Local existence and stability of multivalued solutions to determined analytic first-order systems on the plane (Q1261122)

This paper discusses the local existence and stability of specific types of multivalued solutions to determined, nonlinear, analytic first order PDEs on the plane: \[ {\mathcal F}(x,y,z,w,z_ x,z_ y,w_ x,w_ y)=0,\quad {\mathcal G}(x,y,z,w,z_ x,z_ y,w_ x,w_ y)=0. \] The main result says that at regular points existence is controlled by the classical characteristic variety and, at type-changing points, existence is controlled by transversality conditions of the type encountered in Thom-Boardman singularity theory. As an application, multivalued solutions are used to establish the local existence of specific types of weak single-valued solutions. The graphs of these multivalued solutions correspond to smooth 2-surfaces in 4-spaces with well-behaved nonimmersed points. For some fixed regular PDE the multivalued solutions thus constructed are shown to be stable.