Sharp energy estimates for nonlinear fractional diffusion equations

We study the nonlinear fractional equation (−Δ)su=f(u) in Rn, for all fractions 0<s<1 and all nonlinearities f. For every fractional power s∈(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≤s<1. This result is the analogue of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation −Δu=f(u) in Rn. It remains open for n=3 and s<1/2, and also for n≥4 and all s.