The Aronsson equation for absolute minimizers of L-infinity-functionals associated with vector fields satisfying Hormander's condition

摘要

Given a Carnot-Caratheodory metric space (R-n, d(X)) generated by vector fields {X-i}(i=1)(m) satisfying Hormander's condition, we prove in Theorem A that any absolute minimizer u epsilon W-X(1,infinity) (Omega) to F (v,Omega) = ess sup(x epsilon Omega) f(x, Xv( x)) is a viscosity solution to the Aronsson equation -Sigma X-i=1(i)(f(x,Xu(x))) f(pi) (x, Xu(x)) = 0, in Omega, under suitable conditions on f. In particular, any AMLE is a viscosity solution to the subelliptic infinity-Laplacian equation Delta((X))(infinity)u:-Sigma(m)(i,j=1) X(i)uX(j)uX(i)X(j)u(i) = 0, in Omega, If the Carnot- Carath ' eodory space is a Carnot group G and f is independent of the x- variable, we establish in Theorem C the uniqueness of viscosity solutions to the Aronsson equation A(Xu, (D(2)u)*):= -Sigma(m)(i,j=1) f(pi) (Xu) f(pj)(Xu)X(i)X(j)u = 0, in Omega, u = phi, on partial derivative Omega, under suitable conditions on f. As a consequence, the uniqueness of both AMLE and viscosity solutions to the subelliptic infinity-Laplacian equation is established on any Carnot group G.