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Divergence-Measure Fields, Sets of Finite Perimeter, and Conservation Laws

#### Authors: Gui-Qiang Chen and Monica Torres

#### Title:
Divergence-Measure Fields, Sets of Finite Perimeter, and Conservation Laws

**Abstract**

Divergence-measure fields in $L^\infty$
over sets of finite perimeter are analyzed.
A notion of normal traces over boundaries of sets of finite
perimeter is introduced, and the Gauss-Green formula over
sets of finite perimeter is established for divergence-measure
fields in $L^\infty$.
The normal trace introduced here over a surface of finite
perimeter is shown to be the weak-star limit of the normal traces
introduced in Chen-Frid \cite{CF1} over the Lipschitz deformation
surfaces of the surface, which implies their consistency.
As a corollary, an extension theorem of divergence-measure fields
in $L^\infty$ over sets of finite perimeter is also established.
Then we apply the theory to the initial-boundary value problem
of nonlinear hyperbolic conservation laws over sets of finite
perimeter.

This article has appeared in:

* Arch. Rational Mech. Anal.
* ** vol. **, pages (2004)

This paper is available in the following formats:

A closely related paper is Change me.

**Author Address**
Gui-Qiang Chen
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
gqchen@math.nwu.edu