9 problems

Let $D$ be a bounded body in $\mathbb{R}^d$, and consider the Dirichlet problem with polynomial boundary data on $\partial D$. A polynomial is polynomial boundary data if it is the…

Let $\Omega\subset\mathbb{C}$ be a bounded domain whose boundary consists of finitely many non-intersecting Jordan curves. For each polynomial boundary datum on $\partial\Omega$, c…

Existence and uniqueness conjecture. There exists a unique $k$-admissible solution $u$ satisfying

Let $\mathcal L\in\operatorname{\mathcal S\mathcal E}$, and let $G\subset\mathbb C$ be a bounded simply connected domain. The domain $G$ is $\mathcal L$-regular when the associated…

Let $M$ be a Hadamard manifold and let $\Omega\subset M$ be a mean-convex domain that is not strictly convex at infinity. Write $\partial^{ct}\Omega$ for its ideal and finite bound…

Let $d\geq 2$, let $D\subset\mathbb{R}^d$ be an arbitrary bounded convex set, and let $\varphi$ be the solution of the fractional Dirichlet problem … with the boundary condition fr…

Let $d\geq 3$, and let $D\subset\mathbb{R}^d$ be an arbitrary bounded convex set. Let $\varphi$ denote the solution of the fractional Dirichlet problem … with the boundary conditio…

Global median value conjecture for strictly convex domains. There exists a unique continuous solution $u$ with $u=g$ on $\partial\Omega$, and this solution satisfies $u=u^{*}$. Thi…

Multiple-solutions conjecture. For a rather general family of boundary values, multiple solutions to $(\mathcal{D}_{\epsilon})$ should exist in each component $\mathcal{A}_k(A_0)$,…