Higher Order Convergence Rates in Theory of Homogenization II: Oscillatory Initial Data

0.48 MB·

by  Sunghan Kim

HIGHER ORDER CONVERGENCE RATES IN THEORY OF
HOMOGENIZATION II: OSCILLATORY INITIAL DATA
SUNGHANKIMANDKI-AHMLEE
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a Abstract. Weestablishthehigherorderconvergenceratesinperiodichomoge-
M nizationoffullynonlinearuniformlyparabolicCauchyproblems, accompanied
with rapidly oscillating initial data. The higher order convergence rates are

3 achievedbyconstructingthehigherorderinitiallayercorrectorsandthehigher
orderinteriorcorrectorsthatdescribetheoscillatorybehaviorofthesolutionsto
] thegivenε-problems. Inordertoconstructthesecorrectors,weestablisharegu-
P laritytheoryintheslowvariables,thatis,thenon-oscillatoryphysicalvariables,
A ofsolutionstoeitherthespatiallyperiodicCauchyproblemsorthecellproblems,
basedontheclassicaltheoryforviscositysolutions.
.
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Contents
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v 1. Introduction 2
5
1.1. MainResult 2
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1.2. HistoricalBackground 3
3
3 1.3. HeuristicDiscussionandMainDifficulties 3
0 1.4. Outline 5
. 2. NotationandStandingAssumptions 5
1
0 3. RegularityTheoryinSlowVariables 6
7 3.1. SpatiallyPeriodicCauchyProblem 7
1 3.2. CellProblem 14
:
v 4. HigherOrderConvergenceRate 19
i 4.1. InitialLayerCorrector 19
X
4.2. InteriorCorrector 26
ar 4.3. NonlinearCouplingEffectandtheBootstrapArgument 30
5. FurtherObservations 34
5.1. Non-oscillatoryInitialDataandHigherOrderConvergenceRate 34
5.2. OscillatoryInitialDataandOptimalConvergenceRate 36
References 39
S. Kim has been supported by National Research Foundation of Korea (NRF) grant funded by
the Korean government (NRF-2014-Fostering Core Leaders of the Future Basic Science Program).
K.-A.LeehasbeensupportedbyNRFgrantfundedbytheKoreangovernment(MSIP)(No. NRF-
2015R1A4A1041675). K.-A.LeealsoholdsajointappointmentwiththeResearchInstituteofMathe-
maticsofSeoulNationalUniversity.
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2 SUNGHANKIMANDKI-AHMLEE
1. Introduction
We are interested in higher order convergence rates in periodic homogeniza-
tion of fully nonlinear uniformly parabolic Cauchy problems, accompanied with
rapidly oscillating initial data. We conduct our analysis based on the theory of
viscositysolutions. Readersmayconsult[CIL],[W1],[W2]and[W3]forstandard
existenceandregularitytheoryofviscositysolutions.
1.1. MainResult. Thegoverningproblemunderourconsiderationisformulated
as
uεt = ε12F(cid:18)ε2D2uε,x,t,xε,εt2(cid:19) inRn×(0,T),
(1.1) uε(x,0)= g(cid:18)x,x(cid:19) onRn,
ε
Ourmainresultisstatedasfollows.
Theorem 1.1. Assume F and g verify (2.1) - (2.3) and (2.5) - (2.6) respectively. Let
uε be the viscosity solution to (1.1) for ε > 0. Then for each integer d ≥ 0, there exist
v˜d,k : Rn × [0,T] × Rn × [0,∞) → R, which is periodic in its third argument, and
w˜d,k :Rn×[0,T]×Rn×R→R,whichisperiodicinitsthirdandfourtharguments,for
eachk≥0,suchthatonehas
(cid:12) (cid:12)
(1.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)uε(x,t)−(cid:88)(cid:98)m2(cid:99)m(cid:88)−2dεk+2d(cid:18)v˜d,k(cid:18)x,t,xε,εt2(cid:19)+w˜d,k(cid:18)x,t,xε,εt2(cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)≤Cmεm−1,
(cid:12) d=0 k=0 (cid:12)
forallx∈Rnandall0≤t≤T,and,inparticular,
(cid:12) (cid:12)
(1.3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)uε(x,t)−(cid:88)(cid:98)m2(cid:99)m(cid:88)−2dεk+2dw˜d,k(cid:18)x,t,xε,εt2(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)≤Cmεm−1,
(cid:12) d=0 k=0 (cid:12)
forallx ∈ Rn andallc ε2|logε| ≤ t ≤ T,provided0 < ε ≤ 1 andm ≥ 0,wherec > 0
m 2 m
andC >0dependonlyonn,λ,Λ,α,m,TandK.
m
LetusmakeafewremarksregardingTheorem1.1asfollows.
Remark1.2. Weshallcallv˜d,kthe(d,k)-th(or(k+2d)-thorder)initiallayercorrector.
Wewillseelaterthattheinitiallayercorrectorsdecayexponentiallyfastasthefast
temporal variable s = ε−1t approaches the infinity. This is why the time region
associatedwiththeestimate(1.3)beginswiththelayert(cid:39)ε2|logε|.
Remark 1.3. We shall call w˜d,k the (d,k)-th (or (k+2d)-th order) interior corrector.
We will observe later that w˜d,k for k ∈ {0,1} are constant in the fast variables
(y,s) = (ε−1x,ε−2t), for any d ≥ 0. Thus, the left hand side of the estimate (1.3)
beginswithuε(x,t)−u¯0,0(x,t)−εu¯0,1(x,t)−ε2w0,2(x,t,ε−1x,ε−2t)−···,whichshows
that uε becomes non-oscillatory in ε-scales, as soon as we stay away from the
initialtimelayer; instead, D2uε israpidlyoscillatinginthisregion, whichiswhy
theinteriorcorrectorsbegintooscillateinthefastvariablesfromthesecondorder.
HIGHERORDERCONVERGENCERATES:OSCILLATORYINITIALDATA 3
1.2. HistoricalBackground. Periodichomogenizationof(1.1)(or(1.5),tobemore
exact)isrigorouslyjustifiedin[AB]and[M1];seealsothereferencestherein,and
[Ish]forfirstorderfullynonlinearequationsaswellas[ABM]foriteratedhomog-
enization. Thereisawiderangeofexistingliteratureontherateofconvergencein
thehomogenizationprocessof(1.5),providedthattheinitialdataisnon-oscillatory;
thatis,gisindependentonitssecondargument. Recentdevelopmentcanbefound,
forinstance,in[JK],[M2]and[CM]usingcontinuousdependenceestimates,[Ich]
basedonadifferentapproach,and[L],[LS]instationaryergodicsettings;seealso
thereferencesthereinforclassicalresultsinthisregard.
Higherorderconvergencerateinthetheoryofhomogenizationhasbeenstudied
invarioussettings. Wereferto[CS],[KMP]fordivergencetypeellipticequations,
[MS]forperforateddomainswithmixedboundaryconditions,[CE]forMaxwell
equations, [ABDW] for wave equations, [HO] for some numerical results, and
also the references therein. Recently, the authors proved in [KL] higher order
convergence rate for non-divergence type elliptic equations, based on the theory
ofviscositysolutions.
Asfarasweknow,however,therehasnotyetbeenanyresulton(higherorder)
convergencerateinhomogenizationof(1.1). Inthispaper,weprovehigherorder
convergence rate regarding (1.1). Based on this result, we also achieve higher
orderconvergencerate(inProposition5.2)foruniformlyparabolicequationswith
non-oscillatoryinitialdata,thatis,
 (cid:18) (cid:19)
(1.4) uu¯¯εtε(=x,F0)D=2g¯u¯(εx,)x,t,xε,εt2 oinnRRnn×. (0,T),
Moreover, we achieve optimal convergence rate (in Proposition 5.3) in homoge-
nizationof
 (cid:18) x t (cid:19)
uεt =F D2uε,x,t,ε,ε2 inRn×(0,T),
(1.5) uε(x,0)= g(cid:18)x,xε(cid:19) onRn.
Although(1.1)and(1.5)coincidewitheachotherwhenFislinearintheHessian
variableD2uε,weobserveasignificantdifferencebetweentheminthehomogeniza-
tionprocess, onceFbecomesnonlinear. Theoptimalconvergencerateregarding
thesolutionof(1.5)isestablishedundersomeadditionalconditionsonthebehav-
ior of F at infinity. We believe that these results will enrich our understanding
especiallyonthehighlyoscillatingbehaviorofalowerdimensionalobject.
1.3. Heuristic Discussion and Main Difficulties. Our analysis can be split into
two parts. The first part is about initial layer correctors, and the second part
is about interior correctors. Initial layer correctors will be given by solutions to
certainspatiallyperiodicCauchyproblems. Duetotheperiodicity,wewillobserve
that the initial layer correctors converge to some values as time goes to infinity,
andthattheconvergencewillbeexponentiallyfast. Thelimitvalueswillturnout
tobetheeffectiveinitialdata.
Theexponentialdecayestimateofthedifferencebetweeninitiallayercorrectors
andthecorrespondingeffectiveinitialdataisdeducedfromtheHarnackinequality
forviscositysolutions. Suchandecayestimatecanbeconsideredclassicalifone
thinks of heat equations (or divergence type uniformly parabolic equations) on
4 SUNGHANKIMANDKI-AHMLEE
spatially periodic domain. One may also find some variations in this regard in
several other places. For example, see [JK], [CM] for continuous dependence
estimates,and[AA]forellipticboundarycorrectors.
However, what is new in this paper is that we achieve exponential decay es-
timates for higher order derivatives of initial layer correctors in non-oscillatory
variables. It also provides higher order regularity of the effective initial data.
Moreover,suchanexponentialdecayestimatewillbeusedsignificantlywhenwe
achievehigherorderconvergencerateoftheassociatedhomogenizationprocess.
On the other hand, the analysis on interior correctors mainly follows our pre-
vious work [KL]. Indeed, we find the argument here almost identical to that of
theprescribedpaper. Nonetheless,wewillprovidethedetailsforreaders’conve-
nience.
Themaindifficultiesinachievinghigherorderconvergencerateareduetothe
nonlinearstructureof(1.1)and(1.4). Ifouroperatorislinear(thatis,F(P,x,t,y,s)=
tr(A(x,t,y,s)P)),theconstructionofthehigherordercorrectorsatthecurrentstepis
independentofthatatthepreviousstep. However,whentheoperatorisfullynon-
linear,theoperatorseemsto“record”alltheoscillatorybehaviorofthecorrectors
thatareconstructedinthepreviousstep. Suchaphenomenonmakestheanalysis
muchmorecomplicatedaswemovefurther. Forinstance,seetheproblem(4.55)
thatarisesfromthed-thstep.
An interesting observation in this direction is that the nonlinearity of F pro-
duces the coupling effect near the initial time layer (that is, 0 < t (cid:28) ε2|logε|)
betweentheoscillationduetotheinitialdataandtheoscillation“recorded”from
the previous step. We observe that this nonlinear coupling effect does not affect
the construction of the higher order interior correctors in the current step, since
it dissipates exponentially fast and becomes negligible, once we stay away from
theinitialtimelayer(thatis,whent (cid:38) ε2|logε|). However,itdoesnotmeanthat
thenonlinearcouplingeffectinthecurrentstepcanbedisregardedafterwardsas
well. Werealizethatthenonlinearcouplingeffectbecomeseffectiveandneedsto
betakencareofinthenextstepoftheconstructionofthehigherorderinitiallayer
correctors. Asfarasweknow,ourobservationofsuchanonlinearcouplingeffect
in the construction of the higher order correctors has been unprecedented in the
theoryofhomogenization.
Letusmakethefinalremarkonhomogenizationof(1.5). Thekeydifferencein
thehomogenizationprocessbetween(1.1)and(1.5)isthattheinitiallayercorrector
ofthelatterproblemmaynotbedifferentiableintheslowvariablesingeneral,even
thoughtheoperatorFissmoothandconcave. Incontrast, theinitialcorrectorof
(1.1) is as smooth as is the operator F. The main reason for such a distinction is,
roughly speaking, that the operator of (1.1) oscillates “in coordination with” the
oscillationoftheinitialdata,which“stabilize”theinfluenceofthefullynonlinearity
oftheoperatorneartheinitialtimelayertoacontrollablelevel. Thisensuresthe
base-caseinitiallayercorrector(andthebase-caseeffectiveinitialdata)tobesmooth
enoughintheslowvariablestoinducehigherorderones. However,theoperator
of(1.5)makes“toomuch”impactontheoscillationofthesolutionneartheinitial
layerand,asaresult,defectstheregularityofthebase-caseinitiallayercorrector
inasubstantialway.
We observe that the higher order convergence rate in the framework of (1.5)
is a highly sophisticated matter that requires a thorough analysis on the limiting
HIGHERORDERCONVERGENCERATES:OSCILLATORYINITIALDATA 5
behaviorofthesequence{ε2F(ε12P,x,t,y,s)}ε>0asε→0. Thispartwillbethesubject
ofourforthcomingpaper.
Instead,weprovetheoptimalconvergencerateforcertainuniformlyparabolic
operators F and certain initial data g. We observe that the convergence rate is
determined by that of the sequence {ε2F(ε12P,x,t,y,s)}ε>0 to its limit operator (if
any). Moreover,wefindoutthateitherthestructureofthelimitoperatororthat
oftheinitialdatadeterminesaffectstherateofthehomogenizationprocess.
1.4. Outline. This paper is organized as follows. In Section 2, we introduce the
notationsandthestandingassumptionsthatwillbeusedthroughoutthispaper,
unlessstatedotherwise. InSection3,weestablishtheregularitytheoryintheslow
variables and, in Section 4, we prove our main result, Theorem 1.1. Section 5 is
devotedtoprovingsomeadditionalresults,namelythehigherorderconvergence
rateinhomogenizationof(1.4),andtheoptimalconvergencerateinhomogeniza-
tionof(1.5).
2. NotationandStandingAssumptions
Let n ≥ 1 be the spatial dimension and T > 0 be the terminal time. We will
callx(resp.,t,y,ands)theslowspatial(resp.,slowtemporal,fastspatial,andfast
temporal)variable.
By Sn wedenote thespaceof allreal symmetricmatricesof ordern, endowed
with(L2,L2)-norm;thatis,(cid:107)P(cid:107) = ((cid:80)ni,j=1p2ij)1/2 foranyP ∈ Sn. ByEij = (eikjl)wewill
denotethe(i,j)-thstandardbasismatrixforSnthatiseij =2−1(δ δ +δ δ )withδ
kl ik jl il jk
beingtheKroneckerdelta. Bytr(P)wedenotethetraceofP.
LetFbeasmoothfunctionalonSn. By ∂F(P)wedenotethederivativeofFin
∂pij
directionE atP. ByDkFwedenotethek-thorderderivativeofFonSnsuchthat
ij p
(cid:32) (cid:33) (cid:32) (cid:33)
∂kF ∂kF
DkF(P)= (P) = (P) .
p ∂p ···∂p ···∂p ∂p ···∂p ···∂p
i1j1 iljl ikjk i1j1 jlil ikjk
Fornotationalconvenience,wealsounderstandDkFinthesenseofFre´chetderiva-
p
tives. Thatis,foreachP ∈ Sn,weconsiderDkF(P)asthe(symmetric)multilinear
p
mapfrom(Sn)k toRsuchthat
∂kF
DkF(P)(Q ,··· ,Q )= (P)q1 ···qk ,
p 1 k ∂pi1j1···∂pikjk i1j1 ikjk
for any Q = (ql ) ∈ Sn with 1 ≤ l ≤ k; here and thereafter we use the summation
l ij
conventionforrepeatedindices. Inparticular,wehave
D F(P)(Q)=tr(D F(P)Q).
p p
Wewillusetheparabolicterminologies,suchas|(x,t)|=(|x|2+|t|)1/2. Formore
details,wereferto[W1]. See[CIL]fortheclassicalexistencetheory,thecomparison
principleandthestabilitytheoryofviscositysolutions. Alsosee[W1], [W2]and
[W3] for the basic regularity theory for viscosity solutions, such as the Harnack
inequality,andinteriorandboundaryCk,α-estimates.
Now let us make the standing assumptions throughout this paper. Assume
F:Sn×Rn×[0,T]×Rn×R→RisuniformlyellipticandconcaveonSn. Thatis,
6 SUNGHANKIMANDKI-AHMLEE
thereareconstants0<λ≤Λsuchthat
(2.1) λ(cid:107)Q(cid:107)≤F(P+Q,x,t,y,s)−F(P,x,t,y,s)≤Λ(cid:107)Q(cid:107) (Q∈Sn,Q≥0),
and
ρF(P,x,t,y,s)+(1−ρ)F(Q,x,t,y,s)≤F(ρP+(1−ρ)Q,x,t,y,s) (Q∈Sn,ρ∈[0,1]),
forall(P,x,t,y,s)∈Sn×Rn×[0,T]×Rn×R. WealsoassumethatFisperiodicin
itslasttwoarguments:
(2.2) F(P,x,t,y+k,s+l)=F(P,x,t,y,s) (k∈Zn,l∈Z),
forall(P,x,t,y,s)∈Sn×Rn×[0,T]×Rn×R. SupposefurtherthatF∈C∞(Sn×Rn×
[0,T];Cα(Rn×R))forsome0<α<1,andthereisK>0suchthat
(2.3) (cid:88) (cid:119)(cid:119)(cid:119)(cid:119)(cid:119)DκpDµx∂νtF(P,x,t,·,·)(cid:119)(cid:119)(cid:119)(cid:119)(cid:119)Cα(Rn×R) ≤K(cid:107)P(cid:107)(1−|κ|)+,
|κ|+|µ|+2ν=m
for any (P,x,t) ∈ Sn ×Rn ×[0,T], for any integer m ≥ 0. Let us remark that (2.3)
implies
(2.4) F(0,x,t,y,s)=0,
forany(x,t,y,s)∈Rn×[0,T]×Rn×R. Inotherwords,thereisnosourcetermin
(1.1)and(1.5).
Ontheotherhand,letg:Rn×Rn →Rbeperiodicinitssecondargument;that
is,
(2.5) g(x,y+k)= g(x,y) (k∈Zn),
forany(x,y)∈Rn×Rn. Alsosupposethatg∈C∞(Rn;C2,α(Rn))and
(2.6) (cid:88) (cid:119)(cid:119)(cid:119)(cid:119)(cid:119)Dµxg(x,·)(cid:119)(cid:119)(cid:119)(cid:119)(cid:119)C2,α(Rn) ≤K,
|µ|=m
foranyx∈Rnandeachintegerm≥0.
3. RegularityTheoryinSlowVariables
Supposethatwehaveafamilyofuniformlyparabolicequationsthataredefined
inthefastvariables(y,s)andareparameterizedbytheslowvariables(x,t). Then
weobtainafunctionv=v(x,t,y,s)suchthat(y,s)(cid:55)→v(x,t,y,s)solvestheequation
correspondingto(x,t). Theregularityof(y,s) (cid:55)→ v(x,t,y,s)canbededucedfrom
existing literature concerning uniformly parabolic equations. In this section, we
studytheregularityof(x,t)(cid:55)→v(x,t,y,s),basedonexistingresultsontheregularity
of(y,s)(cid:55)→v(x,t,y,s).
This section is mainly concerned with the following two types of uniformly
parabolic equations, namely the spatially periodic Cauchy problems and the cell
problems. Theformerwillbetheequationsforinitiallayercorrectors,whereasthe
latterarewillbethoseforinteriorcorrectors.
HIGHERORDERCONVERGENCERATES:OSCILLATORYINITIALDATA 7
3.1. SpatiallyPeriodicCauchyProblem. Let f :Rn×[0,T]×Rn×[0,∞)→Rbe
periodicinitsthirdargument,thatis,
(3.1) f(x,t,y+k,s)= f(x,t,y,s) (k∈Zn),
forany(x,t,y,s)∈Rn×[0,T]×Rn×[0,∞). Assumethat f ∈C∞(Rn×[0,T];Cα(Rn×
[0,∞)))andthereisβ>0suchthat
(3.2) (cid:88) (cid:20)(cid:119)(cid:119)(cid:119)(cid:119)(cid:119)Dµx∂νt f(x,t,·,·)(cid:119)(cid:119)(cid:119)(cid:119)(cid:119)Cα(Rn×[0,∞))+eβs(cid:119)(cid:119)(cid:119)(cid:119)(cid:119)Dµx∂νt f(x,t,·,s)(cid:119)(cid:119)(cid:119)(cid:119)(cid:119)Cα(Rn)(cid:21)≤K,
|µ|+2ν=m
forall(x,t,s)∈Rn×[0,T]×[0,∞)andeachintegerm≥0,with0<α<1andK>0
beingthesameconstantsthatareusedin(2.3).
Foreach(x,t) ∈ Rn×[0,T],letusconsiderthefollowingspatiallyperiodicand
uniformlyparabolicCauchyproblem,

(3.3) vvs(x=,tF,y(D,02y)v=,x,gt(,xy,,ys))+ f(x,t,y,s) ionnRRnn×. (0,∞),
Bythestandardexistencetheory[CIL],weknowthatthereexistsauniqueviscosity
solutionv(x,t,·,·)∈BUC(Rn×[0,∞))to(3.3). DuetotheperiodicityofF, f and g,
wededucethatvisalsoperiodiciny,thatis,
(3.4) v(x,t,y+k,s)=v(x,t,y,s) (k∈Zn),
forany(x,t,y,s)∈Rn×[0,T]×Rn×[0,∞).
Weshallbeginwithaneasyobservationthatthespatialoscillationofv(x,t,y,s)
in ydecaysexponentiallyfastass → ∞. Theexponentialratewillturnouttobe
independentof(x,t).
Lemma3.1. Foreach(x,t)∈Rn×[0,T],thereexistsauniqueγ∈Rsuchthat
(3.5) eβ0s|v(x,t,y,s)−γ|≤C,
forany(x,t,y,s)∈Rn×[0,T]×Rn×[0,∞),where0<β <βdependsonlyonn,λ,Λ
0
andβ,andC>0dependonlyonn,λ,Λ,β,β andK.
0
Proof. Since (x,t) will be fixed throughout the proof, let us write v = v(y,s), F =
F(M,y,s), f = f(y,s) and g = g(y) for notational convenience. By S(s), I(s) and
O(s)letusdenotesupRnv(·,s),infRnv(·,s)andrespectivelyoscRnv(·,s). Alsowrite
Y = (0,1)n and 2Y = (0,2)n. By the spatial periodicity (3.4) of v, we have S(s) =
sup v(·,s)=sup v(·,s),andsimilaridentitiesforI(s)andO(s)aswell.
2Y Y
Fixs ≥0. Then
0
(cid:32) (cid:33)
K
∂s S(s0)+ β(e−βs0 −e−βs) =Ke−βs ≥ f(y,s),
for any y ∈ Rn and s ≥ s , due to (3.2). Since we have (2.4), we deduce that
0
S(s0)+ Kβ(e−βs0 −e−βs)isasupersolutionto(3.3)inRn×[s0,∞). Similarly, onecan
observethatI(s0)− Kβ(e−βs0 −e−βs)isasubsolutionto(3.3)inRn×[s0,∞). Thus,by
thecomparisonprinciple[CIL]forviscositysolutions,wededucethat
K K
(3.6) I(s )− (e−βs0 −e−βs)≤v(y,s)≤S(s )+ (e−βs0 −e−βs),
0 β 0 β
foranyy∈Rnands≥s .
0
8 SUNGHANKIMANDKI-AHMLEE
Nowforeachnonnegativeintegerk,letusdefine
K
v (y,s)=v(y,s+k)−I(k)+ e−βk (y∈Rn,s≥0).
k β
From(3.6)withsands replacedbys+kandkrespectively,wededucethat
0
v (y,s)≥0 (y∈Rn,s≥0).
k
Ontheotherhand,weseethatv isa(spatiallyperiodic)viscositysolutionto
k
∂ v =F(D2v ,y,s+k)+ f(y,s+k) in2Y×(0,1).
s k y k
Therefore,wemayapplytheHarnackinequalityinY¯ ×[1,1]anddeducefromthe
2
spatialperiodicityofv that
k
(cid:18) (cid:19) (cid:32) (cid:33)
S k+ 1 −I(k)+ Ke−βk ≤c I(k+1)−I(k)+ Ke−βk ,
2 β 1 β
wherec dependsonlyonn,λandΛ. Utilizing(3.6)withs =k+ 1 ands=k+1,
1 0 2
weobtain that S(k+1) ≤ S(k+ 1)+ Ke−βk. Combining thesetwoinequalities, we
2 β
arriveat
(cid:32) (cid:33)
K
(3.7) S(k+1)−I(k)≤c I(k+1)−I(k)+ e−βk .
1 β
Nowwedefine
K
w (y,s)=S(k)+ e−βk−v(y,s+k) (y∈Rn,s≥0).
k β
Thenby(3.6)and(3.3),w isaspatiallyperiodicnonnegativeviscositysolutionto
k
∂ w =−F(−D2w ,y,s+k)− f(y,s+k) in2Y×(0,1).
s k y k
Notice that the operator −F(−M,y,s) satisfies the same ellipticity condition (2.1).
Hence,wemayinvokeasimilarargumentasaboveandprovethat
(cid:32) (cid:33)
K
(3.8) S(k)−I(k+1)≤c S(k)−S(k+1)+ e−βk .
1 β
Noticethattheconstantc hereisthesameasthatin(3.7).
1
By(3.7)and(3.8),wehave
(3.9) O(k+1)≤ c1−1O(k)+ 2c1K e−βk.
c +1 (c +1)ν
1 1
Iterating(3.9)withrespecttokandusingO(0)=oscRn g≤2K,wearriveat
(3.10) eβ0sO(s)≤c K with0<β <min(cid:18)β,logc1+1(cid:19),
2 0 c −1
1
wherec >0isanotherconstantdependingonlyonn,λ,Λ,βandβ .
2 0
The estimate (3.10) implies that O(s) → 0 as s → ∞. On the other hand, we
knowfrom(3.6)thatbothS(s)andI(s)convergeass → ∞. Combiningthesetwo
observations, we deduce that S(s) and I(s) converge to the same limit, which we
shalldenotebyγ. Then(3.5)followsimmediatelyfrom(3.10). (cid:3)
Remark3.2. TheproofofLemma3.1doesnotinvolvetheperiodicityofFins.
HIGHERORDERCONVERGENCERATES:OSCILLATORYINITIALDATA 9
ByLemma3.1,weareabletodefinev¯ :Rn×[0,T]→Rby
(3.11) v¯(x,t)= limv(x,t,0,s).
s→∞
Thelimitvalueintherighthandsideof(3.11)ispreciselytheuniqueconstantγin
thestatementofLemma3.1. Withv¯athand,(3.5)reads
(3.12) eβ0s|v(x,t,y,s)−v¯(x,t)|≤C,
forany(x,t,y,s)∈Rn×[0,T]×Rn×[0,∞).
OnemaynoticethattheproofofLemma3.1doesnotinvolvetheassumptionson
theconcavityofFinP,theCαregularityofFand f in(y,s)andtheC2,αregularityof
gin y. Assumingtheseconditionsadditionally,weareallowedtousetheinterior
andboundaryC2,α¯ estimates(theso-calledSchaudertheory)forviscositysolutions
(withsome0<α¯ ≤α). Asaresult,weimprovetheestimate(3.12)intermsofC2,α¯
norm.
Lemma 3.3. There exists 0 < α¯ < α, depending only on n, λ, Λ and α, such that
v¯ ∈L∞(Rn×[0,T])andv∈L∞(Rn×[0,T];C2,α¯(Rn×[0,∞)))with
(3.13) (cid:107)v(x,t,·,·)(cid:107)C2,α¯(Rn×[0,∞))+eβ0s(cid:107)v(x,t,·,s)−v¯(x,t)(cid:107)C2,α¯(Rn) ≤C,
forany(x,t,s)∈Rn×[0,T]×[0,∞),whereC>0dependsonlyonn,λ,Λ,β,β andK.
0
Proof. Let us fix (x,t) ∈ Rn ×[0,T] and simply write F(P,y,s), f(y,s), g(y), v(y,s),
andγbyF(P,x,t,y,s), f(x,t,y,s), g(x,y),v(x,t,y,s)and,respectively,v¯(x,t). Letus
denotebyYand2Ythecubes(0,1)nand(0,2)n.
Inviewof(3.3),thefunctionv˜(y,s)=v(y,s)−γisaviscositysolutionto

(3.14) vv˜˜s(y=,0F)(=D2ygv˜(,yy),−s)γ+ f(y,s) oinn22YY.×(0,∞),
α
Since F is uniformly elliptic and concave in P, and since F and f are C while g
isC2,α in(y,s), wemayapplytheboundaryC2,α¯ estimate[W2]to(3.14)forsome
0 < α¯ ≤ α, depending only on n, λ, Λ and α. This yieldsthat v˜ ∈ C2,α¯(Y¯ ×[0,s ])
0
with
(cid:107)v˜(cid:107)C2,α¯(Y¯×[0,s0]) ≤c1(cid:16)(cid:107)v˜(cid:107)L∞(2Y×[0,1))+(cid:119)(cid:119)(cid:119)(cid:119)f(cid:119)(cid:119)(cid:119)(cid:119)Cα(2Y×[0,1))+(cid:119)(cid:119)(cid:119)(cid:119)g(cid:119)(cid:119)(cid:119)(cid:119)C2,α(2Y)(cid:17),
where0 < s ≤ 1 andc > 0dependonlyonn,λ,Λ,αandK. Utilizing(3.5),(3.2)
0 2 1
and(2.6)(withm=0),wederivethat
(3.15) (cid:107)v˜(cid:107)C2,α¯(Y×[0,s0]) ≤c2,
wherec >0isdeterminedonlybyn,λ,Λ,α,β,β andK.
2 0
Nowletusfixanonnegativeintegerkanddefine
v˜ (y,s)=v˜(y,s+k) (y∈Rn,s≥0).
k
Thenfrom(3.14),weknowthatv˜ solves
k
∂ v˜ =F(D2v˜ ,y,s+k)+ f(y,s+k) in2Y×(0,2).
s k y k
Hence,itfollowsfromtheinteriorC2,α¯ estimate(withα¯ beingthesameasthatin
(3.15))thatv˜ ∈C2,α¯(Y¯ ×[s ,s +1])with
k 0 0
(cid:107)v˜k(cid:107)C2,α¯(Y¯×[s0,s0+1]) ≤c3(cid:16)(cid:107)v˜k(cid:107)L∞(2Y×(0,2))+(cid:119)(cid:119)(cid:119)(cid:119)f(cid:119)(cid:119)(cid:119)(cid:119)Cα(2Y×(0,2))(cid:17),
10 SUNGHANKIMANDKI-AHMLEE
where c > 0 depends only on n, λ, Λ, α and K. Utilizing (3.5) and (3.2) (with
3
m=0),wededucethat
(3.16) (cid:107)v˜k(cid:107)C2,α¯(Y¯×[s0,s0+1]) ≤c4e−β0k,
wherec >0isdeterminedonlybyn,λ,Λ,α,β,β andK.
4 0
Iterating (3.16) with respect to k and utilizing (3.15) for the initial case of this
iterationargument,wearriveat(3.13). (cid:3)
LetusremarkthatLemma3.3yieldsthecompactness(in(y,s))ofthesequences
{v(x,t,y,s)}∞ and {v˜(x,t,y,s)}∞ when (x,t) → (x,t). By the stability theory
i i i=1 i i i=1 i i
[CIL]ofviscositysolutions, weobtainthatvandv˜ arecontinuousin(x,t), stated
asbelow. Letusalsopointoutthatthefollowinglemmaisaversionofcontinuous
dependence estimates, and we refer to [JK], [CM] and other literature for more
discussionsinthisregard.
Lemma3.4. Letα¯ betheHo¨lderexponentchoseninLemma3.3. Thenv¯ ∈C(Rn×[0,T])
andv∈C(Rn×[0,T];C2,αˆ(Rn×[0,∞)))forany0<αˆ <α¯.
loc
Proof. As in the proof of Lemma 3.3, we will fix (x,t) ∈ Rn ×[0,T] and continue
withusingthesimplifiednotationforF, f, g,v,γandv˜. Letustakeanysequence
(x,t) → (x,t) as i → ∞. For notational convenience, let us write F(P,y,s) =
i i i
F(P,x,t,y,s), f(y,s)= f(x,t,y,s),g(y)= g(x,y),v(y,s)=v(x,t,y,s),γ =v¯(x,t)
i i i i i i i i i i i i i
andv˜(y,s)=v(y,s)−γ. ByCwedenoteapositiveconstantthatdependsonlyon
i i i
n,λ,Λ,α,β,β andK,andwillletitvaryfromonelinetoanother.
0
Weprovev →vfirst. By(3.13)wehave
i
(cid:107)vi(cid:107)C2,α¯(Rn×[0,∞)) ≤C,
for any i = 1,2,···. Hence, we know from the Arzela-Ascoli theorem that for
any subsequence {w }∞ of {v}∞ , there exist a further subsequence {w }∞ and
j j=1 i i=1 jk k=1
a certain function w ∈ C2,α¯(Rn ×[0,∞)) such that w → w in C2,αˆ(Rn ×[0,∞)) as
jk loc
k→∞,forany0<αˆ <α¯. Onemaynoticethatw solves
jk

∂swjk =Fjk(D2ywjk,y,s)+ fjk(y,s) inRn×(0,∞),
wjk(y,0)= gjk(y) inRn,
intheviscositysense. Duetotheregularityassumptions(2.3),(2.6)and(3.2)onF,
gandrespectively f,weknowthatF → FuniformlyonSn×Rn×[0,∞), g → g
i i
uniformly on Rn and f → f uniformly on Rn ×[0,∞), as i → ∞. Hence, letting
i
k → ∞, we observe from the stability theory [CIL] that the limit function w also
solves

ws =F(D2yw,y,s)+ f(y,s) inRn×(0,∞),
w(y,0)= g(y) onRn,
intheviscositysense. However,theaboveequationisidenticalwiththeequation
(3.3). Sincevistheuniquesolutionto(3.3),wededucethatw=vonRn×[0,∞).
Whatwehaveprovedsofaristhatforanysubsequenceof{v}∞ ,thereexistsa
i i=1
furthersubsequencewhichconvergestov. Thus,v →vasi→∞inC2,αˆ(Rn×[0,∞))
i loc
forany0<αˆ <α¯.