Inserted: 10 mar 2004
Last Updated: 28 jun 2007
Journal: Ann. Fac. Sci. Toulouse Math. (6)
In this paper we study the fine properties and the trace properties for a class of vector fields of the form $C=wB$, where $w$ is a locally bounded scalar function and $B$ is locally bounded and with finite deformation. Assuming also that the distributional divergence of $C$ is a locally finite measure, we relate the (distributional) trace of $C$ on hypersurfaces to the pointwise behaviour of $w$. We study also the behaviour of these traces under the transformation $wB\mapsto h(w)B$, with $h\in C^1$, proving a chain rule for traces.
As a consequence of these results we show that DiPerna--Lions theory can be extended to special vector fields with bounded deformation. In the case when $B$ is locally $BV$ we obtain also estimates on the size of the approximate discontinuity and approximate jump sets of $w$.
Keywords: Functions of Bounded Deformation, Renormalized solutions, Traces