Integration by Parts for Weak Derivative?

Question

We have integration by parts formula: $$\int_U u_{x_i}v\ dx=-\int_{U}u v_{x_i}\ dx$$ for any $u,v\in C^1_c(U)$. Now I try to generalize it to weak case, that is, the formula holds for $u,v\in H^1_0(U)$. My idea is take $\phi_i, \varphi_i \in C^\infty_c(U)$, s.t. $\phi_i\to u, \varphi_i\to v$ in $H^1_0(U)$. But how can I choose $\phi_i, \varphi_i$ to let them fit the formula? Or any other way?

Answer 1

You don't need to choose them in any special way; just having the $\phi_n,\varphi_n$ in $C^\infty_c$ and converging in $H^1_0(U)$ is enough. (I will use $n$ for the index variable instead of $i$, since you are already using $i$ in $x_i$. I also wish you hadn't chosen to use both $\phi$ and $\varphi$, since they are the same letter, but I will follow your notation.)

You know that $\int_U (\phi_n)_{x_i} \varphi_n\,dx = - \int_U \phi_n (\varphi_n)_{x_i}\,dx$ just by elementary calculus. You also know, by virtue of the $H^1$ convergence, that $\phi_n \to u$ and $(\phi_n)_{x_i} \to u_{x_i}$ in $L^2$, and likewise for $\varphi_n$. Now recall or prove the following fact:

If $f_n \to f$ and $g_n \to g$ in $L^2$, then $f_n g_n \to f g$ in $L^1$, and in particular $\int f_n g_n \to \int fg$.