$$\int_{\Omega} \nabla u \cdot \mathbf{n}\, v \, d\Omega,$$
where $\Omega \subset \mathbb{R}^2$ is a bounded domain with Lipschitz continuous and piecewise smooth boundary $\Gamma:=\partial \Omega$, $u, v \in H^1(\Omega)$ and $\mathbb{n}$ is the unit normal vector. In others words, is it possible to apply the Divergence theorem to this integral? What is the solution?