C-INFINITY REGULARITY OF SOLUTIONS OF THE LEVI EQUATION

We will prove the C-infinity regularity of the classical solutions of the equation Lu = q(1 + /gradu/(2))(3/2)/1 + u(t)(2) where Lu = u(xx) + u(yy) + 2u(y) - u(x)u(t)/1 + u(t)(2)u(xt) - 2u(x) + u(y)u(t)/1 + u(t )(2)u(yt) + u(x)(2) + u(y)(2)/1 + u(t)(2)u(tt), q is an element of C-i nfinity(Omega) and q(xi) not equal 0 for every tau is an element of Om ega. This is a second order quasilinear equation, whose characteristic form has zero determinant at every point, and for every function u. H owever we will write it as a sum of squares of nonlinear vector fields , and we will extablish the result by means of a suitable freezing met hod. (C) Elsevier, Paris.