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I'm currently reading a couple of papers, which uses the following identity, which I can't figure out how to prove or see: $\int F(x,t) \Delta_x F(x,t) dx = -1 \int | \nabla _x F(x,t) | ^2 dx $.

Can someone help me figure out why this equality is true?

Thanks !

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May I ask what kind of hypothesis we have about $F$? What about its regularity? I think $F$ should be at least $C^{2}_c(\Omega)$ (I mean with compact support in a open domain $\Omega \subset \mathbb R^{n}$). –  Romeo Jun 30 '12 at 8:46
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up vote 1 down vote accepted

The divergence theorem is lurking here somewhere, since $\mbox{div} (F \nabla F) = F\Delta F + |\nabla F|^2$. You just need some means to conclude $\int \mbox{div} (F \nabla F) = 0$ (which is equivalent to the statement in your posting).

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Thanks a lot ! I'll try it –  joshua Jun 30 '12 at 8:48
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