I am reading p.380 Evans PDE 2nd edition, on the derivation of estimates. I am looking for the proof of the following equality $$\int_{\mathbb{R}^n}(\Delta u)^2 dx=\int_{\mathbb{R}^n}|D^2 u|^2 dx$$

It was stated in the book that the proof is somewhere in the text, but I flip over the text a couple of times and cannot find that proof.

Anyone could help?


We use $u_{ij}, u_{ijj},\cdots$ to denote partial derivatives. First assume that $u\in C^\infty_0(\mathbb R^n)$. Then

$$\begin{split} \int_{\mathbb{R}^n } (\Delta u)^2 dx &= \int_{\mathbb R^n} \left(\sum_{i=1}^n u_{ii} \right) \left(\sum_{j=1}^n u_{jj} \right) dx\\ &= \sum_{i,j=1}^n \int_{\mathbb{R}^n } u_{ii} u_{jj} dx \\ &= -\sum_{i,j=1}^n \int_{\mathbb{R}^n } u_{iij} u_{j} dx \\ &= - \sum_{i,j=1}^n \int_{\mathbb{R}^n } u_{iji} u_{j} dx \\ &=\sum_{i,j=1}^n \int_{\mathbb{R}^n } u_{ij} u_{ij} dx \\ &= \int_{\mathbb R^n } |D^2 u|^2 dx. \end{split}$$

In general for any $u\in W^{2,2}(\mathbb R^n)$, as $W^{2,2}(\mathbb R^n) = W^{2,2}_0(\mathbb R^n)$, there is a sequence $u_m \in C^\infty_0(\mathbb R^n)$ so that $u_m \to u$ in $W^{2,2}(\mathbb R^n)$, thus the result follows by taking limit of

$$\int_{\mathbb{R}^n } (\Delta u_m)^2 dx = \int_{\mathbb{R}^n } |D^2 u_m|^2dx .$$


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