The Aronsson equation for absolute minimizers of L ∞ -functionals associated with vector fields satisfying Hörmander's condition
Abstract
Given a Carnot-Carathéodory metric space
(
R
n
,
d
cc
)
generated by vector fields
{
X
i
}
m
i=1
satisfying Hörmander's condition, we prove in theorem A that any absolute minimizer $u\in W^{1,\infty}_{\hbox{cc}}(\Om)$ to $F(v,\Om)=\sup_{x\in\Om}f(x,Xv(x))$ is a viscosity solution to the Aronsson equation (1.6), under suitable conditions on
f
. In particular, any AMLE is a viscosity solution to the subelliptic
∞
-Laplacian equation (1.7). If the Carnot-Carathédory space is a Carnot group
G
and
f
is independent of
x
-variable, we establish in theorem C the uniquness of viscosity solutions to the Aronsson equation (1.13) under suitable conditions on
f
. As a consequence, the uniqueness of both AMLE and viscosity solutions to the subelliptic
∞
-Laplacian equation is established in
G