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I have a equation something like this:

$\int_U |\Delta u|^2dx \le C\int_U |D^2 u|^2dx$ . where $u \in C_c^{\infty}$ implies that u belongs to a compactly supported smooth function .

I would like to know how i can integrate it using integration by parts. Thank you for your kind help. Its true that when $C=1$ its an equality.

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I don't get the question. Is $\Delta=\nabla^2$ the laplace operator? In that case, is $C=1$? Is $U$ a compact subset of $\mathbb{R}^n$? – akkkk Jun 12 '12 at 9:43
I'm not sure what this is supposed to mean. Usually $\Delta=\nabla^2$; then the inequality (not equation) would be trivially true with $C=1$. What's the $c$ in $C_c^{\infty}$? Perhaps $u$ is a vector quantity, and you mean $\nabla(\nabla\cdot u)$? – joriki Jun 12 '12 at 9:45
@joriki: I bet $C_c^\infty$ stands for the class of regular, compactly supported functions (which is sometimes denoted by $C_0^\infty$ -e.g. on Rudin's books-, or $\mathcal{D}$ -in the Theory of Distributions-). – Pacciu Jun 12 '12 at 10:01
It turns out that the $\nabla$ operator came from an edit from another user, who has now reinstated the original $D$. That raises the question what $D$ denotes. More generally, @Theorem, in the future please take more care to introduce any notation that you use that can't be assumed to obvious to everyone. – joriki Jun 13 '12 at 12:17

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