Properties of the displacement field, assuming only smooth charge distribution and Gauss's theorem

Question

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 ?

Answer 1

I will consider the case $\rho\equiv 0$, as the case of smooth $\rho$ is no different: we can always subtract the convolution of (a restriction of) $\rho$ with the Coulomb potential.

To answer question 2: no, there is no regularity to be claimed from this condition. Consider a vector field on $\mathbb{R}^2$ given by $D(x,y)=(0,f(x))$, where $f$ is arbitrary bounded, Borel measurable function. If $\gamma(t)$ is a parametrized curve, then $D\circ\gamma$ is also bounded and Borel measurable, hence the integral of $D_n$ makes sense, and vanishes for any such curve - to see that, you can for example smooth $f$ out and then pass to the limit.

In general, requiring $D$ to be a vector field (i.e., with coordinates given by functions) that is integrable over any surface is not really a right framework - a right one would be that of vector charges, or currents, and their divergence should be understood in the sense of generalized functions. For example, you can imagine a "flow" that circulates along a smooth closed curve, and vanishes identically outside of it. That is an example of a solenoidal vector charge, i.e., a "generalized vector field" with zero divergence. In general, solenoidal vector charges form a convex set, and thus by Choquet theorem can be decomposed into (uncountably infinite) linear combinations of extremal points of these sets, the "elementary solenoids". These may be more involved that closed curves, even if they are compactly supported: consider an irrational winding on a torus. For more details, see Smirnov, S. K. (1994). "Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional currents" St. Petersburg Mathematical Journal, 5(4), 841-867, MR1246427, Zbl 0832.49024.