One-dimensionality of stable Allen–Cahn solutions with bounded energy density in $\mathbb{R}^n$ for $5\le n\le 7$ (hypothesis $(\star)$)

Hypothesis $(\star)$ of Florit-Simon's 2026 study of phase transitions with bounded Morse index asks whether every entire stable solution of the Allen–Cahn equation $-\Delta u+W'(u)=0$ in $\mathbb{R}^n$ with bounded energy density, i.e. energy growth $E(u,B_R)\le CR^{n-1}$, must be one-dimensional. This classification is known for $n\le3$ (Ambrosio–Cabré; Alberti–Ambrosio–Cabré) and was proved for $n=4$ by Florit-Simon and Serra in 2025, who also identified the precise quantitative obstruction that blocks the argument for $n\ge5$. The 2026 paper's central finite-index classification theorem in dimensions $4\le n\le7$ is conditional on $(\star)$, so settling $(\star)$ in the remaining dimensions $5\le n\le7$ is the single missing input for unconditional finite-index rigidity up to dimension 7. No proof of, or counterexample to, $(\star)$ in any of $n=5,6,7$ is known.

Background

The Allen–Cahn equation $-\Delta u+W'(u)=0\ \ \text{in }\mathbb{R}^n,$ with the double-well potential $W(u)=\frac14(1-u^2)^2$, is the Euler–Lagrange equation of the phase-transition energy $E(u,\Omega)=\int_\Omega\big(\tfrac12|\nabla u|^2+W(u)\big)\,dx$. It models the interface between the two pure phases $u\equiv\pm1$ of a binary material; as the interface thickness $\varepsilon\to0$ in the rescaled equation, transition layers concentrate on minimal hypersurfaces (hypersurfaces of zero mean curvature), making the equation a PDE laboratory for questions from minimal-surface theory. A solution is called one-dimensional if it depends on a single Euclidean variable, $u(x)=\tanh\big((e\cdot x-s_0)/\sqrt2\big)$ for a unit vector $e$ and constant $s_0$ (the profile $\tanh(\cdot/\sqrt2)$ is the unique increasing heteroclinic connecting $-1$ to $1$), or if $u\equiv\pm1$; geometrically, all level sets are parallel hyperplanes. The Morse index of $u$ is the maximal dimension of a linear space of compactly supported perturbations $\eta$ on which the second variation $Q(\eta,\eta)=\int_{\mathbb{R}^n}\big(|\nabla\eta|^2+W''(u)\eta^2\big)\,dx$ is negative; index zero — $Q(\eta,\eta)\ge0$ for all $\eta\in C_c^1(\mathbb{R}^n)$ — is called stability, a linearized notion of local minimality at every scale.

A well-known 1978 conjecture of De Giorgi asserts that bounded solutions that are monotone in one direction (monotone solutions are automatically stable) are one-dimensional in ambient dimension at most 8. It is a theorem in ambient dimensions 2 and 3, was proved in dimensions 4 through 8 by Savin's flat level-set regularity theorem for solutions with $|u|\le1$, $\partial_n u>0$, and $\lim_{x_n\to\pm\infty}u(x',x_n)=\pm1$, and is false in dimension 9 and above by the counterexample of del Pino, Kowalczyk, and Wei. The stronger statement with monotonicity dropped — the stable De Giorgi conjecture — is open even in $\mathbb{R}^3$ without further hypotheses. If one adds the natural codimension-one density bound $E(u,B_R)\le CR^{n-1}\quad\text{for all }R>0,$ which mirrors the area growth of minimal hypersurfaces with bounded area ratios, the stable classification becomes a theorem for $n\le3$ (Ambrosio–Cabré, and Alberti–Ambrosio–Cabré for general nonlinearities) and, very recently, for $n=4$: Florit-Simon and Serra proved in 2025 that entire stable Allen–Cahn solutions in $\mathbb{R}^4$ with bounded energy density are one-dimensional. Their work also identified the concrete obstruction to reaching higher dimensions: the $C^{2,\theta}$ mean-curvature estimates of Wang–Wei for transition layers, after the natural rescaling near a non-flat point, become too weak to initiate the improvement-of-flatness iteration precisely when $n\ge5$.

The minimal-surface analogue of this question — the stable Bernstein theorem, classifying entire stable minimal hypersurfaces as hyperplanes — was completed through $\mathbb{R}^6$ in a sequence of recent works (Chodosh–Li in $\mathbb{R}^4$; Chodosh–Li–Minter–Stryker in $\mathbb{R}^5$; Mazet in $\mathbb{R}^6$) and remains open in $\mathbb{R}^7$; the Allen–Cahn counterpart currently stops at $n=4$. Florit-Simon (2026) proves that any finite-Morse-index solution in $\mathbb{R}^4$ with bounded energy density is one-dimensional (Theorem 1.1), and shows in Theorem 1.2 that the same conclusion holds in every dimension $4\le n\le7$ conditionally on hypothesis $(\star)$: the stable classification above. The dimension restriction is sharp: in $\mathbb{R}^8$ non-planar energy-minimizing — hence stable, finite-index — solutions exist, asymptotic to leaves of the minimal foliation of the Simons cone $\{|x'|=|x''|\}$, a minimal cone that becomes stable exactly above the critical dimension. Thus $(\star)$ in $5\le n\le7$ is the last missing input for unconditional finite-index rigidity up to dimension 7, and the problem below asks exactly for its resolution.

Problem Statement

Let $n\in\{5,6,7\}$ and let $W(u)=\frac14(1-u^2)^2$. Let $u:\mathbb{R}^n\to[-1,1]$ be an entire classical solution of the Allen–Cahn equation $-\Delta u+W'(u)=0\ \ \text{in }\mathbb{R}^n$ which is stable, that is, $\int_{\mathbb{R}^n}\big(|\nabla\eta|^2+W''(u)\,\eta^2\big)\,dx\ \ge\ 0\quad\text{for every }\eta\in C_c^1(\mathbb{R}^n),$ and has bounded energy density $E(u,B_R)=\int_{B_R}\Big(\tfrac12|\nabla u|^2+W(u)\Big)\,dx\ \le\ CR^{n-1}\quad\text{for all }R>0$ for some constant $C$ independent of $R$. Prove that $u$ is one-dimensional, i.e. $u(x)=\tanh\big((e\cdot x-s_0)/\sqrt2\big)$ for some $e\in\mathbb{S}^{n-1}$ and $s_0\in\mathbb{R}$, or else $u\equiv\pm1$; or exhibit a counterexample in some dimension $5\le n\le7$, namely a stable, non-one-dimensional entire solution satisfying the same energy density bound. This is hypothesis $(\star)$ of Florit-Simon (2026) in the dimensions where it remains unresolved; the hypothesis is asserted dimension by dimension.

References

1. Enric Florit-Simon, “Phase transitions with bounded index: Parallels to De Giorgi's conjecture”, arXiv:2602.03136 (2026). DOI: 10.48550/arxiv.2602.03136. https://arxiv.org/abs/2602.03136
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