15 problems

Let $\Omega$ be a Lipschitz domain, and let $u\in H^1$ be a weak solution to the Dirichlet problem … Suppose that the set…

Let $\Omega$ be a bounded domain, strictly starshaped with respect to $0\in\Omega$, and suppose the condition … . Consider the problem with $u=v=0$ on $\partial\Omega$. Lane–Emden…

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 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 $\Omega$ be either $\mathbb{R}^N$ or an exterior domain in $\mathbb{R}^N$, and let $V\in L^{\infty}(\mathbb{R}^N)$ satisfy $\|V\|_{\infty}\leq 1$. Suppose that $u$ solves … and…

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 $\Omega \subset \mathbb R^d$ be a Lipschitz domain and let $\Sigma=B\cap\partial\Omega$ for a ball $B$ centered at a point of $\partial\Omega$. Let $u$ be harmonic in $\Omega$,…

“Qualitative” Landis conjecture. Then $u\equiv 0$.

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…

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…

Huang et al.'s conjecture. Under these assumptions, $f$ vanishes identically.

Let $d\geq 1$, let $E_0,G>0$, let $\delta\in(0,G/2)$, and let $(G,\delta)$-equidistributed sequences define the observation set $S_\delta$. Let $V:\mathds{R}^d\to\mathds{R}$ be mea…

Vanishing of the inner-index conjecture. The inner index vanishes:

Sectorial-projection criterion. There is a criterion for weak unique continuation for $A$ in terms of the range of $P_+(B_0)$.