ON A CLASS OF SEMILINEAR EVOLUTION EQUATIONS FOR VECTOR POTENTIALS ASSOCIATED WITH MAXWELL’S EQUATIONS IN CARNOT GROUPS BRUNO FRANCHI ENRICO OBRECHT EUGENIO VECCHI Abstract. In this paper we prove existence and regularity results for a class of semilinear evolution equations that are satisfied by vector potentials associated with Maxwell’s equations in Carnot groups (con- nected, simply connected, stratified nilpotent Lie groups). The natural setting for these equations is provided by the so-called Rumin’s complex of intrinsic differential forms. 1. Introduction The aim of this paper is to prove existence of strong solutions for a class of higher order semilinear evolution equations in Carnot groups (i.e. connected simply connected stratified nilpotent Lie groups) satisfied by vector poten- tials associated with “intrinsic” Maxwell’s equations in the group. These equations, though not hyperbolic, can still be called “wave equations” be- cause of their origin, as “equations for a vector potential”, from a class of intrinsic Maxwell’s equations, precisely as it holds in the Euclidean set- ting, where the vector potential associated with classical Maxwell’s equations satisfies d’Alembert’s wave equation. Let us remind this procedure in the Euclidean setting. Consider the space-time R × R 3 of special relativity, end we denote by s ∈ R the time variable and by x ∈ R 3 the space variable. If (Ω ∗ ,d) is the de Rham complex of differential forms in R × R 3 , classical Maxwell’s equations can be formulated in their simplest form as follows: we fix the standard volume form dV in R 3 , and we consider a 2-form F ∈ Ω 2 (Faraday’s form), that can be always written as F = ds ∧ E + B, where E is the electric field 1-form and B is the magnetic induction 2-form. Then, if we assume for sake of simplicity all “physical” constants (i.e. magnetic permeability and electric permittivity) equal to 1, classical Maxwell’s equations become (1) dF =0 and d(∗ M F )= J . 1991 Mathematics Subject Classification. 35Q61, 35R03, 58A10, 49J45 . Key words and phrases. Carnot groups, differential forms, semilinear Maxwell’s equa- tions. B.F. is supported by MIUR, Italy, by University of Bologna, Italy, funds for selected research topics and by EC project CG-DICE. E.O. is supported by University of Bologna, Italy, funds for selected research topics. E.V. is supported by Swiss National Found and by EC project CG-DICE.. 1