Hi all. I’m new to the forums; I’m a PhD student in engineering with a Masters in mathematics, and I’m taking a course in PDEs right now. This statement below was claimed to be ‘clear' in our course notes by applying integration by parts, but I’m not seeing how. And, the professor’s out of town and unreachable (of course).

Let $\displaystyle u\in C_{0}^{\infty}(\Omega)$ (i.e. u is smooth with compact support on a region Omega) solve the Poisson equation (i.e. $\displaystyle \nabla(u)(x)=-f(x)$ in $\displaystyle \Omega$). Then, this inequality holds:

$\displaystyle \int_{\Omega}\sum_{k=1}^{n}|\frac{d^{2}u}{dx_{k}^{ 2}}|^{2} dx \leq \int_{\Omega}|f(x)|^{2} dx$

I’m getting really frustrated if it’s as ‘clear’ as the notes claim it is. Help is much appreciated in advance!