Sharp regularity estimates for second order fully nonlinear parabolic equations

0.18 MB·

by  João Vitor da Silva

SHARP REGULARITY ESTIMATES FOR SECOND ORDER FULLY
NONLINEAR PARABOLIC EQUATIONS
6
1
JOÃOVÍTORDASILVAANDEDUARDOV.TEIXEIRA
0
2

n
a
J ABSTRACT. Weprovesharpregularityestimatesforviscositysolutionsoffullynonlinear
2 parabolicequationsoftheform
2 (Eq) ut−F(D2u,Du,X,t)= f(X,t) in Q1,

] whereF isellipticwithrespecttotheHessianargumentand f ∈Lp,q(Q1). Thequantity
P k (n,p,q):= n+2 determinestowhichregularityregimeasolutionof(Eq)belongs. We
p q
A provethatwhen1<k (n,p,q)<2−eF,solutionsareparabolic-Höldercontinuousfora
sharp,quantitative exponent0<a (n,p,q)<1. Preciselyatthecriticalborderlinecase,
.
h k (n,p,q)=1,weobtainsharpLog-Lipschitzregularityestimates.When0<k (n,p,q)<
at 1,solutionsarelocallyofclassC1+s ,1+2s andinthelimitingcasek (n,p,q)=0,weshow
m C1,Log-LipregularityestimatesprovidedFhas“better”aprioriestimates.
Keywords: Fullynonlinearparabolic equations, optimalborderline estimates, sharp
[
moduliofcontinuity.

1
v 1. INTRODUCTION
9
9 Thestudyofsecondorderparabolicequationsplaysafundamentalroleinthede-
0
velopmentof severalfieldsin pureandappliedmathematics,such asdifferentialgeome-
6
0 try,functionalandharmonicanalysis,infinitedimensionaldynamicalsystems,probability,
. as well as in mechanics, thermodynamics, electromagnetism, among others. The non-
1
0 homogeneousheatequation,
6
(1.1) u −D u= f in Q =B ×(−1,0],
1 t 1 1
v: f ∈Lp(Q1), p> n+22, representsthesimplestlinearprototype. Itsmathematicalanalysis
i goesbackto19thcenturyandtheregularitytheoryforsuchanequationisnowadaysfairly
X
complete. The fully nonlinear parabolic theory is quite more recent. The fundamental
r
a worksofKrylovandSafonov,[10],[11]onlinear,non-divergenceformellipticequations
setthebeginningofthedevelopmentoftheregularitytheoryforviscositysolutionstofully
nonlinearparabolicequations.Sincethenthishasbeenacentralsubjectofresearch.Wang
in[15,16]provesHarnackinequalityandC1+a ,1+2a estimatesforfullynonlinearparabolic
equations,andCrandalletalin[2]developanLp-viscositytheory.Krylovin[6,7]obtains
C2+a ,2+2a estimates for solutions to ut−F(D2u)=0, under convexity assumptions, and
CaffarelliandStefanelliin[1]exhibitsolutionstouniformparabolicequationsthatarenot
C2,1.
Non-divergence form parabolic equations involving sources with mixed integrability
conditions f ∈Lp,q(Q ), as in (Eq) have also been fairly well studied in the literature.
1
ExistenceinsuitableparabolicSobolevspaceshasbeenprovenbyKrylov,see[8,9],see
1991MathematicsSubjectClassification. 35K10,35B65.
1
2 JOÃOVÍTORDASILVAANDEDUARDOV.TEIXEIRA
also the sequenceof worksby Kim [4, 5]. Insofarasregularityestimates are concerned,
only qualitative results are available when p and q are sufficient large. Nonetheless, as
ina numberofphysical, geometricandfreeboundaryproblems,obtaininga quantitative
sharpregularityestimateforsolutionsisdecisiveforarefineanalysis. Hence,thepurpose
thispaperis to obtainsharp moduliofcontinuityto solutionsforsecond orderparabolic
equation(Eq),involvingsourceswithmixednorms,whichdependsonlyondimension, p
andq.
Hereafterwedenoteby
n 2
k =k (n,p,q):= + .
p q
Thefirst quantitativeregularityresult we show states that if 1<k (n,p,q)< n+2, where
nP
n+2 <n <n+1isuniversal,thensolutionsareparabolicallya -Höldercontinuousforthe
2 P
sharpexponenta :=2− n+2 .
p q
Intuitively,ask (n,p,q(cid:16))decre(cid:17)ases,oneshouldexpectthatregularityestimatesofsolu-
tionsimprove. Theborderlineisk (n,p,q)=1,whereweprovethatsolutionsareparabol-
ically Log-Lipschitz continuous. The result is a further quantitative improvementto the
factthatu∈Ca ,a2(Q )forany0<a <1.
loc 1
When 0<k (n,p,q)<1, we showthatsolutionsareC1+b ,1+2b , forb .1− n+2 .
p q
Qualitativeresults,whenp=q>n+1,werepreviouslyobtainedbyCrandalleta(cid:16)l[2]an(cid:17)d
Wang[16].
Finally, we deal with the upper borderline case, f ∈BMO(Q ). Under appropriate
1
higher a priori estimates on F, we show that solutions are C1,Log-Lip(Q ). Particularly,
loc 1
u∈C1+a ,1+2a (Q )forany0<a <1.
loc 1
Thetablebelowprovidesaglobalpictureoftheparabolicregularitytheoryforequations
withanisotropicsources,incomparisonwiththesharpellipticestimatefrom[13]:
f ∈Lp(B ) Regularityofu f ∈Lp,q(Q ) Regularityofu
1 1
n−e ≤p<n C0,2−np(B ) 1< n+2 < n+2 CV,V2(Q )
loc 1 p q nP loc 1
p=n C0,Log-Lip(B ) n+2 =1 C0,Log-Lip(Q )
loc 1 p q loc 1
p>n C1,min a −,1−np (B ) 0< n+2 <1 C1+m ,1+2m (Q )
loc n o 1 p q loc 1
BMO!L¥ C1,Log-Lip(B ) BMO!L¥ C1,Log-Lip(Q )
loc 1 loc 1
EllipticTheory X ParabolicTheory
where V :=2− n+2 and m :=min a −,1− n+2 , and a − means a −e for
p q p q
every0<e . (cid:16) (cid:17) n (cid:16) (cid:17)o
It is interesting to note that the parabolic regularity estimates agree with its elliptic
counterpartprovided f ∈Lp,¥ (Q ).
1
Next picture shows the critical surfaces and the regions they define for the optimal
regularityestimatesavailableforsolutionsto(Eq).
SHARPREGULARITYESTIMATESFORSECONDORDERFULLYNONLINEARPARABOLICEQUATIONS 3
q u
0< n+2 <1
p q
1< n+2 < n+2 n+2 =1
p q nP p q
2
n+2 = n+2
p q nP
1
n2 n pu
FIGURE 1. Criticalsurfacesforoptimalregularityestimates.
Acknowledgement. This article is part of the first author’s Ph.D thesis. He would like
tothanktheDepartmentofMathematicsatUniversidadeFederaldoCearáforfosteringa
pleasantandproductivescientificatmosphere,whichhasbenefitedalotthefinaloutcome
ofthiscurrentproject.ThisworkhasbeenpartiallysupportedbyCapesandCNPq,Brazil.
2. DEFINITIONS AND PRELIMINARY RESULTS
ThroughoutthispaperF: Sym(n)×Rn×B (0)×(−1,0]−→Risafullynonlinear
1
uniformlyellipticoperatorwithrespecttotheHessianargumentandLipschitzwithrespect
to gradientdependence. That is, there are constants L ≥l >0 and u ≥0 such that for
allZ,W ∈Rn andM,N ∈Sym(n),spaceofn×nsymmetricmatrices,withM≥N,there
holds
(2.1)
M− (M−N)−u |Z−W|≤F(M,Z,X,t)−F(N,W,X,t)≤M+ (M−N)+u |Z−W|.
l ,L l ,L
Hereafter,M± denotethePucci’sextremaloperators:
l ,L
M+ (M):=l · (cid:229) e +L · (cid:229) e and M− (M):=l · (cid:229) e +L · (cid:229) e
l ,L i i l ,L i i
ei<0 ei>0 ei>0 ei<0
where{e :1≤i≤n}aretheeigenvaluesofM. Wecan(andwill)alwaysassumethatF is
i
normalizedasF(0,0,X,t)=0. AnyoperatorF whichsatisfiesthecondition(2.1)willbe
referredinthisarticleasa(l ,L ,u )-parabolicoperator. Followingclassicalterminology,
anyconstantormathematicaltermwhichdependsonlyondimensionandoftheparabolic
parametersl ,L andu willbecalleduniversal.
4 JOÃOVÍTORDASILVAANDEDUARDOV.TEIXEIRA
Equationsandproblemsstudiedherearedesignedinthe(n+1)-dimensionalEuclidean
space,Rn+1. Thesemi-opencylinderisdenotedbyQ (X ,t )=B (X )×(t −r2,t ]. For
r 0 r 0
simplicitywereferQ (0,0)=Q . TheparabolicdistancebetweenthepointsP =(X ,t )
1 1 1 1 1
andP =(X ,t )∈Q isdefinedby
2 2 2 1
d (P ,P):= |X −X |2+|t −t |.
par 1 2 1 2 1 2
q
Forafunctionu: Q →Rthesemi-normandnormfortheparabolicHölderspaceare
1
definedrespectivelyby
(2.2)
|u(X,t)−u(Y,s)|
[u]Ca ,a2(Q1):=(X,t)s,(uY,ps)∈Q1 dpar((X,t),(Y,s))a and kukCa ,a2(Q1):=kukC0(Q1)+[u]Ca ,a2(Q1).
(X,t)6=(Y,s)
Underfinitenessofsuchanormoneconcludesthatuisa -Hödercontinuouswithrespect
tothespatialvariablesand a2−Höderwithrespecttothetemporalvariable.C1+a ,1+2a (Q1)
isthespaceofuwhosespacialgradientDu(X,t)thereexistsintheclassicalsenseforevery
(X,t)∈Q andsuchthat
1
kukC1+a ,1+2a (Q1) := kukL¥ (Q1)+|uk(XD,utk)L−¥ (uQ(1Y),t )−Du(X,t)·(X−Y)|
+ sup
d1+a ((X,t),(Y,s))
(X,t),(Y,s)∈Q1 par
(X,t)6=(Y,s)
is finite. It is easy to verify that u∈C1+a ,1+2a (Q1) implies every component of Du is
C0,a (Q ),anduis 1+a −Höldercontinuousinthevariablet,seeforinstance[2].
1 2
A function u belongs to the Sobolev space W2,1,p(Q ) if it satisfies u,Du,D2u,u ∈
1 t
Lp(Q ). Thecorrespondingnormisgivenby
1
kukW2,1,p(Q1)=[kukLpp(Q1)+kutkLpp(Q1)+kDukLpp(Q1)+kD2ukLpp(Q1)]1p
ItfollowsbySobolevembeddingthatif p> n+2 thenW2,1,p(Q )iscontinuouslyembed-
2 1
ded inC0(Q ). Also, u∈W2,1,p(Q ) implies that u is twice parabolically differentiable
1 loc 1
a.e.,seeformoredetails[2].
Definition2.1(LP-viscositysolutions). LetG: Sym(n)×Rn×R×B (0)×(−1,0]→Rbe
1
auniformlyellipticoperator,P> n+2 and f ∈LP (Q ). Wesaythatafunctionu∈C0(Q )
2 loc 1 1
isanLP-viscositysubsolution(respectivelysupersolution)to
(2.3) u −G(D2u(X,t),Du(x,t),u(X,t),X,t)= f(X,t) in Q
t 1
ifforallj ∈W2,1,P(Q )whenevere >0andO⊂Q isanopenand
loc 1 1
j −G(D2j (X,t),Dj (x,t),j (X,t),X,t)−f(X,t)≥e (resp.≤−e ) a.e. in O
t
then u−j cannot attains a local maximum (resp. minimum) in O. In an equivalent
manner, u is an LP−viscosity subsolution (resp. supersolution) if for all test function
j ∈W1,2,P(Q )and(X ,t )∈Q atwhichu−j attainalocalmaximum(resp.minimum)
loc 1 0 0 1
onehas
SHARPREGULARITYESTIMATESFORSECONDORDERFULLYNONLINEARPARABOLICEQUATIONS 5
essliminf [j −G(D2j (X,t),Dj (x,t),j (X,t),X,t)−f(X,t)]≤0
t
(X,t)→(X0,t0)
esslimsup [j −G(D2j (X,t),Dj (x,t),j (X,t),X,t)−f(X,t)]≥0
t
(X,t)→(X0,t0)
FinallywesaythatuisanLP-viscositysolutionto(2.3)ifitisbothanLP-viscositysuper-
solutionandanLP-viscositysubsolution.
According to [2] and [15] for a fixed (X ,t )∈Q , we measure the oscillation of the
0 1
coefficientsofF around(X ,t )bythequantity
0
|F(M,0,X,t)−F(M,0,X ,t )|
(2.4) Q (X ,t ,X,t):= sup 0 .
F 0
kMk+1
M∈Sym(n)
Fornotationpurposes,weshalloftenwriteQ (0,0,X,t)=Q (X,t).
F F
Werecallthatafunction f issaidtobelongtotheanisotropicLebesguespace,Lp,q(Q )
1
if
0 qp 1q
kfkLp,q(Q1):= ˆ ˆ |f(X,t)|pdX dt =kkf(·,t)kLp(B1)kLq((−1,0])<+¥ .
−1(cid:18) B1 (cid:19) !
This is a Banach space when endowed with the norm above. When p=q, this is the
standarddefinitionofLp spaces. Thedefinitionarenaturallyextendedwheneither porq
areinfinity.ItisplaintoverifythatLp,q(Q )⊂Ls(Q )fors:=min{p,q}.
1 1
Werecalltheexistenceofaconstantn ,satisfying n+2 ≤n <n+1,forwhichHarnack
P 2 P
inequality holds for LP-viscosity solutions, provided P>n , see for instance [2]. The
P
followingcompactnessresultbecomesthenavailable:
Proposition2.2(Compactnessofsolutions). Letu∈C0(Q )beanLP-viscositysolution
1
to(Eq)undertheassumptionP≥min{p,q}>nP. ThenuislocallyofclassCb ,b2 forsome
0<b <1and
kukCb,b2(Qr)≤C(n,l ,L )r−b (cid:18)kukL¥ (Q2r)+r2−(cid:16)np+q2(cid:17)kfkLp,q(Q2r)(cid:19).
Inthesequel,weobtainaLemmawhichprovidesatangentialpathtowardtheregularity
theoryavailableforconstantcoefficient,homogeneousF-caloricfunctions.
Lemma2.3(F-caloricapproximationLemma). Letu∈C0(Q )beanLP-viscositysolution
1
to(Eq)with|u|≤1and f ∈Lp,q(Q )withP:=min{p,q}>n . Givend >0,thereexists
1 P
h =h (n,L ,l ,d )>0suchthatif
1
P
max ˆ Q PF(X,t) ,kfkLp,q(Q1),u ≤h ,
((cid:18)Q1 (cid:19) )
thenwecanfindafunctionh: Q →Randa(l ,L ,0)−parabolic,constantcoefficients
1/2
operatorF: Sym(n)→R,suchthat
(2.5) h −F(D2h)=0, in Q
t 1/2
intheLP-viscositysense,and,moreover
6 JOÃOVÍTORDASILVAANDEDUARDOV.TEIXEIRA
(2.6) sup |(u−h)(X,t)|≤d .
(X,t)∈Q1/2
Proof. Theproofisbasedonacontradictionargument.Supposethatthereexistsad >0
0
forwhichthethesisofLemma2.6isnottrue.Thatis,wecouldfindasequenceoffunctions
(u ) ,with|u |≤1inQ ,asequenceof(l ,L ,u )-operatorsF : Sym(n)×Rn×Q →R
j j≥1 j 1 j j 1
andasequenceoffunctions(f ) satisfying
j j≥1
(2.7) (u ) −F(D2u ,Du ,X,t)= f (X,t) in Q
j t j j j j 1
intheLP-viscositysense,with
1
P
(2.8) max ˆ Q PFj(X,t) ,kfjkLp,q(Q1),u j =o(1) as j→¥ ,
((cid:18)Q1 (cid:19) )
however
(2.9) sup |(u −h)(X,t)|>d
j 0
(X,t)∈Q1/2
forallhwhichsatisfies(2.5)andall(l ,L ,0)-operatorF. ByHölderregularityofthese-
quence(u ) , Proposition2.2, we mayassume, passingto a subsequenceif necessary,
j j≥1
that u →u locally uniformly in Q . Furthermore, it follows from structural condition
j 0 1
(2.1)ofthesequenceofoperators(F) thatF(M,Z,X,t)→F (M,Z,X,t)locallyuni-
j j≥1 j 0
formlyinthespaceSym(n)×Rnforeach(X,t)∈Q fixed.Moreover,byhypothesis(2.8),
1
F isa(l ,L ,0)constantcoefficientsoperator,seeforinstance[2]and[15]. Toconclude
0
theproof,weusestabilityarguments,see[2,Section6],astodeducethat
(u ) −F (D2u )=0 in Q ,
0 t 0 0 1/2
intheLP-viscositysense,Thisgivesacontradictionto(2.9)to j≫1andtheproofofthe
Lemmaisconcluded. (cid:3)
Weconcludethissectionbycommentingonreductionprocessestobeusedthroughout
theproof.
Remark2.4. [Preservingellipticity]IfF isa(l ,L ,u )-parabolicoperatorthen
−→
−→ M Z
G(M, Z,X,t)=k 2F , X,t
k 2 k
!
isa(l ,L ,ku )-parabolicoperator.
Remark2.5. [Normalizationandscaling]We canalwayssuppose,withoutlossofgener-
ality,thatviscositysolutionsof
u −F(D2u,Du,X,t)= f(X,t) in Q
t 1
skaftkisLfpy,q(kQu1k)L<¥ (2Qe1)0.≤In1f.acAt,lsfoorgkive:=n ae0skmukaL¥ll(Qn1u)e+0mkbfkeLrp,eq(0Q1>) a0n,dwRe>camnaaxlso1s,ueup0po,sdeefithnaitngu +
v(X,t):=k u(R−1X,R−2t) (cid:8) (cid:9)
wereadilyverifythat
SHARPREGULARITYESTIMATESFORSECONDORDERFULLYNONLINEARPARABOLICEQUATIONS 7
(1) kvkL¥ (Q1)≤1;
(2) v −G(D2v,Dv,X,t)=g(X,t)inQ ,intheLP-viscositysense,where
t 1
−→ k R2 R−→ k
G(M, Z,X,t)= F M, Z,R−1X,R−2t and g(X,t)= f(R−1X,R−2t);
R2 k k R2
(cid:18) (cid:19)
(3) Gisa(l ,L ,u )-parabolicoperator,withu <e ;
0
(4) kgkLp,q(Q1)≤e0.
3. OPTIMALCa ,a2 REGULARITY
Our strategy for proving optimalCa ,a2 regularity estimates is based on a refined
compactness method as in [2, 13, 15, 16]. It relies on a control of decay of oscillation
based on the regularity theory available for a nice limiting equation. Roughly speaking
the geometric tangential analysis of the limit arising from of family of fully nonlinear
parabolicoperatorsF aswe areinsmallestregimeonthe sourcetermandonoscillation
i
of coefficients of the respective operators. Next lemma is the key access point for the
approach,asitprovidesthefirststepintheiterationprocesstobeimplemented.
Lemma3.1. Letu∈C0(Q )beanormalizedLP-viscositysolutionfor(Eq),thatis,|u|≤1
1
inQ . Given0<g <1,thereexisth (L ,l ,n,g )>0and0<r (L ,l ,n,g )≪ 1,suchthat
1 2
if
1
P n 2 n+2
max ˆ Q PF(X,t) ,kfkLp,q(Q1),u ≤h with 1< p+q < n
((cid:18)Q1 (cid:19) ) P
then,forsomeV ∈R,with|V |≤C(L ,l ,n)thereholds
(3.1) sup|u−V |≤r g.
Qr
Proof. For a d >0 to be chosen a posteriori, let h be a solution to a homogeneousuni-
formlyparabolicequationwithconstantcoefficients, thatisd -closeto uintheL¥ -norm,
i.e.,
(3.2) h −F(D2h)=0 in Q and sup|(u−h)(X,t)|≤d .
t 1
Q
1/2
Lemma2.3assurestheexistenceofsuchafunction.Onceourchoiceford ,tobesetofthe
endofthisproof,isuniversal,thenthechoiceofh (n,l ,L ,d ) istoouniversal. Fromthe
regularitytheoryavailableforh,seeforinstance[2]or[16],wecanestimate
1
(3.3) |h(X,t)−h(0,0)|≤C(n,l ,L )d ((X,t),(0,0)) ∀ |X|2+|t|< ,
par
3
andalso,
(3.4) |h(0,0)|≤C.
ForV =h(0,0)itfollowsfromequations(3.2)and(3.3)viatriangularinequalitythat
(3.5) sup|u(X,t)−V |≤d +C(n,l ,L )r .
Qr
8 JOÃOVÍTORDASILVAANDEDUARDOV.TEIXEIRA
Wemakethefollowinguniversalselections:
1
(3.6) r := 1 1−g and d := 1r g
2C 2
(cid:18) (cid:19)
whereC >0 is a universal constant from equation (3.3). Let us stress that the choices
abovedependonlyupondimension, ellipticity parametersand the fixed exponent. From
theabovechoicesweobtain
sup|u(X,t)−V | ≤r g
.
Qr
andtheLemmaisconcluded. (cid:3)
Theorem3.2. Letu∈C0(Q )beanLP-viscositysolutionof (Eq)with f ∈Lp,q(Q )with
1 1
n 2 n+2
1< + < .
p q n
P
Thereexistsauniversalconstantq >0suchthatif
0
1
P
sup Q P(Y,t ,X,t) ≤q ,
ˆ F 0
(Y,t)∈Q1/2(cid:18)Q1 (cid:19)
then,forauniversalconstantC>0anda :=2− n+2 ,thereholds
p q
(cid:16) (cid:17)
kukCa ,a2(Q1/2)≤C{kukL¥ (Q1)+kfkLp,q(Q1)}.
Proof. Through normalization and scaling processes, see Remark 2.5, we can suppose
withoutlosinggeneralitythat|u|≤1andkfkLp,q(Q1)≤h ,whereh istheuniversalconstant
from Lemma 3.1 when we set g =x (n,p,q)=2− n+2 . Once selected q =h the
p q 0
goalwillbetoiteratetheLemma3.1. Forafixed(Y,(cid:16)t )∈Q(cid:17)1/2 weclaimthatthereexists
aconvergentsequenceofrealnumbers{V } ,suchthat
k k≥1
(3.7) sup |u(X,t)−V |≤r kx (n,p,q)
k
Qrk(Y,t)
wheretheradius0<r ≪ 1 isgivenbyLemma3.1,upontheselectionofg asabove.
2
The proofof (3.7) will follow by inductionprocess. Lemma3.1givesthe first step of
induction,k=1. Nowsupposeverifiedthekth stepin(3.7). Wedefine
u(Y+r kX,t +r 2kt)
v (X,t)=
k r kx (n,p,q)
and
1 1
F(M,Z,X,t):=r k[2−x (n,p,q)]F M, Z,Y+r kX,t +r 2kt .
k r k[2−x (n,p,q)] r k[1−x (n,p,q)]
(cid:18) (cid:19)
Ascommentedbefore,seeRemark2.4,F is(l ,L ,u )-parabolicoperator,moreoverbythe
k
inductionhypothesis,|v |≤1and
k
(v ) −F(D2v ,Dv ,X,t)=r k.[2−x (n,p,q)]f(Y+r kX,t +r 2kt)=: f (X,t),
k t k k k k
SHARPREGULARITYESTIMATESFORSECONDORDERFULLYNONLINEARPARABOLICEQUATIONS 9
intheLP-viscositysense. Oneeasilycomputes,
t qp q1
kfkkLp,q(Q1) = r k(2−x (n,p,q))r −k(cid:16)np+q2(cid:17)ˆt−r 2k ˆBrk(Y)|f(Z,s)|pdZ! ds .
 
Duetothesharpchoiceofx (n,p,q)=2− n+2 ,wehavethat
p q
kfkkLp,q(Q1)=kfkLp,q(Brk(Y(cid:16))×(t−r 2(cid:17)k,t])≤kfkLp,q(Q1)≤h ,
aswellas
1 1
P P
Q P(X,t) ≤ Q P(X,t) ≤h .
ˆ Fk ˆ F
(cid:18)Q1 (cid:19) (cid:18)Q1 (cid:19)
Inconclusion,weareallowedtoemployedLemma3.1tov ,whichprovidestheexistence
k
ofauniversallyboundedrealnumberV with|V |≤C,suchthat
k k
(3.8) sup|v −V |≤r x (n,p,q).
k k
Qr
Finally,ifweselect
(3.9) V :=V +r kx (n,p,q)V
k+1 k k
and rescale (3.8) back to the unit picture, we obtain the (k+1)th step in the induction
process(3.7). Inaddition,wehavethat
(3.10) |V −V |≤Cr kx (n,p,q),
k+1 k
andhencethesequence{V } isCauchy,andsoitconverges. From(3.7)V →u(Y,t ).
k k≥1 k
Aswellasfrom(3.10)itfollowsthat
(3.11) |u(Y,t )−V |≤ C r kx (n,p,q),
k 1−r x (n,p,q)
Finally,for0<r<r ,letkthesmallestintegersuchthat(X,t)∈Qr k(Y,t )\Qr k+1(Y,t ).
Itfollowsfrom(3.7)and(3.11)that
|u(X,t)−u(Y,t )| |u(X,t)−V |+|u(Y,t )−V |
k k
sup ≤ sup
Qr(Y,t)dpar((X,t),(Y,t ))x (n,p,q) Qr(Y,t) dpar((X,t),(Y,t ))x (n,p,q)
C r kx (n,p,q)
≤ 1+ sup
(cid:18) 1−r x (n,p,q)(cid:19)Qr(Y,t)dpar((X,t),(Y,t ))x (n,p,q)
C 1
≤ 1+ .
1−r x (n,p,q) r x (n,p,q)
(cid:18) (cid:19)
Thelastestimateprovides
kukCx(n,p,q),x(n2,p,q)(Q1/2)≤C
andhencetheproofofTheoremisconcluded. (cid:3)
Remark3.3. TheexponentofHölderregularityofourresultissharp.Thisiscanbeverify
throughoffollowingexamplefrom[14]:Letu∈C ((−1,0];L2 (B ))∩L2 ((−1,0];W1,2(B ))
loc loc 1 loc loc 1
beaweaksolutionto
u −D u= f in Q
t 1
Supposethat1< n+2<2thenfora :=2− 2+n wehavethatu∈Ca ,a2(Q ). Remark
p q p q loc 1
thatinthiscasen = n+2. (cid:16) (cid:17)
P 2
10 JOÃOVÍTORDASILVAANDEDUARDOV.TEIXEIRA
Remark3.4. UnderVMOassumptionofthecoefficientsoftheoperatorF:
Q P(X,t)=o(1),
ˆ F
Qr
asr→0,Theorem3.2holdswithoutthesmallnessoscillationcondition,asitcanalways
beassumeduponanappropriatescaling.
Remark3.5. Undernoassumptionsonthecoefficients,ratherthanellipticity,adjustments
in the proof of previous Theorem yieldsCa ,a2(Q ) where a :=min b −,2− n+2
loc 1 p q
where0<b <1isthemaximalexponentfromPreposition2.2. n (cid:16) (cid:17)o
4. PARABOLIC LOG-LIPSCHITZ TYPEESTIMATES
Inthissectionweaddressthequestionoffindingtheoptimalanduniversalmodulus
ofcontinuityforsolutionsofuniformlyparabolicequationsoftheform(Eq)whoseright
handsideliesintheborderlinespaceLp,q(Q ),when pandqlieonthecriticalsurface:
1
n 2
+ =1.
p q
Such estimate is particularlyimportantto the generaltheoryof fully nonlinearparabolic
equations.Throughasimpleanalysisoneverifiesthatsolutionsof(Eq),withsourcesunder
theaboveborderlineintegrabilityconditionshouldbeasymptoticallyLipschitzcontinuous.
Indeed,as n+2 →1+,solutionsareparabolicallyHöldercontinuousforeveryexponent
p q
0<a <1. The key goal in this section is to obtain the sharp, quantitative modulus of
continuityforu.
Lemma 4.1. Let u∈C0(Q ) be a normalized LP-viscosity solution to (Eq). There exist
1
h (L ,l ,n)>0and0<r (L ,l ,n)≪ 1,suchthatif
2
1
P
(4.1) max ˆ Q PF(X,t) ,kfkLp,q(Q1),u ≤h
((cid:18)Q1 (cid:19) )
underthecondition n+2 =1,then,wecanfindanaffinefunctionL(X,t):=A+hB,Xi,
p q
withuniversallyboundedcoefficients,|A|+|B|≤C(l ,L ,n),suchthat
(4.2) sup|(u−L)(X,t)|≤r .
Qr
Proof. For a d >0 which will be chosen a posteriori, we apply Lemma 2.3 and find a
functionh: Q →Rsatisfying
1
2
h −F(D2h)=0 in Q ,
t 1
2
intheLP-viscositysensesuchthat
(4.3) sup|(u−h)(X,t)|≤d .
Q
1/2
Wenowdefine
(4.4) L(X,t)=h(0,0)+hDh(0,0),Xi,