45 problems

Cone hot spots conjecture. A statement similar to the paper's theorem for cones should hold for these manifolds with boundary.

Let $\Omega$ be a domain with fractal boundary, and let fractal curvatures be the limiting curvature quantities associated with parallel sets of the boundary, with scaling exponent…

Let $F$ be a thermally insulated domain, and consider the heat flow on $F$ with typical initial conditions. Hot spots conjecture. As time passes, the hottest point should move towa…

Let $L=-\epsilon\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)-\frac{\partial}{\partial\theta}$ act on sufficiently smooth periodic functions…

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 ……

Let $\mathcal{M}$ be an almost-Riemannian manifold equipped with a measure, and consider the heat equation associated with its Laplace--Beltrami operator. Denote the minimal observ…

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

Classical solvability conjecture. The occupied heat equation always admits a classical solution whenever $g$ has the stated continuity. Unlike the path-dependent heat equation, who…

The entropy power conjecture. For all $m\geq1$ and all $t>0$,

Let $X$ be a random variable on $\mathbb{R}$ with density $f$, let $Z\sim\mathcal{N}(0,1)$ be independent of $X$, and define $X_t=X+\sqrt{t}Z$ for $t\geq 0$. Write $p(x,t)$ for the…

Let $X$ be a random variable on $\mathbb{R}$ with density $f$, let $Z\sim\mathcal{N}(0,1)$ be independent of $X$, and for $t\geq 0$ define … Let $p(x,t)$ be the density of $X_t$, a…

A generalized Wentzell condition models transmission through a highly conductive layer, and numerical experiments suggest that such a layer may be fractal, meaning that its geometr…

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…

Stabilization-cost divergence conjecture. Even without knowing the exact value of $C_r(\lambda)$, the optimal value of $C_r(\lambda)$, at least among costs obeying the feedback law…

Consider the heat equation on a finite interval with time-periodic Robin-type boundary conditions. The paper expects that solutions will generally become strongly asymptotically pe…

Let $v(x,t)$ satisfy the growth condition that, for sufficiently large $r$, its relevant boundary expression is bounded by $C r^m e^{a r^b}$ with $0\leq b<2$. Growth-condition impr…

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…

The 1/2-log-concavity conjecture. If $\phi\leq 1$ in $\Omega$, then $\phi$ is $1/2$-log-concave in $\Omega$.

Asymptotic existence conjecture. The equation admits a solution $h$ satisfying

Let $\Gamma\subset\mathbb{R}^d$ be an open domain, let $A\in L^2_{\mathrm{loc}}(\Gamma,\mathbb{R}^d)$ and let $V\colon\mathbb{R}^d\to\mathbb{R}$ satisfy…

Let $u_0$ be a bounded nonnegative function in $\mathbf{R}^N$ with compact support. For each $1\leq\alpha<2$, Theorem 3.4 asserts that there exists $T_\alpha>0$ such that…

Let $\Omega\subset\mathbf{R}^N$ be a bounded convex domain, and let $\phi$ be the first positive Dirichlet eigenfunction of $-\Delta$ in $\Omega$. A function $u$ is 2-logconcave wh…