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The critical dimension of the one-phase Bernoulli free boundary problem: is it seven?

Theorem 1.1: every nonzero one-homogeneous global minimizer of the one-phase Bernoulli energy in $\mathbb R^d$ , $1\le d\le6$ , is flat, $u=(x\cdot e)_+$ , and a nonflat one-homogeneous global minimizer exists in $\mathbb R^7$ , so $d_*=7$ . Corollary 1.2: the interior free boundary of a local minimizer in $\mathbb R^n$ is smooth for $n\le6$ , its singular set is locally finite for $n=7$ and has Hausdorff dimension at most $n-7$ for $n\ge7$ , sharp by products of the De Silva-Jerison cone. Method: a Hessian-shape test function and vector field on the spherical link, with an exactly verified polynomial inequality. Not shown: classification of stable but non-minimizing cones, two-phase or vector-valued versions, or rigidity of non-homogeneous entire stable solutions.

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Occurred: Sep 24, 2026

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Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The critical dimension of the one-phase Bernoulli free boundary problem: is it seven? · One-phase Bernoulli critical dimension

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the question: every nonzero one-homogeneous global minimizer in dimensions $1\le d\le6$ is flat and a nonflat one exists in dimension seven, so $d_*=7$ . The new content is flatness in dimensions five and six; existence in dimension seven is De Silva and Jerison's 2009 cone. Lean: lean/ComparatorChallenges/BernoulliNonflatInSeven.json exists with solution module OAI.Analysis.BernoulliCone.Main present, not in the formalization catalogue; the statement read here asserts only the existence of a nonzero, one-homogeneous, nonflat global minimizer in $\mathbb R^7$ , the already known half, and its scope note puts flatness in dimensions at most six outside it. So this entry is unreviewed. The flatness proof rests on a universal tensor inequality verified by an exact finite certificate; the release's reproduction scripts were not run here and, by their own README, do not certify the correspondence between formulas and code.