OpenAI · 367-01 · 2026-09-24 · Partial differential equations
The critical dimension for one-phase Bernoulli minimizers
preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026
Family 367. The critical dimension for the one-phase Bernoulli problem.
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Read against the standing problems
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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 in higher dimensions.
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 , and this bound is sharp. In dimension seven, the singular set is locally finite.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 367: The critical dimension for the one-phase Bernoulli problem. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.