资源描述：
《An Eigenvalue problem for the Infinity-Laplacian》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
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
