Inserted: 13 sep 2012
Last Updated: 6 feb 2018
Journal: Calc. Var. Partial Differential Equations
We study the nonlinear fractional equation $(-\Delta)^su=f(u)$ in $\mathbb R^n,$ 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. They are sharp since they are optimal for solutions depending only on one Euclidian variable. 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 equation $-\Delta u=f(u)$ in $\mathbb R ^n$. It remains open for $n=3$ and $s<1/2$, and also for $n\geq 4$ and all $s$.