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New Monotonicity Formulae for Semi-linear Elliptic and Parabolic Systems

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New Monotonicity Formulae for Semi-linear Elliptic and Parabolic SystemsNew,for,and

    New Monotonicity Formulae for

    Semi-linear Elliptic and Parabolic

    Systems

    (.Ann.Math.

    31B(3),2010,411-432

    DOI:10.1007/s114010080282-8

    C.hineseAnnalsof

    Mathernatics,SeriesB

    ?TheEditorialOfficeofCAMand

    SpringerVerlagBerlinHeidelberg2010

    NewMonotonicityFormulaeforSemilinearElliptic

    andParabolicSystems

    LiMAXianfaSONGLinZHAO

    AbstractTheauthorsestablishageneral

    system

    Aui+(,Ul,

    monotonicityformulaforthefollowingelliptic um)=0inQ

    whereQCCisaregulardomain,(^(,Ul,?-.,))=V~F(x,),F(x,)isagiven smoothfunctionof?忆and("Ul,,um)?m.Thesystemcomesfromunder

    standingthestationarycaseofGinzburg-Landaumode1.Anewmonotonicityformulais

    alsosetupforthefollowingparabolicsystem

    Aui^(,ul,-,u1)=0in(tl,t2)×

    wheretl<t2axetwoconstants,(^(,)isgivenasabove.Thenewmonotonicity formulaearefocusedmoreattentiononthemonotonicityofnonlinearterms.Thenewpoint

    oftheresultsisthatanindexisintroducedtomeasurethemonotonicityofthenonlinear termsintheproblems.Theindexinthestudyofmonotonicityformulaeisusefulin understandingthebehaviorofblow-upsequencesofsolutions.Anothernewfeatureisthat thepreviousmonotonicityformulaeareextendedtononhomogeneousnonlinearities.As applications,theGinzburg-Landaumodelandsomedifferentgeneralizati0nstothefree boundaryproblemsarestudied.

    KeywordsEllipticsystems,Parabolicsystem,Monotonicityformula,

    GinzburgLandaumodel

    2000MRSubjectClassification35Jxx,17B40,17B50

    1Introduction

    Inthispaper,wewillestablishageneralmonotonicityformulaforthefollowingelliptic system:

    Aui+^(,u1,.,u)=0inQ,(1.1)

    whereQCisaregulardomain,((,Ul,?,))=V~F(x,,F(x,)isagivensmooth

    functionofx?and:(1,-,m)(theprecisesmoothnesswillbegivenintheorems).

    Hereweassumethatthesolution?砚c(Q)satisfies(1.1)inthevariationalsensetobedefined inSection2.Weremarkthatsmoothsolutionsto(1.1)satisfy(1.1)inthevariationalsense ManuscriptreceivedJuly18,2008.RevisedOctober20,2008.PublishedonlineApril20,2010.

    DepartmentofMathematicalSciences,TsinghuaUniversity,Beijing100084,China. E-mai1:1ma@math,tsinghua.edu.cnzhaolin05@~mails.tsinghua.edu.cn DepartmentofMathematics,TianjinUniversitTianjin300072,China.E-mail:sxf@yahoo.COrn.ca

    ?

    ProjectsupportedbytheNationalNaturalScienceFoundationofChina(No.10631020)andtheSpecialized

    ResearchFundfortheDoctoraIProgramofHigherEducation(No.20060003002). 412LMa.XF.SongandL.Zhao

    naturally.Ourmotivationforstudyingthesystem(1.1)comesfromunderstandingthestation-

arycaseofGinzburgLandaumodel(see3,20,2l).Weshallalsoestablishamonotonicity

    formulaforregularsolutionsofthefollowingparabolicsystem:

    OtuiAui(z,l,?.-,")=0in(tl,t2)×"(1.2)

    wheretl<2aretwoconstants,(k(x,))isgivenasabove.Onenewpointinourmonotonicity formulaisthatweintroduceanindex,whichmeasuresthemonotonicityofthenonlinear term,=(fl,,,m).ThisindexalsogivesustherateofscaledsequenceoftheblowuD

    processforimpliedsolutions.Anothernewfeatureisthatweextendthepreviousmonotonicitv

    formulaetononhomogeneousnonlinearities.OurmainresultsareTheorems2.12.2.3.1and 3.2below.Asapplicationsofournewmonotonicityformulae,westudytheGinzburg.Landau modelandsomedifferentgeneralizationstothefreeboundaryproblems. Theapplicationsare

    inPropositions4.1

    4.5below.Ourworkismotivatedfromthemonotonicityformulaegivenby G.S.Weiss[29

    32]andthemonotonicityformulagivenbyAlt,Caffarelliandniedmanf21for freeboundaryproblems.Formorebackgroundrelatedtoourwork

    ,onemayseetheappendix

    inSection5.

    Beforewestatethemonotonicityformulae,weintroducesomenotationsandconcepts.As in[a2],wedenotebyx?YtheEuclideaninnerproductin×,bylxltheEuclideannormin

    ,byBr(xo)={?l{x0J<r)theballofcenterx0andradiusr,byQ(0,t0)=(t0

    r,to+r)xBr(x0)thecylinderofradiusrandheight2r.,by_(t0):(t04r.,t0r2)×瓞

    thehorizontallayerfromto?4rtotor,andby(?o)=(t0+r.,t0+4r2)×the

    horizontallayerfromto+toto+4r..

    Wewrite(T)and()asandfor

    notationconvenience.Weuse

    G(t.,.)(t,)=47r(.?t)147r(.t)l--1expffxx01.,

    todenotethebackwardheatkernel,definedin((..,0)u(to,+?))×戚.~rthermore,by

    wewillalwaysrefertotheouterunitnormalonagivensurface.

Wemeanby(Q)and

    H(QT)theusuallocalSobolevspaceandparabolicSobolevspacesrespectivelyasdeftnedin

    [18.

    Roughlyspeaking,ournewmonotonicityformulaefor(1.1)and(1.2)areasfollows.We

    willshowthatforthevariationalsolutionto(1.1),thefunction

    .(r)

    isincreasinginr?

    ;r_f20)(112-2F()

    (0,)ifVr?(0,),

    n-2fl+1

    JoB(0)

    /(2(1)F(x,一厂(,)F(,.x0))o,

    JB0)

    andforthevariationalsolutionto(1.2),thefunctions :=T-2"f(I2_2F())G(T,xo)--)

    and

    r):=V-2fl(Il2_2F()G(T一厶GcT

    (1.3)

    MonotonicityFormulae413

    areincreasinginr?(0,5)providedfortheindex??andtheradialvariableVr?(0,) therehold

    and

    (2(1)F(x,fl;f(x,)F(x,?(X0))G(,.)0

    (2(1)F())-VxF(),X--X0)),.

    (1.4)

    Weremarkthattheconditions(1.3)(1.5)areautomaticallytrueiftheweakerpointwisecon

    ditioniSsatisfied:

    2(?1)F(x,)flfff(x,)VF(x,)?(XX0)0,V?

    WewillgiveillustrationbyexamplesinSection4.Fromtheexpression(1.6)above,itisclear

    thatthenumbermeasuresthemonotonicityofthenonlinearterms,andournewmonotonicity formulaearefocusedmoreattentiononthemonotonicityofnonlinearnonhomogeneousterms.

    Ourmethodcanalsobeusedtostudyellipticandparabolicsystemswithvariablecoefficients. Theremainingpartofthepaperisorganizedasfollows.InSection2,weestablishthe monotonicityformulafor(1.1)andcharacterizethescaledblow

    upsequences.InSection3,we

    establishthemonotonicityformulaforf1.21andcharacterizethescaledblow

    upsequences.In

    Section4.westatetheinterestingapplicationsofourresultsinSection2andSection3tothe Ginzburg

    Landaumode1andvariousextensionsofthefree-boundaryproblemsconsideredby otherauthors(see,forexamples,(29,32).

    2MonotonicityFormulafortheEllipticSystem

    Considertheellipticsystem

    Aui+k(x,1,-,)=0,i=1,,m,inQ

    for=(1,,)?H11(;).Denote=?2,.(=?u.(),IVl=mm

    i=1=1

    

    ?IVuil,V?:(Vul?X,?,Vu?),(V-u)=

    i=1

    m

    ),u-~7ff?u=?~iVUi?Vfor

    t=1

    

    anyvectoru,andVffDCVff=?VuiDCVui.Wesaythat?.(2)ifeverycomponent

    t:1

    .?砚(Q),i=1,,m.

    Definition2.1iscalledasolutionto(2.1)inthesensevariations,orsimplyavaria

    tionalsolutionthefollowingthreeconditionsaresatisfiedsimultaneously."

(1)Ui?月_l10(),u~k(x,),F(x,),F(x,?L1(Q);

    U

    m?汹

    414

    (2)

    (3)

    L.Ma,X.F.SongandL.Zhao

    satisfies(2.1)inthesenseofdistributions; thefirstvariationwithrespecttodomainvariableso1thefunctional

    vanishesat=.i.e..

    G(=(112-2F(z,)

    0=G(+)

    =

    V[2-2F())div2Vff-2VxF()

    ,0rany?(;

    Themainresultinthissectionreadsasfollows Theorem2.1(MonotonicityFormula)AssumethatB~(xo)ccQandisasolutionto

    (2.1)inthesenseofvariations.thereexistsarealnumber?风suchthatVr?(0,), (2(1)F(,).,)F(,.(.))0l,

    B(0)

    thenthefunction

    .(r):=r-n-2+/(Vl.2F(x,)r一一+/

    B(Xo)JOB(XO)

    definedin(0,),satisfiesthemonotonicityformula

    西.()一西.(P)=2ar-n-2fl+l0)(2(_1)一瑚砸)-VxF(?(X--X0)) ,0rall0<P<o<,where

    n2Z+2lJ

    OB(.)

    f.一里1.\r/

    Vg.u-)2:().

Remark2.1Theconditioninvolvingtheintegraltermthat

    /(2(1)F(x,x,VF(x,?(0))0,Vr?(0,)

    I,B(;TO)

    isnotconvenienttoverifysometimes.Wethereforeprefertostateastrongerpoint

    wisecondi

    tion

    2(1)F(x,x,F(x,)-(Xo)0,V?BS(Xo). ProofofTheorem2.1WemayassumethatX0=0byatranslation.Wetakeafter

    approximation?Ex)=()astestfunctioninDefinition2.1(3)forsmallpositivewith

    r

    2

    +0

    >

    MonotonicityFormulae

    ():maX(0,min(1,)),andobtain 0

    /((Iv[2_2F())r/e-2I仉一2V)()

    +

    _/((1[2_2F())X?2Vg?xVff?)

    /(n([Vff[.2F(x,)21V~l.2VF(x,)?)t,B(0) 《一

    |

    JOB(0)

    (1I.?2F(x,2(Vff?))

    fora.e.r?(0,5)asE__+0,i.e.,

    0=(n2)/(II2F(x,)4/Y(x,JB(0)l,B(0)

    2/F(,)?r/(]vllt,B(0)OBr(0) 415

    2F(x,2(Vl?)).(2.2)

    ByDefinition2.1(2),wecanapplymollifierUi,pto(2.1)foreveryUi(i=1,-,m),where P>0,andget

    

    Aui,p=((,u1,?,urn))p

    MultiplyingthisequationbyUiandintegratingoverBr(0),thensendingP0+,wecaneasily

    derivetheidentity

    iBm(o)1=+,识相

    fora.e.r?(0,).Next,multiplying(2.2)byr--n--2+andusing(2.3),weobtain

    (2.3)

    0=(n2)r一一.+/(DVll.2F(x,)+4r一一.+/F(,) JBr(0)JBr(0)

    +2r一一+/VF(,).+r-n-2+/(II.2F(x,?2(vl?/))t,Br(0)JOBr(0) =

    (n2+2)r一一.+/(IVll2F(x,)4(1)r一一.+/F(,JBr(0)JBr(0)

    2VxF()2r-n-2f~+l

    r(0)

    if-u+

    JB0JOBJ

    /Br(o)0

    ),r(O)r()

    +r-n-2~+/(112F(x,)2r一一+./(?)..

    Thatis

    (一礼一2+2)r一?.+/(Ivil2F(x,)+r一一卢+/(Ivll ?B(0)JOB(0)

    =

    2r-n-2~+2/(Vi-v)2r--n--2~+lJOBJ

    /OB0

)(0)()

    +2r一一卢+/(2(1)F(x,)Zif(x,)F(x,?z) Bf01

    =I1+Io.

    

    2F(x,)

    (2.4)

    416L.Ma,X.F,SongandL.Zhao WepointoutthatI1intherighthandsideoftheaboveequalityisjustequalto

    

    0

    .~

    ,

    (3r_n_2fl+lB.)+2r-n-2fl+2B.Vff-v-).

    Thiscanbeeasilyseenfromthenextcomputation

    

    0

    .~

    .

    (3r_n_2fl+lB.if2)=(r2B.,(r))

    :

    232r-2fl

    

    232rn2

    Inserting(2.5)into(2.4),wethenachieve

    namely,

    (2.5)

    /ff2(ry)2/ff(ry)Vff(ry)?Y JOB1(0)|,OB1(0)

    if2+2i3r-n--2fl+l

dOB0jOB0?,,?r()

    (一佗一23+2)r一一卢+/(1I2F(x,))B(0)

    22

    .

    f01(12_2FOB0())(一斛?JOB)()u,r(O)

    =2rn2+

    J

    /,,(?)aB0rf0]\/

    +2rn.fl+/(2(1)F(x,)一厂(,)VF(,)?),

    Bf01

    d,,

    0?2r-n-2lOB(0)

    f,0,2p)

    +2rn.+/(2(1)F(x,)一厂(,)F(,)-)

    B(0)

    >0(2.6)

    fora.e.r?(0,).Integrating(2.6)fromPtoo-,wecanestablishthemonotonicityformulain

    thetheorem.

    Wenowconsidertheblowupanalysisforvariationalsolutionsto(2.1).Letbeafunction

    inB(0).Foragivensequence0<Pk-.?0,wedefinethescaledsequenceas ():=pi(0+Pkz).

    Wewanttoacquiresomeinformationonthe tothenonlinearellipticsystem(2.1).Infact, limit'SbehaviorwheniSavariationa1solution wehavethefollowingtheorem.

    Theorem2.2(Blowup)Supposethat0<pk__?0as..?oo,andisavariational solutionto(2.1)definedinB~(xo)suchthattheconclusionsinTheorem2.1holdtrue.Suppose

    inadditionthatsatisfie8atXothegrowthestimate sup

    ,,

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