The Gauss-Green theorem in stratified groups

Giovanni E. Comi; Valentino Magnani

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The Gauss-Green theorem in stratified groups

Giovanni E. Comi, Valentino Magnani

Differential Geometry · Submitted 8 Jun 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show several properties of divergence-measure fields in stratified groups, ultimately achieving the related Gauss-Green theorem.

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We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Such vector fields form a new family of function spaces, which generalize in a sense the $BV$ fields. They provide the most general setting to establish Gauss-Green formulas for vector fields of low regularity on sets of finite perimeter. We show several properties of divergence-measure fields in stratified groups, ultimately achieving the related Gauss-Green theorem.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups.	contribution 1.00	0.92	0.08	asserted 0.99
s1	Such vector fields form a new family of function spaces, which generalize in a sense the $BV$ fields.	contribution 0.81	0.29	0.05	asserted 0.98
s2	They provide the most general setting to establish Gauss-Green formulas for vector fields of low regularity on sets of finite perimeter.	interpretation 0.66	0.43	0.39	asserted 0.99
s3	We show several properties of divergence-measure fields in stratified groups, ultimately achieving the related Gauss-Green theorem.	contribution 0.54	0.96	0.85	asserted 1.00

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Original abstract

We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Such vector fields form a new family of function spaces, which generalize in a sense the $BV$ fields. They provide the most general setting to establish Gauss-Green formulas for vector fields of low regularity on sets of finite perimeter. We show several properties of divergence-measure fields in stratified groups, ultimately achieving the related Gauss-Green theorem.

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6 arXiv papers cite this in 7 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Cites it as background 5 papers

It seems natural to investigate the possibility to extend the theory of divergence-measure fields and Gauss-Green formulas to non-Euclidean settings, and indeed there have been researches in this direction: in [ref] the authors considered doubling metric measure spaces supporting a Poincaré inequality, while in [ref] the class of horizontal divergence-measure fields in stratified groups is studied.

On $BV$ functions and essentially bounded divergence-measure fields in metric spaces · Introduction · 0.81
More recently, the issue has been attacked also in less regular settings, like metric measure spaces, [ref] , and stratified groups, [ref] .

Rough traces of $BV$ functions in metric measure spaces · Introduction · 0.81
See also [ref] and the references therein for further developments for the theory of divergence-measure fields.

Extended Divergence-Measure Fields, the Gauss-Green Formula, and Cauchy Fluxes · Introduction · 1.00

What it cites · 1 sentence without a confident label

Cites as background 2 papers

We refer for instance to the works [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] and [ref] .

An Anzellotti type pairing for divergence-measure fields and a notion of weakly super-$1$-harmonic functions · Introduction · 0.80
We refer for instance to the works [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] and [ref] .

Rigidity and trace properties of divergence-measure vector fields · Introduction · 0.80

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