Smooth solutions to a class of mixed type Monge–Ampère equations

Abstract

We prove the existence of C∞ local solutions to a class of mixed type Monge–Ampere equations in the plane. More precisely, the equation changes type to finite order across two smooth curves intersecting transversely at a point. Existence of C∞ global solutions to a corresponding class of linear mixed type equations is also established. These results are motivated by and may be applied to the problem of prescribed Gaussian curvature for graphs, the isometric embedding problem for 2-dimensional Riemannian manifolds into Euclidean 3-space, and also transonic fluid flow.