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I'm having some difficulty expanding the following integral:

$$\int_{\Omega}|D^2u|^2(x,t)\;\mathrm{d}x=\int_{\Omega}(D^2u\cdot D^2u)(x,t)\;\mathrm{d}x$$

where $\Omega\subset\mathbb{R}^n$ and $u$ is in $\mathbb{R}^n$. I want to use integration by parts (which I believe is the way to go here) but haven't been able to start anywhere. I'm very familiar with using integration by parts to solve the related integral

$$\int_{\Omega}Du\cdot Dv\;dx=-\int_{\Omega} u\Delta v\;\mathrm{d}x+\int_{\partial\Omega}\frac{\partial v}{\partial\nu}u\;\mathrm{d}s$$

but I haven't been able to achieve the same success in this scenario. Could any of Green's Formulas help? Thanks!

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  • $\begingroup$ What do you mean by $D^{2}u \cdot D^{2}u$? Is this not the definition of $|D^{2}u|^{2}$? What do you mean by "evaluate the integral"? Without this information, it's hard to know what you are trying to do/prove/ask. $\endgroup$
    – user711689
    Apr 3, 2021 at 17:23
  • $\begingroup$ Sorry. I would like to expand the integral in a similar way to how I've expanded $\int_{\Omega}Du\cdot Dv\;\mathrm{d}x$. $\endgroup$ Apr 4, 2021 at 0:23
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    $\begingroup$ If you integrate by parts twice, you can show that $\int_{\Omega} \|D^{2}u(x)\|^{2} \, dx = \int_{\Omega} |\Delta u(x)|^{2} \, dx$ whenever $u$ is smooth and vanishes on $\partial \Omega$ and $\Omega$ has a smooth boundary. I never looked at what happens when $u$ is non-zero on the boundary; however, this should be a useful hint to get you started. $\endgroup$
    – user711689
    Apr 4, 2021 at 1:08
  • $\begingroup$ Is $D$ the Jacobian? $\endgroup$
    – K.defaoite
    Apr 4, 2021 at 3:10
  • $\begingroup$ $D$ is the differential operator $\endgroup$ Apr 4, 2021 at 8:18

1 Answer 1

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Isn't $|D^2u|^2 = \sum_{i,j} (\partial_{x_i}\partial_{x_j} u)^2$? what's your true question?

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