Some Remarks on Vector Potentials for Maxwell's Equations in Space-Time Carnot Groups

Baldi, Annalisa ; Franchi, Bruno

Bollettino della Unione matematica italiana, Série 9, Tome 5 (2012) no. 2, pp. 337-355

Résumé

In this paper we prove a $\Gamma$-convergence result for time-depending variational functionals in a space-time Carnot group $\mathbb{R} \times \mathbb{G}$ arising in the study of Maxwell's equations in the group. Indeed, a Carnot groups $\mathbb{G}$ (a connected simply connected nilpotent stratified Lie group) can be endowed with a complex of ``intrinsic'' differential forms that provide the natural setting for a class of ``intrinsic'' Maxwell's equations. Our main results states precisely that a the vector potentials of a solution of Maxwell's equation in $\mathbb{R} \times \mathbb{G}$ is a critical point of a suitable functional that is in turn a $\Gamma$-limit of a sequence of analogous Riemannian functionals.