The Aronsson Equation for Absolute Minimizers of Supremal Functionals in Carnot-Carath\'eodory Spaces

TL;DR

This paper proves that absolute minimizers of certain supremal functionals in Carnot-Carathéodory spaces are viscosity solutions to the Aronsson equation, extending the theory to spaces with vector fields inducing a continuous distance.

Contribution

It establishes the connection between absolute minimizers and viscosity solutions of the Aronsson equation in Carnot-Carathéodory spaces with $C^2$ vector fields and quasiconvex Hamiltonians.

Findings

Absolute minimizers satisfy the Aronsson equation as viscosity solutions.

Extension of Aronsson equation theory to Carnot-Carathéodory spaces.

Results hold for Hamiltonians with $z$-variable dependence.

Abstract

Given a $C^2$ family of vector fields $X_1,...,X_m$ which induces a continuous Carnot-Carath\'eodory 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 $-\langle X(f(x, u(x), Xu(x))), D_p f(x, u(x), Xu(x))\rangle = 0.$