The critical dimension of the one-phase Bernoulli free boundary problem: is it seven?

Theorem 1.1: every nonzero one-homogeneous global minimizer of the one-phase Bernoulli energy in $\mathbb R^d$, $1\le d\le6$, is flat, $u=(x\cdot e)_+$, and a nonflat one-homogeneous global minimizer exists in $\mathbb R^7$, so $d_*=7$. Corollary 1.2: the interior free boundary of a local minimizer in $\mathbb R^n$ is smooth for $n\le6$, its singular set is locally finite for $n=7$ and has Hausdorff dimension at most $n-7$ for $n\ge7$, sharp by products of the De Silva-Jerison cone. Method: a Hessian-shape test function and vector field on the spherical link, with an exactly verified polynomial inequality. Not shown: classification of stable but non-minimizing cones, two-phase or vector-valued versions, or rigidity of non-homogeneous entire stable solutions.

Theorem 1.1: every nonzero one-homogeneous global minimizer of the one-phase Bernoulli energy in $\mathbb R^d$, $1\le d\le6$, is flat, $u=(x\cdot e)_+$, and a nonflat one-homogeneous global minimizer exists in $\mathbb R^7$, so $d_*=7$. Corollary 1.2: the interior free boundary of a local minimizer in $\mathbb R^n$ is smooth for $n\le6$, its singular set is locally finite for $n=7$ and has Hausdorff dimension at most $n-7$ for $n\ge7$, sharp by products of the De Silva-Jerison cone. Method: a Hessian-shape test function and vector field on the spherical link, with an exactly verified polynomial inequality. Not shown: classification of stable but non-minimizing cones, two-phase or vector-valued versions, or rigidity of non-homogeneous entire stable solutions.