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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 .
CitacióCabre, X.; Cinti, E. Sharp energy estimates for nonlinear fractional diffusion equations. "Calculus of variations and partial differential equations", Gener 2014, vol. 49, núm. 1-2, p. 233-269.