The subelliptic ∞-Laplace system on Carnot-Carathéodory spaces

Abstract

Given a Carnot-Carathéodory space Ω ⊆ ℝn with associated frame of vector fields X = {X<inf>1</inf>,⋯, X<inf>m</inf>}, we derive the subelliptic ∞-Laplace system for mappings u: Ω → ℝN, which reads δX∞u:=(Xu ⊗ Xu +

Xu

2[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<inf>∞</inf>(u, Ω): =

Xu

<inf>L∞(Ω)</inf> for an appropriately defined notion of horizontally ∞-minimal mappings. We also establish a maximum principle for

Xu

for 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.