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1、THENEUMANNPROBLEMFORTHE∞-LAPLACIANANDTHEMONGE-KANTOROVICHMASSTRANSFERPROBLEMJ.GARC´IA-AZORERO,J.J.MANFREDI,I.PERALANDJ.D.ROSSIAbstract.WeconsiderthenaturalNeumannboundaryconditionforthe∞-Laplacian.Westudythelimitasp→∞ofsolutionsof−∆pup=0inadomainΩwit
2、h
3、Dup
4、p−2∂up/∂ν=gon∂Ω.Weobtainanaturalminimizationproblemthatisverifiedbyalimitpointof{up}andalimitproblemthatissatisfiedintheviscositysense.ItturnsoutthatthelimitvariationalproblemisrelatedtotheMonge-Kantorovichmasstransferproblemswhenthemeasuresaresu
5、pportedon∂Ω.1.Introduction.InthispaperwestudythenaturalNeumannboundaryconditionsthatappearwhenoneconsidersthe∞-LaplacianinasmoothboundeddomainaslimitoftheNeumannproblemforthep-Laplacianasp→∞.ThisproblemisrelatedtotheMonge-Kantorovichmasstranferproble
6、mwhentheinvolvedmeasuresaresup-portedontheboundaryofthedomain. Let∆u=div
7、Du
8、p−2Dubethep-Laplacian.The∞-Laplacianisthelimitpoperator∆∞=limp→∞∆pgivenbyXN∂u∂2u∂u∆∞u=∂xj∂xj∂xi∂xii,j=1intheviscositysense.Thisoperatorappearsnaturallywhenoneconsidersabso-l
9、utelyminimizingLipschitzextensionsofaboundaryfunctionf;see[A],[ACJ],and[J].AfundamentalresultofJensen[J]establishesthattheDirichletproblemfor∆∞iswellposedintheviscositysense.WhenconsideringtheNeumannproblem,boundaryconditionsthatinvolvetheouternormal
10、derivative,∂u/∂νhavebeenaddressedfromthepointofviewofviscositysolutionsforfullynonlinearequationsin[B]and[ILi].Inthesereferencesitisprovedthatthereexistviscositysolutionsandcomparisonprinciplesbetweenthemwhenappropriatehypothesisaresatisfied.Inparticu
11、larstrictmonotonicityisneeded,apropertythatdoesnotholdinourcaseofinterest.Date:October21,2005.Keywordsandphrases.Quasilinearellipticequations,Neumannboundaryconditions.2000MathematicsSubjectClassification.35J65,35J50,35J55.JGAandIPsupportedbyprojectMT
12、M2004-02223,M.C.Y.T.Spain.JJMsupportedinpartbyNSFawardDMS-0500983.JDRsupportedbyFundacionAntorchas,CONICETandANPCyTPICT05009.12J.GARC´IA-AZORERO,J.J.MANFREDI,I.PERALANDJ.D.ROSSIWestudytheNeumannproblemforthe∞-Laplacianobtainedasthelimitasp→∞oftheprob
