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Hello,

Initially I thought that the following problem can be solved through Monge-Ampere equation, but with Monge-Ampere, I have not been able to control the support.

The question seems so natural that I'm sure it has been studied somewhere in some form but I have not been able to dig out the right reference. Few minutes in MathScinet did not come up with what I'm looking for. All references seem to handle the situation when $f$ is positive which is clearly not the case here. Any suggestion or reference in this direction is welcome.

Thank you.

QUESTION:

Let $U\subset\mathbb{R}^n$ be open, connected and let $f\in C_{0}^{\infty}(U;\mathbb{R})$ satisfy $$\int_{U}f(x)dx=0.$$ Is it true that there exists a $u\in C_{0}^{\infty}(U;\mathbb{R}^n)$ satisfying $$\operatorname*{det}(\nabla u)=f \text{ in }U?$$ Note that $$C_{0}^{\infty}(U;\mathbb{R}^n)=\{u\in C^{\infty}(U;\mathbb{R}^n):\operatorname*{supp}(u)\text{ is compact and }\operatorname*{supp}(u)\subset U\}.$$

4 added 5 characters in body

Hello,

Initially I thought that the following problem can be solved through Monge-Ampere equation, but with Monge-Ampere, I have not been able to control the support.

The question seems so natural that I'm sure it has been studied somewhere in some form but I have not been able to dig out the right reference. Few minutes in MathScinet did not come up with what I'm looking for. All references seem to handle the situation when $f$ is positive which is clearly not the case here. Any suggestion or reference in this direction is welcome.

Thank you.

QUESTION:

Let $U\subset\mathbb{R}^n$ be open and let $f\in C_{0}^{\infty}(U;\mathbb{R})$ satisfy $$\int_{U}f(x)dx=0.$$ Is it true that there exists a $u\in C_{0}^{\infty}(U;\mathbb{R}^n)$ satisfying $$\operatorname*{det}(\nabla u)=f \text{ in }U?$$ Note that $$C_{0}^{\infty}(U;\mathbb{R}^n)=\{u\in C^{\infty}(U;\mathbb{R}^n):\operatorname*{supp}(u)\text{ is compact and }\operatorname*{supp}(u)\subset U\}.$$

Hello,

The question seems so natural that I'm sure it has been studied somewhere in some form but I have not been able to dig out the right reference. Few minutes in MathScinet did not come up with I'm looking for. All references seem to handle the situation when $f$ is positive which is clearly not the case here. Any suggestion or reference in this direction is welcome.
Let $U\subset\mathbb{R}^n$ be open and let $f\in C_{0}^{\infty}(U;\mathbb{R})$ satisfy $$\int_{U}f(x)dx=0.$$ Is it true that there exists a $u\in C_{0}^{\infty}(U;\mathbb{R}^n)$ satisfying $$\operatorname*{det}(\nabla u)=f \text{ in }U?$$ Note that $$C_{0}^{\infty}(U;\mathbb{R}^n)={u\in C_{0}^{\infty}(U;\mathbb{R}^n)=\{u\in C^{\infty}(U;\mathbb{R}^n):\operatorname*{supp}(u)\text{ is compact and }\operatorname*{supp}(u)\subset U}U\}.$$