Counterexamples to the Local Solvability of Monge-Ampere Equations in the Plane

TL;DR

This paper constructs smooth examples of certain degenerate Monge-Ampere equations in the plane that cannot be solved locally with C^3 regularity, highlighting limitations in local solvability.

Contribution

It provides explicit smooth counterexamples to local C^3 solvability for degenerate hyperbolic and mixed type Monge-Ampere equations in the plane.

Findings

Existence of smooth counterexamples demonstrating non-solvability.

Counterexamples are of degenerate hyperbolic and mixed type.

Showcases limitations of local solution regularity for these equations.

Abstract

In this paper, we present smooth examples of degenerate hyperbolic and mixed type Monge-Ampere equations in the plane, which do not admit a local C^3 solution.