An Eigenvalue problem for the Infinity-Laplacian

An Eigenvalue problem for the Infinity-Laplacian

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1、ANEIGENVALUEPROBLEMFORTHEINFINITY-LAPLACIANTILAKBHATTACHARYAANDLEONARDOMARAZZI1.IntroductionInthiswork,westudyaversionoftheeigenvalueproblemforthenon-homogeneousinfinity-Laplacianonboundeddomains.Inasense,thisisafollowupofworks[5,6]onDirichletproblemsinvolvi

2、ngrighthandsidesthatdependonthesolution.Inordertodescribetheproblembetter,weintroducesomenotations.LetΩ⊂IRn,n≥2,beaboundeddomain,Ωitsclosureand∂Ωitsboundary.Wetakea∈C(Ω)∩L∞(Ω),a>0.Ourgoalistoseekapair(λ,u),λ,real,andu∈C(Ω)thatsolve(1.1)∆u+λa(x)u3=0,inΩandu=

3、0on∂Ω.∞Werefertoλasaneigenvalueof(1.1)anduaneigenfunctioncorrespondingtoλ.Theoperator∆∞istheinfinity-LaplacianandisdefinedasXn∆∞u=DiuDjuDiju.i,j=1Theinfinity-Laplacianisnonlinearandisadegenerateellipticoperator.Asaresult,solutionsaretobeunderstoodintheviscosit

4、ysense.Questionsinvolvingtheinfinity-Laplacianhavebeenattractingconsiderableattentionoflate.Inparticular,existence,uniquenessandlocalregularityhavebeentopicsofgreatinterest.Forgreatermotivationandcontext,wedirectthereadertotheworks[1,4,8,9,18].Ourcurrentwork

5、ismorealongthelinesof[5,6,16,17].Fromhereon,wewilloftenreferto(1.1)astheeigenvalueproblem.arXiv:1211.3074v1[math.AP]13Nov2012Oneofthemajortasksistobeabletocharacterizetheprincipalorthefirsteigenvalueof(1.1).Theseminalwork[3]providesuswithanapproachtoachievin

6、gthisgoal.While[3]treatsthecaseoftheLaplacian,theideasemployedinitaregeneralenoughtobeapplicabletononlinearcase,asshownin[7].Theworkthatcomesclosesttooursisin[13]whichtreatsthecaseofthe1-homogeneousinfinity-Laplacian.Oneofthemajordiscussionin[3,7,13]isthemax

7、imumprinciplewhentheparameterλislessthanthefirsteigenvalueofthecorrespondingoperatorbeingstudied.Thesamethemealsodominatesourworkandcloseanaloguesofsomeoftheresultsin[13]appearhere.12T.BHATTACHARYAANDL.MARAZZIWealsomentionthatthereisgreatinterestinstudyingth

8、eequationthatariseswhenonetakesthelimitp→∞ofthefirsteigenvalueproblemforthep-Laplacian.Theresultingproblemisoftenreferredtoastheinfinity-eigenvalueproblem,seeforinstance,[2,14,15].Theresultsinthiscurrent

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