Stokes theorem and Sobolev spaces. I am interested under which regularity condition is Stokes' theorem is still valid.
For concreteness I am interested in the following problem
Let's consider a domain $\Omega$ in $\mathbb{R}^{3}$  given in cylindrical coordinates by $0\le z\le 1 $,$0\le \theta\le2\pi$,$0\le \rho\le b$. 
Now let $f\in H^{2}(\Omega)$.
If we apply Stokes' theorem for the domain $\Omega_{\epsilon}$ in $\mathbb{R}^{3}$  given in cylindrical coordinates by $0\le z\le 1 $,$0\le \theta\le2\pi$,$\epsilon\le \rho\le b$ we have:
\begin{equation}
\int_{\Omega}\nabla\cdot \nabla fd\Omega=\int_{\partial\Omega_{\epsilon}}\frac{\partial f}{\partial x^{i}} n^{i} dS
\end{equation}
which is well-defined because the trace theorem warranty that $\frac{\partial f}{\partial x^{i}}\in H^{1/2}$.
Now in the case when $\epsilon\rightarrow0$ the boundary integral is:
\begin{equation}
\lim_{\epsilon\rightarrow 0}\int_{\partial\Omega_{\epsilon}}\frac{\partial f}{\partial x^{i}} n^{i} dS=\int\frac{\partial f}{\partial x^{i}}b d\theta dz-\lim_{\epsilon\rightarrow 0}\int\frac{\partial f}{\partial x^{i}}\epsilon d\theta dz
\end{equation}
If $f$ is smooth then the limit vanishes. However in the low differentiability case I have the following questions:
In the limit $\frac{\partial f}{\partial x^{i}}$ has to be restricted to a codimension 2 domain. What can I say about the regularity of the trace in the codimension 2?
In case $f$ has even lower regularity such that the trace in the codimension 2 case is in a negative Sobolev space,should the limit be consider to be a functional?
What's is the lower differentiability required for Stokes' theorem to hold?
 A: To simplify the question, let's just get rid of a derivative by considering $V = \nabla f$. Now when does the divergence theorem hold:
$$ \int_{\partial \Omega}  V \cdot n \, dS = \int_\Omega \nabla \cdot V dx $$
The conditions on the domain are as follows: Suppose $\Omega$ to be a bounded domain with a $C^1$ boundary: $\forall \xi \in \partial \Omega$ there is an open ball $B_r(\xi)$ and a $C^1$ diffeomorphism 
$$\psi: B_r(\xi) \to B_1(0)$$
such that $\psi( \partial \Omega \cap B_r(\xi) ) \subset \{ x \in \mathbb{R}^n : x_n =0 \}$ and $\psi ( \Omega \cap B_r(\xi) ) \subset \{ x \in \mathbb{R}^n : x_n >0\}$. Notice that $\partial \Omega$ is a codimension 1 hypersurface. 
Next the conditions on the vector field $V$ on $\Omega$, we basically need $V_i \in C^1(\bar{ \Omega} )$ for the derivative to be well defined.
Slightly weaker, $\partial \Omega$ is only piecewise $C^1$(i.e. $\partial \Omega$ is a finite union of $C^1$ boundaries). Then we can approximate inside $\Omega$ by taking $V_i \in C^1( \Omega) \cap C ( \bar{\Omega} )$ and it'll still hold if the right hand side converges.
Now we have our conditions, I'll basically say you got it figured out. Either $f$ is good enough, or we can treat it as a functional through a limit. As an example, take one of the many approximations of the delta function, like $f = \ln(r)$ in $\mathbb{R}^2$ 
For a reference , have a look at Evan's or McOwen's PDE Book
