Aronsson’s equations on Carnot–Carathéodory spaces

Abstract

Let (Rⁿ, d_X) be a Carnot–Carathéodory metric space generated by a family of smooth vector fields {X_i}_i=1^m satisfying Hörmander’s finite rank condition, and be the horizontal tangent bundle generated by {X_i}_i=1^m. Assume that is quasiconvex in p-variable. We prove that any absolute minimizer u∈W_X^{1, ∞}(Ω) to F_∞(v, Ω)=ess sup_x∈ΩH(x, Xv(x)) is a viscosity solution of the Aronsson equation