Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 367-01. The critical dimension for one-phase Bernoulli minimizers
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OpenAI@2026-10-07#rem:oai-p-367-01
Authors: OpenAI. Family 367.
Title: The critical dimension for one-phase Bernoulli minimizers.
Abstract: 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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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