Cordes-Nirenberg type estimates for nonlocal parabolic equations

0.28 MB·English

by  Yong-Cheol Kim

CORDES-NIRENBERG TYPE ESTIMATES FOR NONLOCAL
PARABOLIC EQUATIONS
3
1 YONG-CHEOLKIMANDKI-AHMLEE
0
2
Abstract. Inthispaper,weobtainCordes-Nirenbergtypeestimatesfornon-

n localparabolicequationsonthemoreflexiblesolutionspaceL∞T (L1ω)thanthe
a classicalsolutionspaceB(RnT)consistingofallboundedfunctionsonRnT.
J

0
1

] Contents
A
1. Introduction 1
C
1.1. Nonlocal parabolic equations. 1
.
h 1.2. Outline 2
t
a 1.3. Notations 3
m 2. Preliminaries 3
[ 3. Boundary estimates and Global estimates 6

4. Some results by approximation 8
2
5. C1,α-regularityfornonlocalparabolicequationswithvariablecoefficients 14
v
1 6. Cordes-Nirenberg type estimates and Applications 17
9 References 22
5
5
.
2
1 1. Introduction
2
1 1.1. Nonlocal parabolic equations. In this paper, we study Cordes-Nirenberg
: type estimates on the more flexible solution space L∞(L1) for nonlocal parabolic
v T ω
integro-differential equations. In [KL], we obtained interior C1,α-estimates on the
i
X solution space B(Rn) for nonlocal parabolic translation-invariant equations, and
T
r also the reader can refer to [CS1] and [CS2] for the elliptic case.
a
Throughout this paper, we consider the purely nonlocal parabolic Isaacs equa-
tions of the form
Iu(x,t)−∂ u(x,t):= inf sup(L u(x,t)−∂ u(x,t))
t ab t
a∈Ab∈B
c (x,y,t)
(1.1) = inf sup µ (u,x,y)(2−σ) ab dy−∂ u(x,t)
a∈Ab∈B(cid:18)ZRn t |y|n+σ t (cid:19)
=f(x,t) in Ω×(−τ,0]:=Ω , 0<τ ≤T,
τ
whereΩ is a boundeddomainin Rn, µ (u,x,y)=u(x+y,t)+u(x−y,t)−2u(x,t)
t
and {c } is a family of nonnegative functions with indexes a and b in
ab (a,b)∈A×B
arbitrary sets A and B, respectively. We call such L a linear integro-differential
ab
operator and also we simply write L without indices.
2000Mathematics SubjectClassification: 47G20, 45K05,35J60,35B65,35D10(60J75) .
1
2 YONG-CHEOLKIMANDKI-AHMLEE
We denote by ω (y) = 1/(1+|y|n+σ) for σ ∈ (0,2) and we write ω := ω for
σ σ0
some σ ∈ (1,2). Let F denote the family of all real-valued functions defined on
0
RnT :=Rn×(−T,0]. For u∈F, we define the mixed norm kukL∞T(L1ω) by
kukL∞(L1) := sup |u(x,t)|ω(x)dx.
T ω t∈(−T,0]ZRn
Also we denote by L∞T (L1ω)={u∈F:kukL∞T(L1ω) <∞}.
A mapping I:F→F given by u7→Iu is called a nonlocal parabolic operator if
(a) Iu(x,t) is well-defined for any u∈C2(x,t)∩L∞(L1),
x T ω
(b) Iu is continuous on Ω ⊂Rn, whenever u∈C2(Ω )∩L∞(L1).
τ T x τ T ω
We say that a nonlocal operator I is uniformly elliptic with respect to a family L
of linear integro-differential operators if
(1.2) M−v(x,t)≤I(u+v)(x,t)−Iu(x,t)≤M+v(x,t)
L L
where M−Lv(x,t):=infL∈LLv(x,t) and M+Lv(x,t):=supL∈LLv(x,t).
We say that an operator L belongs to L if its corresponding kernel K ∈ K
0 0
satisfies the uniform ellipticity assumption
λ Λ
(1.3) (2−σ) ≤K(x,y,t)≤(2−σ) , 0<σ <2.
|y|n+σ |y|n+σ
We consider the corresponding maximal and minimal operators
Λµ (u,x,y)+−λµ (u,x,y)−
M+ u(x,t)= sup Lu(x,t)= t t dy,
L0 L∈L0 ZRn |y|n+σ
λµ (u,x,y)+−Λµ (u,x,y)−
M− u(x,t)= inf Lu(x,t)= t t dy.
L0 L∈L0 ZRn |y|n+σ
In the final section, we shall consider some of the most interesting applications
as follows;
• if 0<λ ≤c ≤Λ and |∇ c |≤C/|y|, and c (x,y,t) is continuous in (x,t)
ab y ab ab
for a modulus of continuity independent of y, then there is some ε > 0 (with an
estimate depending on kukL∞(L1)) such that a solution u of (1.1) is C1,ε.
T ω
• if 0 <λ≤c ≤ Λ and |∇ c |≤C/|y|, c is constant in (x,t), and Λ−δ ≤
ab y ab ab
1 ≤ λ+δ for a small enough δ > 0, then there is some ε > 0 (with an estimate
depending on kukL∞(L1)) such that the solution u of (1.1) is C2,ε.
T ω
1.2. Outline. InSection2,weintroducethenotionofviscositysolutionsfornonlo-
cal parabolic equations and obtain Ho¨lder regularities and interior C1,α-estimates
of such viscosity solutions by applying the result of [KL] (refer to [CS1] for the
elliptic case). In Section 3, we get boundary estimates and global estimates by
certainparabolicadaptationofthebarrierfunctionwhichwasusedin[CS1]forthe
elliptic case. In Section 4, we establish stability properties of viscosity solutions
and it was proved that if two nonlocal parabolic equations are very close to each
otherincertainsense,thensoarethosesolutions. Theparaboliccasehastimeshift
contrary to the elliptic case, and so this obstacle shall be overcome in this section.
In Section 5, we obtain C1,α-regularity for nonlocal parabolic equations with vari-
ablecoefficients. Finally,inSection6,wefurnishaparabolicversionoftheintegral
Cordes-NirenbergtypeestimatesandvariousapplicationsincludingC2,α-regularity
for nonlocal parabolic equations.
CORDES-NIRENBERG TYPE ESTIMATES 3
1.3. Notations. We write the notations briefly for the reader.
(i) B =B (0) and Rn =Rn×(−T,0] for r >0 and T >0.
r r T
(ii) Q =B ×(−rσ,0] and Q (x,t)=Q +(x,t) for r >0 and (x,t)∈Rn.
r r r r T
(iii) ∂ Ω :=∂ Ω ∪∂ Ω :=∂Ω×(−rσ,0]∪Ω×{−rσ} for a bounded domain
p τ x τ b τ
Ω⊂Rn and τ ∈(0,T].
(iv) For (x,t),(y,s)∈Rn, we define the parabolic distance d by
T
d((x,t),(y,s))=(|x−y|σ+|t−s|)1/σ.
(v) For a,b∈R, we denote by a∨b=max{a,b} and a∧b=min{a,b}.
2. Preliminaries
In this paper, we always impose the following assumptions on ω;
(2.1) 1+|y|∈L1,
ω
(2.2) sup ω ≤C ω(y).
r
Br(y)
Theuniformellipticity(1.2)dependsonaclassLoflinear integro-differential oper-
ators. Such an operator L in L is of the form Lu(x,t)= µ (u,x,y)K(x,y,t)dy
Rn t
for a nonnegative symmetric kernel K satisfying
R
sup (1∧|y|2)K(x,y,t)dy ≤C <∞.
(x,t)∈RnTZRn
Here the symmetric property means that for each (x,t) ∈ Rn, K(x,−y,t) =
T
K(x,y,t) for all y ∈Rn.
Let Π be a (possibly unbounded) region in Rn. Then we say that a function
u:Rn →R is Lipschitz in space on Π×(−1,0] ( we write u∈C0,1(Π×(−1,0]) ),
T x
if there is some constant C >0 (independent of x,y) such that
eeqq--xxlliipp (2.3) sup |u(x,t)−u(y,t)|≤C|x−y|
t∈(−1,0]
for any x,y ∈Π. We denote by [u] the smallest C satisfying (2.3).
Cx0,1(Π×(−1,0])
We say that a function u:Rn →R is in C1,1(Q(x,t)) for (x,t)∈Rn, if there is
T T
a constant M >0 (independent of (y,s)) such that
(2.4) |u(y,s)−u(x,s)−(y−x)·∇ u(x,s)|≤M|x−y|2
x
for all (y,s) ∈ Q(x,t). We denote by the norm kuk the smallest M
C1,1(Q(x,t))
satisfying (2.4).
The followingdefinitionis the parabolicsetting ofthat[CS1]ofthe elliptic case.
Definition 2.1. For a nonlocal parabolic operator I and τ ∈ (0,T], we define kIk
in Ω with respect to some weight ω as
τ
|Iu(x,t)|
kIk= sup
((x,t),u)∈FΩǫτ 1+kukL∞T(L1ω)+kukC1,1(Qǫ(x,t))
where FΩǫτ = {((x,t),u) ∈ Ωτ ×Cx2(x,t) : kukL∞T(L1ω) +kukC1,1(Qǫ(x,t)) < ∞} for
some ǫ>0.
Assumption 2.2. If KL :=supαKα is the supremum of all kernels corresponding
to operators in the class L, then for each r >0 there is a constant C >0 such that
r
sup(x,t)∈Rn KL(x,y,t)≤Crω(y) for any y ∈Rn\Br.
T
4 YONG-CHEOLKIMANDKI-AHMLEE
Assumption 2.3. There is some C >0 such that sup kLk≤C <∞.
L∈L
Remark. Assumption 2.3 implies that kM+k≤C and kM−k≤C.
L L
Definition 2.4. Let I be a uniformly elliptic operator in the sense of (1.2) with
respect to some class L and let f : Rn → R be a continuous function. Then a
T
function u ∈ F is upper (lower) semicontinuous on Ω × J where J := (a,b) ⊂
(−T,0] is said to be a viscositysubsolution(viscositysupersolution) of an equation
∂ u−Iu=f on Ω×J and we write ∂ u−Iu≤f (∂ u−Iu≥f) on Ω×J in the
t t t
viscositysense,ifforeach(x,t)∈Ω×J thereissomeneighborhood Q (x,t)⊂Ω×J
r
of (x,t) such that ∂ ϕ(x,t)−Iv(x,t) ≤ f(x,t) ( ∂ ϕ(x,t)−Iv(x,t) ≥ f(x,t) ) for
t t
v = ϕ1Qr(x,t)+u1RnT\Qr(x,t) whenever ϕ ∈ C2(Qr(x,t)) with ϕ(x,t) = u(x,t) and
ϕ > u ( ϕ < u ) on Q (x,t) \{(x,t)} exists. Also a function u is called as a
r
viscosity solution if it is both a viscosity subsolution and a viscosity supersolution
to ∂ u−Iu=f on Ω×J.
t
Throughout this paper, we denote η ∈ (0,1) by any very small number and we
set ǫ=η/2.
Theorem 2.5. Let σ ∈ (σ ,2) for some σ ∈ (0,2). If u ∈ L∞(L1) is a function
0 0 T ω
satisfying
M+ u−∂ u≥−C and M− u−∂ u≤C in Q ,
L t 0 L t 0 1+η
0 0
then there are some α>0 and C >0 (depending only on λ,Λ,n,η and σ , but not
0
on σ) such that
kukCα(Q1+ǫ) ≤C Qsu1+pη|u|+kukL∞T(L1ω)+C0 .
(cid:0) (cid:1)
Proof. WenotethatuiscontinuousonQ1+η. Setv =u1Q1+η andw=u1RnT\Q1+η.
Since u=v+w and ∂ w ≡0 on Q , we have that
t 1+ǫ
(M+ v−∂ v)+M+ w ≥M+ u−∂ u≥−C in Q ,
L t L L t 0 1+ǫ
0 0 0
(M− v−∂ v)+M− w ≤M− u−∂ u≤C in Q .
L t L L t 0 1+ǫ
0 0 0
So it suffices to show that if (x,t) ∈ Q1+ǫ, then |Lw(x,t)| ≤ CkukL∞(L1) for any
T ω
L∈L , i.e. we have only to show that if (x,t)∈Q , then
0 1+ǫ
u(y,t)K(x,±y,t)dy ≤CkukL∞(L1)
T ω
(cid:12)Z|y|≥1+η (cid:12)
(cid:12) (cid:12)
for any L ∈ L . (cid:12)Indeed, we note that |y| >(cid:12) 2(1+η)|x| for any x ∈ B and
0 (cid:12) (cid:12) 2+η 1+ǫ
y ∈Rn\B , and so |x±y|≥|y|−|x|≥ η |y|. Thus we have the estimate
1+η 2(1+η)
|u(y,t)|
u(y,t)K(x,±y,t)dy ≤C |x±y|n+σ dy ≤CkukL∞T(L1ω).
(cid:12)Z|y|≥1+η (cid:12) Z|y|≥1+η
(cid:12) (cid:12)
This(cid:12)implies that (cid:12)
(cid:12) (cid:12)
M+L v ≥−C0−CkukL∞(L1) and M−L v ≤C0+CkukL∞(L1) in Q1+ǫ.
0 T ω 0 T ω
Hence we complete the proof by applying Theorem 5.1.2 [KL] to v. (cid:3)
We define the class L of operators L with kernels K ∈K satisfying (1.3) such
∗ ∗
that there are some ̺ >0 and a constant C >0 such that
0
(2.5) sup |∇ K(x,y,t)|≤Cω(y) for any y ∈Rn\B .
y ̺0
(x,t)∈Rn
T
CORDES-NIRENBERG TYPE ESTIMATES 5
Theorem 2.6. Let σ ∈ [σ ,2) for some σ ∈ (0,2). Then there is some ̺ > 0
0 0 0
(depending on λ,Λ,σ and n) so that if I is a nonlocal, translation-invariant and
0
uniformlyelliptic operator with respecttoL and u∈L∞(L1)satisfies Iu−∂ u=0
∗ T ω t
in Q , then there are some α > 0 and C > 0 (depending only on λ,Λ,n,η and
1+η
σ , but not on σ) such that
0
kukC1,α(Q1+ǫ) ≤C Qsu1+pη|u|+kukL∞T(L1ω)
(cid:0) (cid:1)
where the constant C also depends on the constant in (2.5).
Remark. We note that it follows from the standard telescopic sum argument [CC]
and Theorem 2.5 that
(2.6) [u]Cx0,1(Q1+η) ≤C Qsu1+pη|u|+kukL∞T(L1ω) .
(cid:0) (cid:1)
Proof. The proof of this theoremgoes along the lines of the proofof Theorem 12.1
in [CS2] by applying Theorem 2.5 to the difference quotients in the x-direction
u(x+h,t)−u(x,t)
wh(x,t)=
|h|β
forβ =α,2α,··· ,1. Writewh =wh+wh wherewh =whχ forr ∈(1+ǫ,1+η).
1 2 1 Qr
By Theorem 2.0.4 [KL], we have that M+ wh−∂ wh ≥0 and M− wh−∂ wh ≤0
L∗ t L∗ t
in Q . Since ∂ wh ≡ 0 in Q for h with |h| < 1+η−r, it follows from uniform
r t 2 r
ellipticity with respect to L that we have that
∗
M+ wh−∂ wh ≥M+ wh−∂ wh ≥M+ wh−M+ wh−∂ wh ≥−M+ wh in Q ,
L0 1 t 1 L∗ 1 t 1 L∗ L∗ 2 t 1 L∗ 2 r
M− wh−∂ wh ≤M− wh−∂ wh ≤M− wh−M− wh−∂ wh ≤−M− wh in Q .
L0 1 t 1 L∗ 1 t 1 L∗ L∗ 2 t 1 L∗ 2 r
If we can show that |M+L∗w2h|∨|M−L∗w2h|≤ckukL∞T(L1ω), then we have that
M+L0w1h−∂tw1h ≥−ckukL∞T(L1ω) and M−L0w1h−∂tw1h ≤ckukL∞T(L1ω) in Qr
for h with |h|<1+η−r. Indeed, it can be obtained from the fact that
|K(x,y,t)−K(x,y−h,t)|
|u(x+y,t)| dy
ZRn\Bρ |h|
+ |u(x+y+h,t)|K(x,y,t)dy ≤CkukL∞(L1)
ZRn\Bρ T ω
for some ρ > 0 (this can be seen by using (2.2) and (2.5)). Hence u admits the
required C1,α-estimates on Q in the x-direction.
1+ǫ
NowwearegoingtoshowthatuisC(1+α)/σ-H¨oldercontinuousintime,following
Lemma 2 in [CW]. For (x ,t )∈Q , we consider
0 0 1+ǫ
u(rx+x ,rσt+t )−u(x ,t )−r∇u(x ,t )·x
0 0 0 0 0 0
w(x,t)=
r1+α
for small r > 0. Without loss of generality, let us assume that [u] ≤ 1.
Cx0,1(Q1+η)
Then w solves the given parabolic equation, and so C1,α-regularity of u in x on
Q implies [w] ≤ 1 and |w(x,0)| ≤ 1 with w(0,0) = 0. Thus as in the
1+η Cx0,1(Q1)
proof of Lemma 2 [CW], it easily follows from comparison principle [KL] that w is
uniformly bounded in Q , which implies the conclusion. (cid:3)
1
6 YONG-CHEOLKIMANDKI-AHMLEE
3. Boundary estimates and Global estimates
Inthissection,werealizethatamodulusofcontinuityontheparabolicboundary
of the domain of some equation makes it possible to obtain another modulus of
continuity inside the domain. This can be established by controlling the growth of
u away from its boundary data through barriers, scaling and interior regularity.
We use a barrier function which was used in [CS1] for the elliptic case and
adaptedtoourparabolicsetting. This barrierfunctionis appropriateasasuperso-
lution of M+ψ ≤ 0 for all values of σ greater than a given σ , where M+ denotes
σ 0 σ
the maximaloperatorM+ . Another wayto saythis wouldbe to define a larger
L0(σ)
class L which is the union of all classes L (σ) for σ ∈ (σ ,2), then M+ψ ≤ 0.
0 0 L
The proof of the following lemma can be achieved by a little modification to our
parabolic setting (refer to [CS1]), and so we leave the proof for the reader.
Lemma 3.1. Let σ ∈ (0,2) be given. Then, for any σ ∈ (σ ,2) and γ ∈ (0,1),
0 0
there are some α > 0 and r > 0 so small that the function g (x,t) = (|x|−1)α
α +
satisfies M+g ≤−1/[(2σ−1)γσ] in (B \B )×(−T,0].
σ α 1+r 1
Corollary 3.2. Let σ ∈(0,2) be given. Then, for any σ ∈(σ ,2) and γ ∈(0,1),
0 0
thereis acontinuousfunctionψ definedonRn suchthat(a)ψ =0inQ , (b)ψ ≥0
T 1
in Rn, (c) ψ ≥γ−σ in Rn\Q , (d) M+ψ−∂ ψ ≤0 and ∂ ψ ≥−[(2σ−1)γσ]−1 in
T T 2 σ t t
Rn \Q .
T 1
Proof. We consider ψ(x,t) = min{γ−σ,C(|x|−1)α}+(t+1) /[(2σ −1)γσ] for
+ −
some large constant C >0 and apply Lemma 3.1. (cid:3)
The function ψ that we obtained in Corollary 3.2 shall be utilized as a barrier
to prove the boundary continuity of solutions to nonlocal parabolic equations. We
observe that ψ is a supersolution outside the parabolic cylinder Q .
1
Throughout this section, we denote η ∈(0,1) by any very small number.
Theorem 3.3. Let σ ∈(σ ,2) for σ ∈(0,2). If u∈L∞(L1) satisfies that
0 0 T ω
M+u−∂ u≥−C and M−u−∂ u≤C in Q ,
σ t σ t 1+η
|u(y,s)−u(x,t)|≤ρ((|x−y|σ+|t−s|)1/σ)
forevery(x,t)∈∂ Q and(y,s)∈Rn\Q ,whereρisamodulusofcontinuity,then
p 1 T 1
thereisanothermodulusofcontinuityρ¯(dependingonlyonρ,λ,Λ,σ0,n,kukL∞(L1),
T ω
η and C, but not on σ) such that
|u(y,s)−u(x,t)|≤ρ¯((|x−y|σ+|t−s|)1/σ)
for every (x,t)∈Q and (y,s)∈Rn.
1 T
Lemma 3.4. Let σ ∈(σ ,2) for σ ∈(0,2). If u∈L∞(L1) is a function such that
0 0 T ω
M+u−∂ u≥−C in Q ,
σ t 1+η
u(x,t)−u(x ,t )≤ρ((|x−x |σ+|t−t |)1/σ)
0 0 0 0
for every (x ,t )∈∂ Q and (x,t)∈Rn \Q , where ρ is a modulus of continuity,
0 0 p 1 T 1
thenthereisanothermodulusofcontinuityρ˜(dependingonlyonρ,λ,Λ,σ0,n,kukL∞(L1),
T ω
η and C, but not on σ) such that
u(x,t)−u(x ,t )≤ρ˜((|x−x |σ+|t−t |)1/σ)
0 0 0 0
for every (x ,t )∈∂ Q and (x,t)∈Rn.
0 0 p 1 T
CORDES-NIRENBERG TYPE ESTIMATES 7
Proof. If we write v =u , then as before we have that
1Q1+η
M+σv−∂tv ≥−cη−kukL∞T(L1ω) and M−σv−∂tv ≤cη+kukL∞T(L1ω) in Q1.
Since u is continuous on Q , we may assume that u∈B(Rn).
1+η T
Sinceσ ≥σ >0,thefunctionp(x,t)= 1(0∨(4−|x|2))+(C+4Λω (2−σ)1−3−σ)t
0 4 n σ
satisfies M+p≤−λω +4Λω (2−σ)1−3−σ in Q , because
σ n n σ 1
Lp(x,t)=− |y|2K(x,y,t)dy+ µ (p,x,y)K(x,y,t)dy
t
ZB1 ZB3\B1
1−3−σ
≤−λω +4Λω (2−σ)
n n
σ
for any L ∈ L (σ) and all (x,t) ∈ Q , where ω denotes the surface measure of
0 1 n
Sn−1. Since ∂ p=C+4Λω (2−σ)1−3−σ in Q , we have that
t n σ 1
(3.1) M+ u−p −∂ u−p ≥M+u−∂ u−M+p+∂ p≥λω ≥0 in Q .
σ t σ t σ t n 1
Let ρ be th(cid:0)e mod(cid:1)ulus o(cid:0)f cont(cid:1)inuity of the function ψ in Corollary 3.2 and let ρ
0 1
be the modulus of continuity of the function p. By the assumption, we see that
u(x,t)−p(x,t)−u(x ,t )+p(x ,t )
0 0 0 0
(3.2)
≤ρ((|x−x |σ+|t−t |)1/σ)+ρ ((|x−x |σ+|t−t |)1/σ)
0 0 1 0 0
for every (x ,t )∈∂ Q and (x,t)∈Rn \Q .
0 0 p 1 T 1
Fix any (x ,t )∈∂ Q . For r >0, we define ρ¯by
0 0 p 1
r
ρ¯(r)= inf ρ(3γ)+ρ (3γ)+ u−p−u(x ,t )+p(x ,t ) ρ .
γ∈(0,1) 1 0 0 0 0 L∞ 0 γ
(cid:18) (cid:18) (cid:19)(cid:19)
(cid:13) (cid:13)
Then we must show that ρ¯ is a mo(cid:13)dulus of continuity. We ea(cid:13)sily see that ρ¯ is
clearly monotonically increasing because ρ is. So we have only to show that for
0
any ε > 0, there is some r > 0 such that ρ¯(r) < ε. Indeed, we choose some
γ ∈ (0,1) such that ρ(3γ)+ρ (3γ) < ε/2, and then choose some r > 0 so that
1
ku−p−u(x0,t0)+p(x0,t0)kL∞ρ0(r/γ) < ε/2. Finally, we show that there is a
modulusofcontinuityρ˜suchthatu(x,t)−u(x ,t )≤ρ˜((|x−x |σ+|t−t |)1/σ)for
0 0 0 0
any (x,t)∈Rn. Take any (x,t)∈Rn. For γ >0, we consider a barrier function
T T
B(x,t)=u(x ,t )−p(x ,t )+ρ(3γ)+ρ (3γ)
0 0 0 0 1
x−x t−t
+γσ u−p−u(x ,t )+p(x ,t ) ψ −x + 0, 0 .
0 0 0 0 L∞ 0 γ γσ
(cid:18) (cid:19)
(cid:13) (cid:13)
By (3.2) and the d(cid:13)efinition of ψ, we have that (cid:13)
B(x,t)≥u(x ,t )−p(x ,t )+ρ(3γ)+ρ (3γ)≥u(x,t)−p(x,t)
0 0 0 0 1
for any (x,t)∈Q (x ,t )∩(Rn \Q ). Also by the definition of ψ, we obtainthat
3γ 0 0 T 1
B(x,t) ≥ u(x,t)−p(x,t) for any (x,t) ∈ (Rn \Q (x ,t ))∩(Rn \Q ). Thus we
T 3γ 0 0 T 1
havethatB ≥u−ponRn\Q . By(d)ofCorollary3.2,weseethatM+ψ−∂ ψ ≤0
T 1 σ t
in Q (γ+1x ,0) because Q (γ+1x ,0)⊂Rn \Q . We observe that
1/γ γ 0 1/γ γ 0 T 1
γ+1
M+ψ−∂ ψ ≤0 in Q x ,0 ⇔ M+B−∂ B ≤0 in Q ,
σ t 1/γ γ 0 σ t 1
(cid:18) (cid:19)
8 YONG-CHEOLKIMANDKI-AHMLEE
by Corollary 3.2. Taking the infimum on γ, it follows from comparison principle
(Theorem 2.0.3 in [KL]) that
u(x,t)−p(x,t)≤B(x,t)≤u(x ,t )−p(x ,t )+ρ¯((|x−x |σ +|t−t |)1/σ)
0 0 0 0 0 0
for all (x,t)∈Rn, because
T
ψ(x,t)≤|ψ(x,t)−ψ(x ,t )|≤ρ ((|x−x |σ+|t−t |)1/σ),∀(x,t)∈Rn.
0 0 0 0 0 T
Hencewehavethatu(x,t)−u(x ,t )≤ρ˜((|x−x |σ+|t−t |)1/σ)forall(x,t)∈Rn,
0 0 0 0 T
where ρ˜=ρ +ρ¯. Therefore we complete the proof. (cid:3)
1
Lemma 3.5. Let σ ∈(σ ,2) for σ ∈(0,2). If u∈L∞(L1) satisfies that
0 0 T ω
M+u−∂ u≥−C and M−u−∂ u≤C in Q ,
σ t σ t 1+η
|u(y,s)−u(x,t)|≤ρ((|x−y|σ+|t−s|)1/σ)
for every (x,t) ∈ ∂ Q and (y,s) ∈ Rn, where ρ is a modulus of continuity, then
p 1 T
thereisanothermodulusofcontinuityρ¯(dependingonlyonρ,λ,Λ,σ0,n,kukL∞(L1),
T ω
η and C, but not on σ) such that
|u(y,s)−u(x,t)|≤ρ¯((|x−y|σ+|t−s|)1/σ)
for every (x,t)∈Q and (y,s)∈Rn.
1 T
Proof. If we set v =u , then as before we have that
1Q1+η
M+σv−∂tv ≥−cη−kukL∞T(L1ω) and M−σv−∂tv ≤cη+kukL∞T(L1ω) in Q1.
SinceuiscontinuousonQ ,wemayassumethatu∈B(Rn). Henceitcaneasily
1+η T
be obtained by an adaptation of Lemma 3 in [CS1] to our parabolic setting. (cid:3)
Proof of Theorem 3.3. We apply Lemma 3.4 to both u and −u to obtain
a modulus of continuity that applies from any point on ∂ Q to any point in Rn.
p 1 T
Then we use Lemma 3.5 to finish the proof. (cid:3)
4. Some results by approximation
In this section, we show that two equations which are very close to each other
in some appropriate way have their solutions which are close by each other on the
unit cube Q .
1
In what follows, for a function u : Rn → R and a quadratic polynomial p we
denotebyupQr(x0,t0) ;p1Qr(x0,t0)+u1RnT\Qr(x0,t0). Thefollowinglemmaisanusual
result in analysis on viscosity solutions, and so we will skip the proof.
Lemma4.1. LetIbeauniformlyelliptic operator in thesenseof (1.2)withrespect
to some class L and let u : Rn → R be a function which is upper semicontinuous
T
on Ω . Then the followings are equivalent.
τ
(a) u is a viscosity subsolution of Iu−∂ u=f in Ω , i.e. Iu−∂ u≥f in Ω .
t τ t τ
(b) If p is a quadratic polynomial satisfying u(x ,t ) = p(x ,t ) and u ≤ p in
0 0 0 0
Q (x ,t ) where Q (x ,t ) ⊂ Ω for some r > 0 and (x ,t ) ∈ Ω , then we have
r 0 0 r 0 0 τ 0 0 τ
that Iup(x ,t )−∂ up(x ,t )≥f(x ,t ) for up =up .
r 0 0 t r 0 0 0 0 r Qr(x0,t0)
We want to show that if I u (x,t) = f (x,t) and I → I, u → u and f → f
k k k k k k
in some appropriate way, then Iu(x,t)=f(x,t).
Inthe elliptic case[CS1], the solutionspaceL1 is enoughforthe weaklyconver-
ω
gence of operators I . But, in the parabolic case, the possible substitute L∞(L1)
k T ω
for the solution space L1 is not enough for it, because solutions for the parabolic
ω
CORDES-NIRENBERG TYPE ESTIMATES 9
casehavethetimeshift. ThusweimposetheadditionalconditionC(Rn)togetthe
T
weakly convergence of operators I . This makes it possible to obtain the stability
k
properties for the nonlocal parabolic case.
Definition 4.2. We say that I converges weakly to I in Ω with respect to ω
k τ
(and we denote by lim I = I in Ω ), if for any (x ,t ) ∈ Ω there is some
k→∞ k ω τ 0 0 τ
Q (x ,t )⊂Ω such that
r 0 0 τ
(4.1) lim I up =Iup
k r r
k→∞
uniformly in Q (x ,t ) for any function up of the form up = up where p
r/2 0 0 r r Qr(x0,t0)
is a quadratic polynomial and u∈L∞(L1)∩C(Rn).
T ω T
Lemma 4.3. Let {I } be a sequence of uniformly elliptic operators with respect to
k
some class L. Assume that Assumption 2.3 holds. Let {u } be a sequence of lower
k
semicontinuous functions in Ω such that
τ
(a) I u −∂ u ≤f in Ω , (b) lim u =u in the Γ sense in Ω ,
k k t k k τ k→∞ k τ
(c) limk→∞kuk−ukL∞(L1) =0, (d) limk→∞Ik =ω I in Ωτ,
τ ω
(e) lim f =f ∈C(Ω ) locally uniformly in Ω ,
k→∞ k τ τ
(f) sup sup |u |≤C <∞.
k∈N Ωτ k
Then we have that Iu−∂ u≤f in Ω .
t τ
Proof. Let p be a quadratic polynomial touching u from below at a point (x,t) in
a neighborhood V ⊂ Ω . Since {u } Γ-converges to u in Ω , there are a cylinder
τ k τ
Q (x,t)⊂V and a sequence {(x ,t )}⊂Q (x,t) with lim d((x ,t ),(x,t))=
r k k r k→∞ k k
0 such that p touches u from below at (x ,t )(refer to [GD]). Without loss of
k k k
generality, we may assume that Q (x,t) is a cylinder so that (4.1) holds for the
r
point (x,t).
If we set (u )p = (u )p , then we have I(u )p(x ,t ) ≤ f(x ,t ). If we set
k r k Qr(x,t) k r k k k k
up =up , then we see that up(z,q)=(u )p(z,q) and ∂ (u )p(z,q)=∂ up(z,q)
r Qr(x,t) r k r t k r t r
for any k ∈ N and (z,q) ∈ Q (x,t). Take any (z,q) ∈ Q (x,t). Then we have
r r/4
that
|I (u )p(z,q)−∂ (u )p(z,q)−Iup(z,q)+∂ up(z,q)|
k k r t k r r t r
≤|I (u )p(z,q)−I up(z,q)|+|I up(z,q)−Iup(z,q)|
k k r k r k r r
≤|M+((u )p−up)(z,q)|∨|M+(up−(u )p)(z,q)|+|I up(z,q)−Iup(z,q)|
L k r r L r k r k r r
≤ sup|L((u )p−up)(z,q)|+|I up(z,q)−Iup(z,q)|
k r r k r r
L∈L
≤ |µ ((u )p−up,z,y)|K(x,y,t)dy+|I up(z,q)−Iup(z,q)|
q k r r k r r
ZRn\Br/2
≤C |((u )p−up)(z+y,q)|+|((u )p−up)(z−y,q)| ω(y)dy
r k r r k r r
ZRn\Br/2
(cid:8) +|I up(z,q)−Iup(z,q(cid:9))|
k r r
≤C 2|(u )p(y,q)−up(y,q)| sup ω(y+z)dy+|I up(z,q)−Iup(z,q)|
k r r k r r
ZRn z∈Br/4
≤Ckuk−ukL∞τ (L1ω)+|Ikupr(z,q)−Iupr(z,q)|→0
ask →∞,byusingAssumption2.3and(2.2). Since(u )p ∈C2(Q (x,t))∩L∞(L1)
k r r τ ω
for all k ∈N andlimk→∞kuk−ukL∞τ (L1ω) =0, we see upr ∈C2(Qr(x,t))∩L∞τ (L1ω),
10 YONG-CHEOLKIMANDKI-AHMLEE
and thus Iup is continuous in Q (x,t). Thus by (4.1) we have that
r r
|I (u )p(x ,t )−∂ (u )p(x ,t )−Iup(x,t)+∂ up(x,t)|
k k r k k t k r k k r t r
≤|I (u )p(x ,t )−∂ (u )p(x ,t )−Iup(x ,t )+∂ up(x ,t )|
k k r k k t k r k k r k k t r k k
+|I up(x ,t )−Iup(x ,t )|+|Iup(x ,t )−Iup(x,t)|
k r k k r k k r k k r
+|∂ up(x ,t )−∂ up(x,t)|→0
t r k k t r
ask →∞. Sincelim d((x ,t ),(x,t))=0andlim f =f ∈C(Ω )locally
k→∞ k k k→∞ k τ
uniformly in Ω , we have that f (x ,t ) → f(x,t). Therefore this implies that
τ k k k
Iup(x,t)−∂ up(x,t)≤f(x,t). Hence we conclude that Iu−∂ u≤f in Ω . (cid:3)
r t r t τ
Lemma 4.4. Let up = up where p is a quadratic polynomial and u ∈ L∞(L1)∩
C(Rn). If {I } is arsequeQnrce of uniformly elliptic operators with respectTto sωome
T k
class L satisfying Assumptions 2.2 and 2.3, then there is a subsequence {I } such
kj
that I up converges uniformly in Q .
kj r r/2
Proof. We have only to find a uniform modulus of continuity for I up in Q so
kj r r
that the lemma follows from Arzela-Ascoli Theorem.
Take any two (x,t),(y,s) ∈ Q with d((x,t),(y,s)) < r/8. Then there exists
r/2
some large R>0 so that the function ϕ ∈C∞(Rn) defined by
R c T
1 on B ×(−T +2,0]
R
ϕ =
R (0 on RnT \(BR+1×(−T +1,0])
satisfiesku−vkL∞(L1) <(|x−y|σ+|s−t|)1/σ where v =ϕRu. Sowe mayassume
ω
that u ∈ L∞(L1)∩C (Rn). We denote by τ u(x,t) = u(x+z,t) for z ∈ Rn. By
T ω c T z
the uniform ellipticity of I , we have that
k
(4.2) I up(x,t)−I up(y,t)≤M+(up−τ up)(x,t).
k r k r L r y−x r
Also we see that up−τ up =0 in B (x)×t, because
r y−x r r/4
p(x+z,t)+p(x−z,t)−2p(x,t)−p(y+z,t)−p(y−z,t)+2p(y,t)=0
for any z ∈B . From (2.2) and Assumption 2.3, for any L∈L we have that
r/4
L(up−τ up)(x,t)= µ (up−τ up,x,z)K(x,z,t)dz
r y−x r t r y−x r
ZRn\Br/4
= up(x+z,t)+up(x−z,t)−2up(x,t) K(x,z,t)dz
r r r
ZRn\Br/4
(cid:2) (cid:3)
− up(y+z,t)+up(y−z,t)−2up(y,t) K(x,z,t)dz
r r r
ZRn\Br/4
(cid:2) (cid:3)
≤Cr |p(x,t)−p(y,t)|+kτxupr −τyuprkL∞T (L1ω)
≤C (cid:0) sup |p(ξ,t)−p(η,t)|+ |τ u(cid:1)p(z,t)−up(z,t)| sup ω dz
r ξ−η r r
|ξ−η|≤|x−y|(cid:18) ZRn Br(z) (cid:19)
ξ,η∈Br/2 (cid:0) (cid:1)
≤m (|x−y|)
1
where m is defined as m (̺)=sup m (̺) and
1 1 t∈(−r/2,r/2] t
m (̺)=C sup |p(ξ,t)−p(η,t)|+ |τ up(·,t)−up(·,t)|ω(z)dz .
t r ξ−η r r
|ξ−η|≤̺(cid:18) ZRn (cid:19)
ξ,η∈Br/2