Maxwell's equations in Carnot groups

Bruno Franchi

(Università di Bologna)

Event: ERC Workshop on Geometric Analysis on sub-Riemannian and Metric Spaces

Date: Oct 12, 2011, time: 11:40

Place: Centro De Giorgi, Scuola Normale Superiore

Abstract. In this talk we define Maxwell's equations in the setting of the intrinsic complex of differential forms in Carnot groups introduced by M. Rumin. It turns out that these equations are higher order equations in the horizontal derivatives. In addition, when looking for a vector potential, we have to deal with a new class of higher order evolution equations that replace usual wave equations of the Euclidean setting and that are no more hyperbolic. We prove equivalence of these equations with the "geometric equations'' defined in the intrinsic complex, as well as existence and properties of solutions. Finally, we prove that, in spite of their higher order, these equations can be seen as the limit (in a suitable sense) of usual Maxwell's equations.