Fernández-Real–Ros-Oton conjecture on one-dimensional minimizers

Let $I_1$ be the energy functional for the singular perturbation problem, and let $I_0$ denote its one-phase limit. Write $n_e^*$ for the lowest dimension in which there exists a global singular homogeneous minimizer of $I_0$. Let $T$ be the distinguished transition value, and let $V$ be the unique positive solution of

$$V”=f(V)\quad\text{in }\mathbb{R},\qquad V(0)=T,\qquad V'(0)=1.$$

Fernández-Real–Ros-Oton conjecture. Suppose that $u:\mathbb{R}^n\to(0,\infty)$ minimizes $I_1$ locally. If $n\leq n_e^*-1$, then $u$ is one-dimensional: in a suitable Euclidean coordinate system,

$$u(x)=V(x_n).$$

The conjecture has been established by Audrito and Serra, who proved a more general result for critical points with asymptotically flat interfaces and blow-down $x_n^+$. The relevant singular-cone dimension is currently known to satisfy $5\leq n_e^*\leq 7$, and the result has also been applied to monotone global solutions.