In physics, the displacement field satisfies Gauss's theorem:
$$
\int_{\partial \Omega} {\bf D}\ {\bf n}\operatorname{d\!}S = \int_{\Omega} \rho\operatorname{d\!}V,
$$ where 
* $\Omega$ is a bounded connected set with smooth boundary $\partial \Omega$, and 
* $\rho$ is a distribution of charge. 

This formulation is more general than the local Maxwell formulation, as the field $\bf D$ may be not differentiable in the classic sense.
It follows in particular that the field $\bf D$ must be integrable on every smooth bounded surface of $\bf R^3$. 

**q 1:** How to characterize a vector field $\bf D\in \mathbb R^3$ which satisfies Gauss's theorem, for $\rho$ smooth?

**q 2:** If the required characterization turns out to be difficult, could such a $\bf D$ be shown to be essentially continuous ? weakly differentiable ? other property ?