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Traces and fine properties of a \(BD\) class of vector fields and applications. (English) Zbl 1091.35007
Summary: We study the fine properties and the trace properties of 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 pointwisc 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 approximation discontinuity and approximate jump sets of \(w\).

MSC:
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35F05 Linear first-order PDEs
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