Sharp energy estimates for nonlinear fractional diffusion equations

We study the nonlinear fractional equation $(-\mathrm{\Delta }{)}^{s}u=f(u)$ 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 $-\mathrm{\Delta }u=f(u)$ in ${\mathbb{R}}^n$. It remains open for $n=3$ and $s<1\mathrm{/}2$, and also for $n\ge 4$ and all $s$.