44 problems

Lions's conjecture. For every $p>\frac{n}{r}$, an $L^p$ estimate should hold for $D^2u$ whenever $f\in L^p$.

Mass-growth conjecture. Analogous mass-growth and boundedness results should hold at least for the $\alpha$-stable operator (fractional Laplacian) $\mathcal L=(-\Delta)^{\alpha/2}$…

Let … where the $U_j(x)$ are fixed non-random potentials of class $C^1$ satisfying condition, and $(a_j^\omega(t),1\leq j\leq N)$ is a realization of a stationary vector-valued ran…

Generic finite-graph conjecture. For a generic such function $A(x)$, the closure of the singular set of $u$ is a finite graph.

Consider the setting of Theorems --, with the convergence rates established there. Optimality conjecture. The convergence rates obtained therein are optimal up to slowly varying fa…

Let $\mathcal Q$ be the space of paths used to parameterize overlaps, let $q=(q_1,q_2)\in\mathcal Q^2$, and let $\psi_2:\mathcal Q^2\to\mathbb R$ be the initial free-energy functio…

Let $\xi:\mathbb{R}^D_+\to\mathbb{R}$ be the model's interaction function, and let $S_t\chi$ denote the operator defined according to equation … . Theorem … are the Hopf–Lax-type a…

Let $p>1$ and let denote the diffusive Hamilton–Jacobi problem studied in the source. An entire solution is a solution defined for all $t\in\mathbb{R}$; it is nonstationary if it d…

Positive-limsup conjecture. The assumption $(Q2\text{-}2n)$ can be weakened and replaced by

Nonradial asymptotic conjecture. Radiality can be dropped, and every singular solution satisfies

Let $u(\{x_i\},t)$ be the solution of the Hamilton–Jacobi–Bellman equation, let $t_0<t_1<\ldots<t_J$ be a uniform discretization of $[0,t]$ with grid spacing $\delta=t/J$, and let…

Let $D$ be the spin dimension, let $S^D_+$ denote the cone of positive semidefinite symmetric $D\times D$ matrices, and let … be the collection of increasing paths. Set…

Optimal-control formulation. Following the control formulation for Hamilton–Jacobi equations, the solution should satisfy

Parisi PDE conjecture. The enriched free energy $F_N$ should converge to a function $f:\mathbb{R}_+\times\mathcal Q^2\to\mathbb{R}$ solving

Let $P_1^\uparrow(\mathbb{R}^D_+)$ be the relevant space of probability measures, let $X_D$ be the set of functions of the form … where each $chi_d$ is $1$-Lipschitz, nondecreasing…

Let $\overline\xi$ be the interaction function on $S^D_+$, and let $f$ denote the viscosity solution of the Hamilton–Jacobi equation associated with the model. When $\overline\xi$…

Let $f(t,\mu)$ denote the limit of the re-scaled mutual information between the signal and the observation, parametrized by $(t,\mu)\in\mathbb{R}_{\geq 0}\times\mathcal{M}_+$. Domi…

Let spacetime have dimension $D$, and consider different $(D-1)$-dimensional foliations with leaves on which the covariant Hamilton–Jacobi functions are integrated. A foliation and…

Mourat's Hamilton–Jacobi conjecture. While neither $\xi$ nor $\psi$ need be convex nor concave in general, $\overline F_N$ converges to the solution of this Hamilton–Jacobi equatio…

Chow's conjectured Hopf–Lax formula. The solution of this equation should be given by the Hopf–Lax type formula proposed by the authors of Chow. State-dependent Hamiltonians are im…

Generalized Hopf–Lax conjecture. The solution is given by

Let $p,q$ satisfy $pq>1$, and consider system (1) in $\mega=\mathbb{R}^{N}$, without imposing a sign condition on the solutions. Liouville conjecture. Every solution of this system…

Let $\overline{F}_N$ be the finite-$N$ free energy and let $f$ be the viscosity solution of the associated infinite-dimensional Hamilton–Jacobi equation. Before characteristic line…

Let $f:[0,\infty)\times\mathcal{M}_+\to\mathbb{R}$ be the solution of the infinite-dimensional Hamilton–Jacobi equation … where $\mathcal{M}_+$ is the space of measures used in the…

Let $\overline{F}_N$ be the modified free energy, and let $f(t,\mu)$ solve the infinite-dimensional Hamilton–Jacobi equation specified by the model's enriched free energy. Free-ene…