Stable De Giorgi conjecture for the classical Allen–Cahn equation: one-dimensional symmetry of bounded stable solutions (open for 4 ≤ d ≤ 7)

The stable De Giorgi conjecture predicts that every bounded stable solution of the classical Allen–Cahn equation $-\Delta u=u-u^3$ in $\mathbb{R}^d$ is a one-dimensional layer for every $d\le7$. It is known for $d=2$ (Ghoussoub–Gui 1998) and, since 2026, for $d=3$ (Chan–Fernández-Real–Figalli–Florit-Simon–Serra 2026; Liu–Luo–Wang–Wei–Wei–Wu 2026). In the surrounding literature, the monotone version of De Giorgi's conjecture is settled in low dimensions and false for $d\ge9$; fractional-order analogues are resolved in neighboring regimes (Gui–Li 2020; Cabré–Cinti–Serra 2025); and the 2025–2026 work settles only conditional or variant statements — stable solutions in $\mathbb{R}^4$ under a bounded-energy-density hypothesis, finite-index analogues, and a free-boundary variant in 3D. The problem — prove that every bounded stable solution in $\mathbb{R}^d$, $4\le d\le7$, is a one-dimensional layer, or construct a stable non-one-dimensional example in some dimension in that range — is open.

Background

The Allen–Cahn equation

$$-\Delta u = u - u^3$$

is the Euler–Lagrange equation of the Ginzburg–Landau energy $\int\left(\tfrac{1}{2}|\nabla u|^2+\tfrac{1}{4}(1-u^2)^2\right)dx$, the basic continuum model of phase coexistence between two preferred states $u\approx\pm1$: a bounded solution with values in $(-1,1)$ describes a diffuse interface whose mid-surface is the level set $\{u=0\}$. Under rescaling, the associated phase-transition energies converge to the perimeter functional, so interfaces of low-energy solutions behave like minimal hypersurfaces, that is, hypersurfaces with zero mean curvature.

In 1978 Ennio De Giorgi conjectured that every bounded solution of $-\Delta u=u-u^3$ in $\mathbb{R}^d$ that is monotone in one direction, $\partial_{x_d}u>0$, has only planar level sets (for $d\le8$). A solution with only planar level sets is one-dimensional: $u(x)=\phi(e\cdot x)$ for a unit vector $e$ and a bounded increasing function $\phi$ solving the ordinary differential equation $\phi''=\phi^3-\phi$; such solutions are called layer solutions. The monotone conjecture is known in $d=2$ and $d=3$ (Ghoussoub–Gui 1998; Ambrosio–Cabré 2000), in $4\le d\le8$ under the additional condition $\lim_{x_d\to\pm\infty}u=\pm1$ (Savin 2009), and is false for $d\ge9$, where del Pino, Kowalczyk, and Wei (2011) constructed monotone solutions with non-planar level sets.

Monotone solutions are stable, meaning the second variation of the Ginzburg–Landau energy at $u$ is nonnegative:

$$\int_{\mathbb{R}^d}\left(|\nabla\xi|^2+(3u^2-1)\xi^2\right)dx\ \ge\ 0\quad\text{for every }\xi\in C_c^\infty(\mathbb{R}^d).$$

Stability is strictly weaker than monotonicity or minimality, and the corresponding stronger statement — the stable De Giorgi conjecture — asserts that every bounded stable solution of $-\Delta u=u-u^3$ in $\mathbb{R}^d$, $d\le7$, is one-dimensional. It is known for $d=2$ (Ghoussoub–Gui 1998) and for $d=3$ (Chan–Fernández-Real–Figalli–Florit-Simon–Serra 2026; Liu–Luo–Wang–Wei–Wei–Wu 2026). The threshold $d\le7$ mirrors minimal-surface theory: complete stable minimal hypersurfaces in $\mathbb{R}^3$ are planes (Fischer-Colbrie–Schoen 1980), and stable minimal cones are hyperplanes up to dimension 7 (Simons 1968), while stable non-one-dimensional solutions exist for $d\ge8$ (Pacard–Wei 2013); the monotone counterexamples above are likewise stable but begin only at $d\ge9$. The first open case is now $d=4$.

Figalli–Serra (2020) settled the analogue of this conjecture for the half-Laplacian boundary-reaction equation $(-\Delta)^{1/2}u+f(u)=0$ in $\mathbb{R}^3$, where $(-\Delta)^{1/2}$ is the fractional differential operator of order $1/2$, realizable as the Dirichlet-to-Neumann map of a harmonic extension to the half-space $\mathbb{R}^{4}_+$ (hence 'dimension 4+1' in the title). Their proof rests on energy estimates for stable solutions in every dimension (their Proposition 1.7) that lose a logarithmic factor, which they remove by a bootstrap exploiting the nonlocal structure. They note explicitly that these methods do not resolve the classical local equation, whose $d=3$ case was settled only in 2026 by independent arguments (Chan–Fernández-Real–Figalli–Florit-Simon–Serra 2026; Liu–Luo–Wang–Wei–Wei–Wu 2026). The classification is still open in the whole remaining range $4\le d\le7$: the strongest unconditional results either assume bounded energy density (an a priori energy-growth hypothesis) or concern fractional and free-boundary variants of the equation.

Problem Statement

Prove, or disprove, the stable De Giorgi conjecture for the classical Allen–Cahn equation in its open range $4\le d\le7$; the first open case is $d=4$.

Let $u\in C^2(\mathbb{R}^4)$, $|u|\le1$, solve

$$-\Delta u = u - u^3$$

and be stable, i.e.

$$\int_{\mathbb{R}^4}\left(|\nabla\xi|^2+(3u^2-1)\xi^2\right)dx\ \ge\ 0\quad\text{for all }\xi\in C_c^\infty(\mathbb{R}^4).$$

Must $u$ be one-dimensional, $u(x)=\phi(e\cdot x)$ for some $e\in S^3$ and some bounded increasing $\phi$ solving $\phi''=\phi^3-\phi$? The same question is open in every dimension $4\le d\le7$ (for $d=2$ and $d=3$ the answer is affirmative; stable non-one-dimensional solutions exist for $d\ge8$, so the range $d\le7$ is the conjectural boundary). The conjecture is settled either by a proof that all bounded stable solutions in $\mathbb{R}^d$, $4\le d\le7$, are one-dimensional — the conjectured answer, with $\mathbb{R}^4$ the critical first case — or by an explicit construction of a bounded, stable, non-one-dimensional solution in some dimension in that range, which would refute the conjecture. The analogous statement for general balanced bistable nonlinearities (derivatives of double-well potentials with wells at $\pm1$) is a natural strengthening of the same classification problem; this question concerns the classical equation alone.

References

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