Abstract
Abstract Chen, Torres and Ziemer ([9], 2009) proved the validity of generalized Gauss–Green formulas and obtained the existence of interior and exterior normal traces for essentially bounded divergence measure fields on sets of finite perimeter using an approximation theory through sets with a smooth boundary. However, it is known that the proof of a crucial approximation lemma contained a gap. Taking inspiration from a previous work of Chen and Torres ([7], 2005) and exploiting ideas of Vol’pert ([29], 1985) for essentially bounded fields with components of bounded variation, we present here a direct proof of generalized Gauss–Green formulas for essentially bounded divergence measure fields on sets of finite perimeter which includes the existence and essential boundedness of the normal traces. Our approach appears to be simpler since it does not require any special approximation theory for the domains and it relies only on the Leibniz rule for divergence measure fields. This freedom allows one to localize the constructions and to derive more general statements in a natural way.
Highlights
Chen, Torres and Ziemer ([9], 2009) proved the validity of generalized Gauss-Green formulas and obtained the existence of interior and exterior normal traces for essentially bounded divergence measure fields on sets of finite perimeter using an approximation theory through sets with a smooth boundary
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 ⊂ Rn and for subdomains E ⊂⊂ Ω of finite perimeter in 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 Rn 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
Summary
We wish to set the notations we will use and present the necessary preliminaries for the main results . The following corollary (for which we refer to Theorem 3.2 in [25]), in the case p = ∞, is one of the pillars on which the proof of generalized Gauss-Green theorems for essential bounded divergence measure fields rests. Recall that F · Dg is the weak-star limit of F · ∇gδ as δ → 0, where gδ is a mollification of g One tests this sequence of Radon measures on a test function φ ∈ Cc1(Ω) and some straightforward calculations yield φ dF · Dg = φF ∗ · dDg. The density of Cc1(Ω) in Cc(Ω) implies the identity F · Dg = F ∗ · Dg in M(Ω), and the consistency of the two product rules.
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