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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!

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There is a very deep paper of Stanislav Smirnov about such vector measures: Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows. (Russian) Algebra i Analiz 5 (1993), no. 4, 206–238; translation in St. Petersburg Math. J. 5 (1994), no. 4, 841–867. You can find it on his homepage http://www.unige.ch/~smirnov/.

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  • $\begingroup$ Thank you Jochen, this seems to be a god starting point indeed. However, Smirnov always works in the whole space and never mentions boundaries or traces. As far as I understood in his setting solenoidal fields are essentially (superpositions of) circulation along closed curves. However in bounded domains the circulation of an open curve starting and ending on the boundary gives zero when tested on compactly supported functions, hence becomes solenoidal, so the corresponding result becomes unclear to me. Any further insight about normal trace especifically? $\endgroup$ – leo monsaingeon Dec 19 '19 at 15:38

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