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The Nonlinear Singularly Perturbed Problems for Elliptic Equations with Boundary Perturbation

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The Nonlinear Singularly Perturbed Problems for Elliptic Equations with Boundary PerturbationThe,for,with,the

    The Nonlinear Singularly Perturbed

    Problems for Elliptic Equations with

    Boundary Perturbation

    烹取'NORTHEAST.MATH.J

    23(4)(2007),293--297

    TheNonlinearSingularlyPerturbedProblems

    forEllipticEquationswithBoundary

    Perturbation

    MOJia-qi1.2(莫嘉琪),ZHANGWei-jiang2,3(张伟江)andCHENXian-feng2,3(陈贤峰)

    (.DepartmentofMathematics,AnhuiNormalUniversity,Wuhu,DD)

    (2.DivisionofComputationalScience,E-InstitutesofShanghaiUniversitiesatSJTU, Shanghai,2oo24

    (3.DepartmentofMathematics,ShanghaiJiaotongUniversity,Shanghai,2DD) Abstract:Thenonlinearsingula$lyperturbedproblemsforellipticequationswith boundaryperturbationareconsidered.Undersuitableconditions,byusingthetheory ofdifferentialinequalitiestheasymptoticbehaviorofsolutionsfortheboundaryvalue problemsisstudied.

    Keywords:nonlinear,ellipticequation,singularperturbation,boundaryperturba- tion

    2000MRsubjectclassification:35B25.35J60

    CLCnumber:0175.29

    Documentcode:A

    ArticleID:1000-1778(2007)04-0293-05

    1Introduction

    Thenonlinearsingularlyperturbedproblemisaveryattractiveobjectofstudyinthe internationalacademiccircles(see).Duringthepastdecademanyapproximatemethods

    havebeendevelopedandrefined,includingthemethodofaveraging,boundarylayermethod, methodsofmatchedasymptoticexpansionandmultiplescales.Recently,manyscholarssuch asAkhmetov,LavrentievmandSpigler[2],

    BellandDeng[3],Hwangm[4],Zhang[,Ammari,

    KangandTouibi[6],

    KhasminskiiandYin[7],

    Marques[8andBobkova[9havedoneagreat

    dealofworksontheseproblems.UsingthedifferentialinequalitiesandothermethodsMo eta1.consideredalsoaclassofnonlinearsingularlyperturbedproblems(Seelo-2o]).In

    thispaper,usingaspecialandsimplemethod,westudyboundaryvalueproblemsforaclass ofsingularlyperturbednonlinearellipticequationswithboundaryperturbation. Nowweconsiderthefollowingnonlinearsingularlyperturbedproblem: Eu=f(r,,u,E),(r,)?s,(1.1)

    2;

    

    

    .王‰

    

    啪?

    _

    M

    藏一

    

NoRTHEAST.MATH.JVoL.23

    =9(,)),Of/e:r=(,)),(1.2)

    where)isasmallpositiveparameter,

    :m

    +2(r'+

1ir+2a12re+a22(+),>0

    isauniformlyellipticoperator,(r;)?Q=..(7.,)l0r(,)))denotesabounded

    convexregioninR,andesignifiesthesmoothboundaryofQProblem(1.1)(1.2)

    isaDirichletboundaryvalueproblemforanellipticequation.Thispaperinvolvesaclass ofnonlinearsingularlyperturbedproblemswihboundaryperturbation.Weconstructthe

    asymptoticexpansionofasolutionanddiscussitsasymptoticbehavior. Weneedthefollowinghypotheses:

    [H1Thecoefficientsin,,gandaresufficientlysmoothfunctionswithrespectto theirvariablesincorrespondingdomains.

    2r,,,))c1>0,a(0,)a0>0,wherec1anda0areconstants.

    2FormalAsymptoticSolution

    Wenowconstructtheformalasymptoticexpansionforthesolutionoftheproblem(1.1)- (1.2).;

    Thereducedproblemoftheproblem(1.1)(1.2)is

    J(r,,,0)=0,(r,)?fi0.(2.1)

    Obviously,thereexistsasufficientlysmoothsolutionu0fortheequation(2.1). LettheformalexpansionoftheoutersolutionUfortheoriginalproblem(1.1)(1.2)be

    

    ?.(2.2)

    i=0

    Substituting(2.2)into(1.1),developing,in),andequatingcoefficientsofthesame

    powersof)respectively,fort=1,2,weobtain

    =

    F,/(r,,Uo,0),(r,)?Q{,(2.3)

    where,i=1,2,,aredeterminedfunctions.

    From(2.2)weobtaintheoutersolutionUfortheoriginalproblem.Butitmaynotsatisfy theboundarycondition(1.2),SOthatweneedtoconstructaboundarylayercorrectiveterm .

    Weintroduceastretchedvariable(see[1):

    r

p'

    Andletthesolutionoftheoriginalproblem(1.1)-(1.2)be

    =

    U(r,,))+v(p,,)).(2.4)

    Substituting(2.4)into(1.1)(1.2),wehave

    e2LV=l(ep,,U+))f(ep,,)),()?Q.,

    V=9(,))(,)),P=(,))/E.

    (2.5)

    (2.6)

NO.4MO3Q.eta1.NONLINEARSINGULARLYPERTURBEDPROBLEMS295

    (2.7)

    Substituting(2.4),(2.2)and(2.7)into(2.5)(2.6),expandingnonlineartermsin),and equatingthecoefficientsofthesamepowersof),weobtain .(02V0

    ,(0,,+.,.)+f(o,,,.)=.,(r,?Qs,

    0=(,0)u0(0,),P=(,O)/).

    Fori=1,2,,wehave

    .(.,)等一九(.,,+.)+=0-()?Qt,

    V=glmGi,P=(,O).

    (2.8)

    (2.9)

    (2.1O)

    (2.11)

    Obviously,Fi,Giaresuccessivelydeterminedfunctions.

    Fromtheproblems(2.8)(2.9)and(2.10)(2.II),wecanobtainV0andVi,i=1,2,, whichsatisfy

    =

    D(eXp{p))=O(exp{一‰)),o<)《1,

    whereki?ki1,i=0,1,2,,arepositiveconstants. Thenwecanconstructthefollowingformalasymptoticexpansionofthesolutionfor

    theoriginalproblem(1.1)(1.2):

    

    ?+vi]e',0<)《1.(2.12)

    3TheMainTheorem

    Nowweprovethat(2.12)isauniformlyvalidasymptoticexpansions.

    Theorem3.1Underthehypotheses[H1and[n2],thereexistsasolutionolthenonlin earsingularlyperturbedproblem(1.1)(1.2),whichsatisfiesholdstheuniformlyvalidasymp

    toticexpansion(2.12),0r(r,)?(Q+012)).

    Proo1.WefirstconstructtheauxiliaryfunctionsQand

    Q=一)+

    ,

    =+)+l,

    (3.1)

    (3.2)

    whereisapositiveconstantlargeenough,whichwillbedecidedbelow,and

    ym三?[+.i=0

    Obviously,wehave

    Q,(r,?(Q+Q).(3.3)

    ?

    p

    ??:i

    (

    t

    e

    L

NORTHEAST.MATH.JVOL.23

Andtherearepositiveconstants

    ,MIsuchthatforr:(,s), a=ymsm+=?+?一如m+1

    -!m

    +g(,0)%(o,)+?一+MISm+ls'1 (,s)+(+)sm+1.

    Thusselecting(+),wehave (,),(r,)?.(3.

    4)An

    alogously,wecanprovethat,

    ?夕(,),(r')?Qs.(3.

    5)Nowweprovethat, s口一,(r',,)?0,(r')?Q,(3.

    61

    s,(r',,)0,(r,?)?Q.(3.

    7)

    By

    

    thehyp.the,for.smallen.ughand0<s6

    0,thereisapos~ivec0ttM2, suchthat

    62La,(r,,a,)

    =s2rms+】一/(r,,a,s) =s2三【ymJf(r,,ym,s)+[,(r,,ym,s)/(r,,a,s) ?/(r,,uo,o)+?(r'Uo,o)阢一只

    +)-/(o,Uo+vo,0)+,(0,,uo,0)J

    .

    +0(0嚣(,,+)+</