Regularity results for the equation $u_{11}{u}_{22}=1$

Abstract

We study the equation $u_{11}{u}_{22}=1$ in ${\mathbb{R}}^2$. Our results include an interior $C^2$ estimate, classical solvability of the Dirichlet problem, and the existence of non-quadratic entire solutions. We also construct global singular solutions to the analogous equation in higher dimensions. At the end we state some open questions.