De Giorgi's conjecture for the Allen–Cahn equation in intermediate dimensions $5\le N\le 8$

De Giorgi's 1978 conjecture predicts that every bounded entire solution of the Allen–Cahn equation $\Delta u+u-u^3=0$ in $\mathbb{R}^N$ that is monotone in one direction, $\partial_{x_N}u>0$, is one-dimensional (all level sets are parallel hyperplanes), at least for $N\le 8$. It is proven for $N\le 3$ (Ghoussoub–Gui; Ambrosio–Cabré) and for $N=4$ (Chan–Fernández-Real–Figalli–Florit-Simon–Serra, 2026), and refuted for $N\ge 9$ (del Pino–Kowalczyk–Wei, via the Bombieri–De Giorgi–Giusti minimal graph), and Savin proved the intermediate range $4\le N\le 8$ only under an additional limit hypothesis at infinity. The problem is the surviving open core: decide, for each $5\le N\le 8$ and without any extra hypothesis, whether such a solution must be one-dimensional, or construct a monotone bounded non-one-dimensional entire solution in some intermediate dimension. Recent 2025–2026 work settles adjacent stable- and finite-index classifications under bounded energy density but explicitly leaves the remaining dimensions $5\le N\le 8$ open.

Background

The Allen–Cahn equation is the semilinear elliptic PDE

$\Delta u + u - u^3 = 0 \quad \text{in } \mathbb{R}^N,$

introduced as a phase-field model for binary alloys: bounded solutions taking values in $(-1,1)$ describe a diffuse interface between the two pure phases $u=\pm1$. It is the Euler–Lagrange equation of the Modica–Mortola energy $\int \big(\tfrac12|\nabla u|^2+\tfrac14(1-u^2)^2\big)\,dx$, whose Gamma-convergence as the interface thickness tends to zero links phase transitions to minimal hypersurfaces. In one space dimension the increasing heteroclinic profile $H(t)=\tanh(t/\sqrt2)$, suitably translated, is the unique bounded increasing solution, and a solution of the PDE in $\mathbb{R}^N$ is called one-dimensional if $u(x)=H(x\cdot a-b)$ for some unit vector $a$ and constant $b$, i.e. all its level sets are parallel hyperplanes.

Motivated by the analogy with Bernstein's theorem for minimal graphs, E. De Giorgi conjectured in 1978 that if $u$ is a bounded entire solution with $\partial_{x_N}u>0$ (a monotone solution), then $u$ is one-dimensional, at least for $N\le 8$; the dimensional threshold mirrors the Simons cone singularity in dimension 8 and the Bombieri–De Giorgi–Giusti (BDG) nonplanar area-minimizing graph, which exists exactly in dimension 9 and above. Monotone solutions are stable, meaning the second variation quadratic form $\int(|\nabla\varphi|^2+(3u^2-1)\varphi^2)\,dx$ is nonnegative, since $\pi=\partial_{x_N}u>0$ solves the linearized equation and the existence of a positive solution implies stability; the Morse index counts the maximal dimension of a subspace on which this form is negative. The conjecture is proven for $N=2$ (Ghoussoub–Gui, via Hamiltonian identities) and $N=3$ (Ambrosio–Cabré, via energy and curvature estimates valid only in low dimensions), and Savin proved it for $4\le N\le 8$ under the additional hypothesis that $u(x',x_N)\to\pm1$ as $x_N\to\pm\infty$ (pointwise in $x'$); with uniform limits this is Gibbons' conjecture, settled in all dimensions. Savin's improvement-of-flatness method crucially uses the limit hypothesis, which together with monotonicity makes the solution a local energy minimizer. In sharp contrast, del Pino, Kowalczyk and Wei constructed, for every $N\ge 9$, a bounded monotone entire solution whose level sets are perturbations of the BDG minimal graph, refuting the conjecture there; nontrivial energy-minimizing solutions also exist in dimension 8 (Liu–Wang–Wei). What remains untouched is the intermediate range $5\le N\le 8$ with no hypothesis beyond boundedness and monotonicity, since $N=4$ was settled in 2026 (Chan–Fernández-Real–Figalli–Florit-Simon–Serra). The same authors proposed a parallel classification program for solutions of finite Morse index (helicoid solutions in $\mathbb{R}^3$, catenoidal ends, and the conjecture that an index-1 solution of the Allen–Cahn equation in $\mathbb{R}^3$ with $\nabla u\ne 0$ outside a compact set must be axially symmetric), which remains open in its unrestricted form and sharpens the picture of how rigid monotone and low-index solutions really are.

Problem Statement

Decide De Giorgi's conjecture in its original, unrestricted form for the intermediate dimensions. Precisely: fix $N$ with $5\le N\le 8$, and let $u\in C^2(\mathbb{R}^N)$ be a bounded entire solution of

$\Delta u + u - u^3 = 0, \qquad |u|<1, \qquad \partial_{x_N}u>0 \ \text{ in } \mathbb{R}^N.$

Must $u$ be one-dimensional, that is, $u(x)=\tanh\big((x\cdot a-b)/\sqrt2\big)$ for some unit vector $a$ with $a_N>0$ and some $b\in\mathbb{R}$, so that every level set $\{u=\lambda\}$ is a hyperplane? The problem is to settle this question, dimension by dimension, for all $5\le N\le 8$ without any additional hypothesis (no pointwise or uniform limits at infinity, no energy growth bound, no minimality or symmetry assumption): either prove one-dimensional symmetry in full generality for each such $N$, or construct a counterexample — a bounded, monotone, non-one-dimensional entire solution — in at least one dimension $5\le N\le 8$. The restriction to $5\le N\le 8$ is the surviving core of De Giorgi's original statement for $N\le 8$, since the cases $N\le 4$ are proven and the conjecture is refuted for $N\ge 9$.

References

1. E. De Giorgi, "Convergence problems for functionals and operators", in Recent Methods in Nonlinear Analysis, Proc. International Meeting (Rome, 1978), Pitagora Editrice, Bologna, 1979, pp. 131–188 (zbMATH record 3629689), https://zbmath.org/3629689
2. N. Ghoussoub and C. Gui, "On a conjecture of De Giorgi and some related problems", Mathematische Annalen 311 (1998) 481–491, DOI 10.1007/s002080050196, https://doi.org/10.1007/s002080050196
3. L. Ambrosio and X. Cabré, "Entire solutions of semilinear elliptic equations in R^3 and a conjecture of De Giorgi", Journal of the American Mathematical Society 13 (2000) 725–739, DOI 10.1090/s0894-0347-00-00345-3, https://doi.org/10.1090/s0894-0347-00-00345-3
4. G. Alberti, L. Ambrosio and X. Cabré, "On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property", Acta Applicandae Mathematicae 65 (2001) 9–33, DOI 10.1023/A:1010602715526, https://doi.org/10.1023/A:1010602715526
5. O. Savin, "Regularity of flat level sets in phase transitions", Annals of Mathematics (2) 169 (2009) 41–78, DOI 10.4007/annals.2009.169.41, https://doi.org/10.4007/annals.2009.169.41
6. M. del Pino, M. Kowalczyk and J. Wei, "On De Giorgi's conjecture in dimension N≥9", Annals of Mathematics 174 (2011) 1485–1569, DOI 10.4007/annals.2011.174.3.3, arXiv:0806.3141, https://arxiv.org/abs/0806.3141
7. M. del Pino, M. Kowalczyk and J. Wei, "On De Giorgi's conjecture and beyond", Proceedings of the National Academy of Sciences USA 109 (2012) 6845–6850, DOI 10.1073/pnas.1202687109, https://doi.org/10.1073/pnas.1202687109
8. M. del Pino, M. Kowalczyk and J. Wei, "Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in R^3", Journal of Differential Geometry 93 (2013), DOI 10.4310/jdg/1357141507, arXiv:0902.2047, https://arxiv.org/abs/0902.2047
9. Y. Liu, K. Wang and J. Wei, "Global minimizers of the Allen-Cahn equation in dimension n≥8", Journal de Mathématiques Pures et Appliquées 108 (2017) 818–840, DOI 10.1016/j.matpur.2017.05.006, arXiv:1606.05315, https://arxiv.org/abs/1606.05315
10. C. Gui, K. Wang and J. Wei, "Axially symmetric solutions of the Allen-Cahn equation with finite Morse index", Transactions of the American Mathematical Society 373 (2020) 3649–3668, DOI 10.1090/tran/8035, arXiv:1901.11263, https://arxiv.org/abs/1901.11263
11. A. Figalli and J. Serra, "On stable solutions for boundary reactions: a De Giorgi type result in dimension 4+1", Inventiones Mathematicae 219 (2020) 153–177, DOI 10.1007/s00222-019-00904-2, arXiv:1705.02781, https://arxiv.org/abs/1705.02781
12. E. Florit-Simon and J. Serra, "On stable solutions to the Allen–Cahn equation with bounded energy density in R^4", arXiv:2509.02739 (2025), https://arxiv.org/abs/2509.02739
13. E. Florit-Simon, "Phase transitions with bounded index: Parallels to De Giorgi's conjecture", arXiv:2602.03136 (2026), https://arxiv.org/abs/2602.03136
14. G. Ferla, "Some remarks on the Allen–Cahn equation in R^n", arXiv:2603.29060 (2026), https://arxiv.org/abs/2603.29060
15. H. Chan, X. Fernández-Real, A. Figalli, E. Florit-Simon and J. Serra, "Stable solutions of the Allen–Cahn equation in dimension three are one-dimensional", arXiv:2609.30194 (2026), https://arxiv.org/abs/2609.30194