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

TL;DR

This paper derives the subelliptic infinity-Laplace system on Carnot-Carathéodory spaces, characterizes it variationally, and establishes maximum principles, extending Euclidean results to subelliptic geometries.

Contribution

It introduces the subelliptic infinity-Laplace system, links it to a variational principle, and proves maximum principles in the subelliptic setting, extending prior Euclidean work.

Findings

Derived the subelliptic infinity-Laplace PDE from the limit of p-Laplacian as p approaches infinity.

Identified the variational principle as the Euler-Lagrange PDE of a supremal functional.

Established a maximum principle for solutions to the subelliptic infinity-Laplace system.

Abstract

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.