Let $\Omega\subset\mathbb{R}^n$ be a bounded open set . Show that if $u,v\in H_0^2(\Omega)$ , then $$\int_\Omega\Delta u\Delta v=\sum_{i,j=1}^n\int_\Omega\partial_{ij}u\partial_{ij}v$$
By Green's theorem \begin{align*}\int_\Omega\Delta u\Delta v&=\sum_{i,j=1}^n\int_\Omega\partial_{ii}u\partial_{jj}v\\&=-\sum_{i,j=1}^n\int_\Omega\partial_iu\color{red}{\partial_{ijj}v}+\sum_{i,j=1}^n\int_{\partial\Omega}\gamma_1(u)\gamma_0(\partial_{jj}v)\\&=\sum_{i,j=1}^n\int_\Omega\partial_{ji}u\partial_{ij}v+\sum_{i,j=1}^n\int_{\partial\Omega}\gamma_1(u)\gamma_0(\partial_{ij}v)\\&=\sum_{i,j=1}^n\int_\Omega\color{red}{\partial_{ij}u}\partial_{ij}v\end{align*} where $\gamma_0,\gamma_1$ are the trace maps and I have used $\gamma_1(u)=0$ . But I have a doubt that whether we can perform the thrice differentiation on $v$ as in the second step since we only have $v\in H_0^2(\Omega)$ . Also in the last line , can we draw the conclusion that derivatives have the symmetry since we don't have sufficiently smooth $u$ ?
In short , is that approach correct ? It will be helpful if someone checks its correctness . Regards .
As you work with $H^2_0$ you can just take $u, v \in C^\infty_0$ (in which all your computation are valid) and then conclude by density.