Math Conjectures

De Giorgi conjecture in dimensions 44 through 88

OPENLandmarkConjectureProposed 1978 · Standard version

Canonical statement

Let n∈{4,5,6,7,8}n\in\{4,5,6,7,8\}. If u∈C2(Rn)u\in C^2(\mathbb R^n) is bounded, satisfies
Δu=u3−uand∂xnu>0on Rn, \Delta u=u^3-u\quad\text{and}\quad \partial_{x_n}u>0\qquad\text{on }\mathbb R^n,
then uu is one-dimensional: there are ν∈Sn−1\nu\in S^{n-1} and h∈C2(R)h\in C^2(\mathbb R) such that u(x)=h(ν⋅x)u(x)=h(\nu\cdot x) for every x∈Rnx\in\mathbb R^n.
View source LaTeX
Let \(n\in\{4,5,6,7,8\}\). If \(u\in C^2(\mathbb R^n)\) is bounded, satisfies \[ \Delta u=u^3-u\quad\text{and}\quad \partial_{x_n}u>0\qquad\text{on }\mathbb R^n, \] then \(u\) is one-dimensional: there are \(\nu\in S^{n-1}\) and \(h\in C^2(\mathbb R)\) such that \(u(x)=h(\nu\cdot x)\) for every \(x\in\mathbb R^n\).
PDE-009:[DeGiorgi1979ConvergenceProblems][AmbrosioCabre2000DeGiorgi][Savin2009FlatLevelSets][DelPinoKowalczykWei2011DeGiorgi]
Partial differential equationsAnalysis and functional analysisMathematical physics

Notes

The assertion is proved in dimensions 22 and 33, and in dimensions through 88 under an additional two-ended limit hypothesis. Monotone counterexamples exist in every dimension at least 99, while the standard monotone problem above remains open exactly in dimensions 44 through 88.
No hypothesis that u(x′,xn)→±1u(x',x_n)\to\pm1 as xn→±∞x_n\to\pm\infty is imposed; adding it gives the version proved by Savin for n≤8n\le8.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.