# An introduction to completely exceptional $2^{\textrm{nd}}$ order scalar PDEs

## TL;DR

This paper reviews the concept of completely exceptional second-order scalar PDEs, highlighting their geometric characterization and their equivalence to Monge-Ampère equations, based on historical and mathematical insights.

## Contribution

It provides a unified geometric perspective on completely exceptional PDEs and clarifies their equivalence to Monge-Ampère equations.

## Key findings

- Complete exceptionality coincides with Monge-Ampère equations.
- Historical development of the concept from Lax and Boillat.
- Geometric framework unifies the understanding of these PDEs.

## Abstract

In his 1954 paper about the initial value problem for 2D hyperbolic nonlinear PDEs, P. Lax declared that he had "a strong reason to believe" that there must exist a well-defined class of "not genuinely nonlinear" nonlinear PDEs. In 1978 G. Boillat coined the term "completely exceptional" to denote it. In the case of $2^{\textrm{nd}}$ order (nonlinear) PDEs, he also proved that this class reduces to the class of Monge-Amp\`ere equations. We review here, against a unified geometric background, the notion of complete exceptionality, the definition of a Monge-Amp\`ere equation, and the interesting link between them.