The de Giorgi conjecture asserts that bounded entire solutions of the stationary Allen-Cahn equation, strictly monotone in
one direction, depend on one Euclidean coordinate,
so their level sets are parallel hyperplanes
in dimensions at most eight. The normalization is 
with 
.
de Giorgi Conjecture
See also
Allen-Cahn Equation, Partial Differential EquationExplore with Wolfram|Alpha
References
del Pino, M.; Kowalczyk, M.; and Wei, J. "On De Giorgi's Conjecture and Beyond." Proc. Nat. Acad. Sci. USA 109, 6845-6850, 2012. https://doi.org/10.1073/pnas.1202687109.Cite this as:
Weisstein, Eric W. "de Giorgi Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/deGiorgiConjecture.html