A classification of isolated singularities of elliptic Monge-Amp\`ere equations in dimension two

TL;DR

This paper classifies isolated singularities of solutions to elliptic Monge-Ampère equations in two dimensions, linking solution spaces to convex curves and describing asymptotic behaviors.

Contribution

It provides a complete classification of isolated singularities for elliptic Monge-Ampère equations and describes the asymptotic behavior of solutions.

Findings

Solution space characterized by convex Jordan curves

Asymptotic behavior of solutions detailed

Existence theorem for singularities in generalized equations

Abstract

Let $\mathcal{M}_1$ denote the space of solutions $z(x,y)$ to an elliptic, real analytic Monge-Amp\`ere equation ${\rm det} (D^2 z)=\varphi(x,y,z,Dz)>0$ whose graphs have a non-removable isolated singularity at the origin. We prove that $\mathcal{M}_1$ is in one-to-one correspondence with $\mathcal{M}_2\times Z_2$, where $\mathcal{M}_2$ is a suitable subset of the class of regular, real analytic strictly convex Jordan curves in $R^2$. We also describe the asymptotic behavior of solutions of the Monge-Amp\`ere equation in the $C^k$-smooth case, and a general existence theorem for isolated singularities of analytic solutions of the more general equation ${\rm det} (D^2 z +\mathcal{A}(x,y,z,Dz))=\varphi(x,y,z,Dz)>0$.