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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.


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. – anon May 2 '12 at 4:29

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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Could you be more precise on how to work out the integral equations? – newbie May 2 '12 at 4:44
$$ \int_{{\mathbb R}^2} \overline{u} v = \int_{{\mathbb R}^2} \overline{{\cal F} u} \ {\cal F} v$$ – Robert Israel May 2 '12 at 7:53
What do you mean the bar $\bar{u}$? – newbie May 2 '12 at 10:02
Complex conjugate – Robert Israel May 2 '12 at 18:04

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