a r X i v :m a t h /0307198v 1 [m a t h .A P ] 14 J u l 2003The Aronsson equation for absolute minimizers of L ∞-functionals associated with vector fields satisfying H¨o rmander’s conditionChangyou WangDepartment of Mathematics,University of KentuckyLexington,KY 40506Abstract .Given a Carnot-Carath´e odory metric space (R n ,d cc )generated by vector fields {X i }m i =1satisfying H¨o rmander’s condition,we prove in theorem A that any absolute mini-mizer u ∈W 1,∞cc (Ω)to F (v,Ω)=sup x ∈Ωf (x,Xv (x ))is a viscosity solution to the Arons-son 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-Carath´e dory 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 .§1.Introduction Variational problems in L ∞are very important because of both its analytic difficulties and their frequent appearance in applications,see the survey article [B]by Barron .The study began with Aronsson’s papers [A1,2].The simplest model is to consider minimal Lipschitz extensions (or MLE):for a bounded,Lipschitz domain Ω⊂R n and g ∈Lip(Ω),find u ∈W 1,∞(Ω),with u |∂Ω=g ,such that Du L ∞(Ω)≤ Dw L ∞(Ω),∀w ∈W 1,∞0(Ω),with w |∂Ω=g.(1.1)Since MLE’s may be neither unique nor smooth,Aronsson [A1]introduced the notation of absolutely minimizing Lipschitz extensions(or AMLE for short),and proved that any C 2AMLE solves the ∞-Laplacian equation∆∞u :=−n ij =1∂u ∂x j ∂2u
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