42 problems

Let $u\in C^2(\mathbb{R}^n)$ be a bounded entire solution of … where $W:\mathbb{R}\to\mathbb{R}$ is a double-well potential satisfying the stated regularity and sign conditions, an…

Let $u:\mathbb{R}^n\to\mathbb{R}$ be a bounded solution with … uniformly with respect to $x'\in\mathbb{R}^{n-1}$. A solution is one-dimensional when it depends on only one spatial…

Let $u\colon\mathbb{R}^{n}\to[-1,1]$ be a solution of the Allen–Cahn equation in $\mathbb{R}^{n}$. Call $u$ stable when the second variation of the Allen–Cahn energy is non-negativ…

Finite index De Giorgi conjecture. If $4\leq n\leq 7$, then $u$ is one-dimensional.

Del Pino–Kowalczyk–Wei conjecture. Outside a large ball, each level set of $u$ has finitely many components, each asymptotic either to a plane or to a catenoid; after a rotation of…

Finite index implies finite ends. For $n\geq 3$, every finite Morse index solution has finitely many ends.

Wang–Wei classification conjecture. If $n\leq 7$, then $u$ is one-dimensional: either $u\equiv \pm1$, or

Stable Allen–Cahn regularity conjecture. If $n \leq 7$, then the curvatures of the level sets of $u_{\varepsilon_j}$ in these interfacial regions are uniformly bounded along the se…

Consider the gradient descent scheme with convex–concave splitting for the Allen–Cahn equation, and its energy-dissipation estimate, which includes the energy decrease and a neglec…

Consider the convex–concave splitting scheme for the Allen–Cahn equation with either box constraints or an $L^1$ penalty, in the setting where the source establishes the correspond…

Geometric lower-semicontinuity conjecture. A geometric condition on $\omega(u_0;r)$ could be used to improve the Hölder exponent in the estimate for the displacement of the iterate…

Extension conjecture. The result should extend to all

Let $k\geq 2$, $r_0>0$, and define … where $G(t)=\int_0^t g(s)\,ds$. In Theorem 2, there are positive numbers $\underline{r_0}$, $r_0^{**}$ and $\overline{r_0}$ satisfying the stat…

Let $k\geq 2$ and let $u$ be the unique radial solution in Theorem 1 for initial value $u(0)=\xi$, with positive thresholds $\underline{\xi}$, $\xi^{**}$ and $\overline{\xi}$ satis…

Let $\phi: \mathbb{R}^{n-1} \times \mathbb{R}^+ \to \mathbb{R}$. Suppose that $\phi(s,0) \equiv 0$. Let $f: \mathbb{R}^{n-1} \to \mathbb{R}$, and suppose … Here $g$ is the one-dime…

Nonlocal De Giorgi conjecture. Is $u$ one-dimensional, at least under suitable restrictions on $\alpha$ and $n$? This is the fractional analogue of De Giorgi's one-dimensional symm…

Let $u\in C^2(\mathbb{R}^{n})$ satisfy $|u|<1$ and be a stable critical point of $E_{s,\varepsilon}$. Stability conjecture. If $n\leq7$, then $u$ is one-dimensional provided…

Projected Wiener dynamics conjecture. For $T_\varepsilon\leq c\varepsilon\eta_\varepsilon^{-1}\wedge\tau$, one expects

Interface annihilation conjecture. When two interfaces in the stochastic Allen–Cahn equation become separated by a distance of order $\varepsilon$, both interfaces annihilate.

Let $u=u(y,z)$ denote the saddle solution on the domain $\Omega^{\ast}$, with $u_{yy}$ its second derivative in the $y$ direction. Concavity conjecture. … The source notes that thi…

Let $u=u(s,t)$ denote the saddle solution in the relevant quadrant, with subscripts denoting partial derivatives. Hessian-sign conjecture. The following inequalities should hold: ……

Let $F$ be a double well potential, and consider an Allen–Cahn type equation … Let the saddle solution in dimension $8$ be the solution with the saddle symmetry associated with the…

Let $U_m$ be the saddle solution of the Allen–Cahn equation in $\mathbb{R}^{2m}$, and let the corresponding energy functional be the Allen–Cahn energy. Minimizer conjecture. The so…

Let $U_m$ be the saddle solution of the Allen–Cahn equation … vanishing exactly on the cone $C_{m,m}$. Cabré's stability conjecture. The saddle solution $U_m$ should be stable for…

Let $U_m$ denote the saddle solution of the Allen–Cahn equation in $\mathbb{R}^{2m}$, and let the corresponding energy functional be the Allen–Cahn energy. Global-minimizer conject…