Application of equivalence method to Monge-Amp\`ere equations: Elliptic case

TL;DR

This paper applies the equivalence method to classify elliptic Monge-Ampère systems, presenting the associated symmetry group as a complex group, and distinguishes it from parabolic and hyperbolic cases.

Contribution

It provides a detailed analysis of the elliptic case of Monge-Ampère equations using the equivalence method and characterizes the symmetry group as a complex group.

Findings

Classification of Monge-Ampère systems into three types

Presentation of the elliptic case's symmetry group as a complex group

Differentiation of elliptic case from parabolic and hyperbolic cases

Abstract

The application of equivalence method to classify Monge-Amp\`ere system leads to three orbits, parabolic case, hyperbolic case and elliptic case wich correspond to three types of Monge-Amp\`ere systems. In this paper we will study the elliptic case and give a presentation of the group as a complex group.