The Subelliptic $\mathrm{\infty }$-Laplace System on Carnot-Carath\'eodory Spaces

Given a Carnot-Carath'eodory space $\text{Om}\subset {\mathbb{R}}^n$ with associated vector fields $X=\{X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_m\}$, we derive the subelliptic $\mathrm{\infty }$-Laplace system for mappings $u:\text{Om}\text{larrow}{\mathbb{R}}^N$, which reads $$\begin{array}{cccc}& \text{label}1{\text{De}}_{\mathrm{\infty }}^{X}u:=(Xu\text{ot}Xu+\mathrm{\parallel }Xu{\mathrm{\parallel }}^2[Xu{]}^{\mathrm{\perp }}\text{ot}I):XXu=0& & \text{(1)}\end{array}$$ in the limit of the subelliptic $p$-Laplacian as $p\text{ri}\mathrm{\infty }$. Here $Xu$ is the horizontal gradient and $[Xu{]}^{\mathrm{\perp }}$ is the projection on its nullspace. Next, we identify the Variational Principle characterizing \eqref{1}, which is the “Euler-Lagrange PDE” of the supremal functional $$\begin{array}{cccc}& \text{label}2E_{\mathrm{\infty }}(u,\text{Om}):=\mathrm{\parallel }Xu{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\text{Om})}& & \text{(2)}\end{array}$$ for an appropriately defined notion of \emph{Horizontally $\mathrm{\infty }$-Minimal Mappings}. We also establish a maximum principle for $\mathrm{\parallel }Xu\mathrm{\parallel }$ for solutions to \eqref{1}. These results extend previous work of the author \cite{K1, K2} on vector-valued Calculus of Variations in $L^{\mathrm{\infty }}$ from the Euclidean to the subelliptic setting.