A note on existence of smooth solution to Jacobian equation for compactly supported smooth data in ${\mathbb{R}}^2$

Abstract

In this note, we give an explicit construction of global solutions to the prescribed Jacobian equation \[ \det(\nabla u)=f\mbox{ in }\mathbb{R}^2, \] for a class of data. For every $f\in C_c^1({\mathbb{R}}^2)$ and $p>1$, we construct a solution $u\in {\dot{W}}^{1,2p}({\mathbb{R}}^2)\cap L^{\infty }({\mathbb{R}}^2)$. In particular, no sign condition or integral constraint is imposed on the compactly supported data. For $f\in C_c^{\infty }({\mathbb{R}}^2)$, the construction yields a smooth solution. We also consider rapidly decaying data and prove the existence of global solutions with bounded gradient for every $f\in \mathcal{S}({\mathbb{R}}^2)$. Finally, we extend the construction to compactly supported data that are measurable in one variable and $C^1$ in the other. Our proof relies on a similar idea as in [Moser, Trans. Amer. Math. Soc. 1965].