Compensated compactness for differential forms in Carnot groups and applications

Abstract

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Differential forms in Carnot groups: a Γ-convergence approach

“…∧ θ n . Starting from * g and * g, by left translation, we can define now two families of vector bundles (still denoted by * g and * g) over G (see [2] for details). Sections of these vector bundles are said respectively vector fields and differential forms.…”

“…3 to make the paper self-consistent. For a more exhaustive presentation, we refer to original Rumin's papers, as well as to the presentation in [2]. The main properties of (E * 0 , d c ) can be summarized in the following points:…”

“…Indeed, we are able to prove that the L 2 -energies associated with ε −κ d ε on 1-forms -converge, as ε → 0, to the energy associated with d c . We stress that intrinsic 1-forms in groups appear in several applications, like H -convergence of elliptic operators on groups ( [2,3]) and Maxwell's equations in Carnot groups ([4,14,15]). …”

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

“…. , i n ) is a multi-index, we set X I = X i 1 1 · · · X in n . By the Poincaré-Birkhoff-Witt theorem (see, e.g.…”

“…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…”