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

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!

• 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.
– user711689
Apr 3, 2021 at 17:23
• Sorry. I would like to expand the integral in a similar way to how I've expanded $\int_{\Omega}Du\cdot Dv\;\mathrm{d}x$. Apr 4, 2021 at 0:23
• 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.
– user711689
Apr 4, 2021 at 1:08
• Is $D$ the Jacobian? Apr 4, 2021 at 3:10
• $D$ is the differential operator Apr 4, 2021 at 8:18

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