95 problems

For the three-dimensional axially symmetric steady incompressible cavity-flow free-boundary problem, determine whether the cavity solutions whose existence was established by Garab…

Let the Laplacian operator be the operator defined by the jump conditions in --, and let $h_1$ denote the function in Theorem. Uniqueness conjecture. The unique, up to multiplicati…

Let $K\subset\mathbb{C}$ be compact and connected with more than one point. Let $M_K$ be the smallest number such that, whenever $p$ has degree $n\geq1$ and $p=p_1\cdots p_m$, … He…

Let $\Sigma_0$ be the class of univalent meromorphic functions $F$ in the exterior of the unit disk, normalized by $F(z)=z+O(z^{-1})$ as $z\to\infty$. Let $K=F(\mathbb{B}^c)^c$ con…

Let $K\subset\mathbb{C}$ be compact and connected, with logarithmic capacity $\operatorname{cap}(K)=1$, $0\in K$, and conformal centroid at the origin. Let $\mu_K$ be the equilibri…

Let $K\subset\mathbb{C}$ be compact and connected, with logarithmic capacity $\operatorname{cap}(K)=1$, $0\in K$, and conformal centroid at the origin. Let $g$ be the Green's funct…

Complex-charge equilibrium zero conjecture. The meromorphic function

Let $\{a_i\}_{i\in\mathbb{N}}$ be positive with finite sum, and let $\{z_i\}_{i\in\mathbb{N}}$ be distinct complex numbers such that … for some $\rho>0$. Borel-series zero conjectu…

Let $P$ be a critical operator in a Riemannian manifold $\mathcal{M}$, and let $\varphi$ denote its ground state. Let $\mathcal{H}_P(\mathcal{M}\times\mathbb{R})$ be the cone of po…

Let $\Gamma$ be a metrized graph, with total length $\ell(\Gamma)$ and tau constant $\tau(\Gamma)$. The ratio $\tau(\Gamma)/\ell(\Gamma)$ is invariant under scaling all edge length…

One-dimensional Maxwell bound. For any values of the charges, the function $V_{\alpha}^*(x)$ has at most $2l-1$ real critical points. This is a stronger one-dimensional version of…

Mother-body conjecture. The root-counting measures of $Sp_n$ converge weakly to a positive measure $\mu_{Q_k}$ depending only on $Q_k$. Its support is a finite planar tree…

Spectral-tree and mother-body conjecture. The measures $\mu_n$ converge weakly to a probability measure $\mu_Q$ depending only on $Q_k$. Its support is a finite planar tree contain…

Let $\lambda\in\mathbb C$ be an affine coordinate on $\mathbb{CP}^1$, let … and let $G(q)$ denote the logarithmic energy of the elliptic law with elliptic parameter $\sqrt q$. Let…

Strict descent conjecture. Every convex polygon satisfies the strict descent condition. The condition is introduced to construct electrostatic skeletons where different Schwarz ref…

Green-function normal-derivative conjecture. As $4y4$ approaches $4x4$ along the normal,

Strip energy ball comparison conjecture. If

Polynomial separation conjecture. If

Let $K\subset {\mathbb{C}}\setminus \{0\}$ be a compact set. A polynomial of the indicated form is $\sum_{k=n/\tau_n}^{n}a_kz^k$, where $\tau_n\to 1$. Polar-set characterization co…

Let $K$ be a non-empty compact subset of the complex plane, and let $\delta_n$ and $C$ denote the quantities defined in the preceding results. Dilation asymptotic conjecture. … Thi…

Asymptotic empirical-current conjecture. The total variation measures $\lVert T^{(r)}\rVert$ converge to $\mu_E$ in the weak$^*$ topology of measures supported on $E$. Moreover, if…

Let $K$ be a $C^{1+}$ Jordan arc, let $z0 in \overline{\mathbb{C}}backslash K$, and let $\rho:Kto[0,\infty)$ be upper-semicontinuous. Writing $tn(\rho,z0)$ for the weighted residual-pol…

Let $\Omega$ be a bounded domain, let $\Omega^*$ denote the comparison domain of the same area, and let $B_R$ be the comparison ball for a domain whose transfinite diameter is $R$.…

Let $\Omega$ be a bounded domain and let $H$ be a polarizer. The reverse Faber–Krahn conjecture. … The inequality is immediate in the cases discussed in the source when the transfi…

Let $E$subset ${\mathbb R}^2$ be compact and let $H$ be a polarizer. The polarization monotonicity conjecture. … Moreover, equality holds only if $P_H(E)\cong E$ or…