We study the nonlinear fractional equation
$(-\Delta)^su=f(u)$ in $\re^n,$
for all fractions $0<s<1$ and all nonlinearities $f$. For every fractional power
$s\in (0,1)$,
we obtain sharp energy estimates for bounded global minimizers and for bounded
monotone solutions.
As a consequence,
we deduce the one-dimensional symmetry of bounded global minimizers and of bounded
monotone solutions in dimension $n=3$ whenever $1/2\leq s<1$. This result is the
analogue of a conjecture of De
Giorgi on one-dimensional symmetry for the classical Allen-Cahn equation. This is a
work in collaboration with Xavier Cabre.