On a long standing conjecture of E. De Giorgi: old and recent results

This paper studies a conjecture made by E. De Giorgi in 1978 and concerning the one-dimensional character (or symmetry) of the solutions of semilinear elliptic equation Du = f(u) which are defined on the entire n-dimensional Euclidean space and are increasing in one direction. We extend to all nonlinearities f of class C2 the symmetry result in dimension n=3 previously established by the second and the third authors for a special class of nonlinearities f. The extension of the present paper is based on a new energy estimates which follow from a local minimality property of u. In addition, we establish a symmetry result for semilinear equations in the 4-dimensional halfspace. Finally, we prove that an asymptotic version of the conjecture of De Giorgi is true when the dimension does not exceed 8, namely that the level sets of u are flat at infinity.