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 ≥ 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.
Cite
@misc{OAI:The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026,
author = {{OpenAI}},
title = {{The critical dimension for one-phase Bernoulli minimizers}},
howpublished = {OpenAI Math Release preprint
\href{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}{OAI:The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026}},
year = {2026}
}