Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups

Rumin, Michel

doi:10.1016/s0764-4442(00)88624-3

Cited by 33 publications

(74 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…It is not known whether the symmetry assumption is necessary for this inequality to hold [34,Open problem 1]. This result was used to obtain the Gagliardo-Nirenberg inequality for forms in the Rumin complex [78,79,80,81] and for forms on the Heisenberg groups H 1 and H 2 [10]. Since the Rumin complex contains higher-order differential operators, the higher-order estimates play a crucial role in the proof.…”

Section: Noncommutative Situationsmentioning

confidence: 99%

Limiting Bourgain–Brezis estimates for systems of linear differential equations: Theme and variations

Schaftingen

2014

J. Fixed Point Theory Appl.

View full text Add to dashboard Cite

To Haïm Brezis who has taught me so much and has asked me many inspiring questions; on the occasion of his 70th birthday, with admiration and gratitude Abstract. J. Bourgain and H. Brezis have obtained in 2002 some new and surprising estimates for systems of linear differential equations, dealing with the endpoint case L 1 of singular integral estimates and the critical Sobolev space W 1,n (R n ). This paper presents an overview of the results, further developments over the last ten years and challenging open problems. Mathematics Subject Classification. 35A23 26D15 46E35.Keywords. Critical Sobolev spaces, Gagliardo-Nirenberg-Sobolev inequality, canceling differential operator, overdetermined elliptic systems, underdetermined systems. Theme Limiting Hodge theory for Sobolev formsThe study of limiting estimates for systems of linear differential equations starts from the following problem: given a function g ∈ L n (R n ; R n ), find the best regularity that a vector field u : R n → R n can have such that div u = g in R n .(1.1)If n ≥ 2, equation (1.1) is strongly underdetermined. The standard way of finding a solution u consists in lifting the underdeterminacy by solving the system div u = g in R n , curl u = 0 in R n , (1.2)

show abstract

Section: Noncommutative Situationsmentioning

confidence: 99%

Limiting Bourgain–Brezis estimates for systems of linear differential equations: Theme and variations

Schaftingen

2014

J. Fixed Point Theory Appl.

View full text Add to dashboard Cite

show abstract

“…The notion of intrinsic form in Carnot groups is due to M. Rumin ( [22,24]). A more extended presentation of the results of this section can be found in [2,15].…”

Section: Differential Forms In Carnot Groupsmentioning

confidence: 99%

“…In fact, we need a more sophisticated notion of "intrinsic" exterior differential to obtain a complex of differential forms that reflects the lack of commutativity of the group. It turns out that such a complex (E * 0 , d c ) , with E * 0 ⊂ * , has been defined and studied by M. Rumin in [24] and [22] ( [21] for contact structures). Rumin's theory needs a quite technical introduction that is sketched in Sect.…”

mentioning

confidence: 99%

Differential forms in Carnot groups: a Γ-convergence approach

Baldi

Franchi

2011

Calc. Var.

View full text Add to dashboard Cite

Carnot groups (connected simply connected nilpotent stratified Lie groups) can be endowed with a complex (E * 0 , d c ) of "intrinsic" differential forms. In this paper we prove that, in a free Carnot group of step κ, intrinsic 1-forms as well as their intrinsic differentials d c appear naturally as limits of usual "Riemannian" differentials d ε , ε > 0. More precisely, we show that L 2 -energies associated with ε −κ d ε on 1-forms -converge, as ε → 0, to the energy associated with d c .

show abstract

“…Set X := ∂ x + 2y∂ t , Y := ∂ y − 2x∂ t , T := ∂ t . The stratification of its algebra h is given by h = V 1 In a series of papers ( [21], [22], [23], [19]), M. Rumin developed a theory of intrinsic forms in Carnot groups (see also [2], [1], [3]). The definition of these classes of forms is quite technical, and will be sketched in Section 5; let us remind the basic points of Rumin's result: there exists a complex (E * 0 , d c ) such that, if we denote by (Ω * , d) the usual de Rham complex of differential forms on G identified with R n , then…”

Section: Introductionmentioning

confidence: 99%

“…The following definition of intrinsic covectors (and therefore of intrinsic forms) is due to M. Rumin ([23], [22]). Since it is easy to see that E 1 0 = span {θ 1 , .…”

mentioning

confidence: 99%

Faraday’s Form and Maxwell’s Equations in the Heisenberg Group

Franchi

Tesi

2009

Milan J. Math.

View full text Add to dashboard Cite

In this note we present a geometric formulation of Maxwell's equations in Carnot groups (connected simply connected nilpotent Lie groups with stratified Lie algebra) in the setting of the intrinsic complex of differential forms defined by M. Rumin. Restricting ourselves to the first Heisenberg group H 1 , we show that these equations are invariant under the action of suitably defined Lorentz transformations, and we prove the equivalence of these equations with differential equations "in coordinates". Moreover, we analyze the notion of "vector potential", and we show that it satisfies a new class of 4th order evolution differential equations. Mathematics Subject Classification (2000). 43A80, 58A10, 83C22 .

show abstract

Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups

Cited by 33 publications

References 6 publications

Limiting Bourgain–Brezis estimates for systems of linear differential equations: Theme and variations

Limiting Bourgain–Brezis estimates for systems of linear differential equations: Theme and variations

Differential forms in Carnot groups: a Γ-convergence approach

Faraday’s Form and Maxwell’s Equations in the Heisenberg Group

Contact Info

Product

Resources

About