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Math-2493 · mathematics · Pde

Rigidity of Stable One-Phase Solutions

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $u$ be a global stable solution of the one-phase free-boundary problem in $\mathbb R^n$. Prove that $u$ is one-dimensional when $n\leq 6$.

Formulation references: [S1], [S2].

Short English statement

Let $u$ be a global stable solution of the one-phase free-boundary problem in $\mathbb R^n$. Prove that $u$ is one-dimensional when $n\leq 6$.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2493-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2493

Critical Bernoulli dimension: The flatness theorem covers one-homogeneous global minimizers in dimensions at most six, which are stable solutions in that special class. The catalogue asks all global stable solutions without homogeneity or global-minimality assumptions.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Rigidity of Stable One-Phase Solutions

OpenAI's claim

Critical Bernoulli dimension: The claimed first nonflat minimizing Bernoulli cone occurs in dimension seven; all homogeneous global solutions that minimize in lower dimensions are flat. Dimensions at most six have smooth local free boundaries, with the announced sharp singular-set bounds in higher dimensions.

Mathematical statement of the claimed result

Mathematical claim 1.
For nonnegative $u\in H^1_{\mathrm{loc}}(\mathbb R^d)$, call it a global one-phase minimizer if on every ball it minimizes \[J(u;B)=\int_B(|\nabla u|^2+\mathbf1_{\{u>0\}})\,dx\] against all nonnegative Sobolev competitors with the same boundary trace. The claim states that every nonzero one-homogeneous global minimizer, $u(rx)=ru(x)$, in $1\le d\le6$ is $(x\cdot e)_+$ with $|e|=1$, and some nonflat one-homogeneous global minimizer exists in $\mathbb R^7$. Thus the least nonflat minimizing-cone dimension is $d_*=7$. This concerns global minimizers rather than all stable critical points.

Outline of the argument

Critical Bernoulli dimension: A first hypothetical nonflat minimizer below dimension seven is reduced to a spherical component in dimension six. Tensor identities and stability produce a flux comparison whose algebraic estimate is verified with exact coefficient data. Removing cutoffs contradicts the spherical configuration; the known nonflat seven-dimensional construction gives the other side and the sharpness consequences.

Formal verification

Family 367 describes Lean coverage for existence of a nonflat homogeneous minimizer in dimension seven; the lower-dimensional flatness and singular-set classifications are separate.

Critical Bernoulli dimension: The manuscript links BernoulliNonflatInSeven.lean.

The documented selection does not cover the entire manuscript assertion relevant here.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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