C ∞ regularity of solutions of an equation of Levi’s type in R 2 n +1

Abstract

We study the regularity of the solutions u of a class of P.D.E., whose prototype is the prescribed Levi curvature equation in ℝ2 n +1. It is a second-order quasilinear equation whose characteristic matrix is positive semidefinite and has vanishing determinant at every point and for every function u∈C 2. If the Levi curvature never vanishes, we represent the operator ℒ associated with the Levi equation as a sum of squares of non-linear vector fields which are linearly independent at every point. By using a freezing method we first study the regularity properties of the solutions of a linear operator, which has the same structure as ℒ. Then we apply these results to the classical solutions of the equation, and prove their C ∞ regularity.