22 problems

Continuity conjecture. The map

A maximal solution is obtained from the stochastic optimization problem described above: for initial law $\mu\in {\mathcal{S}}_{\delta}(U)$, Brownian motion $W_t$, first exit time…

Existence conjecture. $\Sigma_d^2\neq\emptyset$ for an open set of initial data.

Asymptotic barrier-speed conjecture. Such a result should hold: it should be possible to have

Barrier-height convergence conjecture. Under a suitable rescaling, the barrier height satisfies that $\xi_t/\sqrt{t}$ converges to some distribution.

Let $w$ be a minimal solution of the arrival-time problem described above. Non-jump-to-infinity conjecture. A minimal solution $w$ cannot jump to infinity; consequently, $w$ is con…

Let $P(\mu,f,u)$ be the optimal-stopping problem considered in the paper, with admissible target measures $\nu$ constrained by the initial data $\mu$, obstacle $\f$, and cost \u.…

Let $\eta$ be a weak solution of the supercooled Stefan problem, and let $\Sigma(\eta)$ be its transition zone. A weak solution is maximal if, for every other weak solution…

Let $\eta_0$ be the initial data, with initial domain ${\eta_0>0}$. Condition $(C)$ means that $\eta_0$ is strictly subharmonically ordered with $\nu$ with respect to ${\eta_0>0}$…

Sensitivity conjecture. The map

Let $\Lambda$ be a probabilistic solution of the radial Stefan problem, and let the physicality condition be … which constrains the jumps of the solution. Uniqueness conjecture. Th…

The particle system is considered with its initial condition as in Theorem 1.4 of the paper. Initial-condition perturbation conjecture. The perturbation in the initial condition of…

Consider locally bounded weak solutions of the Stefan problem … with $\beta$ the enthalpy graph and $p\ge2$. The modulus … for some $c,\sigma>0$ is conjectured to be optimal as a s…

Urbano's optimality conjecture. The modulus $r\mapsto |\ln (cr)|^{-\sigma}$ should be optimal for the interior continuity of the temperature in the Stefan problem. The cited work i…

Optimality conjecture. The logarithmic type modulus of continuity should be optimal as a structural property for weak solutions to such partial differential equations. The paper es…

Fix … be a limit point of an external multi-particle diffusion-limited aggregation process, with associated absorbed Brownian motion density $u$ and occupied region … partialt u=fr…

True-order conjecture. The second term on the right-hand side of the convergence estimate gives the true order in $\Delta$ of

Delay-mismatch sensitivity conjecture. The delay-compensated controller might exhibit greater sensitivity to delay mismatch when the delay is underestimated rather than overestimat…

A Stefan-type problem is obtained by replacing the elliptic free boundary equation with a parabolic one, so the relevant free boundary operator should satisfy a space-time comparis…

Monotonicity conjecture for $F_{2\alpha}$. The function $F_{2\alpha}$ is increasing, with

Optimal-modulus conjecture. The modulus of continuity $\omega(r)$ is optimal for solutions of the $p$-degenerate two-phase Stefan problem. The paper derives this modulus, including…

Let $j>0$ and let $(h_\epsilon,m_\epsilon)$ be the antisymmetric stationary profiles described in Theorem 2.3, which solve the stationary problem on $\epsilon^{-1}(-\ell,\ell)$ for…