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Math-2374 · mathematics · Pde

De Giorgi Conjecture for Allen–Cahn

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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.
If a bounded solution of $$\Delta u+(1-u^2)u=0\quad\text{in }\mathbb R^N$$ satisfies $\partial_{x_N}u>0$, must all level sets of $u$ be hyperplanes when $N\leq8$?

Formulation references: [S1], [S2].

Short English statement

If a bounded solution of $$\Delta u+(1-u^2)u=0\quad\text{in }\mathbb R^N$$ satisfies $\partial_{x_N}u>0$, must all level sets of $u$ be hyperplanes when $N\leq8$?

Sources

Catalogue entry

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

Contributions 1

Math-2374-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2374

De Giorgi rigidity in dimension eight: The dimension-eight monotone theorem and the trivial extension of any lower-dimensional solution to unused coordinates establish hyperplane level sets for every bounded monotone solution in N<=8. This matches the duplicated De Giorgi formulation, not only a subclass with asymptotic boundary conditions.

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

De Giorgi Conjecture for Allen–Cahn

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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