Smooth Solutions to a Class of Mixed Type Monge-Ampere Equations

TL;DR

This paper establishes the existence of smooth local and global solutions for a class of mixed type Monge-Ampere equations, with applications to geometry and fluid dynamics.

Contribution

It proves the existence of smooth solutions to mixed type Monge-Ampere equations and related linear equations, advancing understanding in geometric analysis and fluid mechanics.

Findings

Existence of C^{ } solutions in the plane.

Solutions applicable to prescribed Gaussian curvature problems.

Results relevant to transonic fluid flow.

Abstract

We prove the existence of C^{\infty} 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^{\infty} 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.