The Subelliptic $\infty$-Laplace System on Carnot-Carathéodory Spaces

Nicholas Katzourakis

Published 2013-03-01, updated 2013-04-11Version 2

Given a Carnot-Carath\'eodory space $\Om \sub \R^n$ with associated vector fields $X=\{X_1,...,X_m\}$, we derive the subelliptic $\infty$-Laplace system for mappings $u : \Om \larrow \R^N$, which reads \[ \label{1} \De^X_\infty u \, :=\, \Big(Xu \ot Xu + \|Xu\|^2 [Xu]^\bot \ot I \Big) : XX u\, = \, 0 \tag{1} \] in the limit of the subelliptic $p$-Laplacian as $p\ri \infty$. Here $Xu$ is the horizontal gradient and $[Xu]^\bot$ 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 \[ \label{2} E_\infty(u,\Om)\ := \ \|Xu\|_{L^\infty(\Om)} \tag{2} \] for an appropriately defined notion of \emph{Horizontally $\infty$-Minimal Mappings}. We also establish a maximum principle for $\|Xu\|$ for solutions to \eqref{1}. These results extend previous work of the author \cite{K1, K2} on vector-valued Calculus of Variations in $L^\infty$ from the Euclidean to the subelliptic setting.