$$\int_{\Omega} \nabla u \cdot \mathbf{n}\, v \, d\Omega,$$
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?
Are you sure about your integral? The normal derivative is defined on the boundary. Green's theorem would give you (with some extra regularity): $$ \int_{\Gamma} (\nabla u \cdot n) v = \int_{\Omega} \nabla u \cdot \nabla v - \int_{\Omega} \Delta u v. $$