{"id":"openai-math/The-critical-dimension-for-one-phase-Bernoulli-minimizers","pdf":"https://github.com/openai/math/blob/main/preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026","tags":["free-boundary-problems","geometric-measure-theory","calculus-of-variations"],"score":0,"title":"The critical dimension for one-phase Bernoulli minimizers","domain":"Partial differential equations","family":"367","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.AP","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":false,"family_title":"The critical dimension for the one-phase Bernoulli problem"}