One-dimensionality conjecture for minimizers of the one-phase free boundary energy

Assume (A1)–(A2): $\beta$ is non-negative and $C^1$ with $\textrm{supp}\,\beta=[0,1]$ and $\int_0^1\beta=1$, while $u\geq0$ satisfies $\|\nabla u\|_{L^\infty(\mathbb{R}^n)}<\infty$. Let

$$J(u,E)=\int_E\left\{|\nabla u|^2+\Phi(u)\right\}\,dx,$$

where $\Phi'=\beta$, and call $u$ a minimizer in $\mathbb{R}^n$ if $J(u,E)\leq J(u+v,E)$ for every compact set $E\subset\mathbb{R}^n$ and every $v\in H^1_0(E)$. One-dimensionality conjecture. If $u$ is any minimizer of $J$ in $\mathbb{R}^n$ and $n\leq4$, then $u$ is one-dimensional.

The conjecture follows in $\mathbb{R}^2$, while the cases $\mathbb{R}^3$ and $\mathbb{R}^4$ remain open. Its dimensional range is connected to the classification of minimal cones for the one-phase problem; extending it to $n\leq6$ would follow from the conjectured corresponding improvement, whereas counterexamples are known in dimension $7$.