The critical dimension for the one-phase Bernoulli problem

Establishes seven as the first dimension admitting a nonflat, one-homogeneous global minimizer of the one-phase Bernoulli energy. Consequently, minimizing free boundaries are smooth through dimension six, and their singular sets have dimension at most n-7 in higher dimensions. (Lean)

Manuscript

The critical dimension for one-phase Bernoulli minimizers

September 24, 2026 · source · cite as \cite{OAI:The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026}

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 ≥ 7, its singular set has Hausdorff dimension at most n-7, and this bound is sharp. In dimension seven, the singular set is locally finite.

Result 367 · Partial differential equations. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.