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

Gui-Qiang Chen and Monica Torres

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.

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Author(s):
Gui-Qiang Chen
Monica Torres, <torres@math.northwestern.edu>
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