Contact geometry of multidimensional Monge-Ampère equations: Characteristics, intermediate integrals and solutions

Abstract

We study the geometry of multidimensional scalar 2 nd order PDEs (i.e. PDEs with n independent variables), viewed as hypersurfaces ε in the Lagrangian Grassmann bundle M (1) over a (2n +1)-dimensional contact manifold (M, C). We develop the theory of characteristics of ε in terms of contact geometryand of the geometry of Lagrangian Grassmannian and study their relationshipwith intermediate integrals of ε. After specializing such results to general Monge-Ampère equations (MAEs), we focus our attention to MAEs of type introduced by Goursat in 1899: (Equation presented) We show that any MAE of this class is associated with an n-dimensional subdistribution D of the contact distribution C, and viceversa. We characterize these Goursat-type equations together with their intermediate integrals in terms of theircharacteristics and give a criterion of local contact equivalence. Finally, we developa method to solve Cauchy problems for this kind of equations.