5 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 $F(u)=\frac{1}{4}(1-u^2)^2$, so that $F'(u)=u^3-u$, and consider the reaction-diffusion equation … in $\mathbb{R}^n\times[0,\infty)$. Suppose that $u$ satisfies $|u|\leq 1$ and…

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

De Giorgi's level-set conjecture. The set $\{u>t\}$ should itself be a half space for every $t$, even when $u$ is only monotone in one direction and is not a minimizer. This is the…

Nonlocal stable De Giorgi conjecture. If $d\le 7$, then $u$ is a $1$-D solution. Stable solutions include local minimizers and monotone stationary solutions, and the conjecture con…