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Is there a way to write $\int_\Omega (\Delta u)^2$ or more generally, $\int_\Omega \Delta u \Delta v$ more nicely (possibly after integrating by parts)? I want something like $\int \nabla f\cdot \nabla g$ for some functions $f$ and $g$.

I can write it in terms of divergence of course but that is not very nice. So my question is does anyone have a different way of writing the above expressions? Thanks.

1 Answers 1

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Yes and no.

When integrating by parts the total number of derivatives is conserved. When you start with two Laplacians, that makes four derivatives, so you cannot express the end result as something with "only" two derivatives total.

That is, unless you "hide" some derivatives.

Let $f = \nabla u$ the vector valued function representing the gradient of $u$, and $g = \nabla v$ the vector valued function representing the gradient of $v$. We have that

$ \int_\Omega \Delta u \Delta v ~~\mathrm{d}x = \sum_{i,j}\int_\Omega \partial_i\partial_i u \partial_j\partial_j v~~ \mathrm{d}x $

Integrate by parts twice, and using that partial derivatives commute we have

$ = \sum_{i,j} \int_{\Omega} \partial_i \partial_j u ~ \partial_i\partial_j v ~~\mathrm{d}x + \text{boundary terms} $

The first term we can write as

$ \sum_{i} \int_{\Omega} \partial_i f \cdot \partial_j g~~ \mathrm{d}x$

which we can abuse notation and write as

$ \int_{\Omega} \nabla f \cdot \nabla g ~~\mathrm{d}x$