Entire Solutions of Semilinear Elliptic Equations in ^3 and a Conjecture of De Giorgi

We consider bounded entire solutions of semilinear

elliptic equations of the form

u = F'(u) in R³,

satisfying

_x3 u > 0 in

R³ and

u(x₁,x₂,x₃)

1 as x₃

.

We do not assume these limits to be uniform

with respect to (x₁,x₂). Under the hypothesis that

F min{F(1),F(-1)} in [-1,1]

, we prove that the level

sets of u are planes. That is, u is a function of

one variable only. This establishes in dimension three a

conjecture formulated by De Giorgi in 1978. In dimension two,

this conjecture was recently proved by Ghoussoub and Gui, and

similar results were obtained by Berestycki, Caffarelli

and Nirenberg. The conjecture remains open in dimension

n 4.

AMS Classification: 35B05, 35B40, 35B45, 35J60

Keywords: Semilinear elliptic PDE,

Symmetry and monotonicity properties, Energy estimates,

Liouville theorems