest side and is always closer to the foot of the altitude onto that side than to the midpoint of the side.

Ramaswamy's conjecture. The fail points occur either at the contact points of the largest inscribed circle or at points of minimal curvature.

Saint-Venant's conjecture. Fail points occur at the contact points of the largest inscribed circle.

Convex planar eigenvalue-gap conjecture.

Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain with disconnected boundary. Consider the overdetermined Neumann problem … and suppose that the solution is locally consta…

Planar annulus conjecture. Is it true that every doubly connected planar domain $\Omega$ has

Generalized Payne conjecture. The same conclusion should hold for any bounded planar domain.

Multiplicity-two conjecture.

Let $\Omega\subset\mathbb C$ be an $m$-connected domain without degenerate boundary components. For each $z\in\Omega$ and $j=0,\ldots,m-1$, let $f_{z,j}$ be the unique conformal ma…

Let $\Lambda_{\epsilon}$ be a bounded connected planar annulus with two boundary components, and let $\lambda_{1}(\Lambda_{\epsilon})$ be its first nontrivial Steklov eigenvalue, g…

Let $\Lambda_{\epsilon}$ denote the planar annulus with outer radius $1$ and inner radius $\epsilon\in(0,1)$, let $|\partial\Lambda_{\epsilon}|$ be its perimeter, and let…

Let $\Omega$ be a bounded planar domain with one hole, meaning that its complement has two connected components, one bounded and one unbounded, and assume that $\Omega$ has Lipschi…

Let $D$ be a planar regular region whose boundary consists of $n$ analytic Jordan curves. Let $\hat{R}(z,\bar{w})$ be its conjugate Hardy $H^{2}$ kernel and $B(z,\bar{w})$ its Berg…

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with piecewise smooth boundary $\partial\Omega$, and let $\lambda_i$ be the $i$-th Dirichlet eigenvalue. Let…