9 problems

Uniform observability conjecture. For every $T>0$, there exists a constant $C=C(X,S,T)>0$ such that the observability inequality holds. Moreover, the constant can be chosen to depe…

For each $d\geq 1$, let $Y_0$ be the closure space for Schrödinger waves introduced in the paper, and let $\tilde{Y}_0\subset Y_0$ be its time-independent subspace. Time-independen…

For each $d\geq 1$, let $Y$ be the normed function space introduced in the paper for Schrödinger-wave observability, and let $Y_0$ denote the closure of continuous functions in $Y$…

Let $d\geq 1$, let $\mathbb{T}^d$ be the $d$-dimensional torus, and let $V\in L^\infty_x(\mathbb{T}^d)$. Let $\mathcal{H}_V=-\Delta+V$, and let $f$ be any eigenfunction of…

Let $m_d(r)$ be the optimal observability constant for Laplacian eigenfunctions on the $d$-dimensional torus observed on a ball of radius $r$. For $d\geq 4$, let $C_d$ and $C'_d$ b…

Let $\tau>0$ and let $\function{P}_\tau$ denote the spectral projector associated with $-\Delta+|x|^\tau$. Suppose that $\omega\five \subset\real^d$ is measurable and that, for some…

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,