Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 33, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
MORREY ESTIMATES FOR SUBELLIPTIC p-LAPLACE TYPE
SYSTEMS WITH VMO COEFFICIENTS IN CARNOT GROUPS
HAIYAN YU, SHENZHOU ZHENG
Abstract. In this article, we study estimates in Morrey spaces to the horizontal gradient of weak solutions for a class of quasilinear sub-elliptic systems
of p-Laplace type with VMO coefficients under the controllable growth over
Carnot group if p is not too far from 2. We also show a local Hölder continuity
with an optimal exponent to the solutions.
1. Introduction
Let G be a Carnot group of step r ≥ 2, that is, a simply connected Lie group
with Lie algebra g admits a decomposition g = ⊕ri=1 Vj such that [V1 , Vj ] = Vj+1
for 1 ≤ P
j ≤ r − 1 and [V1 , Vr ] = 0. The homogeneous dimension of G is defined
r
as Q = i=1 imi , where mi = dim Vi is the topological dimension with m1 = m.
For a family of vector fields X = (X1 , X2 , . . . , Xm ) satisfying the Hörmander’s
finite rank condition rank Lie[X1 , X2 , . . . , Xm ] = r, we assume thatPeach compon
nent bij (x) (i = 1, 2, . . . , m; j = 1, 2, . . . , n) of vector-field Xi =
j=1 bij (x)∂j
is a smooth function defined in Carnot group G for i = 1, 2, . . . , m. Therefore,
Xu = (X1 u, X2 u, . . . , Xm u) may be called the horizontal gradient of u, which is
understood as Xi u = hXi , ∇ui = Σnj=1 bij (x)∂j u for i = 1, 2, . . . , m if u ∈ C 1 (G),
see [11, 15, 21, 7, 29]. To describe our assumptions and main results better, we first
recall some relevant notations and basic facts.
Definition 1.1. An absolutely continuous path γ : [0, T ] → G is said X-subunit if
γ̇(t) =
m
X
ci (t)Xi (γ(t))
i=1
with
Pm
i=1 ci (t)
≤ 1, for almost every t ∈ [0, T ].
With X-subunit in hand, we can define the Carnot-Caratheodory’s metric (the
C-C distance) dX (x, y) as follows, see [19, 27].
dX (x, y) = inf{T > 0 : ∃γ : [0, T ] → G X-subunit with γ(0) = x, γ(T ) = y}.
2010 Mathematics Subject Classification. 35H20, 35B65, 35D30.
Key words and phrases. Subelliptic p-Laplace; VMO coefficients; controllable growth;
Morrey regularity; Carnot group.
c
2016
Texas State University.
Submitted August 16, 2015. Published January 21, 2016.
1
2
H. YU, S. ZHENG
EJDE-2016/33
Note that these vector-fields (X1 , . . . , Xm ) are free up to the order r, then there
exists a positive constant C > 0 satisfying the following relation between the C-C
distance and the Euclidean metric, see [27, 10];
C −1 |x − y| ≤ dX (x, y) ≤ C|x − y|1/r .
In this context, all balls centered at x of radius R with respect to dX (x, y)
are called metric balls and denoted still by BR (x). The distance function dX (·, ·)
satisfies the local doubling property, that is, for B2R (x) b G there exists a positive
constant R0 depending on vector fields X and G such that for all 0 < R ≤ R0 there
holds
|B2R (x)| ≤ Cd |BR (x)|,
(1.1)
where the number Q = log2 Cd is called the local homogeneous dimension of G
with respect to the vector fields X1 , X2 , . . . , Xm . In fact, Q will play a role of the
dimension in the local analysis involving what we are considering problems.
Let Ω be a bounded open subset in Carnot group G. Let us recall the following horizontal Sobolev space with respect to given a family of vector fields
X = (X1 , X2 , . . . , Xm ). For any 1 < p < ∞ and k ∈ N, we define
HW k,p (Ω) := {u ∈ Lp (Ω) : Xi1 . . . Xik u ∈ Lp (Ω) for all {i1 , . . . , ik } ⊂ {1, . . . , m}}
with the norm kukHW k,p (Ω) = kukLp (Ω) + kX k ukLp (Ω) . Furthermore, the closure of
C0∞ (Ω) in HW k,p (Ω) is denoted by HW0k,p (Ω).
In this article, we consider the estimates in Morrey spaces to the horizontal
gradient of weak solutions in HW01,p (Ω) for the following degenerate subelliptic
systems of p-Laplace type.
m
X
p−2
(1.2)
−
Xi hA(x)Xi u, Xi ui 2 A(x)Xi u = B(x, u, Xu), a. e. x ∈ Ω,
i=1
T
where A(x) ∈ V M O L∞ (Ω) and B(x, u, Xu) satisfies a controllable growth. In
order to more precisely impose structural assumptions on A(x), B(x, u, Xu) and
state our main results, we need recall two useful notations (see[2, 17, 28]).
Definition 1.2 (BMO functions). Let Ω(x, r) = Ω ∩ Br (x). For any 0 < s < +∞,
we say u ∈ L1loc (Ω) belongs to BM O(Ω) if
Z
1
Mu (s) :=
sup
|u(y) − ūΩ(x,r) |dy < +∞,
x∈Ω,0<r<s |Ω(x, r)| Ω(x,r)
where
ūΩ(x,r)
Z
=–
1
u(y)dy =
|Ω(x,
r)|
Ω(x,r)
Z
u(y)dy.
Ω(x,r)
Definition 1.3 (VMO functions). Let Mu (s) be defined as above. We say u ∈
BM O(Ω) belongs to V M O(Ω) if
lim Mu (s) = 0,
s→0
where Mu (s) is called the V M O modulus of u.
Definition 1.4 (Morrey space). Let p ≥ 1, λ > 0. For u ∈ Lploc (Ω), if
Z
1/p
rλ
kukLp,λ (Ω) :=
< +∞,
sup
|u|p dx
X
x0 ∈Ω, 0<r≤d0 |Ω(x0 , r)| Ω(x0 ,r)
(1.3)
EJDE-2016/33
SUBELLIPTIC p-LAPLACE TYPE SYSTEMS
3
then u ∈ Lp,λ
X (Ω), where d0 = min{diam(Ω), RD }, and the norm of u is kukLp,λ (Ω) .
X
Lploc (Ω)
Definition 1.5 (Campanato space). Let p ≥ 1, λ > −p. If u ∈
satisfies
Z
1/p
rλ
|u|Lp,λ (Ω) :=
sup
|u − ux0 ,r |p dx
< +∞,
(1.4)
X
x0 ∈Ω, 0<r≤d0 |Ω(x0 , r)| Ω(x0 ,r)
then u ∈ Lp,λ
X (Ω), and has norm kukLp,λ (Ω) := kukLp (Ω) + |u|Lp,λ (Ω) .
X
X
On the basis of the above notation, we are now in a position to impose some
structure assumptions on A(x) and B(x, u, Xu) as follows.
(H1) (Uniform ellipticity) For A(x) = (aαβ
ij (x)), there exist L and ν, L ≥ ν > 0,
such that for a.e. x ∈ Ω and for any ξ ∈ RnN we have
α β
2
ν|ξ|2 ≤ aαβ
ij (x)ξi ξj ≤ L|ξ| ;
T
∞
(H2) aαβ
V M OX ;
ij (x) ∈ L (Ω)
(H3) (Controllable growth) The inhomogeneity B(x, u, Xu) satisfies
1
|B(x, u, Xu)| ≤ µ(|Xu|p(1− γ ) + g(x)),
where
(
γ=
pQ
Q−p ,
any γ ≥ p,
g(x) ∈ Lq,µ (Ω, RnN ) with q >
γ
γ−1 ,
1<p<Q
p ≥ Q,
and Q is the homogenous dimension.
We say that u ∈ HW 1,p (Ω, RnN ) is a weak solution of (1.2), if
Z
Z
p−2
hA(x)Xu, Xui 2 A(x)Xu, Xϕ dx =
B(x, u, Xu)ϕ dx,
Ω
(1.5)
Ω
for all ϕ ∈ HW01,p (Ω).
Recently several studies on subelliptic PDEs arising from non-commuting vector
fields have been well developed based on the Hörmander’s fundamental work [18];
see [4, 3, 10, 11, 15, 23, 24, 27, 22, 26, 29, 30]. Many important results about the
fundamental solution to subelliptic operators and the Harmonic analysis theory on
stratified nilpotent Lie groups have been obtained by Folland [15], Rothschild-Stein
[24] and Nagel-Stein-Wainger [23]. These results laid a solid foundation for further
investigation of subelliptic Partial Differiential Equations theory. Up to the 90s,
the function theory and harmonic analysis tools on Carnot groups, such as the
Sobolev embedding inequality of X-gradient and the isoperimetric inequality, become increasingly mature, cf. [3, 15, 16, 23, 24]. In fact, such subelliptic problems
have received continuous attention due to their significant applications in geome∂
try and physics [1, 24]. In the case of Euclidean spaces (i.e. m = n, Xi = ∂x
),
i
it was an important observation by Uhlenbeck [25] that there exists the interior
C 1,α -regularity by using the classical De Giorgi-Moser-Nash iteration to the homogeneous p-harmonic systems as a prototype. However, it is not true for subelliptic
systems of p-Laplacian with p > 1. Actually, Domokos-Manfredi in [11, 12] and
Domokos in [13, 14] have derived Γ1,α regularity for p-harmonic systems only while
p is in a neighborhood of 2 in Heisenberg group and in Carnot group, respectively.
Very recently, Zheng-Feng [29, 30] also got the estimates and an application of the
Green functions for subelliptic A-harmonic operators, and Γ1,α regularity for weak
solutions to subelliptic p-harmonic systems under the subcritical growth with p
4
H. YU, S. ZHENG
EJDE-2016/33
close to 2, respectively. Notice that Fazio-Fanciullo [8] and Dong-Niu [9] recently
established the estimates of the gradient in Morrey spaces to nonlinear subelliptic systems for p = 2. Therefore, this is a natural thought what happens if one
consider a regularity of the gradient in Morrey spaces to subelliptic A-harmonic
systems. In this article, we are devoted to local Morrey regularity of the horizontal
gradient to a class of subelliptic A-harmonic systems with VMO coefficients under
the controllable growth. More precisely, we have the following result.
Theorem 1.6. Let u ∈ HW 1,p (Ω, RN ) is a weak solution of (1.2) with p close to
nN
2. Suppose A(x) and B(x, u, Xu) satisfy (H1)–(H3). Then Xu ∈ Lp,λ
);
X (Ω, R
moreover, there exists a constant C > 0 such that for any Ω0 b Ω we have
1
),
kXukLp,λ (Ω0 ,RnN ) ≤ C(kXukLp (Ω,RnN ) + kgkLp−1
q,µ
(Ω,RnN )
X
(1.6)
X
where
(
λ=
p µ−q
p−1 q ,
γ
γ−1
any λ ∈ (0, Q),
< q < µ,
q ≥ µ.
Here, we employ a classical disturbance argument which is compared with subelliptic Laplacian due to the lack of regularity to subelliptic systems of p-Laplace.
Indeed, our approach on the singularity (1 < p < 2) and degeneracy (p > 2) for
A-harmonic systems (1.2) was essentially influenced by way of a comparison with
sub-Laplacian while p is close to 2 because of the Cordes conditions. This is an
important technique to consider PDEs with wild coefficients, also see [6].
With Theorem 1.6 in hand, as a direct consequence we can obtain an interior
Hölder continuity of weak solutions of subelliptic systems (1.2) while Q−n < λ < p.
Let us first recall the concept of Hölder space under the Carnot-Caratheodory
metric.
Definition 1.7 (Hölder space). Let Ω b G and 0 < α < 1. We say that u ∈ Γ0,α
X (Ω)
has norm kukΓ0,α (Ω) , if
X
kukΓ0,α (Ω) := sup |u| + sup
X
Ω
Ω
|u(x) − u(y)|
< ∞.
[dX (x, y)]α
(1.7)
Now we state the interior Hölder continuity of weak solutions with a sharp index
to subelliptic systems (1.2) as follows.
Theorem 1.8. Let u ∈ HW 1,p (Ω, RN ) is a weak solution of (1.2) with p close to
2. Suppose A(x) and B(x, u, Xu) satisfy H1-H3. If Q − n < λ < p, we have
N
u ∈ Γ0,α
X,loc (Ω, R )
with α = 1 − λp .
The remainder of this paper is organized as follows. In Section 2, we present
some preliminaries concerning the sub-elliptic setting and some several technical
lemmas. In Section 3, we are devoted to proving our main results.
2. Preliminaries
We adopt the usual convention of denoting by C a general constant, which may
vary from line to line in the same chain of inequalities. Now let us first recall
the Sobolev embedding inequality with respect to the horizontal vector fields, see
[3, 5, 16].
EJDE-2016/33
SUBELLIPTIC p-LAPLACE TYPE SYSTEMS
5
Qp
Lemma 2.1. Let 1 ≤ p < Q and 1 ≤ q ≤ Q−p
, where Q is the homogeneous
dimension of X in G. Then we have:
(1) If u(x) ∈ HW 1,p (BR0 , X), then there exists a positive constant
C = C(p, q, Q, X) such that for any 0 < R < R0 ,
1
1
ku − ūR kLq (BR ) ≤ CR1+Q( q − p ) kXukLp (BR ) ,
R
where ūR = |B1R | BR udx.
(2) If u ∈ HW01,q (BR0 , X), then there exists a positive constant
C = C(p, q, Q, X) such that for any 0 < R < R0 ,
Z
1/q
Z
1/p
q
– |u|
≤ CR – |Xu|p
.
BR
(2.1)
(2.2)
BR
To obtain the higher integrability of the horizontal gradients of solutions for
(1.2) we recall the following reverse Hölder inequality originated from Gehring’s
celebrating work on quasiconformal mappings, see [17, Theorem 2.3 of Chapter 5].
Lemma 2.2. Suppose that h(x) and f (x) are nonnegative measurable functions
satisfying h(x) ∈ Lt (Ω) and f (x) ∈ Ls (Ω) with t > s > 1. If for all x0 ∈ Ω and all
R : 0 < R < R0 ≤ dist(x0 , ∂Ω) there holds
Z
Z
nZ
os Z
hs dx + θ–
f s dx,
(2.3)
f s dx ≤ b –
f dx + –
–
B R (x0 )
BR (x0 )
BR (x0 )
BR (x0 )
2
with constants b > 1 and 0 ≤ θ < 1, then there exist positive constants δ =
δ(b, Q, q, s) and C = C(b, Q, q, r) such that f ∈ Ltloc (Ω) for any t ∈ [s, s + δ) and
nZ
o1/s
nZ
o1/t
o1/t
nZ
s
t
–
≤C –
f dx
+C –
ht dx
.
(2.4)
f dx
B R (x0 )
BR (x0 )
BR (x0 )
2
With the reverse Hölder inequality above in hand, We can obtain the following
higher integrability of the horizontal gradients to systems (1.2).
Lemma 2.3 (Higher integrability). Let u ∈ HW 1,p (Ω) be any weak solution of
quasilinear subelliptic systems (1.2) with A(x), B(x, u, Xu) satisfying assumptions
(H1) and (H3). Then, there exists a higher exponent r : p < r < p + δ such that
for Ω0 b Ω, we have Xu ∈ HW 1,r (Ω0 ). Moreover, there exists a positive constant
C = C(Q, L, ν, p) such that for any BR (x0 ) b Ω with the estimate
Z
1/r
–
|Xu|r )dx
B R (x0 )
2
Z
≤C –
BR (x0 )
p
|Xu| dx
1/p
Z
+C R –
1
1/q p−1
.
|g(x)| dx
(2.5)
q
BR (x0 )
Proof. Given any x0 ∈ Ω, we take R > 0 such that BR := BR (x0 ) b Ω. Let η be a
cutting-off function with η ∈ C0∞ (BR ) such that 0 ≤ η(x) ≤ 1, η = 1 for x ∈ BR/2 ,
p
η = 0 for x ∈ Rn \ B R and |Xη| ≤ C
R . Let us take a test function ϕ = η (u − ūR )
in (1.2), it follows from (1.5) that
Z D
E
p−2
hA(x)Xu, Xui 2 A(x)Xu, η p Xu + pη p−1 (u − ūR )Xη dx
Ω
Z
=
B(x, u, Xu)η p (u − ūR )dx,
Ω
6
H. YU, S. ZHENG
EJDE-2016/33
By considering the uniformly ellipticity (H1) and the controllable growth, it yields
Z
ν p/2
η p |Xu|p dx
Z Ω
≤
η p hA(x)Xu, Xuip/2 dx
Ω
Z
p−2
= − hhA(x)Xu, Xui 2 A(x)Xu, pη p−1 (u − ūR )Xηidx
ZΩ
(2.6)
+
B(x, u, Xu)η p (u − ūR )dx
Ω
Z
Z
1
p/2
≤ pL
|ηXu|p−1 |(u − ūR )Xη|dx + µ
η p |u − ūR ||Xu|p(1− γ ) dx
Ω
Z Ω
+µ
η p |u − ūR ||g(x)|dx
Ω
:= pLp/2 I1 + µI2 + µI3 .
Next, we estimate I1 , I2 and I3 . For I1 , using Young inequality with ε1 > 0 and
Sobolev inequality we have
Z
I1 =
|ηXu|p−1 |(u − ūR )Xη|dx
Ω
Z
Z
≤ ε1
|ηXu|p dx + C(ε1 )
|(u − ūR )Xη|p dx
Ω
Ω
Z
Z
(2.7)
C(ε1 )
p
p
|u
−
ū
|
dx
≤ ε1
|Xu| dx +
R
Rp
BR
BR
Z
Z
Q+p
Qp
C(ε1 )
Q
Q+p dx
≤ ε1
|Xu|p dx +
|Xu|
.
p
R
BR
BR
For estimating I2 , by Sobolev inequality and Hölder inequality, it follows that
Z
1
I2 =
η p |u − ūR ||Xu|p(1− γ ) dx
Ω
1/γ Z
1− γ1
Z
≤
|u − uR |γ dx
|Xu|p dx
(2.8)
BR
BR
Z
p1 − γ1 Z
1
1
≤ CR1+Q( γ − p )
|Xu|p dx
|Xu|p dx ,
BR
BR
is defined as the assumption H3 with 1 + Q( p1 − γ1 ) ≥ 0.
to estimate I3 we use Hölder inequality, Young inequality with ε2 > 0
inequality; it yields
where γ ≥ p
Similarly,
and Sobolev
Z
I3 =
η p |u − ūR ||g(x)|dx
Ω
≤
Z
γ
|u − ūR | dx
BR
≤ CR
γ−1
γ
γ
|g(x)| γ−1 dx
1/γ Z
BR
1+Q( γ1
1
−p
)
Z
1/p Z
|Xu|p dx
BR
Z
≤ ε2
BR
γ
|g(x)| γ−1 dx
γ−1
γ
BR
p
|Xu|p dx + C(ε2 )R p−1 (1+Q( γ − p ))
1
1
Z
BR
p
γ−1
γ
γ · p−1
|g(x)| γ−1 dx
.
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SUBELLIPTIC p-LAPLACE TYPE SYSTEMS
7
Now let us put the estimates of I1 , I2 , I3 together into (2.6), we obtain
Z
|ηXu|p dx
BR
≤
C(L, p, ε1 ) Rp
+ C(µ, ε2 )R
Z
Qp
|Xu| Q+p dx
Q+p
Q
BR
p
γ−1
γ
γ · p−1
|g(x)| γ−1 dx
Z
p
p−1
1
1+Q( γ1 − p
)
BR
p1 − γ1 o Z
|Xu| dx
n
Z
1
1+Q( γ1 − p
)
p/2
+ µε2 + pL ε1 + µCR
|Xu|p dx .
p
BR
BR
Let us write
1
1
ϑ = µε2 + pLp/2 ε1 + µCR1+Q( γ − p )
Z
|Xu|p dx
p1 − γ1
.
BR
Notice that from the absolute continuity of the Lebesgue integral, we have that
1
1 R
R1+Q( γ − p ) BR |Xu|p → 0 as R → 0. Consequently we can take small R > 0 such
that 0 < ϑ < 1, and
Z
Z
Q+p
Qp
C Q
p
|Xu| dx ≤ p
|Xu| Q+p dx
R
BR
BR
2
Z
p
γ−1
Z
γ
p
1
1
γ · p−1
|g(x)| γ−1 dx
+ϑ
|Xu|p dx,
+ CR p−1 (1+Q( γ − p ))
BR
which implies
Z
Z
p
– |Xu| dx ≤ C –
BR
|Xu|
Qp
Q+p
BR
Q+p
Z
Q
dx
+C –
BR
γ0
|Rg(x)| dx
p00
γ
Z
+ ϑ–
BR
|Xu|p dx,
BR
2
with p0 =
p
p−1
and γ 0 =
γ
γ−1 .
Therefore, we obtain
Z
1/p
≤C –
|Xu|p dx
Z
–
Qp
|Xu| Q+p dx
Q+p
pQ
Z
+C –
BR
BR
(|Rg(x)|γ
0
/p p
) dx
p1 p00
γ
BR
2
Z
+ ϑ1/p –
1/p
|Xu|p dx
.
BR
Using the reverse Hölder inequality of Lemma 2.2, it yields
Z
1/r
Z
1/p
Z
r1 p00
γ
r
p
γ 0 /p r
– |Xu| dx
≤ C – |Xu| dx
+ C – (|Rg(x)|
) dx
,
BR
BR
(2.9)
BR
2
for some p < r ≤
Z
–
pq(γ−1)
γ
due to q >
) dx
γ 0 /p r
(|Rg(x)|
0
1 p
r γ0
γ
γ−1 .
=R
Note that
1
p−1
Z
–
1
p−1
Z
–
BR
1
pr · γ−1
γ
r
γ · p−1
|g(x)| γ−1 · p dx
BR
≤R
1
q1 · p−1
|g(x)|q dx
,
BR
because r ≤
pq(γ−1)
,
γ
then we obtain (2.5) which completes the proof.
8
H. YU, S. ZHENG
EJDE-2016/33
The following elementary inequalities concerning A(x) are useful to our main
proof, see [20].
Lemma 2.4. Suppose that A = (Aij ) is a symmetric matrix and satisfies uniform
ellipticity (H1). Then there exists a positive constant C = C(p, ν, L) such that for
1 < p < ∞ we have
p−2
p−2
p−2
hAξ, ξi 2 Aξ − hAη, ηi 2 Aη, ξ − η ≥ C(|ξ|2 + |η|2 ) 2 |ξ − η|2 .
(2.10)
In addition, for p ≥ 2 there holds
p−2
p−2
hAξ, ξi 2 Aξ − hAη, ηi 2 Aη, ξ − η ≥ ν p/2 |ξ − η|p ;
(2.11)
and for 1 < p < 2, there exists C = C(p, ν, L) such that for every 0 < ε < 1 we
have
p−2 p−2
p−2
(2.12)
|ξ − η|p ≤ Cε p hAξ, ξi 2 Aξ − hAη, ηi 2 Aη, ξ − η + ε|η|p .
To use Campanato’s freezing argument, we obverse the following local Dirichlet
problems of homogeneous elliptic systems with constant coefficients.
−X ∗ (hAR Xv, Xvi
p−2
2
v = u,
AR Xv) = 0,
x ∈ ∂BR ,
x ∈ BR
(2.13)
R
where AR = |B1R | BR A(x)dx is the integral average of A(x). Now we recall [30,
Lemma 3.4] while p is close to 2, and have the following perturbation estimates.
Lemma 2.5. Let v ∈ HW 1,p (Ω) be a weak solution to the Dirichlet problems (2.13)
with p close to 2. Then for any u ∈ HW 1,p (Ω), there exists C > 0 such that for
any x0 ∈ Ω we have
Z
Z
Z
ρ Q
|Xu|p dx ≤ C
|Xu|p dx + C
|Xu − Xv|p dx, (2.14)
R
Bρ (x0 )
BR (x0 )
BR (x0 )
for all 0 < ρ < R ≤ R0 .
Moreover, by a direct calculation we obtain the following conclusion, see [30,
Lemma 6].
Lemma 2.6. Let v ∈ HW 1,p (Ω) be any weak solution to the Dirichlet problem
(2.13) with any x0 ∈ Ω and 0 < R ≤ R0 . Then we have
Z
Z
|Xv|p dx ≤ C
|Xu|p dx.
BR (x0 )
BR (x0 )
In addition, we need the following iteration lemma from [17] in the proof of our
main theorem.
Lemma 2.7. Let Φ(ρ) be a non-negative and non-decreasing function on (0, R).
Suppose that
ρ α
Φ(ρ) ≤ A
+ Φ(R) + BRβ , ∀ 0 < ρ < R ≤ R0 = dist(x0 , ∂Ω),
R
with non-negative constants A, B, α and β, and α > β. Then there exist two constants 0 = 0 (A, α, β) and C = C(A, α, β) such that for any 0 < < 0 we have
ρ β
Φ(R) + Bρβ ,
Φ(ρ) ≤ C
R
for any 0 < ρ < R ≤ R0 = dist(x0 , ∂Ω).
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SUBELLIPTIC p-LAPLACE TYPE SYSTEMS
9
Finally, the following equivalence of spaces is useful to prove a local Hölder
continuity of the weak solutions based on the main Theorem, see [9, 8].
Lemma 2.8. If 0 < λ < Q, the Campanato space Lp,λ
X (Ω) is isomorphic to the
Morrey space Lp,λ
(Ω).
If
−p
<
λ
<
0,
then
the
Campanato
space Lp,λ
X
X (Ω) is
0,α
isomorphic to the Hölder space ΓX (Ω) with α = −λ/p.
3. Proof of the main results
Proof of theorem 1.6. Let u(x) ∈ HW 1,p (Ω) be any weak solution of (1.2). To
obtain an interior estimate for the horizontal gradients to the solutions, for any
fixed point x0 ∈ Ω let us take a ball BR (x0 ) b Ω, and write BR =: BR (x0 ) in the
context. Suppose that v(x) is a weak solution of the local Dirichlet problem (2.13).
Computing the difference between (1.2) and (2.13) yields
p−2
p−2
− X ∗ hA(x)Xu, Xui 2 A(x)Xu − hAR Xu, Xui 2 AR Xu
p−2
p−2
− X ∗ hAR Xu, Xui 2 AR Xu − hAR Xv, Xvi 2 AR Xv
= −B(x, u, Xu).
Let us take ϕ = u − v as a test function in the weak sense to the equations above,
then we have
Z
p−2
p−2
hAR Xu, Xui 2 AR Xu − hAR Xv, Xvi 2 AR Xv, Xu − Xv dx
BR
Z
p−2
p−2
=
hAR Xu, Xui 2 AR Xu − hA(x)Xu, Xui 2 A(x)Xu, Xu − Xv dx (3.1)
BR
Z
+
hB(x, u, Xu), u − vidx.
BR
Note that (2.14) implies that if p is close to 2, for any 0 < ρ < R there holds
Z
Z
Z
ρ Q
|Xu|p dx ≤ C
|Xu|p dx + C
|Xu − Xv|p dx. (3.2)
R
Bρ (x0 )
BR (x0 )
BR (x0 )
R
Next, we focus on the estimate of BR (x0 ) |Xu − Xv|p dx. To that end, we will
estimate it by dividing into two cases.
Case 1: p ≥ 2. Let us put an elementary inequality (2.11) and (H3) into the
equation (3.1) above it follows that
Z
p/2
ν
|Xu − Xv|p dx
BR
Z
p−2
p−2
≤
hAR Xu, Xui 2 AR Xu − hAR Xv, Xvi 2 AR Xv, Xu − Xv dx
B
Z R
p−2
p−2
=
hAR Xu, Xui 2 AR Xu − hA(x)Xu, Xui 2 A(x)Xu, Xu − Xv dx
BR
Z
+
hB(x, u, Xu), u − vidx
B
Z R
Z
1
≤C
|AR − A(x)||Xu|p−1 |Xu − Xv|dx + µ
|Xu|p(1− γ ) · |u − v|dx
BR
BR
10
H. YU, S. ZHENG
EJDE-2016/33
Z
|g| · |u − v|dx
+µ
BR
:= J1 + J2 + J3 .
(3.3)
For estimating J1 , the Hölder inequality and Young inequalities with any ε > 0
yield
Z
|AR − A(x)||Xu|p−1 |Xu − Xv|dx
BR
Z
1− p1 Z
|Xu| dx
1/p
|Xu − Xv|p dx
|AR − A(x)|
BR
BR
Z
Z
p
p
p−1
|Xu| dx + ε4
≤ C(ε4 )
|Xu − Xv|p dx
|AR − A(x)|
BR
BR
1− pr Z
p/r
Z
p
r
· r−p
p−1
dx
– |Xu|r dx
≤ C(ε4 )|BR | – |AR − A(x)|
BR
BR
Z
+ ε4
|Xu − Xv|p dx,
≤
p
p−1
p
BR
where an r > p is the same integrable index as that in Lemma 2.3. Setting t =
pr
(p−1)(r−p) , and by a higher integrability from Lemma 2.3 we obtain
p Z
1− pr Z
Z
q1 · p−1
p
t
p
p−1
J1 ≤ C(ε4 )|BR | – |AR − A(x)| dx
– |g|q dx
– |Xu| dx + R
BR
BR
BR
Z
p
+ ε4
|Xu − Xv| dx
BR
Z
p
p
q−Q
p
|Xu|p dx + CRQ+ p−1 · q kgkLp−1
≤ C(ε4 )MA (R)1− r
q
BR
Z
+ ε4
|Xu − Xv|p dx.
BR
To estimate J2 , we employ Höder inequality and Young inequality again, and obtain
Z
1− γ1 Z
1/γ
J2 ≤ µ
|Xu|p dx
|u − v|γ dx
B
BR
Z R
Z
p
≤ ε5
|Xu| dx + C(µ, Q, p, ε5 )
|u − v|γ dx
BR
BR
Z
≤ ε5
p
|Xu| dx + CR
γ+Q(1− γ
p)
Z
p
|Xu − Xv| dx
BR
BR
γ
Observing δ(R) := Rγ+Q(1− p )
holds
Z
R
J2 ≤ Cδ(R)
|Xu − Xv|p dx.
BR
|Xu − Xv|p dx
BR
p
γ−p
p
→ 0 as R → 0, then there
Z
|Xu − Xv| dx + ε5
BR
Z
γ−p
p
|Xu|p dx.
(3.4)
BR
To estimate J3 , by using Hölder inequality, Sobolev embedding inequality and
Young inequality it follows that
Z
γ−1
Z
1/γ
γ
γ
J3 ≤ µ
|g| γ−1 dx
|u − v|γ dx
BR
BR
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≤
SUBELLIPTIC p-LAPLACE TYPE SYSTEMS
Z
γ−1
γ
1
1
γ
|g| γ−1 dx
R1+Q( γ − p )
Z
BR
p
|Xu − Xv| dx + C(ε6 )R
p
1
p−1 [1+Q( γ
1
−p
)]
BR
|Xu − Xv|p dx + CR
≤ ε6
p
γ−1
γ
γ · p−1
|g| γ−1 dx
Z
BR
Z
1/p
BR
Z
≤ ε6
|Xu − Xv|p dx
11
p
Q+ p−1
· q−Q
q
BR
p
p−1
Lq
kgk
.
Putting estimates of J1 , J2 and J3 together in (3.3), one deduces
Z
p/2
ν
|Xu − Xv|p dx
BR
Z
Z
p
1− p
r
+ ε5
≤ C δ(R) + ε4 + ε6
|Xu − Xv| dx + C(ε4 )MA (R)
p
+ CRQ+ p−1 ·
q−Q
q
|Xu|p dx
BR
BR
p
kgkLp−1
q .
Therefore, by choosing arbitrary positive constants ε4 , ε6 and 0 < R < R0 small
enough that C(δ(R) + ε4 + ε6 ) < ν p/2 we obtain
Z
|Xu − Xv|p dx
BR
Z
p
p
q−Q
1− p
r
≤ C(ε4 )MA (R)
|Xu|p dx + CRQ+ p−1 · q kgkLp−1
+ ε5
(3.5)
q
BR
Z
p
p
q−µ
p
≤ C(ε4 )MA (R)1− r + ε5
|Xu|p dx + CRQ+ p−1 · q kgkLp−1
q,µ .
X
BR
1− p
r
+ ε5 in (3.5), and put the estimate (3.5) into (3.2),
Let us set $ = C(ε4 )MA (R)
then for any 0 < ρ < R we have
Z
Z
p
ρ Q
q−µ
p
p
|Xu| dx ≤ C
(3.6)
+$
|Xu|p dx + CRQ+ p−1 · q kgkLp−1
q,µ .
X
R
Bρ
BR
p
While q ≥ µ such that Q + p−1
· q−µ
q ≥ Q, it follows from Lemma 2.7 that
Z
Z
p
ρ Q−λ
|Xu|p dx ≤ C
|Xu|p dx + CρQ−λ kgkLp−1
,
q,µ
X (BR )
R
Bρ
BR
(3.7)
0
0
for any 0 < λ < Q. This implies Xu ∈ Lp,λ
X,loc (Ω ) for Ω b Ω with the estimate
1
kXukLp,λ (Ω0 ) ≤ C kXukLp (Ω) + kgkLp−1
.
q,µ
(Ω)
X
While
that
Z
γ
γ−1
X
< q < µ such that Q +
|Xu|p dx ≤ C
Bρ
p µ−q
ρ Q− p−1
q
R
p,
p
p−1
which implies Xu ∈ LX,loc
kXuk
p
p−1
p,
·
q−µ
q
Z
< Q, then we deduce from Lemma 2.7
p
|Xu|p dx + CρQ− p−1 ·
p
p
kgkLp−1
q,µ ,
X
BR
µ−q
q
LX p−1
µ−q
q
(Ω0 ) with the estimate
1
µ−q
≤ C kXukLp (Ω) + kgkLp−1
.
q,µ
(Ω)
q
(Ω0 )
X
(3.8)
12
H. YU, S. ZHENG
EJDE-2016/33
Case 2: 1 < p < 2. Using inequality (2.12) yields
Z
|Xu − Xv|p dx
Bρ
≤ Cε
Z
p−2
2
hAR Xu, Xui
p−2
2
AR Xu − hAR Xv, Xvi
p−2
2
AR Xv, Xu − Xv dx
Bρ
Z
|Xu|p dx
+ε
Bρ
Z
= C(ε)
p−2
p−2
hAR Xu, Xui 2 AR Xu − hA(x)Xu, Xui 2 A(x)Xu, Xu − Xv dx
Bρ
Z
|Xu|p dx
+ B(x, u, Xu) + ε
Bρ
Z
p−1
≤ C(p, ν, L)
|AR − A(x)||Xu| |Xu − Xv|dx + µ
BR
Z
Z
+µ
|g| · |u − v|dx + ε
|Xu|p dx
BR
BR
Z
p
|Xu| dx.
:= J1 + J2 + J3 + ε
Z
1
|Xu|p(1− γ ) · |u − v|dx
BR
BR
Considering the estimates of J1 , J2 , J3 in Case 1, and for any ε > 0 we obtain
Z
Z
p
ρ Q
p
q−µ
|Xu|p dx ≤ C
|Xu|p dx + CRQ+ p−1 · q kgkLp−1
+ $0
(3.9)
q,µ ,
X
R
Bρ
BR
p
with $0 = C(ε4 )MA (R)1− r + ε5 + ε. While q ≥ µ, it follows form Lemma 2.7 as
the same as Case 1 that
Z
Z
p
ρ Q−λ
|Xu|p dx ≤ C
|Xu|p dx + CρQ−λ kgkLp−1
q (B ) ,
R
R
Bρ
BR
0
0
for any 0 < λ < Q. It yields Xu ∈ Lp,λ
X,loc (Ω ) for any Ω b Ω with the estimate
1
.
kXukLp,λ (Ω0 ) ≤ C kXukLp (Ω) + kgkLp−1
q,µ
(Ω)
X
While
X
γ
γ−1
< q < µ, by Lemma 2.7 we obtain
Z
p
p µ−q
p
µ−q
ρ Q− p−1
q
|Xu|p dx ≤ C
|Xu|p dx + CρQ− p−1 · q kgkLp−1
q,µ .
X
R
Bρ
BR
Z
p,
p
p−1
which implies Xu ∈ LX,loc
kXuk
p,
µ−q
q
p
LX p−1
(Ω0 ) with the estimate
1
µ−q
≤ C kXukLp (Ω) + kgkLp−1
.
q,µ
(Ω)
q
(Ω0 )
X
This completes the proof.
Proof of Theorem 1.8. For any 0 < ρ < R with BR (x0 ) b Ω, by Poincare inequality
we have
Z
Z
|u − uρ |p dx ≤ Cρp
|Xu|p dx ≤ Cρp ρQ−λ kXukp p,λ
.
(3.10)
Bρ
Bρ
LX (Bρ )
EJDE-2016/33
SUBELLIPTIC p-LAPLACE TYPE SYSTEMS
13
Note that by Theorem 1.6,
kXukp p,λ (Bρ ) ≤ C
LX
1 Q−λ
R
Z
BR
p
|Xu|p dx + CkgkLp−1
.
q,µ
(BR )
(3.11)
X
Therefore, combining (3.10) and (3.11) yields
Z
Z
p
ρλ−p
1 Q−λ
,
|u − uρ |p dx ≤ C
|Xu|p dx + CkgkLp−1
q,µ
X (BR )
|Bρ | Bρ
R
BR
(3.12)
which implies
p,λ−p
u ∈ LX,loc
(Ω).
Note that Q − n < λ < p implies −p < Q − n − p < λ − p < 0, it follows from
Lemma 2.8 that
u ∈ Γ0,α
X,loc (Ω),
λ
with α = − λ−p
p = 1 − p.
Remark 3.1. We would like to point out that Theorems 1.6 and 1.8 are valid
only under the assumption of p close to 2 when we consider subelliptic (1.2) in
Carnot group. Indeed, we employ the perturbation inequality (2.14) in our main
proof, which is attained by a local Lipschitz boundedness of subelliptic p-harmonic
only if p close to 2. However, it is not necessary to limit p close to 2 if X =
∂
∂
∂
n
∂x1 , ∂x2 , . . . , ∂xn is a classical usual gradient in the Euclidian spaces R .
Acknowledgments. This work is supported by NSF of China under 11371050,
and by the NSF of China under grants 2013AA013702 (863) and 2013CB834205
(973).
References
[1] Bensoussan, A.; Frehse, J.; Regularity results for nonlinear elliptic systems and applications.
Applied Math. Sci. Vol., 151, Springer-Verlag, Berlin, 2002.
[2] Bramanti, M.; Brandolini,L.; Lp -estimates for nonvariational hypoelliptic operators with
VMO coefficients. Trans. Amer. Math. Soc., 352 (2002), 781-822.
[3] Capogna, L.; Danielli, D.; Garofalo, N.; An embedding theorem and the Harnack inequality
for nonlinear subelliptic equations. Comm. Partial Differential Equations, 18(9-10)(1993),
1765-1794.
[4] Capogna, L.; Garofalo, N.; Regularity of minimizers of the calculus of variations in Carnot
groups via hypoellipticity of systems of Hörmander type. J. Eur. Math. Soc., 5(1)(2003),1-40.
[5] Capogna, L.; Han, Q.; Pointwise Schauder estimates for second order linear equations in
Carnot groups. Proceedings for the AMS-SIAM Harmonic Analysis conference in Mt. Holyhoke, 2001.
[6] Caffarelli, L. A.; Peral, I.; On W 1,p estimates for elliptic equations in divergence form.
Comm. Pure Appl. Math., 51(1)(1998), 0001-0021.
[7] Citti, L.; Garofalo, N.; Lanconelli, E.; Harnacks inequality for sum of squares of vector fields
plus a potential. Amer. J. Math., 115(1993), 699-734.
[8] Di Fazio, G.; Fanciullo, M. S.; Gradient estimates for elliptic systems in Carnot-Carathéodory
spaces. Comment. Math. Univ. Caroline., 43 (4)(2002), 605-618.
[9] Dong, Y., Niu, P. C.; Estimates in Morrey spaces and Hölder continuity for weak solutions
to degenerate elliptic systems. Manuscripta Math., 138(2012), 419-437.
[10] Di Fazio, G.; Zamboni,P.; Hölder continuity for quasilinear subelliptic eauations in CarnotCarathéodory spaces. Math. Nachr., 272(2004), 3-10.
[11] Domokos, A.; Manfredi, J. J.; Subelliptic Cordes estimates. Proc. Amer. Math. Soc.,
133(2005), 1047-1056.
[12] Domokos, A.; Manfredi, J. J.; C 1,α -regularity for p-harmonic functions in the Heisenberg
group for p near 2. Contemp. Math., 370 (2005), 17-23.
14
H. YU, S. ZHENG
EJDE-2016/33
[13] Domokos, A.; On the regularity of p-harmonic functions in the Heisenberg group. Doctoral
Dissertation, University of Pittsburgh, 2004.
[14] Domokos, A.; On the regularity of subelliptic p-harmonic functions in Carnot groups. Nonlinear Analysis, 69(2008), 1744-1756.
[15] Folland, G.; A fundamental solution for a subelliptic operator. Bull. Amer. Math. Soc.,
79(1973), 373-376.
[16] Garofalo, N.; Nhieu, D. M.; Isoperimetric and Sobolev inequalities for Carnot-Carathéodory
spaces and the existence of minmal surfaces. Comm. Pure Appl. Math., 49(10)(1996), 10811144.
[17] Giaquinta, M.; Multiple integrals in the calculus of variations and nonlinear elliptic systems.
in ’Ann. Math. Stud.’, Princeton University Press, Vol. 105, 1983.
[18] Hörmander, L.; Hypoelliptic second order differential equations. Acta Math., 119(1967), 147171.
[19] Jost, J.; Xu, C. J.; Subelliptic harmonic maps. Trans. Amer. Math. Soc., 350(11)(1998),
4633-4649.
[20] Kinnunen, J.; Zhou, S. L.; A local estimate for nonlinear equations with discontinuous coefficients. Comm. Part. Diff. Equ., 24(11-12)(1999), 2043-2068.
[21] Lu, G.; The sharp Poincaré inequality for free vector fields: An endpoint result. Rev. Mat.
Ibe., 10(1994), 453-466.
[22] Mingione, G.; Zatorska-Goldstein, A.; Zhong, X.; Gradient regularity for elliptic equations
in the Heisenberg group. Adv. Math., 222 (2009) 62-129.
[23] Nagel, A.; Stein, E.; Wainger, S.; Balls and metrics defined by vector fields I: basic properties.
Acta Math., 155(1985), 103-147.
[24] Rothschild, L.; Stein, E.; Hypoelliptic operators and nilpotent Lie groups. Acta Math.,
137(1977), 247-320.
[25] Uhlenbeck, K.; Regularity for a class of nonlinear elliptic systems. Acta Math., 138(1977),
219-240.
[26] Wang, J.; Liao, D.; Optimal partial regularity for sub-elliptic systems with sub-quadratic
growth in Carnot groups. Nonlinear Analysis, 75(2012), 2499-2519.
[27] Xu, C.; Zuily, C.; Higher interior regularity for quasilinear subelliptic systems. Calc. Var.
Partial Differential Equations, 5(1997), 323-343.
[28] Yu, H.; Zheng, S.; Optimal partial regularity for quasilinear ellipyic systems with VMO
coefficients based on A-harmonic approximations. Electronic Journal of Differential Equation,
16(2015), 1-12.
[29] Zheng, S.; Feng, Z.; Green functions for a class of nonlinear degenerate operators with Xellipticity. Trans. Amer. Math. Soc., 364(7)(2012), 3627-3655.
[30] Zheng, S.; Feng, Z.; Regularity of subelliptic p-harmonic systems with subcritical growth in
Carnot group. J. Differential Equations, 258(2015), 2471-2494.
Haiyan Yu
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China.
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, 028043,
China
E-mail address: 12118381@bjtu.edu.cn
Shenzhou Zheng (corresponding author)
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China
E-mail address: shzhzheng@bjtu.edu.cn