Aronsson’s equations on Carnot–Carathéodory spaces

Abstract

Let $(\mathbf{R}^n, d_X)$ be a Carnot–Carathéodory metric space generated by a family of smooth vector fields $\{X_i\}_{i=1}^m$ satisfying Hörmander's finite rank condition, and $\mathcal{H}_X= \{(x, \sum_{i=1}^m a_i X_i(x))| x\in\mathbf{R}^n, (a_i)_{i=1}^m\in\mathbf{R}^m\}$ be the horizontal tangent bundle generated by $\{X_i\}_{i=1}^m$. Assume that $H=H(x,p)\in C^1(\mathcal{H}_X)$ is quasiconvex in $p$-variable. We prove that any absolute minimizer $u\in W^{1,\infty}_X(\Omega)$ to $F_\infty(v,\Omega)=\operatorname{ess \sup}_{x\in\Omega} H(x, Xv(x))$ is a viscosity solution of the Aronsson equation \[ \mathcal{A}^{X}[u]:= X(H(x,Xu(x)))\cdot H_p(x, Xu(x)) =0 \quad\hbox{in } \Omega. \]