On the equation $\det\nabla \varphi=f$ prescribing $\varphi=0$ on the boundary

Abstract

We discuss the existence of a regular map $\varphi$ satisfying $$ \left\{ \begin{array}{cl} \det\nabla \varphi=f & \text{in $\Omega$}\\ \varphi=0 & \text{on $\partial \Omega,$} \end{array} \right. $$ where $\Omega$ is a bounded smooth domain and $f$ is a regular function satisfying $ \int_{\Omega}f=0$.