The subelliptic ∞-Laplace system on Carnot-Carathéodory spaces

Given a Carnot-Carathéodory space Ω ⊆ ℝn with associated frame of vector fields X = {X1,⋯, Xm}, we derive the subelliptic ∞-Laplace system for mappings u: Ω → ℝN, which reads δX∞u:=(Xu ⊗ Xu +Xu2[Xu]⊥ ⊗ I): X Xu = 0 in the limit of the subelliptic p-Laplacian as p → ∞. Here Xu is the horizontal gradient and [Xu]⊥ is the projection on its nullspace. Next, we identify the variational principle characterizing the subelliptic ∞-Laplacian system, which is the "Euler-Lagrange PDE" of the supremal functional E∞(u, Ω): =XuL∞(Ω) for an appropriately defined notion of horizontally ∞-minimal mappings. We also establish a maximum principle forXufor solutions to the subelliptic ∞-Laplacian system. These results extend previous work of the author [J. Differential Equations 253 (2012), no. 7, 2123-2139; Proc. Amer. Math. Soc, to appear] on vector-valued calculus of variations in L∞ from the Euclidean to the subelliptic setting