On a long-standing conjecture of E.!De Giorgi:: Symmetry in 3D for general nonlinearities and a local minimality property

This paper studies a conjecture made by De Giorgi in 1978 concerning the one-dimensional character (or symmetry) of bounded, monotone in one direction, solutions of semilinear elliptic equations Deltau = F'(u) in all of R-n. We extend to all nonlinearities F is an element of C-2 the symmetry result in dimension n = 3 previously established by the second and third authors for a class of nonlinearities F which included the model case F'(u) = u(3) - u. The extension of the present paper is based on new energy estimates which follow from a local minimality property of u. In addition, we prove a symmetry result for semilinear equations in the halfspace R-+(4). Finally, we establish that an asymptotic version of the conjecture of De Giorgi is true when n less than or equal to 8, namely that the level sets of u are flat at infinity.