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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 $\xi = 0$?

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    Taking $\xi=0$ won't help because the integrand in the desired claim involves a *product* of double partial derivatives, not just $u$. Also, it'd make more sense to write $\mathcal{F}(u)(\xi)$ rather than $\mathcal{F}(u(\xi))$ in my opinion.2012-05-02

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Hint: use Plancherel's theorem twice, once on $\iint_{{\mathbb R}^2} u_{xx} u_{yy}$ and once on $\iint_{{\mathbb R}^2} u_{xy}^2$

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    Complex conjugate2012-05-02