The Aronsson Equation for Absolute Minimizers of Supremal Functionals in Carnot-Carathéodory Spaces

Abstract

Given a $C^2$ family of vector fields $X_1,...,X_m$ which induces a continuous Carnot-Carathéodory distance, we show that any absolute minimizer of a supremal functional defined by a $C^2$ quasiconvex Hamiltonian $f(x,z,p)$, allowing $z$-variable dependence, is a viscosity solution to the Aronsson equation $-⟨X(f(x,u(x),Xu(x))),D_{p}f(x,u(x),Xu(x))⟩=0.$