37 problems

Let $u$ solve … where $V$ is a real-valued bounded potential. Landis' conjecture. If, for some $C>0$ and any $\epsilon>0$, … then $u$ is trivial. The conjecture concerns quantitati…

Let $u$ solve … where $V\in W^{1,\infty}$, and let the rate of vanishing order refer to the maximal vanishing order of $u$. Kukavica's conjecture. The rate of vanishing order of…

Let $u(x,t)$ be a solution with bounded growth of … where $W$ and $V$ are bounded, and $A(x,t)$ is uniformly elliptic and bounded. Landis–Oleinik conjecture. Under suitable conditi…

Let $(\mathcal M,\textsl{g})$ be a Lorentzian manifold satisfying the hypotheses (H1) and (H3), and consider the unique continuation theorem in the exterior of the double null cone…

Let $\Omega\subset\mathbb{R}^d$ be a Lipschitz domain, let $u$ be a bounded harmonic function in $\Omega$, and let $U\subset\partial\Omega$ be relatively open. Lin's conjecture. If…

Simon’s conjecture. Unique continuation should hold for this inequality in all dimensions: if $u$ vanishes on a nonempty open subset of $U$, then $u$ vanishes everywhere on $U$. Sa…

Let $M^n$ be a complete Riemannian manifold with nonnegative Ricci curvature and maximal volume growth, and let $\mathscr{F}_u(r)$ denote the frequency function of a harmonic funct…

Let $\mathcal{L}$ be an elliptic operator of order $2m$ on an open domain $\Omega\subset\mathbb{R}^N$, with $N\geq2$. For an open portion $\Gamma$ of $\partial\Omega$, impose homog…

Let $\mathcal{M}$ be a rescaled mean curvature flow in $\mathbb{R}^{n+1}$ without boundary. The source distinguishes asymptotic cases for such flows, including a case of super-expo…

Let $V$ be a stationary $m$-dimensional varifold in $U\subseteq\mathbb{R}^{m+n}$, let $h$ be harmonic on $V$, meaning that for every $\varphi\in C_c^1(U)$, … and let…

Let $V$ be a stationary varifold, let $x_0\in{\rm spt}(V)\cap U$, and let $\mathcal{M}$ be a classical smooth minimal $m$-dimensional surface such that … The unique-continuation co…

Let $h>0$ and let $u\in\ell^2((h\mathbb{Z})^n)$ solve the stationary discrete Schrödinger equation … under the finiteness assumptions on $\|q\|_\infty$ and $\|u\|_2$ appearing in t…

Let $u$ be a nontrivial solution of a uniformly elliptic partial differential equation with bounded coefficients in the whole space or in an exterior domain. The coefficients are a…

Let $u\in C^1(\overline{\mathbb{R}^d_+})$ be a harmonic function in the upper half-space $\mathbb{R}^d_+$, and let $U\subset\partial\mathbb{R}^d_+$ have positive surface measure. L…

Let $\Omega$ be a domain whose boundary $\partial\Omega$ is Lipschitz, possibly with a large Lipschitz constant, and let $E$ denote the relevant boundary zero Cauchy data set for a…

Let $u$ be a super-exponentially decaying solution of a second-order periodic, or more generally an appropriate class of periodic elliptic, equation. Landis-type conjecture. Such a…

Let $u$ and $v$ satisfy the hypotheses of the preceding strong Laplace Hopf-lemma conjecture, including $u_t(0,0)=0$. Finite-order vanishing conjecture. If, in addition, $u(t,0)$ a…

Let $\Omega$ be a Lipschitz domain with boundary $\partial \Omega$, and let $\Sigma \subset \partial \Omega$ be an open subset. Let $u$ be a harmonic function in $\Omega$ that is c…

Optimal minimal scale conjecture. The optimal minimal scale for estimates like

Let $h$ solve a divergence-form elliptic equation in a ball $B$, with $A$ an elliptic matrix-valued function with smooth coefficients. Landis conjecture. The question about whether…

Let $\Omega\subset\mathbb{R}^n$ be a Lipschitz domain and let $\Sigma\subset\partial\Omega$ be relatively open with respect to $\partial\Omega$. Let $u$ be harmonic in $\Omega$ and…

Yau's conjecture. There exist constants $C_1$ and $C_2$, which depend on $M$, such that

Let $d$ be the dimension of space. Assume that the external and interaction potentials belong locally to $L^p(\mathbb{R}^d)$ and the magnetic potentials belong locally to…

Fractional Landis conjecture. There is a constant $C$ such that, for the relevant radii $r$, one has

For $s=1$, let $q$ be a potential and let $u$ be a solution with $|q(x)|\leq 1$ and $|u(x)|\leq C_0$. Kondratev–Landis conjecture. If … then $u\equiv 0$. The conjecture is a Landis…