Homogenization of Hamilton-jacobi Equations in Carnot Groups

We study an homogenization problem for Hamilton-Jacobi equations in the geometry of Carnot Groups.The tiling and the corresponding notion of periodicity are compatible with the dilatations of the Group and use the Lie bracket generating property. 1. Background and motivations Consider a Hamilton-Jacobi equation: u+H(ξ,∇u) = 0 in R where the Hamiltonian H(ξ, p) : R × RN×N → R is not coercive in p. The lack of coerciveness of the Hamiltonian can be overcome by changing the underlying geometry with a suitable family of vector fields. More precisely, we are able to consider the case when H(ξ, p) = H(ξ, σ(ξ)q), where σ(ξ) is a m × N matrix , m < N , H is coercive in q. Here the rows of the matrix σ(ξ) will be considered as coefficients of vector fields satisfying Hörmander condition and that generate a Carnot Group, therefore σ(ξ)∇u will be the horizontal gradient in a Carnot Group, denoted by Dhu, see section 2. We shall consider homogenizations problems for Hamilton-Jacobi equations of the form : (1.1) u(ξ) +H(ξ, ξ ε ,Dhu (ξ)) = 0 in a Carnot Group G, where the Hamiltonian H is G-periodic in the second variable and ξε will be interpreted in the geometry of the group. Let us revise first some key results in the Euclidean setting, for Z -periodic Hamiltonians. The pioneering paper on homogenization of Hamilton-Jacobi equations is due to P.L. Lions, Papanicolau and Varadhan. They proved that the asymptotic behaviour, as ε tends to zero, of the solutions u is governed by the equation: (1.2) u(ξ) + H̄(ξ,Du(ξ)) = 0 The effective Hamiltonian H̄ is obtained by solving a cell problem: (1.3) H(x, ξ, p+Dv(ξ)) = λ for every (x, p) fixed and putting H̄(x, p) = λ. 1991 Mathematics Subject Classification. 35B27,35H05.