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

TL;DR

This paper investigates the equation u_{11}u_{22} = 1 in two dimensions, establishing regularity, solvability, and solutions, and extends some results to higher dimensions with open questions.

Contribution

It provides new regularity results, solvability conditions, and constructs solutions for the nonlinear PDE u_{11}u_{22} = 1, including higher-dimensional singular solutions.

Findings

Interior C^2 estimate established

Classical solvability of the Dirichlet problem proven

Existence of non-quadratic entire solutions demonstrated

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.