Let $u,v\in H^1(\Omega)$, where $\Omega$ is a Lipschitz domain in $\Bbb R^n$. It is my understanding that the divergence theorem tells us
$$\int_\Omega\nabla u\cdot\nabla v\,dx=\int_{\partial\Omega}\frac{\partial u}{\partial n}v\,dS(x)-\int_\Omega v\Delta u \,dx,$$
where $n$ is the unit outward normal to $\partial\Omega$.
My issue with this theorem (which I believe to be true) is that for $u\in H^1(\Omega)$ we do not necessarily have that $\frac{\partial u}{\partial n}\in H^{-\frac{1}{2}}(\partial\Omega)$ or $\Delta u\in H^1(\Omega)^*$ (the dual of $H^1(\Omega)$), so what exactly is meant by the integration on the right hand side? Is it in some sense a duality pairing of the form $$\langle\frac{\partial u}{\partial n}-\Delta u|\cdot\rangle_{H^1(\Omega)^*}:H^1(\Omega)\to\Bbb R,$$
i.e., we are able to deduce that the sum of the two are in the dual space but not the separate entities?
The reason this is of interest to me is that through the divergence theorem I believe one can easily prove that $\frac{\partial u}{\partial n}\in H^{-\frac{1}{2}}(\partial\Omega)$ if and only if $\Delta u\in H^1(\Omega)^*$.
Best Answer
Here is the most general divergence theorem that I am aware of:
Theorem. Let $\Omega$ be an open subset of $\mathbb{R}^N$ of class $C^1$ with a bounded frontier $\Gamma$ and $v \in W^{1,1}(\Omega,\mathbb{R}^N)$. Then $$ \int_\Omega \operatorname{div} v\, dx = \int_\Gamma \gamma_0(v)\cdot \nu \, d\gamma $$ where $\gamma_0$ is the trace operator.
This is Theorem 6.3.4 of Michel Willem's book Functional analysis. It seems to me that if you want to take $v = u \nabla w$ some strong assumption on the regularity of $u$ and $w$ should be made.