10 problems

Let $\Sigma\subset \mathbb{R}^d$ be a compact set that is not a removable set for $\Delta$ in $L^2(\mathbb{R}^d)$. A self-adjoint extension of…

Finite-time nontriviality conjecture. For every $m\in\mathbb{N}$, this problem admits nontrivial solutions, meaning solutions whose domains are not Euclidean balls.

Exterior-domain rigidity conjecture. Complements of closed balls are the only exterior domains whose solution satisfies the discrete-time overdetermined boundary condition.

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 ……

Generic heat-control conjecture. Generically with respect to the lengths $(\ell_i)_{i=1,\dots,N}$, the heat equation is small-time null-controllable.

Let $G$ be the point-symmetry pseudogroup under consideration, with splitting $G=G^{\rm ess}\ltimes G^{\rm lin}$ and Lie algebras $\mathfrak g$ and $\mathfrak g^{\rm ess}$. For…

Let $\Omega$ be an open connected subset of $\mathbb{R}^d$. Let $H_0$ and $H_+$ be two self-adjoint operators in $L^2(\Omega)$ such that … Assume that there is a positive function…

Let $({\mathcal M},g)$ be a Riemannian manifold and let $\omega\subset {\mathcal M}$ satisfy $\overline{\omega}\neq {\mathcal M}$. Define … Let ${\mathfrak K}_{heat}(\omega)$ denot…

Let $L>0$ and $T>0$, and let $C_{\int}(T,L)$ be a constant for which the finite-time weighted observability estimate … holds for the relevant solutions of the one-dimensional heat e…

Ervedoza–Zuazua conjecture. For every $L>0$ and every $K>L^2/4$, there exists some $C(K)>0$ such that, for every $T$ sufficiently small,