6 problems

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…