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

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