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

Torres, Mónica

doi:10.1007/s00205-004-0346-1

Cited by 57 publications

(56 citation statements)

References 17 publications

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“…In section 5, we present two gluing constructions for building DM ∞ (Ω; R n )-fields out of a pair of DM ∞ -fields whose domains decompose Ω, with or without essential overlap. These results are similar to results presented in [7] and [9]. Ultimately, we will use these constructions also to obtain Gauss-Green and integration by parts formulas up to the boundary of open bounded sets with regular enough boundary in Corollary 5.5.…”

Section: (E) = σ(∂E)supporting

confidence: 79%

“…We notice that this new proof also adjusts a dubious point in the proof of the Gauss-Green formula in [7]; indeed, the formula (44) of [7], which states χ E F · Dχ E = χ * E F · Dχ E , is false in general for a vector field F ∈ DM ∞ (Ω; R n ) and a set E ⊂⊂ Ω of finite perimeter in Ω (see also Remark 3.4). In addition, this method of proof yields immediately many relevant consequences, such as the representation formula for divF on the reduced boundary of sets of finite perimeter, integration by parts formulas and various results on gluing constructions which come from the ability to directly localize constructions as one does not need to pass through an approximation procedure.…”

Section: (E) = σ(∂E)mentioning

confidence: 66%

“…The method is analogous to the one Vol'pert used in order to prove his integration by parts theorem and it is based on the product rule established by Chen and Frid [5] and re-presented in [7] and [18]. The results are similar to those presented in the paper of Chen, Torres and Ziemer [9], but here we are not using their theory concerning the one-sided approximation of sets of finite perimeter by sets with smooth boundary.…”

Section: Gauss-green Formulas and Consistency Of Normal Tracesmentioning

confidence: 79%

“…Since there are no a priori compatibility assumptions made on the pair of fields, the results provide a wealth of DM ∞ extensions of a given DM ∞ field. The two theorems presented here are similar to Theorem 3 of Chen-Torres [7] and Theorem 8.5 and Corollary 8.6 of Chen-Torres-Ziemer [9] respectively; however, we have removed some of their assumptions on domains and modified and completed the proofs. In particular, we make use of the integration by parts formula on the complement of sets of finite perimeter compactly contained in the domain (Proposition 4.6) which justifies the treatment of the term with an unbounded domain in Corollary 8.6 of [9].…”

Section: Gluing Constructions and Extension Theoremsmentioning

confidence: 98%

“…As already mentioned, the main idea is to present a new proof of the Gauss-Green formula for essentially bounded divergence measure fields F on Ω for sets of finite perimeter E ⊂⊂ Ω. Carefully studying the paper of Chen and Torres [7], we noticed that it was possible to work directly with E along the lines of Vol'pert's proof for essentially bounded BV -vector fields, which was sketched above. Hence we are able to avoid the need to approximate E from the interior by smooth domains.…”

Section: (E) = σ(∂E)mentioning

confidence: 99%

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

Comi

Payne

2017

Advances in Calculus of Variations

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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 GaussGreen 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.

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Section: (E) = σ(∂E)supporting

confidence: 79%

Section: (E) = σ(∂E)mentioning

confidence: 66%

Section: Gauss-green Formulas and Consistency Of Normal Tracesmentioning

confidence: 79%

Section: Gluing Constructions and Extension Theoremsmentioning

confidence: 98%

Section: (E) = σ(∂E)mentioning

confidence: 99%

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

Comi

Payne

2017

Advances in Calculus of Variations

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

Chen

Torres

Ziemer

2008

Comm Pure Appl Math

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125

167

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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.

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Compensated Compactness

2005

Grundlehren Der Mathematischen Wissenschaften

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

Cited by 57 publications

References 17 publications

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

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

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

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