Cauchy Fluxes Associated with Tensor Fields Having Divergence Measure

Degiovanni, Marco; Marzocchi, Alfredo; Musesti, Alessandro

doi:10.1007/s002050050149

Cited by 50 publications

(60 citation statements)

References 17 publications

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“…4 Schuricht actually treats divergence tensor fields F ∈ DM 1 loc (Ω; R n×m ) and uses the opposite orientation with respect to our choice. See also the related Theorems 5.2 and 5.4 in [13].…”

Section: Discussionmentioning

confidence: 97%

“…However, one knows that it is possible to recover such formulas also in the case F ∈ DM p loc (Ω; R n ) \ DM ∞ loc (Ω; R n ), provided that F and the set E of finite perimeter in Ω satisfy some additional assumptions. We refer to DegiovanniMarzocchi-Musetti [13], Schuricht [24] andŠilhavý [25] for a complete treatment of this theory. Here we only discuss how their results are consistent with ours in the case p = ∞.…”

Section: Discussionmentioning

confidence: 99%

“…We begin with the question of the consistency of normal traces with the classical dot product even when F is not continuous, provided that F and E satisfy two additional conditions which were in introduced in [13] and exploited in greater generality in [24]. These conditions are (6.7) |divF |(∂ * E) = 0 and…”

Section: Discussionmentioning

confidence: 99%

“…Degiovanni, Marzocchi and Musesti in [13] and later Schuricht in [24] sought to prove the existence of normal traces under weak regularity hypotheses in order to achieve a representation formula for Cauchy fluxes, contact interactions and forces in the context of the foundations of continuum physics. In particular, a justification of Cauchy's stress theorem under weak regularity assumptions is a main unifying ingredient in much of the divergence measure field literature, as is well explained in the introduction of [24].…”

Section: (E) = σ(∂E)mentioning

confidence: 99%

“…In particular, a justification of Cauchy's stress theorem under weak regularity assumptions is a main unifying ingredient in much of the divergence measure field literature, as is well explained in the introduction of [24]. While the resulting Gauss-Green formulas (and justifications of the stress theorem) obtained in [13] and [24] are valid for DM p (Ω; R n )-fields for any p ≥ 1, the subdomains E cannot be taken to be arbitrary sets of finite perimeter. Instead, E must be chosen to lie in a suitable subalgebra of sets which are related to the particular vector field F .…”

Section: (E) = σ(∂E)mentioning

confidence: 99%

See 4 more Smart Citations

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: Discussionmentioning

confidence: 97%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: (E) = σ(∂E)mentioning

confidence: 99%

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

123

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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Notes on Metric Independent Analysis of Classical Fields

Segev

2012

Math Methods in App Sciences

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A metric independent geometric framework for some fundamental objects of continuum mechanics is presented. In the geometric setting of general differentiable manifolds, balance principles for extensive properties are formulated and Cauchy's theorem for fluxes is proved. Fluxes in an n-dimensional space are represented as differential .n 1/-forms. In an analogous formulation of stress theory, a distinction is made between the traction stress, enabling the evaluation of the traction on the boundaries of the various regions, and the variational stress, which acts on the derivative of a virtual velocity field to produce the virtual power density. The relation between the two stress fields is examined as well as the resulting differential balance law. As an application, metric-invariant aspects of electromagnetic theory are presented within the framework of the foregoing flux and stress theory.

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Cauchy Fluxes Associated with Tensor Fields Having Divergence Measure

Abstract: Cauchy fluxes induced by locally summable tensor fields with divergence measure are characterized. The equivalence between integral formulations involving subsets of finite perimeter and much more restricted classes of subsets is proved.

Cited by 50 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

Notes on Metric Independent Analysis of Classical Fields

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