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Math-2858 · mathematics · Pde

Classification of every bounded stable Allen–Cahn solution in dimension seven

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Must every stable C2 solution $v:R^7\to [-1,1]$ of Delta $v=v^3-v$ be identically +1 or $-1$, or have the planar form $tanh((a\cdot x-b)/\operatorname{sqrt}(2))$ for a unit vector a and real b, without an energy-growth or asymptotic end-state assumption?

Short English statement

Must every stable C2 solution $v:R^7\to [-1,1]$ of Delta $v=v^3-v$ be identically +1 or $-1$, or have the planar form $tanh((a\cdot x-b)/\operatorname{sqrt}(2))$ for a unit vector a and real b, without an energy-growth or asymptotic end-state assumption?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2858-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2858

De Giorgi rigidity in dimension eight: The stable classification uses the scalar double-well equation in dimension seven. It assumes neither growth of the energy nor prescribed states at infinity and permits only the two constant wells or the explicitly planar transition profile.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Classification of every bounded stable Allen–Cahn solution in dimension seven

OpenAI's claim

De Giorgi rigidity in dimension eight: The paper claims that every strictly direction-monotone entire solution of the scalar Allen--Cahn equation in $\mathbb R^8$ is a planar heteroclinic. It also classifies stable solutions in $\mathbb R^7$ without an energy-growth assumption.

Mathematical statement of the claimed result

Mathematical claim 1.
The monotone theorem claims that if $u\in C^2(\mathbb R^8,(-1,1))$ solves $\Delta u=u^3-u$ and $\partial_8u>0$, then \[u(x)=\tanh((e\cdot x-c)/\sqrt2),\qquad e\in S^7,\quad e_8>0.\] No limits at infinity are assumed. Its stable seven-dimensional theorem states that every $v\in C^2(\mathbb R^7,[-1,1])$ solving the same equation and satisfying $\int(|\nabla\phi|^2+(3v^2-1)\phi^2)\ge0$ for compactly supported smooth $\phi$ is either $1$, $-1$ or a one-dimensional transition $\tanh((a\cdot x-b)/\sqrt2)$ with $a\in S^6$. No energy-growth assumption is used; lower-dimensional monotone cases follow by adding unused coordinates.

Outline of the argument

De Giorgi rigidity in dimension eight: Finite-density stability first forces a planar profile. To remove that density assumption in dimension seven, penalized reach controls both interface bending and neighboring sheets, and stability bounds a fourth curvature moment. An ambient radial test and matched sheet amplitudes contradict unbounded interface density; the stable directional limits of the eight-dimensional monotone solution then provide the well limits needed for the planar conclusion.

Formal verification

De Giorgi rigidity in dimension eight: The pinned catalogue lists no Lean formalization for this result.

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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