I was reading N. V. Krylov's book "Lectures on Elliptic and Parabolic Equations in Sobolev Spaces" and decided to work through some of the exercises I left behind and got easily stuck on
3. Exercise By using the Fourier transform, prove that, if $d = 2$ and $u \in C^2_0$, then $$\int_{\mathbb{R}^d} \det u_{xx} dx =0 $$
I can show this using integration by parts and an approximation by polynomial argument, but I am unsure where to use the Fourier transform. I have tried the following:
By Parseval's identity and the exchange formula we have $$ \|u_{xy}\|_2^2 = \|\tilde{u}_{xy}\|_2^2 = \int_{\mathbb{R}^2} \xi_1^2\xi_2^2 |\tilde{u}(\xi)|^2 d\xi_1d\xi_2$$ which kind of deals with the $u_{xy}$ term, but this probably leads to nowhere.
Somewhere in the argument I have to use the fact that $u$ is compactly supported, but I am not sure how.
On a related note, the book has a section on "Integrating the determinants of Hessians" which I skipped, but looking back it seemed quite interesting. Unfortunately I was not able to find any source online that connects the exercise
1. Exercise Let $\Omega$ be a connected bounded domain with smooth boundary and let $F, G: \Omega \to \mathbb{R}^d$ be $C^1(\bar{\Omega})$ mappings such that $F = G$ on $\partial \Omega$, then $$ \int_{\Omega} \det \frac{\partial F}{\partial x} dx = \int_{\Omega} \det \frac{\partial G}{\partial x} dx$$
to the theory of nonlinear PDEs (as mentioned in the book, but no reference is given whatsoever). I am guessing this has to do with certain Monge–Ampère equations. Any reference on this and any hint on the first exercise will be extremely appreciated!
First exercise:
$$\int u_{xx} u_{yy} dxdy = c \int \xi_1^2 \hat{u}(\xi) \xi_2^2 \hat{u}(\xi) d \xi_1 d\xi_2 = c\int \xi_1 \xi_2 \hat{u}(\xi) \xi_1 \xi_2 \hat{u}(\xi) d \xi_1 d\xi_2 = \int u_{xy} u_{xy} dxdy.$$
Second exercise:
So far I just think of some possible solution, which I am not totally sure. Set $\vec{H}:=\vec{F}- \vec{G}.$ Construct some $C_0^2 (\Omega)$ function $\varphi$ such that $\vec{H} =\nabla \varphi$ in $\Omega.$ For instance, consider the problem $$ \Delta \varphi = div \vec H \,\,\,\text{in} \,\,\,\Omega, \quad \varphi =0 \,\,\,\text{on} \,\,\,\partial \Omega.$$
Then if the first exercise holds for general dimension $d \geq 2,$ then we end up with $$0 =\int_{\Omega} \det (\nabla^2 \varphi) dx = \int_{\Omega} \det(\nabla^{\top}\vec{H})dx =\int_{\Omega} \det(\nabla^{\top}\vec{F})dx - \int_{\Omega} \det(\nabla^{\top}\vec{G})dx.$$