Abstract. We analyze a class of weakly differentiable vector fields F : R N → R N with the property that F ∈ L ∞ and div F is a Radon measure. These fields are called bounded divergencemeasure fields. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence-measure field F over the boundary of an arbitrary set of finite perimeter, which ensures the validity of the Gauss-Green theorem. To achieve this, we first develop an alternative way to establish the Gauss-Green theorem for any smooth bounded set with F ∈ L ∞ . Then we establish a fundamental approximation theorem which states that, given a Radon measure µ that is absolutely continuous with respect to H N−1 on R N , any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure-theoretic interior of the set with respect to the measure µ . We employ this approximation theorem to derive the normal trace of F on the boundary of any set of finite perimeter, E, as the limit of the normal traces of F on the boundaries of the approximate sets with smooth boundary, so that the Gauss-Green theorem for F holds on E.With these results, we analyze the Cauchy fluxes that are bounded by a Radon measure over any oriented surface (i.e. an (N − 1)-dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measurevalued source terms from the formulation of balance law. This framework also allows the recovery of Cauchy entropy fluxes through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws.

No abstract

We establish the interior and exterior Gauss-Green formulas for divergence-measure fields in L p over general open sets, motivated by the rigorous mathematical formulation of the physical principle of balance law via the Cauchy flux in the axiomatic foundation, for continuum mechanics allowing discontinuities and singularities. The method, based on a distance function, allows to give a representation of the interior (resp. exterior) normal trace of the field on the boundary of any given open set as the limit of classical normal traces over the boundaries of interior (resp. exterior) smooth approximations of the open set. In the particular case of open sets with continuous boundary, the approximating smooth sets can explicitly be characterized by using a regularized distance. We also show that any open set with Lipschitz boundary has a regular Lipschitz deformable boundary from the interior. In addition, some new product rules for divergence-measure fields and suitable scalar functions are presented, and the connection between these product rules and the representation of the normal trace of the field as a Radon measure is explored. With these formulas at hand, we introduce the notion of Cauchy fluxes as functionals defined on the boundaries of general bounded open sets for the rigorous mathematical formulation of the physical principle of balance law, and show that the Cauchy fluxes can be represented by corresponding divergence-measure fields.

In this note we present a new proof of a one-sided approximation of sets of finite perimeter introduced in [2], in order to fill a gap in the original proof. By the polar decomposition of measures, we can write DχWe define the perimeter of E asDate: June 2016. Key words and phrases. sets of finite perimeter, one-sided approximation, tangential properties of the reduced boundary.

No abstract