Calculus of Variations and Geometric Measure Theory
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L. Ambrosio - G. Crippa - S. Maniglia

Traces and fine properties of a $BD$ class of vector fields and applications

created on 10 Mar 2004
modified by crippa on 28 Jun 2007

[BibTeX]

Published Paper

Inserted: 10 mar 2004
Last Updated: 28 jun 2007

Journal: Ann. Fac. Sci. Toulouse Math. (6)
Volume: 14
Number: 4
Pages: 527-561
Year: 2005

Abstract:

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


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