An exercise asks to show the following integration by parts formula: let $D$ be any open subset of $\mathbb{R}^n$, $u\in H^1_0(D)$ and $\varphi=(\varphi_1,\ldots,\varphi_n)\in C^\infty(\mathbb{R}^n,\mathbb{R}^n)$, then \begin{equation}\label{eq.inbyparts} \int_D\varphi\cdot\nabla (u^2)\:dx=-\int_D u^2\text{div }\varphi\:dx. \end{equation}
Here $C^\infty(\mathbb{R}^n,\mathbb{R}^n)$ only refers to smoothness, no norm $L^\infty$ assumed to be bounded, in fact a second part of the exercise asks to use in particular $\varphi(x)=x$.
I fail to see how this can be true as stated.
I can show
- that the formula holds for $u\in C^\infty_c(D)$;
- that if $u\in H^1_0(D)$ then $u^2\in W^{1,1}_0(D)$ and $\nabla (u^2)=2u\nabla u$. This has to be done "by hand" since the standard chain rule for Sobolev spaces (as stated for instance in chapter 7 of Gilbarg and Trudinger) does not hold in this case;
- that the formula holds for $u\in H^1_0(D)$ as soon as $\varphi$ and $\text{div }\varphi$ are bounded. This is done using the previous setp and, for instance, weak convergence.
Now, without any furher assumption on $\varphi$ it is not clear for me how to proceed to the general case nor if the result is true at all. I thought of exauhsting $\mathbb{R}^n$ with balls of increasing radius, since it when $D$ is bounded one can get a bound on $\varphi$ and $\text{div }\varphi$ but that would be a technical argument.
My concern is that $u^2\text{div }\varphi$ and $\varphi\cdot \nabla(u^2)$ are not necesarilly integrable in general.
Am I missing something?
Your doubts are justified. Take $u\in H^1_0(D)$ such that $u \nabla u \not \in L^2(D)$, which is possible if $n>1$. Now take a sequence $(\phi_k)$ of smooth functions $\phi_k \in C_c^\infty(D)$, such that $\phi_k \to u \nabla u$ pointwise a.e. on $D$.
Then the integrals explode: $\int_D |\phi_k u \nabla u | \to \infty$ by Fatous Lemma.