ENERGY ESTIMATES AND 1-D SYMMETRY FOR NONLINEAR EQUATIONS INVOLVING THE HALF-LAPLACIAN

We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation (-Delta)(1/2)u = f(u) in R(n). Our energy estimates hold for every nonlinearity f and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable. As a consequence, in dimension n = 3, we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation -Delta u = f(u) in R(n).