A Liouville theorem for solutions of degenerate Monge-Amp\`ere equations

TL;DR

This paper provides a new proof of a classical theorem characterizing convex solutions of the Monge-Ampère equation, and extends Liouville-type results to degenerate cases with solutions having specific polynomial forms.

Contribution

It introduces a novel proof technique avoiding complex analysis and establishes Liouville theorems for a class of degenerate Monge-Ampère equations with explicit solution forms.

Findings

Classical solutions are second order polynomials in the non-degenerate case.

Convex solutions of degenerate equations have explicit polynomial forms.

The method applies to a broader class of degenerate Monge-Ampère equations.

Abstract

In this paper, we give a new proof of a celebrated theorem of J\"orgens which states that every classical convex solution of \[ \det\nabla^2 u (x)=1\quad {in} \mathbb{R}^2 \] has to be a second order polynomial. Our arguments do not use complex analysis, and can be applied to establish such Liouville type theorems for solutions of a class of degenerate Monge-Amp\`ere equations. We prove that every convex generalized (or Alexandrov) solution of \[ \det \nabla^2 u(x_1,x_2)=|x_1|^{\alpha} \quad {in} \mathbb{R}^2, \] where $\alpha>-1$, has to be \[ u(x_1,x_2)= \frac{a}{(\alpha+2)(\alpha+1)}|x_1|^{2+\alpha}+\frac{a b^2}{2}x_1^2 +bx_1x_2+ \frac{1}{2a} x_2^2+\ell(x_1,x_2) \] for some constants $a>0$, $b$ and a linear function $\ell(x_1,x_2)$. This work is motivated by the Weyl problem with nonnegative Gauss curvature.