The conjecture that the critical dimension in the Alt–Caffarelli problem is six

Let $u$ be a minimizer of the Alt–Caffarelli problem in a domain $\Omega\subset\mathbb{R}^d$. Decompose its free boundary as

$$\Gamma(u)=\operatorname{Reg}(u)\cup\operatorname{Sing}(u).$$

Define the critical dimension

$$d^*:=\max\{d\in\mathbb{N}:\text{ homogeneous minimizers are rotations of }(x_d)_+\text{ in }\mathbb{R}^d\}.$$

Critical-dimension conjecture. The critical dimension $d^*$ is $6$. In particular, for any minimizer $u$ of the Alt–Caffarelli problem in $\Omega\subset\mathbb{R}^d$ with $d\leq 6$, its free boundary $\Gamma(u)$ is analytic. This conjecture would settle the currently known bounds $4\leq d^*\leq 6$ and, through the dimension estimates for the singular part, would establish analyticity of the free boundary in dimensions at most six. The upper bound is supported by the De Silva–Jerison cone in $\mathbb{R}^7$, while the lower bound follows from work of Caffarelli–Jerison–Kenig and Jerison–Savin.