Sharp Energy Estimates for Nonlinear Fractional Diffusion Equations

We study the nonlinear fractional equation (−∆)u = f(u) in R, 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 −∆u = f(u) in R. It remains open for n = 3 and s < 1/2, and also for n ≥ 4 and all s.