# Regularity and normal trace of “Hdiv” measures

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!