The critical dimension for one-phase Bernoulli minimizers

We prove that seven is the critical dimension for the one-phase Bernoulli problem: every nonzero one-homogeneous global minimizer in dimensions at most six is flat, while a nonflat one-homogeneous global minimizer exists in dimension seven. It follows that the interior free boundary of a local minimizer is smooth in dimensions at most six. In dimension $n\ge7$, its singular set has Hausdorff dimension at most $n-7$, and this bound is sharp. In dimension seven, the singular set is locally finite.