The Aronsson equation for absolute minimizers of $\{L^\infty \}$-functionals associated with vector fields satisfying Hörmander’s condition

Abstract

Given a Carnot-Caratheodory metric space (R, d X ) generated by vector fields {X i } m i=1 satisfying Hormander's condition, we prove in Theorem A that any absolute minimizer u ∈ W 1,∞ X (Ω) to F(v, Q) = ess sup x∈Ω f(x, Xv(x)) is a viscosity solution to the Aronsson equation - Σ X i (f(x, X u (x)))f pi (x, X u (x)) = 0, in Ω, under suitable conditions on f. In particular, any AMLE is a viscosity solution to the subelliptic oo-Laplacian equation Δ (X) ∞ u:= - Σ X i uX j uX i X j u = 0, in Ω. If the Carnot-Caratheodory space is a Carnot group G and f is independent of the x-variable, we establish in Theorem C the uniqueness of viscosity solutions to the Aronsson equation A(Xu,(D 2 u)*):= - Σ f pi (Xu)f pj (Xu)X i X j u = 0, in Ω, u = Φ, on ∂Ω, under suitable conditions on f. As a consequence, the uniqueness of both AMLE and viscosity solutions to the subelliptic oo-Laplacian equation is established on any Carnot group G.