# Addendum to: Dacorogna-Moser theorem on the Jacobian determinant equation with control of support

## TL;DR

This paper extends the Dacorogna-Moser theorem by demonstrating that solutions to the Jacobian determinant equation can have their support controlled based on initial data, while maintaining optimal regularity, thus generalizing previous results.

## Contribution

It provides a generalization of the Dacorogna-Moser theorem, allowing support control of solutions to the Jacobian PDE with preserved regularity, addressing an open problem from the original work.

## Key findings

- Support of solutions can be controlled from initial data.
- The regularity of solutions remains optimal.
- The result generalizes previous specific cases.

## Abstract

In Dacorogna-Moser theorem on the pullback equation $\varphi^* (g)=f$ between two prescribed volume forms (with the same total volume), control of support of the solutions can be obtained from that of the initial data, while keeping optimal regularity. This result answers a problem implicitly raised on page 14 of Dacorogna-Moser's original article ("On a partial differential equation involving the Jacobian determinant", Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire 7 (1990), 1-26), and fully generalizes the solution to the particular case of $g\equiv 1$ (prescribed Jacobian PDE, $\text{det}\,\nabla\varphi=f$) given in the author's paper "Dacorogna-Moser theorem on the Jacobian determinant equation with control of support", Discrete Cont. Dyn. Syst. 37 (2017), 4071-4089.