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We analyze a class of weakly differentiable vector fields (\FF \colon \rn \to\rn) with the property that (\FF\in L^{\infty}) and (\div \FF) is a Radonmeasure. The primary focus of our investigation is to introduce a suitablenotion of the normal trace of any divergence-measure field $\FF$ over theboundary of an arbitrary set of finite perimeter, which ensures the validity ofthe Gauss-Green theorem. To achieve this, we establish a fundamentalapproximation theorem which states that, given a Radon measure $\mu$ that isabsolutely continuous with respect to $\mathcal{H}^{N-1}$ on $\rn$, any set offinite perimeter can be approximated by a family of sets with smooth boundaryessentially from the measure-theoretic interior of the set with respect to themeasure $\|\mu\|$. With this approximation theorem, we derive the normal traceof $\FF$ on the boundary of any set of finite perimeter, (E), as the limit ofthe normal traces of $\FF$ on the boundaries of the approximate sets withsmooth boundary, so that the Gauss-Green theorem for $\FF$ holds on (E). Withthese results, we analyze the Cauchy fluxes that are bounded by a Radon measureover any oriented surface (i.e. an $(N-1)$-dimensional surface that is a partof the boundary of a set of finite perimeter) and thereby develop a generalmathematical formulation of the physical principle of balance law through theCauchy flux. Finally, we apply this framework to the derivation of systems ofbalance laws with measure-valued source terms from the formulation of balancelaw. This framework also allows the recovery of Cauchy entropy fluxes throughthe Lax entropy inequality for entropy solutions of hyperbolic conservationlaws.

Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws