Homogenization and Convergence of Correctors in Carnot Groups
Bruno Franchi, Cristian E. Gutiérrez, Truyen V. Nguyen · Communications in Partial Differential Equations · 2005
We consider homogenization of differential operators of the form where is a family of linearly independent vector fields in ℝ N that by commutation generate the Lie algebra of a Carnot group, a ij (ξ) are periodic functions in the sense of the group, and δ1/ε are the dilations in the group. We establish Meyers type estimates for the horizontal gradients Xu = (X 1 u,…,X m u) of solutions to equations defined with general vector fields satisfying Hörmander's condition, and use them to prove convergence of the horizontal gradients of correctors in L 2+Θ, Θ > 0.