In physics, the displacement field satisfies Gauss's theorem: $\int_{\partial \Omega} {\bf D}\ {\bf n} dS = \int_{\Omega} \rho$, 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 continuous ? weakly differentiable ? other property ?