Singularity propagation of completely nonlinear equations in a complex domain

Abstract

We study completely nonlinear and noncharacteristic Cauchy problems of order two for singular initial values in a complex domain. We shall prove that if the characteristic roots are distinctive, then the singularities of the solution propagate toward characteristic directions. For this purpose, we generalize the classical theory of hodograph transformation. We also mention applications to the theory of perfect irrotational fluids and Monge–Ampère equations.