gpt 6 astra ultra attempt on Math-239

De Giorgi Conjecture for the Allen–Cahn Equation

Definitions and mathematical statement

Fix $4\le N\le8$ and let $u\in C^2({\mathbb{R}}^N)$ take values in $(-1,1)$. Assume \[ \Delta u=u^3-u,\qquad \partial_eu>0\text{ everywhere for some unit vector }e. \] The assertion is that $u$ is one-dimensional: there exist a unit vector $a$ and $g:{\mathbb{R}}\to(-1,1)$ with $u(x)=g(a\cdot x)$. This describes solutions on all of Euclidean space, not a bounded domain, and strict directional monotonicity is the selected hypothesis. Limits $u\to\pm1$ in the monotone direction, energy-growth restrictions, and minimizing properties are not assumed unless independently derived. Adding them would restrict the conjecture.

Formulation and concept references: [S1].

Short English statement

In dimensions four through eight, must every bounded entire Allen–Cahn solution that is strictly monotone in one direction depend on only one linear coordinate?

Research attempt: a ground-state identity and a conditional rigidity theorem

Status and source audit. We derive stability from the stated strict monotonicity and prove one-dimensionality under an additional explicit integral condition. That condition is not part of the question and is not established in dimensions four through eight. The linked source [S1] studies a free-boundary Allen--Cahn/Bernoulli problem in dimension three; its title and abstract do not constitute a solution of the classical entire equation selected here.

1. Monotonicity produces a positive ground state

Let $q=\partial_eu>0$. Standard interior elliptic bootstrapping for the polynomial equation makes $u$ smooth, so differentiating is legitimate. With $V=3u^2-1$, \[-\Delta q+Vq=0.\] For $\phi\in C_c^\infty(\mathbb R^N)$, write $\phi=q\eta$. Expansion and integration by parts give \[\int (|\nabla\phi|^2+V\phi^2) =\int q^2|\nabla\eta|^2+\int\{\nabla q\cdot\nabla(q\eta^2)+Vq^2\eta^2\} =\int q^2|\nabla(\phi/q)|^2\ge0.\tag{1}\] There is no assumption that $q$ is bounded away from zero at infinity: division occurs only on the compact support, where positivity and continuity suffice. Equation $1$ proves stability under compactly supported variations for every solution in the target.

2. Quotients of derivatives and a quantitative sufficient condition

For each coordinate let $w_i=\partial_i u$ and $\sigma_i=w_i/q$. Both $w_i$ and $q$ solve the same linearized equation, hence \[\operatorname{div}(q^2\nabla\sigma_i)=q\Delta w_i-w_i\Delta q=0.\tag{2}\] Testing $2$ with $\eta^2\sigma_i$, where $\eta$ has compact support, yields \[\int q^2\eta^2|\nabla\sigma_i|^2 \le4\int q^2\sigma_i^2|\nabla\eta|^2=4\int w_i^2|\nabla\eta|^2.\tag{3}\] Suppose additionally that, for every $i$, \[\int_{B_{2R}\setminus B_R}w_i^2=o(R^2)\qquad(R\to\infty).\tag{4}\] Choose cutoffs equal to one on $B_R$, zero outside $B_{2R}$, with gradient bounded by $C/R$. The right side of $3$ tends to zero. For each fixed ball the integral of $q^2|\nabla\sigma_i|^2$ is therefore zero, so $\sigma_i$ is constant. Write these constants as a vector $c$. Then $\nabla u=qc$ and $e\cdot c=1$, so $c\ne0$. Every derivative tangent to $c^\perp$ vanishes. Integrating along lines in that hyperplane proves $u(x)=g(a\cdot x)$ with $a=c/|c|$.

3. Classification after the reduction

The resulting $g$ satisfies $g''=g^3-g$, is bounded in $(-1,1)$, and, orienting $a$ appropriately, is strictly increasing. Its limits $\ell_-<\ell_+$ exist. The ODE and bounded monotonicity imply $g'\to0$ and $\ell_\pm^3-\ell_\pm=0$: otherwise bounded second derivative together with a persistent nonzero slope or acceleration contradicts the finite limits. Multiplication by $g'$ gives \[\tfrac12(g')^2-\tfrac14(1-g^2)^2=C.\] The endpoint potentials must agree. Among the possible limits $-1,0,1$, the only distinct ordered pair with equal potential is $(-1,1)$. Consequently $C=0$, $g'=(1-g^2)/\sqrt2$, and separation of variables gives \[g(t)=\tanh((t-t_0)/\sqrt2).\] Thus the additional hypothesis $4$ gives the full expected profile, not merely a weaker local flatness statement.

4. Verification and gap

The sign in the linearized operator is fixed by $\Delta u=u^3-u$, and the cancellation in $1$ uses $-\Delta q+Vq=0$. Inequality $3$ uses $2ab\le\tfrac12a^2+2b^2$. Condition $4$ is sufficient, not necessary: even planar fronts in higher dimensions can have derivative energy of order $R^{N-1}$ and need not satisfy it. In particular, a uniform pointwise gradient bound only gives an $O(R^N)$ volume bound and cannot imply $4$. We have no argument forcing the derivative ratios to be constant under the original hypotheses alone. That missing Liouville or geometric rigidity step remains the conjectural part.

Sources

• S1 Hardy Chan, Xavier Fernández-Real, Alessio Figalli, and Joaquim Serra, arXiv:2503.21245. https://arxiv.org/abs/2503.21245. Formulation source.