# Motivation of Monge-Ampere equation

I have a question about Monge-Ampere equation. I read a book The shape of inner space by Shing-Tung Yau.

Chapter 5 in this book, Complex-Monge-Ampere equation's explaination was so interesting. In the beginning, Monge researched this equation during french revolution. after a long time, Ampere researched the equation succeed to Monge.

But there were still more preliminaries to be done before Cheng and I could attack complex Monge-Ampère equations. We started working on
the Dirichlet problem, named after the German mathematician Lejeune
Dirichlet. It’s what we call a boundary value problem, and tackling
such problems is usually the first thing we do when trying to solve
an elliptic differential equation. We discussed an example of a
boundary value problem in Chapter 3, the Plateau problem, which is
often visualized in terms of soap films and which stated that for an
arbitrary closed curve, one could find a minimal surface stretching
across that same boundary. Every point on that surface is also a
solution to a particular differential equation. (The shape of inner space p.113-114)

This is part of the book. When I read this part, there are various terminologies such as Plateau problem, Dirichlet problem and so on. So I want to know about Monge-Ampere equation's historical and mathematical background for understanding 'flow of the idea' above terminologies. So I searched Google about this.

According to my research, Monge-Ampere equation is a non-linear generalization of the Poisson equation. Is this right?

• "appropriate source" what I wrote means "historical motivation and background" written by English. Commented Jun 24, 2023 at 5:53
• you can edit your own question no need to add self-comments Commented Jun 24, 2023 at 11:13
• Thank you for your advice!! Commented Jun 24, 2023 at 11:44