Regularity of flat level sets in phase transitions

We consider local minimizers of the Ginzburg-Landau energy functional integral 1/2 vertical bar del u vertical bar(2) + 1/4(1-u(2))(2)dx and prove that, if the 0 level set is included in a flat cylinder then, in the interior, it is included in a flatter cylinder. As a consequence we prove a conjecture of De Giorgi which states that level sets of global solutions of Delta u = u(3) - u such that vertical bar u vertical bar <= 1, partial derivative(n)u > 0, lim(xn ->+/-infinity) u(x', x(n)) = +/- 1 are hyperplanes in dimension n <= 8.