In order to fix the ideas let me consider an open, smooth, bounded domain $\Omega\subset \mathbb R^d$. I am wondering what can be said about a vector-valued measure $v\in \mathcal M^d(\Omega)$ with finite mass $|v|(\Omega)<+\infty$ and zero distributional divergence, $$ \operatorname{div} v=0\qquad \mbox{in }\mathcal D'(\Omega), $$ in particular:
is it possible to give a meaning whatsoever to the normal trace $\gamma_\nu(v)=v\cdot \nu|_{\partial\Omega}$?
(being $\nu(x)$ the outer unit normal to $\Omega$)
Of course I am aware of the classical trace theory in the Sobolev context, e.g. $$ H_{div}=\{v\in L^2,\quad \operatorname{div}v \in L^2\} \qquad\mbox{with}\qquad \gamma_\nu(H_{div})\to H^{-1/2}(\partial\Omega). $$ However, I am interested in vector-fields that are really measures, not Lebesgue $L^p$ functions. I suspect that density arguments might give something useful, in particular the fact that my measures do not charge the boundary might help to use for example the density of $\mathcal D(\Omega)\subset \mathcal D'(\Omega) $. Before I spend too much time thinking on it: is this studied somewhere? If not, is there a good reason why? (perhaps this is completely pointless) For now I'm looking for any useful reference, pointers, insights, etc.
Thank you in advance!
