The problem requests to use Fourier transformation, which I totally have no clue how. It states as following:
$u\in C^2_0$, prove $\int_{\mathbb{R}^2}u_{xx}u_{yy}-u_{xy}^2 \,dx = 0$
Any comments would be welcome. Cheers.
------update-------
One can show $-\xi^i \xi^j \mathcal{F}(u)(\xi) = \int_{\mathbb{R^2}}e^{-i\xi x}u_{x^i x^j}(x)\, dx$.
Compose the LHS by the above equation, it shows the result is zero. But I still have the term $e^{-i\xi x}$, should I take $\xi = 0$?