R E S E A R C H
Open Access
Interior regularity of obstacle problems for
nonlinear subelliptic systems with VMO
coefficients
Guangwei Du
*and Fushan Li
*Correspondence:
guangwei87@mail.nwpu.edu.cn School of Mathematical Sciences, Qufu Normal University, Qufu, China
Abstract
This article is concerned with an obstacle problem for nonlinear subelliptic systems of second order with VMO coefficients. It is shown, based on a modification of
A-harmonic approximation argument, that the gradient of weak solution to the corresponding obstacle problem belongs to the Morrey spaceL2,X,λloc.
MSC: 35J70; 35B65; 35J50
Keywords: Subelliptic systems; Obstacle problem; VMO;L2,X,λloc-regularity
1 Introduction
In this paper we consider weak solutions to an obstacle problem for the following
nonlin-ear subelliptic system in a bounded domainof Euclidean spaceRn:
Xα∗
aαβij (x)Xβuj
=Bi(x,u,Xu) +Xα∗g α
i(x,u,Xu), i= 1, 2, . . . ,N, (1.1)
whereX= (X1, . . . ,Xm) (m≤n) is a system of smooth real vector fields satisfying the
Hör-mander’s rank condition,Xα∗is the formal adjoint ofXα.
If we setA(x) ={aαβij (x)},B= (Bi),g= (gαi), then (1.1) reads
–X∗A(x)Xu=B(x,u,Xu) –X∗g(x,u,Xu).
Given two vector-valued functionsψ= (ψ1, . . . ,ψN) andθ= (θ1, . . . ,θN) withθ(x)≥ψ(x) a.e. on∂(i.e.θi(x)≥ψi(x) a.e. on∂,i= 1, 2, . . . ,N), we define the set
Kθψ=v∈S1,2X ,RN:v≥ψa.e. in,v–θ∈S1,2
X,0
,RN.
Here the functionsψ andθ are called obstacle and boundary datum, respectively. The
functionu∈Kθψ is called a weak solution to the obstacle problem related to (1.1) if
A(x)Xu·Xϕdx≥
B(x,u,Xu)ϕdx+
g(x,u,Xu)·Xϕdx (1.2)
holds for allϕ∈C0∞(,RN) withϕ+u≥ψa.e.x∈.
As we know, the uniform ellipticity requirement on coefficients is not sufficient to get the local boundedness of solutions even for one single equation in the Euclidean metric (see [1]). Therefore some additional assumptions on the coefficients is needed to ensure the regularity results. In [2–4], Campanato obtained theL2,λ-regularity and Hölder continuity
for the weak solutions of elliptic systems with continuous coefficients. See also [5–8] for related results.
Since the functions of vanishing mean oscillation (VMO) can have some kind of dis-continuities, regularity results under a VMO assumption have been established by many authors; see, for example, [9–12] for elliptic systems, and [13–17] for subelliptic systems constructed by Hörmander’s vector fields. Huang in [9] established the gradient estimates in the generalized Morrey spaces of weak solutions to the linear elliptic systems with VMO coefficients. Similar results for the nonlinear elliptic systems were obtained by Daněček and Viszus in [10] and [11]. In [15] and [16] Di Fazio and Fanciullo proved that the local gradient estimates in [9] still hold true for the subelliptic systems structured on Höman-der’s vector fields. Dong and Niu [14] established the Morrey and Campanato regularity for weak solutions to the nondiagonal subelliptic systems. The direct methods were mainly used to prove the desired results in the papers mentioned above. An important step of this kind of methods is to establish the higher integrability of gradients of weak solutions. These arguments were also used to prove the Morrey regularity and Hölder continuity for weak solutions to the obstacle problems associated with a single elliptic equation with constant coefficients or continuous coefficients; see [18–22].
Recently, another method calledA-harmonic approximation has been widely applied to
prove the optimal partial regularity for nonlinear elliptic systems or subelliptic systems in the Heisenberg group and Carnot groups; see [23–29]. This method is based on Simon’s technique of harmonic approximation ([30]) and generalized by Duzaar and Grotowski in [31] in order to deal with partial regularity for nonlinear elliptic systems. The key point is to show that a function which is “approximately harmonic”, i.e. a function closes
suf-ficiently to some harmonic function inL2. Making use of this method, one can simplify
the proof avoiding the proof of a suitable reverse Hölder inequality for the gradient of a weak solution. We also mention that Daněček-John-Stará [32] proved the Morrey space regularity for weak solutions of Stokes systems with VMO coefficients by using a mod-ifiedA-harmonic approximation lemma. Inspired by this work, Yu and Zheng [33] ob-tained optimal partial regularity for quasilinear elliptic systems with VMO coefficients by
a modification ofA-harmonic approximation argument.
In the present paper we study the interior regularity of weak solutions to the obstacle
problem related to the system (1.1) by the technique ofA-harmonic approximation, which
implies that these solutions have the same kind of regularity as the weak solutions of (1.1). Throughout this article, we make the following assumptions.
(H1) The coefficientsaαβij are bounded measurable and such that, for some suitable λ> 0and> 0,
λ|ξ|2≤aαβij (x)ξαiξβj ≤|ξ|2, x∈Rn,ξ∈RmN;
(H2) The functionsBi,giα:Rn×RN×RmN→Rare both Carathéodory functions and
for almostx∈and all(u,ξ)∈RN×RmN, there existsL> 0such that
gαi(x,u,ξ)≤fiα(x) +L|ξ|γ,
where1≤γ0<QQ+2,0≤γ < 1, and
f∈L2XQ/(Q+2),λQ/(Q+2),RN, f˜∈L2,Xλ,RmN, Q–n<λ<Q.
HereQis the homogeneous dimension relative toandf= (fi),f˜= (fiα).
We are now in the position to state our main result.
Theorem 1.1 Suppose that(H1)–(H2)hold and that aαβij ∈VMO()for i,j= 1, 2, . . . ,N,
α,β= 1, 2, . . . ,m.Let u∈Kθψ be a weak solution to the obstacle problem for system(1.1) with Xψ ∈L2,Xλ(,RmN),then Xu∈LX2,λ,loc(,RmN).Moreover, if Q–n<λ< 2then u∈ CX0,(2–λ)/2(,RN).
The paper is organized as follows. In the next section we recall some concepts and facts
associated to Carnot–Carathéodory spaces and give the proof of the modifiedA-harmonic
approximation lemma for vector fields. In Sect. 3, we consider the following linear subel-liptic system with VMO coefficients:
X∗A(x)Xu=X∗A(x)Xψ,
and we prove a comparison principle and a Morrey type estimate for weak solutions of
the above system by a modification ofA-harmonic approximation argument. Section 4 is
devoted to the proofs of Theorem 1.1. On the basis of the Morrey type estimate established for linear subelliptic system, we can first prove theL2,Xλ,loc-regularity for weak solutions of the obstacle problems and then interior Hölder continuity is obtained by virtue of the equivalence between the Campanato space and the Hölder continuity function space (see [34, 35]).
In what follows, we usecto denote a positive constant that may vary from line to line.
2 Some notations and preliminaries
Let
Xα=
n
k=1 bαk
∂ ∂xk
, bαk∈C∞,α= 1, 2, . . . ,m
be a family of vector fields inRnsatisfying Hörmander’s condition ([36]):
rankLie{X1, . . . ,Xm}
=n.
We considerXαas a first order differential operator acting onu∈Lip(Rn) defined as
Xαu(x) = Xα(x),∇u(x)
, α= 1, 2, . . . ,m.
We denote byXu= (X1u, . . . ,Xmu) the gradient ofuand hence|Xu(x)|= (
m
α=1|Xαu(x)|2)
1 2.
An absolutely continuous curveγ : [a,b]→Rnis said to be admissible if
γ (t) =
m
α=1 cα(t)Xα
for some functionscα(t) satisfying
m
α=1cα(t)2≤1. The Carnot–Carathéodory distance
d(x,y) generated byXis defined by
d(x,y) =infT> 0 : there is an admissible curveγ,γ(0) =x,γ(T) =y.
Forx∈RnandR> 0 we let
BR(x) =B(x,R) =
y∈Rn:d(y,x) <R.
In what follows, ifσ> 0 andB=B(x,R) we writeσBto indicateB(x,σR). Furthermore, if E⊂Rnis a Lebesgue measurable set with Lebesgue measure|E|, we setuE=–
Eu dxthe
integral average ofuonE.
In [37], it was proved that for every connectedK⊂there exist constantsC1,C2> 0 and 0 <λ< 1 such that
C1|x–y| ≤d(x,y)≤C2|x–y|λ, x,y∈K.
Moreover, there areRd> 0 andCd≥1 such that, for anyx∈KandR≤Rd,
B(x, 2R)≤CdB(x,R). (2.1)
Property (2.1) is the so-called “doubling condition” which is assumed to hold on the spaces
of homogeneous type. The best constantCdin (2.1) is called the doubling constant. We
call thatQ=log2Cdis the homogeneous dimension relative to. As a consequence of
(2.1), we have
|BtR| ≥C–2d tQ|BR|, ∀R≤Rd,t∈(0, 1). (2.2)
We now introduce the relevant Sobolev spaces. Given 1≤p<∞, the Sobolev space
S1,Xp(,RN) is the Banach space
S1,Xp,RN=u∈Lp,RN:X
αu∈Lp
,RN,α= 1, 2, . . . ,m
endowed with the norm
uS1,p
X (,RN)=uL p(,RN)+
m
α=1
XαuLp(,RN).
Here,Xαuis the distributional derivative ofu∈L1loc(,RN) defined by
Xαu·φdx=
u·Xα∗φdx, ∀φ∈C0∞,RN,
where
Xα∗= –
n
k=1
∂ ∂xk
is the formal adjoint ofXα, not necessarily a vector field in general. The spaceSX1,p,0(,RN)
is defined as the completion ofC∞0 (,RN) under the norm ·
S1,Xp(,RN).
In addition, we also need the following Sobolev inequalities for vector fields.
Theorem 2.1([38, 39]) For every compact set K⊂,there exist constants C> 0andR¯> 0
such that,for any metric ball B=B(x0,R)with x0∈K and0 <R≤ ¯R,for any f∈S1,Xp(BR),
–
BR
|f –fR|κpdx
1 κp
≤CR
–
BR
|Xf|pdx 1
p
,
where1≤κ≤Q/(Q–p),if1≤p<Q; 1≤κ<∞,if p≥Q.Moreover,
–
BR
|f|κpdx 1
κp
≤CR
–
BR
|Xf|pdx 1
p
,
whenever f ∈S1,X,0p(BR).
Now we define the Morrey spaces, the Campanato spaces, VMO and the Hölder spaces with respect to the Carnot–Carathéodory metric. To simplify our description, we intro-duce the following notation:
(x,R) =∩B(x,R), fx,R=
1
|(x,R)|
(x,R) f(y)dy,
and
d0=min{diam,Rd}.
Definition 2.2 For 1 <p<∞andλ≤Q, we say thatf ∈Lploc(,RN) belongs to the Morrey
spaceLpX,λ(,RN) if
fLp,λ
X (,RN)= sup
x∈,0<ρ<d0
ρλ |(x,ρ)|
(x,ρ) f(y)p
dy 1
p
<∞;
f∈Lploc(,RN) belongs to the Campanato spaceLp,λ
X (,RN) if
fLp,λ
X (,RN)= sup
x∈,0<ρ<d0
ρλ
|(x,ρ)|
(x,ρ)
f(y) –fx,ρpdy 1
p
<∞.
Definition 2.3 Forα∈(0, 1), the Hölder spaceC0,Xα(¯,RN) is the collection of functions
f :¯ →RNsatisfying
fC0,α
X (¯,RN)=sup |f|+sup¯
|f(x) –f(y)| d(x,y)α <∞.
We say thatf is locally Hölder continuous, i.e.f ∈CX0,α(,RN), iff ∈C0,α
X (K,RN) for every
Definition 2.4 We say thatf∈L1
loc(,RN) belongs to BMO(,RN) if
f∗= sup
x∈,0<ρ<d0
1
|(x,ρ)|
(x,ρ)
f(y) –fx,ρdy<∞;
f belongs to VMO(,RN) iff ∈BMO(,RN) and
ηr(f) = sup x∈,0<ρ<r
1
|(x,ρ)|
(x,ρ)
f(y) –fx,ρdy→0, r→0.
The integral characterization for a Hölder continuous function was shown in [35] and [34].
Lemma 2.5 If–p<λ< 0,thenLXp,λ(,RN)C0,α
X (,RN),α= –
λ
p.
3 Morrey type estimate for a subelliptic system
In this section we will prove by the modifiedA-harmonic approximation technique a
Mor-rey type estimate for the subelliptic system
X∗A(x)Xu=X∗A(x)Xψ. (3.1)
Let us first recall that a functionh∈SX1,2(,RN) is calledA-harmonic forA∈Bil(RmN) if
hsatisfies
A(Xh,Xϕ)dx= 0, ∀ϕ∈C01,RN.
We cite theA-harmonic approximation lemma for vector fields as follows ([24, 31]).
Lemma 3.1 Consider fixed positiveλand,and m,N∈Nwith m≥2.Then for any
givenε> 0there existsδ=δ(m,N,λ,,ε)with the following property:for any A∈Bil(RmN)
satisfying
A(ξ,ξ)≥λ|ξ|2, for allξ∈RmN, (3.2)
and
A(ξ,ξ˜)≤|ξ||˜ξ|, for allξ,ξ˜∈RmN, (3.3)
for any g∈S1,2X (Bρ(x0),RN) (for someρ> 0,x0∈Rn)satisfying
–
Bρ(x0)
|Xg|2dx≤1 (3.4)
and –
Bρ(x0)
A(Xg,Xϕ)dx≤δ sup
Bρ(x0)
|Xϕ|, ∀ϕ∈C0∞Bρ(x0),RN
there exists an A-harmonic function h∈S1,2X (Bρ(x0),RN)such that
–
Bρ(x0)
|Xh|2dx≤1 and 1
ρ2–
Bρ(x0)
|g–h|2dx≤ε.
Similarly to [32] and [33], we can prove the following modification of theA-harmonic
approximation lemma.
Lemma 3.2 Let0 <λ<and m∈Nwith m≥2be fixed.Then,for anyε> 0,there exists
a constant k=k(m,N,λ,,ε)such that the following holds:for any A∈Bil(RmN)satisfying conditions(3.2), (3.3)and for any u∈S1,2X (Bρ(x0),RN),there exists an A-harmonic function
h∈S1,2X (Bρ(x0),RN)such that
Bρ(x0)
|Xh|2dx≤
Bρ(x0)
|Xu|2dx (3.6)
and,moreover,there existsϕ∈C0∞(Bρ(x0),RN)such that
XϕL∞(Bρ(x0),RN)≤
1
ρ (3.7)
and
Bρ(x0)
|u–h|2dx
≤ερ2
Bρ(x0)
|Xu|2dx+k(ε)
ρ4
|Bρ(x0)|
Bρ(x0)
A(Xu,Xϕ)dx 2
. (3.8)
Proof For any givenε> 0 andu∈S1,2X (Bρ(x0),RN), we takeδ(ε) as in the above Lemma 3.1
and set
g=
–
Bρ(x0)
|Xu|2dx –12
u.
Then (3.4) holds. Assume that forgthe inequality (3.5) is true. From Lemma 3.1, there is anA-harmonic functionwsatisfying
–
Bρ(x0)
|Xw|2dx≤1, 1
ρ2–
Bρ(x0)
|w–g|2dx≤ε
and thus the functionh= (–
Bρ(x0)|Xu|
2dx)12wsatisfies (3.6). Moreover, we have
|u–h|2= –
Bρ(x0)
|Xu|2dx· |g–w|2,
which implies
Bρ(x0)
|u–h|2dx≤
Bρ(x0)
|Xu|2dx·–
Bρ(x0)
|g–w|2dx≤ερ2
Bρ(x0)
If, vice versa, there is a nonconstant functionϕ˜∈C0∞(Bρ(x0),RN) such that
–
Bρ(x0)
A(Xg,Xϕ˜)dx>δ(ε) sup
Bρ(x0) |Xϕ˜|.
Settingϕ=ρsup ϕ˜
Bρ(x0)|Xϕ˜|, it follows that ρ
δ(ε) –
Bρ(x0)
A(Xg,Xϕ)dx> 1.
We now take h = uρ. Using the Poincaré inequality and the fact that Xg =
(–
Bρ(x0)|Xu|
2dx)–12Xuwe deduce
Bρ(x0)
|u–h|2dx=
Bρ(x0)
|u–uρ|2dx
≤cρ2
Bρ(x0)
|Xu|2dx
≤ cρ4 δ2(ε)
–
Bρ(x0)
A(Xg,Xϕ)dx 2
Bρ(x0)
|Xu|2dx
≤cρ4|Bρ(x0)|
δ2(ε) –
Bρ(x0)
A(Xu,Xϕ)dx 2
≤ cρ4 δ2(ε)|B
ρ(x0)|
Bρ(x0)
A(Xu,Xϕ)dx 2
. (3.10)
Combining (3.9) and (3.10) and takingk(ε) = c
δ2(ε) complete the proof.
Now we are in a position to establish the Morrey type estimate for gradient of weak solution to (3.1) based on Lemma 3.2.
Lemma 3.3 Suppose that A(x)satisfies(H1)and u∈SX1,2,loc(,RN)is a weak solution to the
system(3.1),i.e.,
A(x)Xu·Xϕdx=
A(x)Xψ·Xϕdx, ∀ϕ∈C0∞,RN.
Then for any x0∈there exists a constant c> 0such that,for all Bρ(x0)⊂BR(x0)⊂,
R<Rd,
Bρ(x0)
|Xu|2dx≤c
ρ
R Q
+ε+ηR(A) BR(x0)
|Xu|2dx+c
BR(x0)
|Xψ|2dx. (3.11)
Proof For fixedx0∈and 0 <R<Rd, denoteBR:=BR(x0). Letηbe a cut-off function on
BRrelative toBρ, i.e.η∈C∞0 (BR,RN) and satisfies
0≤η(x)≤1, η(x) = 1 inBρ, Xη(x)≤
Taking the functionϕ=η2(u–u
R) as a test function, it follows that
BR
η2A(x)Xu·Xu dx
= –2
BR
A(x)η(u–uR)Xu·Xηdx+
BR
η2A(x)Xψ·Xu dx
+ 2
BR
A(x)η(u–uR)Xψ·Xηdx.
From (H1) and Young’s inequality
λ
BR
η2|Xu|2dx≤2
BR
|ηXu||u–uR||Xη|dx
+
BR
|ηXψ||ηXu|dx+ 2
BR
|ηXψ||u–uR||Xη|dx
≤ε
BR
η2|Xu|2dx+ cε (R–ρ)2
BR
|u–uR|2dx+cε
BR
|Xψ|2dx.
Choosingε<λ, it follows that
Bρ
|Xu|2dx≤ c (R–ρ)2
BR
|u–uR|2dx+c
BR
|Xψ|2dx. (3.12)
Next we defineAR=–
BRA(x)dx. By Lemma 3.2, there exists anAR-harmonic function
h∈S1,2X (BR,RN) such that (3.6)–(3.8) hold. Moreover, by standard results of the subelliptic
system with constant coefficients (see for example [34]), we have
Bρ
|Xh|2dx≤c
ρ
R Q
BR
|Xh|2dx, ∀0 <ρ≤R.
Therefore, from (3.12) and (3.6) it follows that for any 0 <ρ<R/2
Bρ
|Xu|2dx
≤ c ρ2
B2ρ
|u–u2ρ|2dx+c
B2ρ
|Xψ|2dx
≤ c ρ2
B2ρ
u–u2ρ– (h–h2ρ)
2 dx+
B2ρ
|h–h2ρ|2dx
+c
B2ρ
|Xψ|2dx
≤ c ρ2
B2ρ
|u–h|2dx+c
B2ρ
|Xh|2dx+c
B2ρ
|Xψ|2dx
≤ c ρ2
B2ρ
|u–h|2dx+c
ρ
R Q
BR
|Xh|2dx+c
BR
|Xψ|2dx
≤ c ρ2
B2ρ
|u–h|2dx+c
ρ
R Q
BR
|Xu|2dx+c
BR
For the first term in the right-hand side, we have from Lemma 3.2
c
ρ2
B2ρ
|u–h|2dx≤cε
B2ρ
|Xu|2dx+ck(ε) ρ 2
|B2ρ|
B2ρ
ARXu·Xϕdx
2
≤cε
BR
|Xu|2dx+cε
ρ2 |BR|
BR
ARXu·Xϕdx
2
, (3.14)
whereϕ∈C0∞(B2ρ,RN) satisfiesXϕL∞(B2ρ,RN)≤
1
R <
1
2ρ. Sinceuis a weak solution to
(3.1), it follows that
BR
ARXu·Xϕdx
2 =
BR
(AR–A)Xu·Xϕdx+
BR
AXu·Xϕdx 2
≤2
BR
(AR–A)Xu·Xϕdx
2 + 2
BR
AXu·Xϕdx 2
:=I1+I2.
From Hölder’s inequality, using (H1), we have
I1≤ 1 2ρ2
BR
|Xu|2dx·
BR
|AR–A|2dx
≤|BR|
ρ2 1
|BR|
BR
|AR–A|dx·
BR
|Xu|2dx
≤|BR|
ρ2 ηR(A)
BR
|Xu|2dx
and
I2≤
2
2ρ2
BR
|Xψ|dx 2
≤2|BR|
2ρ2
BR
|Xψ|2dx.
Hence
BR
ARXu·Xϕdx
2
≤c|BR|
ρ2
ηR(A)
BR
|Xu|2dx+
BR
|Xψ|2dx
. (3.15)
Combining (3.15), (3.14) and (3.13), we have, for any 0 <ρ<R/2,
Bρ
|Xu|2dx≤c
ρ
R Q
+ε+ηR(A) BR
|Xu|2dx+c
BR
|Xψ|2dx.
ForR/2≤ρ≤R, obviously
Bρ
|Xu|2dx≤
BR
|Xu|2dx≤2Q
ρ
R Q
BR
|Xu|2dx.
A combination of these two cases leads to (3.11) for 0 <ρ≤R.
Lemma 3.4 Suppose that u,ψ∈SX1,2(BR,RN)satisfy
X∗A(x)Xu=X∗A(x)Xψ
where A(x)satisfies(H1).Ifψ≤u on∂BR,thenψ≤u a.e.in BR.
Proof For anyϕ∈C0∞(BR,RN) we have
BR
A(x)Xu·Xϕdx=
BR
A(x)Xψ·Xϕdx. (3.16)
Setu+=max{u, 0}. Sinceψ≤uon∂BR, we conclude that (see [40, Lemma 6]) (ψ–u)+∈
S1,2X,0(BR,RN) and
X(ψ–u)+= ⎧ ⎨ ⎩
X(ψ–u), ψ>u,
0, ψ≤u.
Choosingϕ= (ψ–u)+in (3.16) gives
BR
A(x)X(ψ–u)·X(ψ–u)+dx= 0,
which implies
BR∩{ψ>u}A(x)X(ψ–u)·X(ψ–u)dx= 0.
From (H1) we have
BR
X(ψ–u)+ 2
dx=
BR∩{ψ>u}
X(ψ–u)2dx
≤1 λ
BR∩{ψ>u}A(x)X(
ψ–u)·X(ψ–u)dx= 0.
Thus from Poincaré inequality, we obtain
BR
(ψ–u)+2dx≤cR2
BR
X(ψ–u)+2dx= 0,
which implies (ψ–u)+= 0, orψ≤ua.e. inBR. The proof is complete.
4 Proof of main result
In this section we are going to prove our main result. To this end, we need a generalized iteration lemma, which can be found in [9, Proposition 2.1].
Lemma 4.1 Let H be a nonnegative almost increasing function on the interval[0,T]and
(1) for any0 <ρ≤R≤T,there existA,B,εanda> 0such that
H(ρ)≤
A
ρ
R a
+ε
H(R) +BF(R);
(2) there existsτ∈(0,a)such thatFρ(τρ) is almost increasing in(0,T].Then there exist positive constantsε0andCsuch that,for any0≤ε≤ε0,
H(ρ)≤CF(ρ)
F(R)H(R) +CB·F(ρ), 0 <ρ≤R≤T,
whereε0depends only onA,aandτ.
Proof of Theorem1.1 LetBR=B(x0,R)⊂⊂be an arbitrary ball aroundx0of radiusR
and letu∈Kθψ be a weak solution to the obstacle problem related to (1.1). InBRwe splitu
asu=w+ (u–w), wherew∈SX1,2(BR,RN) is the weak solution to the following system:
⎧ ⎨ ⎩
X∗(A(x)Xw) =X∗(A(x)Xψ) inBR,
w=u on∂BR.
(4.1)
Sincew=u≥ψa.e. on∂BR, it follows from Lemma 3.4 thatw≥ψ a.e. inBR.
By the definition of weak solutions, we have
BR
A(x)Xw·X(w–u)dx=
BR
A(x)Xψ·X(w–u)dx.
From (H1) and Young’s inequality one gets
λ
BR
|Xw|2dx≤
BR
A(x)Xw·Xw dx
≤
BR
|Xw||Xu|dx+
BR
|Xψ||Xw–Xu|dx
≤ε
BR
|Xw|2dx+cε
BR
|Xu|2dx+cε
BR
|Xψ|2dx.
Choosingε<λleads to
BR
|Xw|2dx≤c
BR
|Xu|2dx+c
BR
|Xψ|2dx. (4.2)
On the basis of (4.2), it follows from Lemma 3.3 that for any 0 <ρ≤R
Bρ
|Xu|2dx
≤2
Bρ
|Xw|2dx+ 2
Bρ
X(u–w)2dx
≤c
ρ
R Q
+ε+ηR(A) BR
|Xw|2dx+c
BR
|Xψ|2dx+ 2
Bρ
≤c
ρ
R Q
+ε+ηR(A) BR
|Xu|2dx
+c
BR
|Xψ|2dx+ 2
BR
X(u–w)2dx. (4.3)
Note thatw–uis admissible as a test function in the definition of weak solutions to the obstacle problem due tow–u∈S1,2X,0(BR,RN) andw≥ψa.e. inBR. Applyingw–uto (1.2)
leads to
A(x)Xu·X(u–w)dx≤
B(x,u,Xu)(u–w)dx+
g(x,u,Xu)·X(u–w)dx.
From (H1)–(H2) and Hölder’s inequality, we have
λ
BR
|Xu–Xw|2dx
≤
BR
A(x)X(u–w)·X(u–w)dx
≤–
BR
A(x)Xw·X(u–w)dx+
BR
B(x,u,Xu)|u–w|dx
+
BR
g(x,u,Xu)|Xu–Xw|dx
≤–
BR
A(x)Xψ·X(u–w)dx+
BR
|f|+L|Xu|γ0|u–w|dx
+
BR
|˜f|+L|Xu|γ|Xu–Xw|dx
≤c
BR
|Xu–Xw|2dx 1
2
BR
|Xψ|2dx 1 2 + BR
|f|+|Xu|γ02Q/(Q+2)dx
Q+2 2Q
+c
BR
|Xu–Xw|2dx 1
2
BR
|˜f|+|Xu|γ2dx 1 2 , which means BR
|Xu–Xw|2dx≤c
BR
|Xψ|2dx+c
BR
|˜f|+|Xu|γ2dx
+c
BR
|f|+|Xu|γ02Q/(Q+2)dx
Q+2
Q
. (4.4)
In view of 1≤γ0<QQ+2, 0≤γ < 1, it follows by Hölder’s inequality that
BR
|Xu|2γ0Q/(Q+2)dx
Q+2
Q
=
BR
|Xu|2γ0Q/(Q+2)dx
Q+2 2γ0Q2γ0
≤
|BR|
Q+2 2γ0Q–12
BR
|Xu|2dx 1
22γ0
=|BR|
Q+2
Q –γ0
BR
|Xu|2dx γ0
and
BR
|Xu|2γdx≤
|BR|
1 2γ–12
BR
|Xu|2dx 1
22γ
≤ |BR|1–γ
BR
|Xu|2dx γ
≤ε
BR
|Xu|2dx+cε|BR|. (4.6)
Combine (4.4)–(4.6) to deduce
BR
|Xu–Xw|2dx≤cω(R) +ε
BR
|Xu|2dx+c
BR
|Xψ|2dx
+c
BR
|f|2Q/(Q+2)dx Q+2
Q
+c
BR
|˜f|2dx+cε|BR|, (4.7)
whereω(R) =|BR|
Q+2
Q –γ0(
BR|Xu|
2dx)γ0–1.
From (4.7) and (4.3), we find, for any 0 <ρ≤R(we may supposeR< 1),
Bρ
|Xu|2dx≤c
ρ
R Q
+ε+ηR(A) +ω(R) BR
|Xu|2dx+c
BR
|Xψ|2dx
+c
BR
|f|2Q/(Q+2)dx (Q+2)/Q
+c
BR
|˜f|2dx+c|BR|
≤c
ρ
R Q
+ϑ(R)
BR
|Xu|2dx+c˜|BR| Rλ ,
whereϑ(R) =ε+ηR(A) +ω(R),c˜=˜c(f2L2Q/(Q+2),λQ/(Q+2)+˜f2L2,λ+Xψ2Lp,λ). By the absolute
continuity of Lebesgue integral, we know thatω(R)→0 asR→0. Finally, we can takeR< R0such thatηR(A) is small enough due to the VMO property ofA(x). If we takeF(ρ) =|Bρρ|λ,
0 <Q–λ<τ <Q, we claim that Fρ(τρ) =ρ|Bτ+λρ| is almost increasing. In fact, it follows from (2.2) that, for anyt∈(0, 1),
(tρ)τ
F(tρ)/
ρτ
F(ρ)= (tρ)τ+λ
|Btρ|
/ρ
τ+λ
|Bρ|
=t
τ+λ|B ρ|
|Btρ| ≤
C2dtτ+λ–Q≤Cd2.
By Lemma 4.1, we obtain, for 0 <ρ≤R,
Bρ
|Xu|2dx≤c|Bρ|
ρλ , (4.8)
which shows thatXu∈L2,Xλ,loc(,RmN).
On the other hand, from Poincaré inequality and (4.8) we see that
Bρ
|u–uρ|2dx≤cρ2
Bρ
|Xu|2dx≤c|Bρ|
which impliesu∈L2,Xλ,loc–2(,RN) and sou∈C0,(2–λ)/2
X (,RN) according to Lemma 2.5. The
proof is finished.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 11201258), National Science Foundation of Shandong Province of China (ZR2011AM008, ZR2011AQ006, ZR2012AM010).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 13 December 2017 Accepted: 23 February 2018
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