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

Abstract

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.

On locally essentially bounded divergence measure fields and sets of locally finite perimeter

“…We are principally motivated by the paper of Chen-Torres-Ziemer [9] that examines the validity of the divergence theorem for essentially bounded divergence measure fields F on an open set Ω ⊂ R n and for subdomains E ⊂⊂ Ω of finite perimeter in Ω. Such vector fields are those F ∈ L ∞ (Ω; R n ) whose distributional divergence is a real finite Radon measure on Ω and such sets have characteristic functions χ E which are of bounded variation; that is, they are L 1 and have distributional gradients which are R n -valued Radon measures on Ω.…”

“…In [9], in order to prove the Gauss-Green formula and to extract interior and exterior normal traces in the context specified above, the authors make use of an approximation theory for sets E of finite perimeter in R n in terms of a family of smooth subsets which is well calibrated to any fixed Radon measure µ that is absolutely continuous with respect to the Hausdorff measure H n−1 (see Theorem 4.10 of [9]). They first prove the result for sets with smooth boundary and then pass to a limit by exploiting their approximation theorem and a result ofŠilhavý [25] which shows that if F is an essentially bounded divergence measure field then the total variation measure µ = |divF | is absolutely continuous with respect to H n−1 .…”

“…The principal aim of this paper is to show that this approximation step is not needed; that is, one can obtain the main result (Theorem 5.2 of [9]) directly by following the lines of Vol'pert's proof for essentially bounded BV vector fields and sets of finite perimeter (see [28] and [29]). To do so, one combines the aforementioned absolute continuity result of [25] with the Leibniz formula of Chen-Frid [5] (for the product of an essentially bounded function of bounded variation and an essentially bounded divergence measure field) and performs some elementary calculations of geometric measure theory.…”

“…To do so, one combines the aforementioned absolute continuity result of [25] with the Leibniz formula of Chen-Frid [5] (for the product of an essentially bounded function of bounded variation and an essentially bounded divergence measure field) and performs some elementary calculations of geometric measure theory. One might also note that in the aforementioned approximation result of [9], there was a known gap in the proof, which motivated our alternative method in the first place and has however been removed in the recent paper [10] by the first author and Torres. On the other hand, one should note that the approximation result is of independent interest and shows that for any set of finite perimeter E there exist sequences of smooth sets E k,i and E k,e converging to it from the interior and from the exterior in a measure theoretic sense (see Theorems 3.1 and 4.1 and Remark 4.1 of [10]).…”

“…On the other hand, one should note that the approximation result is of independent interest and shows that for any set of finite perimeter E there exist sequences of smooth sets E k,i and E k,e converging to it from the interior and from the exterior in a measure theoretic sense (see Theorems 3.1 and 4.1 and Remark 4.1 of [10]). For instance, these sequences have been used by Chen-Torres in the proof of Lemma 4.3 in [8] and by Chen-Torres-Ziemer [9] for showing that the integrals of the generalized normal traces are indeed the limits of the integrals of the classical normal traces over the smooth sets which approximate E. A more detailed discussion of this point is given in Remark 3.11.…”