The Aronsson equation for absolute minimizers of $\{L^\infty \}$-functionals associated with vector fields satisfying Hörmander’s condition

Abstract:

Given a Carnot-Carathéodory metric space $(R^n, d_{\textrm {X}})$ generated by vector fields $\{X_i\}_{i=1}^m$ satisfying Hörmander’s condition, we prove in Theorem A that any absolute minimizer $u\in W^{1,\infty }_{\textrm {X}}(\Omega )$ to $F(v,\Omega )= \textrm {ess sup}_{x\in \Omega }f(x,Xv(x))$ is a viscosity solution to the Aronsson equation \begin{equation*}-\sum _{i=1}^{m} X_{i}(f(x,Xu(x))) f_{p_{i}}(x,Xu(x)) = 0, \ \text { in }\ \Omega , \end{equation*} under suitable conditions on $f$. In particular, any AMLE is a viscosity solution to the subelliptic $\infty$-Laplacian equation \begin{equation*}\Delta _{\infty }^{(X)} u: =-\sum _{i,j=1}^{m} X_{i} u X_{j} u X_{i} X_{j} u = 0, \ \text { in } \ \Omega . \end{equation*} If the Carnot-Carathéodory space is a Carnot group ${\mathbf {G}}$ and $f$ is independent of the $x$-variable, we establish in Theorem C the uniqueness of viscosity solutions to the Aronsson equation \begin{align*} A(Xu, (D^{2}u)^{*}):= -\sum _{i, j=1}^{m} f_{p_{i}}(Xu)f_{p_{j}}(Xu)X_{i} X_{j} u &= 0, \ \text { in }\ \Omega , u & = \phi , \ \text { on } \partial \Omega , \end{align*} under suitable conditions on $f$. As a consequence, the uniqueness of both AMLE and viscosity solutions to the subelliptic $\infty$-Laplacian equation is established on any Carnot group ${\mathbf {G}}$.