One-dimensional symmetry of bounded H-monotone entire solutions of the elliptic system $\Delta u=\nabla H(u)$ in dimensions $4\le N\le 8$ (Fazly–Ghoussoub system analogue of the De Giorgi conjecture)

Fazly and Ghoussoub (Calc. Var. Partial Differential Equations 47 (2013) 809–823) extended De Giorgi's conjecture for the Allen–Cahn equation to gradient systems $\Delta u=\nabla H(u)$, $u:\mathbb{R}^N\to\mathbb{R}^m$, through the notion of H-monotone solutions, and proved that every bounded H-monotone entire solution has one-dimensional components in dimensions $N\le 3$. Whether the conjecture holds in dimensions $4\le N\le 8$ — the range in which even the scalar ($m=1$) case is open without a limit condition at infinity — remains unresolved. This problem asks for a resolution of that conjecture: either a proof that every bounded H-monotone entire solution of $\Delta u=\nabla H(u)$ in each dimension $4\le N\le 8$ has one-dimensional components, or an explicit counterexample. The companion question from the same paper — a Modica-type pointwise gradient bound for systems — has since been refuted in its general form by counterexamples of Smyrnelis and Sourdis (see references 11 and 12).

Background

The Allen–Cahn equation $\Delta u+u-u^3=0$ models phase coexistence: bounded entire solutions (defined on all of $\mathbb{R}^N$) with $|u|\le 1$ describe diffuse interfaces. A solution is called one-dimensional if it has the form $u(x)=g(a\cdot x)$ for a profile $g$ and a fixed direction $a$, i.e. its level sets are parallel hyperplanes. In 1978 E. De Giorgi conjectured — the conjecture is documented in the works of Ghoussoub–Gui cited below — that any bounded entire solution of the Allen–Cahn equation that is monotone in one direction, $\partial u/\partial x_N>0$, must be one-dimensional, at least for $N\le 8$. The threshold 8 parallels the Bernstein problem for minimal hypersurfaces: interfaces of solutions formally converge to minimal hypersurfaces, which are hyperplanes for $N\le 8$ but not for $N\ge 9$.

The scalar conjecture is known for $N=2$ (Ghoussoub–Gui 1998, via a Liouville theorem for equations $\mathrm{div}(\varphi^2\nabla\sigma)=0$ in divergence form), for $N=3$ (Ambrosio–Cabré 2000), for $N=4,5$ under an antisymmetry condition (Ghoussoub–Gui 2003), and for all $4\le N\le 8$ under the additional hypothesis that $u(x',x_N)\to\pm 1$ as $x_N\to\pm\infty$ (Savin 2009, by a nonvariational improvement-of-flatness argument). It is false for $N\ge 9$ (del Pino–Kowalczyk–Wei 2011, who built monotone non-one-dimensional solutions by infinite-dimensional gluing). With monotonicity alone, the scalar problem is still open for $4\le N\le 8$.

Fazly and Ghoussoub (2013) studied the gradient system

$\Delta u=\nabla H(u)\quad\text{in }\mathbb{R}^N,$

where $u=(u_i)_{i=1}^m:\mathbb{R}^N\to\mathbb{R}^m$, $H\in C^2(\mathbb{R}^m)$ and $\nabla H(u)=(H_{u_i}(u))_{i=1}^m$; for $m=1$ this is the scalar semilinear equation $\Delta u=H'(u)$. Such systems model multi-phase transitions and coupled condensates with vector-valued order parameters. Because different components of a solution need not be simultaneously increasing, the scalar monotonicity is replaced by H-monotonicity: each component $u_i$ is strictly monotone in the $x_N$-variable and the sign condition $H_{u_iu_j}\,\partial_N u_i\,\partial_N u_j\le 0$ holds for all $i<j$, which forces the mixed second derivatives of $H$ to compensate the (possibly opposite) monotonicity directions of the components; for $m=1$ it reduces to ordinary monotonicity. A related structural notion is that of an orientable system, meaning there exist nonzero $C^1$ functions $\theta_k$ of fixed sign with $H_{u_iu_j}\theta_i\theta_j\le 0$ for $i<j$; the existence of an H-monotone solution forces orientability, and for orientable systems the variational notion of stability (nonnegative second variation of the energy $\int (\frac12|\nabla u|^2+H(u))\,dx$) coincides with the pointwise (spectral) one. Fazly–Ghoussoub proved a Liouville theorem for the linearized systems, a geometric Poincaré inequality for systems, rigidity of bounded stable solutions in the plane (their components have parallel gradients), and their main Theorem 5: every bounded H-monotone entire solution has one-dimensional components for $N\le 3$. Their Conjecture 2 asserts — stated per component, so that the directions of different components need not coincide — that the level sets of each component of a bounded H-monotone entire solution are hyperplanes, 'at least in dimensions $N\le 8$'; the range $4\le N\le 8$ is the open part, and it contains the classical scalar conjecture as the special case $m=1$. Note that the conjecture imposes no limit condition at $x_N\to\pm\infty$ and no hypothesis on $H$ beyond smoothness and the sign conditions induced by H-monotonicity.

A companion analytic tool in the scalar theory is Modica's gradient bound (Modica 1985): a bounded entire solution of $\Delta u=H'(u)$ satisfies the pointwise estimate $|\nabla u|^2\le 2H(u)$ (when $H\ge 0$ is normalized appropriately), which implies that the scaled energy $E_R(u)/R^{N-1}$, with $E_R(u)=\int_{B_R}(\frac12|\nabla u|^2+H(u)-\inf H)\,dx$, is increasing in $R$. Fazly–Ghoussoub's Remark 2 records that this pointwise estimate was not known for systems, only the weaker monotonicity of $E_R(u)/R^{N-2}$ under $H\ge 0$ (Caffarelli–Lin 2008; Alikakos 2011). The general system version of this bound has since been refuted by explicit counterexamples of Smyrnelis and Sourdis (see references 11 and 12), so the higher-dimensional symmetry conjecture is the surviving open core of the source's acknowledged open problems.

Problem Statement

Determine the validity of Conjecture 2 of Fazly–Ghoussoub in the remaining open dimensions. Let $N$ satisfy $4\le N\le 8$, let $m\ge 1$, let $H\in C^2(\mathbb{R}^m)$, and let $u=(u_i)_{i=1}^m:\mathbb{R}^N\to\mathbb{R}^m$ be a bounded entire solution of the gradient system $\Delta u=\nabla H(u)$ that is H-monotone, i.e. each component $u_i$ is strictly monotone in the $x_N$-direction (that is, $\partial_N u_i\ne 0$ with constant sign on $\mathbb{R}^N$) and $H_{u_iu_j}\,\partial_N u_i\,\partial_N u_j\le 0$ for all $1\le i<j\le m$. Must each component be one-dimensional, that is, for each $i$ there exist a profile $g_i\in C^2(\mathbb{R})$ and a direction $a_i\in\mathbb{R}^{N-1}$ (possibly different for different components) such that $u_i(x)=g_i(a_i\cdot x'-x_N)$, where $x=(x',x_N)$ — equivalently, must the level sets of every component be parallel hyperplanes?

The conclusion is per component: Conjecture 2 as stated by Fazly–Ghoussoub asserts that the level sets of each component are hyperplanes, and the directions of different components need not coincide. This reading is forced by the conjecture itself: for a decoupled potential $H(u)=W_1(u_1)+\cdots+W_m(u_m)$ with scalar potentials $W_k$ whose monotone heteroclinic profiles solve the component equations, a solution whose components are one-dimensional along different directions is bounded, H-monotone (the mixed derivatives $H_{u_iu_j}$ vanish, so the sign condition holds vacuously), and exists in every dimension, including the range $N\le 3$ where the conjecture is proved. A formulation demanding one shared direction for all components is therefore not the conjecture and is refuted for trivial reasons.

The problem is settled either by a rigorous proof that the answer is affirmative for every $N$ with $4\le N\le 8$ (for all $m\ge 1$ and all $H\in C^2(\mathbb{R}^m)$ as above, with no unstated hypotheses such as limits of $u$ as $x_N\to\pm\infty$; a proof covering only a proper subset of the dimensions $4\le N\le 8$, or only special classes of potentials or special $m$, is progress but not a settlement), or by an explicit counterexample: a bounded H-monotone entire solution of $\Delta u=\nabla H(u)$ in some dimension $4\le N\le 8$ of which at least one component is not one-dimensional. Known partial results that do not settle the problem: the case $N\le 3$ (Fazly–Ghoussoub, Theorem 5), the scalar case $m=1$ in $4\le N\le 8$ under the additional limit condition $u\to\pm 1$ as $x_N\to\pm\infty$ (Savin), and the low-dimensional or special-system extensions in the literature, e.g. two-component degenerate elliptic systems in $\mathbb{R}^2$ (Dipierro 2013), fractional gradient systems $(-\Delta)^s u=\nabla H(u)$ in $n=2,3$ (Fazly–Sire 2015), quasilinear symmetric systems in $N=2,3$ (Fazly 2015), and two-component Rabi-coupled condensate systems in $\mathbb{R}^3$ (Wu–Zhang 2025).

References

1. Mostafa Fazly, Nassif Ghoussoub, De Giorgi type results for elliptic systems, Calculus of Variations and Partial Differential Equations 47 (2013) 809–823, DOI 10.1007/s00526-012-0536-x, arXiv:1203.6114, https://arxiv.org/abs/1203.6114
2. Nassif Ghoussoub, Changfeng 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. Luigi Ambrosio, Xavier Cabré, Entire solutions of semilinear elliptic equations in $\mathbb{R}^3$ 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. Nassif Ghoussoub, Changfeng Gui, On De Giorgi's conjecture in dimensions 4 and 5, Annals of Mathematics (2) 157 (2003) 313–334, DOI 10.4007/annals.2003.157.313, https://doi.org/10.4007/annals.2003.157.313
5. Ovidiu 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. Manuel del Pino, Michał Kowalczyk, Juncheng Wei, On De Giorgi's conjecture in dimension $N\ge 9$, Annals of Mathematics (2) 174 (2011) 1485–1569, DOI 10.4007/annals.2011.174.3.3, arXiv:0806.3141, https://arxiv.org/abs/0806.3141
7. Luciano Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Communications on Pure and Applied Mathematics 38 (1985) 679–684, DOI 10.1002/cpa.3160380515, https://doi.org/10.1002/cpa.3160380515
8. Stanley Alama, Lia Bronsard, Changfeng Gui, Stationary layered solutions in $\mathbb{R}^2$ for an Allen-Cahn system with multiple well potential, Calculus of Variations and Partial Differential Equations 5 (1997) 359–390, DOI 10.1007/s005260050071, https://doi.org/10.1007/s005260050071
9. Nicholas Alikakos, Some basic facts on the system $\Delta u-W_u(u)=0$, Proceedings of the American Mathematical Society 139 (2011) 153–162, DOI 10.1090/s0002-9939-2010-10453-7, https://doi.org/10.1090/s0002-9939-2010-10453-7
10. L. Caffarelli, Fang-Hua Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, Journal of the American Mathematical Society 21 (2008) 847–862, DOI 10.1090/s0894-0347-08-00593-6, https://doi.org/10.1090/s0894-0347-08-00593-6
11. Panayotis Smyrnelis, Gradient estimates for semilinear elliptic systems and other related results, Proceedings of the Royal Society of Edinburgh Section A 145 (2015) 1313–1330, DOI 10.1017/s0308210515000347, arXiv:1401.4847, https://arxiv.org/abs/1401.4847
12. Christos Sourdis, Counterexamples to Modica's gradient estimate for systems arising in multi-phase transitions, arXiv:1510.03651 (2015), DOI 10.48550/arXiv.1510.03651, https://arxiv.org/abs/1510.03651
13. Nassif Ghoussoub, Brendan Pass, Decoupling of DeGiorgi-Type Systems via Multi-Marginal Optimal Transport, Communications in Partial Differential Equations 39 (2014) 1032–1047, DOI 10.1080/03605302.2013.849730, https://doi.org/10.1080/03605302.2013.849730
14. Mostafa Fazly, Entire solutions of quasilinear symmetric systems, arXiv:1506.02731 (2015), DOI 10.48550/arXiv.1506.02731, https://arxiv.org/abs/1506.02731
15. Serena Dipierro, Geometric inequalities and symmetry results for elliptic systems, Discrete and Continuous Dynamical Systems 33 (2013) 3473–3496, DOI 10.3934/dcds.2013.33.3473, arXiv:1207.4435, https://arxiv.org/abs/1207.4435
16. Mostafa Fazly, Yannick Sire, Symmetry results for fractional elliptic systems and related problems, Communications in Partial Differential Equations 40 (2015) 1070–1095, DOI 10.1080/03605302.2014.1003073, arXiv:1402.1193, https://arxiv.org/abs/1402.1193
17. Alberto Farina, Berardino Sciunzi, Nicola Soave, Monotonicity and rigidity of solutions to some elliptic systems with uniform limits, Communications in Contemporary Mathematics 22 (2020), no. 5, article 1950044, DOI 10.1142/s0219199719500445, arXiv:1704.06430, https://arxiv.org/abs/1704.06430
18. Leyun Wu, Chilin Zhang, Phase transitions in two-component Bose–Einstein condensates with Rabi frequency (I): The De Giorgi conjecture for the local problem in $\mathbb{R}^3$, arXiv:2509.19124 (2025), DOI 10.48550/arXiv.2509.19124, https://arxiv.org/abs/2509.19124
19. 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