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- TẠP CHÍ KHOA HỌC HO CHI MINH CITY UNIVERSITY OF EDUCATION
TRƯỜNG ĐẠI HỌC SƯ PHẠM TP HỒ CHÍ MINH JOURNAL OF SCIENCE
Tập 18, Số 9 (2021): 1359-1367 Vol. 18, No. 9 (2021): 1359-1367
ISSN:
2734-9918 Website: http://journal.hcmue.edu.vn
Research Article *
BẤT ĐẲNG THỨC CACCIOPOLI CÓ TRỌNG
CHO NGHIỆM CỦA PHƯƠNG TRÌNH P-LAPLACE
Trần Quang Vinh
Trường Đại học Sư phạm Thành phố Hồ Chí Minh, Việt Nam
Corresponding author: Tran Quang Vinh – Email: tranvinh3111@gmail.com
Received: June 01, 2021; Revised: August 16, 2021; Accepted: September, 2021
TÓM TẮT
Không gian Sobolev cấp phân số có trọng có nhiều ứng dụng trong phương trình đạo hàm
riêng. Trong bài báo này, chúng tôi khảo sát lớp không gian Sobolev cấp phân số có trọng, ứng với
hàm trọng là hàm khoảng cách đến biên của miền xác định. Lớp không gian này được sử dụng để
thu được một dạng bất đẳng thức dạng Cacciopoli có trọng cho bài toán p-Laplace với dữ liệu độ
đo. Kết quả của chúng tôi là mở rộng của bất đẳng thức Cacciopoli trong bài báo gần đây (Tran &
Nguyen, 2021b).
Từ khoá: bất đẳng thức dạng Cacciopoli; phương trình đạo hàm riêng; phương trình p-
Laplace; không gian Sobolev cấp phân số có trọng
1. Introduction
In this paper, we are interested in the following Dirichlet problem with measure data
)) µ
−div ( ( x, ∇u= in Ω,
(1.1)
u = 0 on ∂Ω,
where the domain Ω ⊂ n is open and bounded, and the given data µ is a Borel measure with
p −2
finite mass in Ω . The operator is close to the operator ξ ξ ξ , ξ ∈ n , this means
g1 ( ξ ) Id n ≤ ∂ξ ( , ξ ) ≤ g 2 ( ξ ) Id n ,
where g1 ( ξ ) ≈ g 2 ( ξ ) ≈ ξ . It is well-known that when p = 2 , if the data µ belongs to
p −2
the Lebesgue space Lqloc ( Ω ) then ∇u belongs to the Sobolev space Wloc
1, q
(Ω) :
µ ∈ Lqloc ( Ω ) ⇒ ∇u ∈ Wloc
1, q
( Ω ) , 1 < q < ∞. (1.2)
We hope that (1.2) still true for q = 1 , but instance, in the recent paper by Avelin et
al. in (Avelin, Kuusi & Mingione, 2018), the authors showed that the result just holds for
the fractional Sobolev spaces. More precisely, they proved that
Cite this article as: Tran Quang Vinh (2021). Using creative methodology to explore factors influencing
teacher educator identity. Ho Chi Minh City University of Education Journal of Science, 18(9), 1359-1367.
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- HCMUE Journal of Science Vol. 18, No. 9 (2021): 1359-1367
µ ∈ L1loc ( Ω ) ⇒ ∇u ∈ Wloc
σ ,1
( Ω ) , 0 < σ < 1. (1.3)
Moreover, also in the same paper, authors gave a very important regularity result
1
when 2 − < p ≤ 2 . Let us recall the following theorem for the reader's convenience:
n
Theorem 1.1. (Avelin, Kuusi & Mingione, 2018) Let Ω be an open subset of n and
1 1,max{1, p −1}
p > 2 − . Assume that u ∈ Wloc (Ω) is a SOLA solution to (1.1). Then for any
n
σ ∈ (0,1) one has
σ ,1
(∇u ) ∈ Wloc (Ω). (1.4)
constant C C (c , σ , n, p ) > 0 such that
Moreover, there exists a=
1 ( ∇u ( x ) ) − ( ∇u ( y ) )
µ ( BR /2 ) ∫BR /2 ∫BR /2 n +σ
dxdy
x− y
(1.5)
C 1 µ ( BR )
≤ σ ∫ ( ∇u ( x ) ) dx + n −1 ,
R µ ( BR ) BR R
for every ball BR Ω .
We remark that the weak solution to the measure data problem (1.1) may be not
unique. To ensure the existence and uniqueness of solution to (1.1), we deal with the
SOLA solution which has been defined in (Benilan et al., 1995) and (Maso et al., 1999).
There are lots of interesting results related to regularity for solutions to the measure data
problem (1.1), such as (Mingione, 2007), (Tran & Nguyen, 2019, 2020a, 2021a), (Balci et
al., 2020), etc.
Recently, Tran & Nguyen established the global regularity result of (1.4) in (Tran &
Nguyen, 2021). However, they only proved that (∇u ) belongs to the weighted fractional
Sobolev space, even for the smooth domain Ω . In the present article, we improve the
result in (Tran & Nguyen, 2021) by proving the inequality similar to (1.5), where the
weights are both on the left-hand and right-hand side. In other word, we prove the
following inequality
α
( ∇u ( x ) ) − ( ∇u ( y ) )
∫Ω ∫Ω d ( x )d β ( y ) n +σ
dxdy
x− y (1.6)
≤C (∫Ω )
d γ ( x ) ( ∇u ( x ) ) dx + µ ( Ω ) ,
where =
d ( x ) : dist( x, ∂Ω) defines the distance from x to the boundary of the domain. Here
the result holds for every α , β > 0 and γ ≥ 0 satisfying α > γ , β > γ , α + β − γ > σ .
Motivated by these works, we first consider some basic properties of the weighted
fractional Sobolev spaces, which the weights are the power of distances to the boundary.
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- HCMUE Journal of Science Tran Quang Vinh
Then we prove the weighted Cacciopoli type inequality (1.6) which corresponds to SOLA
solution to the measure data problem (1.1).
The rest of the article will be organized as follows. In the next section, we introduce
the weighted fractional Sobolev spaces by introducing some basic notation, definitions and
some properties of weighted fractional Sobolev spaces. Then, we end up with a section that
introduces the main results and proving the main results in this paper, and it allows us to
conclude a weighted approach for Cacciopoli inequality for solutions to p-Laplace
equations (1.1).
2. Preliminaries
2.1. Basic notation
In this article, the constant depends on real numbers α , β and γ will be denoted by
C (α , β , γ ) . From now on, Bρ (ζ ) stands for the ball with radius ρ and centered at ζ ∈ Ω .
Finally, for 1 ≤ p < ∞ , we will denote by Lp ( Ω ) the usual Lebesgue spaces; and the
Sobolev spaces is signed as W s , p (Ω) .
2.2. Fractional Sobolev spaces
We now introduce the definition of fractional Sobolev spaces, see (Avelin, Kuusi &
Mingione, 2018) and (Di Nezza, Palatucci & Valdinoci, 2012) for instance.
Definition 2.1. (The fractional Sobolev space) Assume that Ω ⊂ n is an open set wth
n ≥ 2 , s is the fractional in (0,1) and p ∈ [1, +∞) . Then, the fractional Sobolev space
WGs , p (Ω) is defined as follow
| u( x ) − u( y ) |
W (Ω) := u ∈ L (Ω) :
G
s, p p
n
∈ L (Ω × Ω) ,
p
(2.1)
+s
| x − y |p
with the natural norm
1
u ( x) − u ( y )
p p
u W s , p ( Ω ) ∫ u ( x) dx + ∫ ∫
p
= n + sp
dxdy . (2.2)
G Ω ΩΩ x− y
The Gagliardo semi-norm of u is defined by
1
| u ( x) − u ( y ) | p p
[u ]W s , p (Ω) := ∫ ∫ n + sp
dxdy . (2.3)
G Ω Ω | x − y |
Furthermore, we defined WGσ,loc ,1
(Ω) as
WGσ,loc
,1
(Ω)=: {v ∈W σ ,1
G (Ω1 ) : ∀Ω1 ⊂ Ω, Ω1 is compact . } (2.4)
Let us introduce some properties of weighted fractional Sobolev spaces
Lemma 2.2. Assume that Ω ⊂ n is an open domain, p ∈ [1, +∞) and u : Ω → . Then
u W s , p ( Ω ) ≤ u W t , p ( Ω ) , for all t ∈ ( s,1) .
G G
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It follows that
WGt , p ( Ω ) ⊆ WGs , p ( Ω ) , for all t ∈ ( s,1) .
If we have Ω is the bounded Lipschitz domain, then we have the following lemma.
Lemma 2.3. Assume that Ω ⊂ n is an open bounded and Lipschitz domain, p ∈ [1, +∞)
and u : Ω → . Then
WG1, p ( Ω ) ⊆ WGs , p ( Ω ) , for all s ∈ (0,1) .
Proof of Lemma 2.2 and Lemma 2.3 can be found in (Di Nezza, Palatucci & Valdinoci,
2012).
2.3. Weighted fractional Sobolev spaces
Since the main content of the article uses some properties of weighted fractional
Sobolev space where the weights are the distance functions to the boundary of the domain.
We will introduce weighted fractional Sobolev spaces via the following definition.
Definition 2.4. (Weighted fractional Sobolev space) Assume that Ω ⊂ n is an open
bounded and Lipschitz domain, q ∈ [1, ∞) , s ∈ (0,1) and α , β ≥ 0 . Then, we define the
weighted fractional Sobolev space as
u ( x) − u ( y )
p
( Ω;α , β )=: u ∈ L ( Ω ) : ∫ ∫ d ( x)d ( y )
WGs , p p α β
n + sp
dxdy < ∞ , (2.5)
ΩΩ x− y
with the natural norm
1
u ( x) − u ( y )
p p
u=WG ( Ω;α , β )
s, p ∫ u ( x) p dx + ∫ ∫ d α ( x)d β ( y ) dxdy . (2.6)
n + sp
Ω ΩΩ x− y
where =
d ( x) dist( x, ∂Ω) .
Similar to the non-weight spaces, the weighted Gagliardo semi-norm of WGs , p ( Ω; α , β ) is
defined by
1
α β | u ( x) − u ( y ) | p p
[u ]W s , p (Ω;α ,β ) := ∫ ∫ d ( x)d ( y ) dxdy . (2.7)
G
Ω Ω | x − y |n+ sp
Let us introduce some properties of weighted fractional Sobolev space, which similar to
fractional Sobolev space.
Lemma 2.5. Assume that v : Ω → is measurable function. Then, there exists a constant
C ≥ 1 such that
u W s , p ( Ω;α ,β ) ≤ C u W t , p ( Ω;α ,β ) , for all t ∈ ( s,1) .
G G
In particular,
WGt , p (Ω; α , β ) ⊆ WGs , p (Ω; α , β ), for all t ∈ ( s,1).
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- HCMUE Journal of Science Tran Quang Vinh
The proof is similar in spirit to the proof of Lemma 2.2. Now, we establish the connection
between fractional Sobolev space and weighted fractional Sobolev space by the following
lemma.
Lemma 2.6. For every α , β ≥ 0 we have
α +β
[v]W s , p (Ω;α ,β ) ≤ ( diam(Ω) ) q [v]W s , p (Ω) ,
G G
and it yields
WGs , p (Ω) ⊂ WGs , p (Ω; α , β ).
That means that weighted fractional Sobolev space is the expansion of fractional Sobolev
space, and the result we have obtained is more general. In the following section, we
introduce the main results and prove the main results.
3. Main results
In this section, we state our main results and their proofs.
1
Theorem 3.1. Let p > 2 − , σ ∈ (0,1) and Ω be an open bounded and smooth domain in
n
n . Assume that u ∈ W 1,max{1, p −1} (Ω) is a SOLA solution to (1.1). Then for every α ,
β > 0 and γ ≥ 0 satisfying α > γ , β > γ , α + β − γ > σ , there exists a constant C > 0
such that
( ∇u ( x ) ) − ( ∇u ( y ) )
∫Ω ∫Ω d α ( x )d β ( y ) n +σ
dxdy
x− y (3.1)
≤C (∫Ω
d γ ( x ) ( ∇u ( x ) ) dx + µ ( Ω ) )
θ
where d= ( x) : [dist( x, ∂Ω)]θ .
1
In this section, we always assume that p > 2 − , σ ∈ (0,1) , Ω ⊂ n be an open
n
bounded and smooth domain. Furthermore, u ∈ W 1,max{1, p −1} (Ω) is a SOLA solution to
(1.1). Denote by D(Ω) the diameter of Ω , this means D(Ω) =sup d ( x, y ) .
x , y∈Ω
First, suppose that 0 < R0 < D(Ω) / 2 , let
R
Ω0=: x ∈ Ω | 0 < d ( x) ≤ 0 ,
2
be the set of points near ∂Ω . We define Ω k as
Ω k=: {x ∈ Ω | r k +1 < d ( x) ≤ rk ,}
rk 2− k R0 , ∀k ∈ . It is clear that
with=
∞
Ω0 = Ω
k =1
k (see Figure 1).
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Figure 1. The sets of points near the boundary
To facilitate the proof of main results, we introduce the following function.
A(∇u ( x)) − A(∇u ( y ))
( x , y ) : d α ( x ) d β ( y ) n +σ
, x, y ∈ Ω, x ≠ y.
x− y
Let us introduce some lemmas that necessary for later use.
Lemma 3.2. Assume that α , β > 0 , γ ≥ 0 ; α > γ , β > γ and α + β − γ > σ . Then, there
exists a constant C > 0 such that
(x, y )dxdy ≤ C ∫
∫Ω \ Ω ∫Ω \ Ω d γ ( x ) ( ∇u ( x ) ) dx + µ ( Ω \ Ω0 ) . (3.2)
0 0 Ω \ Ω0
Proof of Lemma 3.2. First, let us establish
( ) := ∫ ∫ ( x, y )dxdy.
Ω \ Ω0 Ω \ Ω0
We remark that Ω \ Ω0 can be covered by actually finite balls centered at zk with radius r1
, k = 1, N , i.e
N
Ω \ Ω0 ⊂
Br1 ( zk ) = Br1 ( zk ).
= k 1 zk ∈Ω \ Ω0
Let P be the set of all centers, i.e.
=: {zk ∈ Ω \ Ω0 : k ∈ {1, 2, …, N }}.
P
Now, we estimate ( ) as follows
=( ) ∫Ω \ Ω ∫Ω \ Ω ( x, y )dxdy ≤ ∑ ∫B r1 ( zk )
∫B
r1 ( zl )
( x, y )dxdy.
0 0
zk , zl ∈P
Let Pzk be the set of all centers that are closed to zk , which means
Pzk=: {zl ∈ P : B3r /2 ( zl ) ∩ B3r /2 ( zk ) ≠ ∅} ,
1 1
It is clear that
Br1 ( zl ) ⊂ B3r1 /2 ( zl ) ⊂ B4 r1 ( zk ), ∀zl ∈ Pzk (see Figure 2).
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- HCMUE Journal of Science Tran Quang Vinh
Figure 2. The centers are closed to zk .
Furthermore, the cardinality of Pzk is finite, i.e. there exists C > 0 such that Pzk ≤ C
. So, we can decompose the integral Ω \ Ω0 × Ω \ Ω0 as follows
∫Ω \ Ω ∫Ω \ Ω ( x, y )dxdy ≤ ∑ ∫B r1 ( zk )
∫Br1 ( zl )
( x, y )dxdy
0 0
zk , zl ∈P
≤ ∑ ∑ ∫B r1 ( zk )
∫B r1 ( zl )
( x, y )dxdy + ∑ ∑ ∫B r1 ( zk )
∫Br1 ( zl )
( x, y )dxdy. (3.3)
zk ∈P zl ∈Pzk zk ∈P zl ∈P \ Pzk
With the first term on the right-hand side of (3.3), we get
∑ ∑ ∫B r1 ( zk )
∫Br1 ( zl )
( x, y )dxdy ≤ C ∑ ∫B 4 r1 ( zk )
∫B4 r1 ( zk )
( x, y )dxdy. (3.4)
zk ∈P zl ∈Pzk zk ∈P
Applying (1.5) in Theorem 1.1, we have
∫B 4 r1 ( zk )
∫B
4 r1 ( zk )
( x, y )dxdy
A(∇u ( x)) − A(∇u ( y ))
≤ 4α + β −γ ⋅ r1α + β −γ ∫ ∫ d γ ( x) n +σ
dxdy
B4 r1 ( zk ) B4 r1 ( zk )
x− y
≤ C.r1α + β −γ −σ ∫ d γ ( x) A(∇u ( x)) dx + r1[ µ ( B8r1 )] . (3.5)
B8 r1 ( zk )
Combining between (3.4) and (3.5), we reach that
∑ ∑ ∫B r1 ( zk )
∫Br1 ( zl )
( x, y )dxdy
zk ∈P zl ∈Pzk
≤ C.r1α + β −γ −σ ∑ ∫ d γ ( x) A(∇u ( x)) dx + r1 ∑ µ ( B8r1 ) . (3.6)
z ∈P B8 r1 ( zk )
k zk ∈P
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Notice that there is a constant = C C (n) > 0 such that
∑ χ B8r ( zk ) (ξ ) ≤ C χΩ \ Ω0 (ξ ), ∀ξ ∈ Ω.
1
zk ∈P
therefore, for all f ∈ L1loc ( n ) , we reach that
∑∫
= f (ξ )dξ
B8 r1 ( zk ) ∑ ∫ n χ B8 r ( zk ) (ξ ) f (ξ )dξ ≤ C ∫
1 Ω \ Ω0
f (ξ )dξ . (3.7)
zk ∈P zk ∈P
Substituting (3.7) to (3.6), we obtain that
∑ ∑ ∫B r1 ( zk )
∫B r1 ( zl )
( x, y )dxdy
zk ∈P zl ∈Pzk
≤ C.r1α + β −γ −σ ∫ d γ ( x) A(∇u ( x)) dx + r1[ µ (Ω \ Ω0 )] . (3.8)
Ω \ Ω 0
Moreover, it's clear that for any x ∈ Br1 ( zk ) , y ∈ Br1 ( zl ) , with zk ∈ P and zl ∈ P \ Pzk , we
get x − y ≥ 3r1 . It is easy for us to check that
A(∇u( x ))
∑ ∫B ( z ) ∫B
d α ( x )d β ( y ) n +σ
dxdy
zl ∈P \ Pzk r1 k r1 ( zl ) x− y
1
∫Br1 ( zk ) ∑ ∫Br1 ( zl ) x − y n +σ dy d ( x ) A(∇u( x )) dx
α + β −γ α + β −γ γ
≤3 ⋅ r1
zl ∈P \ Pzk
1
≤ C.r1α + β −γ −σ ∫ ∫ d ξ d γ ( x) A(∇u ( x)) dx
Br1 ( zk ) { ξ ≥1} n +σ
ξ
≤ C.r1α + β −γ −σ ∫ d γ ( x) A(∇u ( x)) dx.
Br1 ( zk )
Now we estimate the last term in (3.3) as
∑ ∑ ∫B ∫
r1 ( zk ) Br1 ( zl )
( x, y )dxdy ≤ C.r1α + β −γ −σ ∑ ∫B r1 ( zk )
d γ ( x) A(∇u ( x)) dx
zk ∈P zl ∈P \ Pzk zk ∈P
≤ C.r1α + β −γ −σ ∫ d γ ( x) A(∇u ( x)) dx. (3.9)
Ω \ Ω0
Applying (3.8), (3.9) to (3.3), we reach that
( ) = ∫ ∫ ( x, y )dxdy
Ω Ω0 Ω Ω0
≤ C. r1α + β −γ −σ ∫ d γ ( x) A(∇u ( x)) dx + r1α + β −γ −σ +1 µ (Ω \ Ω0 )
Ω \ Ω 0
≤ C. ∫ d γ ( x) A(∇u ( x)) dx + µ (Ω \ Ω0 ) r1α + β −γ −σ
Ω \ Ω0
≤ C. ∫ d γ ( x) A(∇u ( x)) dx + µ (Ω \ Ω0 ) , (3.10)
Ω \ Ω0
which leads to the desired result.
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- HCMUE Journal of Science Tran Quang Vinh
Lemma 3.3. Assume that α , β > 0 , γ ≥ 0 ; α > γ , β > γ and α + β − γ > σ . Then, there
exists a constant C > 0 such that
α β A(∇u ( x)) riα −γ r jβ γ
∫Ωi ∫Ω j d ( x)d ( y) x − y n+σ dxdy ≤ C (r + r )σ ∫Ωi d ( x) A(∇u ( x)) dx. (3.11)
i j
Proof of Lemma 3.3. First, for any x ∈ Ωi , y ∈ Ω j , i − j ≥ 2 , we get
r rj ri + rj
x − y ≥ max i , ≥ (see Figure 3).
4 4 8
Figure 3.
it yields
A(∇u ( x)) A(∇u ( x))
∫Ω ∫Ω d α ( x)d β ( y ) dxdy = ∫ d α −γ ( x)d β ( y ) d γ ( x)dxdy
n +σ Ω ∫Ω n +σ
i j x− y i j x− y
1
≤ riα −γ r jβ ∫ ∫ ri + r j n+σ dξ d γ ( x) A(∇u ( x)) dx
Ωi ξ ≥
8 ξ
α −γ β
ri r j 1
≤ 8σ σ ∫Ωi ∫{ ξ ≥1} n +σ
dξ d γ ( x) A(∇u ( x)) dx
(ri + r j ) ξ
α −γ β
ri r j
≤C σ ∫Ωi
d γ ( x) A(∇u ( x)) dx. (3.12)
(ri + r j )
1
Notice that the fraction n +σ
is integrable since n + σ > n .
ξ
Lemma 3.4. Assume that α , β > 0 , γ ≥ 0 ; α > γ , β > γ and α + β − γ > σ . Then, there
exist constants C1 > 0 and C2 > 0 such that
riα −γ r jβ riα r jβ −γ
∑ σ ∫Ω
γ
d ( x) A(∇u ( x)) dx + ∫
(ri + r j )σ Ω j
d γ
( y ) A(∇ u ( y )) d y
i − j ≥ 2 ( ri + r j ) i
∞
≤ C1.∫ d γ ( x) A(∇u ( x)) dx ∑ rαj + β −γ −σ , (3.13)
Ω0
j =1
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and
riα −γ r jβ riα r jβ −γ
∑ (r + r )σ ∫Ω d γ ( x) A(∇u ( x)) dx + ∫
(ri + r j )σ Ω j
d γ
( y ) A(∇ u ( y )) d y
j −i ≥ 2 i j i
∞
≤ C2 .∫ d γ ( y ) A(∇u ( y )) dy ∑ riα + β −γ −σ . (3.14)
Ω0
i =1
Proof of Lemma 3.4. First, let us establish
riα −γ r jβ riα r jβ −γ
= ( )11 : ∑
σ ∫Ωi
γ
d ( x) A(∇u ( x)) dx + ∫Ω j d γ
( y ) A(∇ u ( y )) d y ,
i − j ≥ 2 ( ri + r j ) (ri + r j )σ
and
riα −γ r jβ riα r jβ −γ
= ( )12 : ∑
σ ∫Ωi
γ
d ( x) A(∇u ( x)) dx + ∫Ω j d γ
( y ) A(∇ u ( y )) d y .
j −i ≥ 2 ( ri + r j ) (ri + r j )σ
We have
∞ ∞ riα −γ r jβ
( )11 ∑∑ σ ∫Ω d
γ
( x) A(∇u ( x)) dx
j = 1 i= j + 2 r σ i
i + 1 r j
rj
∞ ∞ riα r jβ −γ
+∑ ∑ σ ∫Ω d γ ( y ) A(∇u ( y )) dy
j = 1 i= j + 2 r σ j
i + 1 r j
rj
∞ ∞
riα −γ
∑ rj β −σ
∑ j −i
+ 1)σ Ωi
∫ d γ ( x) A(∇u ( x)) dx
j= 1 i = j + 2 (2
∞ ∞
riα
+ ∑ r jβ −γ −σ ∫ d γ ( y ) A(∇u ( y )) dy ∑
j= 1
Ωj
i = j + 2 (2
j −i
+ 1)σ
∞ ∞ ∞ ∞
≤ ∑ rαj + β −γ −σ ∑ ∫Ω d γ ( x) A(∇u ( x)) dx + ∑ r jβ −γ −σ ∫ d γ ( y ) A(∇u ( y )) dy ∑ riα
Ωj
j=
1 i=
j +2 i j=
1 i=
j +2
∞ ∞
≤∫ d γ ( x) A(∇u ( x)) dx ∑ rαj + β −γ −σ + C1.∑ rαj + β −γ −σ ∫ d γ ( y ) A(∇u ( y )) dy
Ω0 Ωj
=j 1 =j 1
∞
≤ C1.∫ d ( x) A(∇u ( x)) dx ∑ rαj + β −γ −σ ,
γ
Ω0
j =1
and similarly, we get
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- HCMUE Journal of Science Tran Quang Vinh
riα −γ r jβ riα r jβ −γ
=( )12 ∑ σ ∫Ω
γ
d ( x) A(∇u ( x)) dx + ∫ d γ
( y ) A(∇ u ( y )) d y
j −i ≥ 2 ( ri + r j ) i (ri + r j )σ Ω j
∞
≤ C2 .∫ d γ ( y ) A(∇u ( y )) dy ∑ riα + β −γ −σ ,
Ω0
i =1
which provides us (3.13) and (3.14).
Lemma 3.5. Assume that α , β > 0 , γ ≥ 0 ; α > γ , β > γ and α + β − γ > σ . Then, there
exists constant C > 0 such that
∞
∑ ∫Ω ∫Ω ( x, y )dxdy ≤ C ∫
Ω0
d γ ( x) A(∇u ( x)) dx ∑ rαj + β −γ −σ . (3.15)
i − j ≥2 i j
j=
1
Proof of Lemma 3.5. In this proof, let us set
( )1 = ∑ ∫Ω ∫Ω ( x, y )dxdy.
i − j ≥2 i j
Applying (3.11) in Lemma 3.3, we get
( )1 = ∑ ∫Ω ∫Ω ( x, y )dxdy
i − j ≥2 i j
A(∇u ( x)) A(∇u ( y ))
≤ ∑ ∫Ω ∫Ω d α ( x)d β ( y ) n +σ
dxdy + ∫
Ωi Ω j∫ d α ( x)d β ( y ) n +σ
dxdy
i − j ≥2 i j x− y x− y
riα −γ r jβ riα r jβ −γ
≤ ∑ σ ∫Ωi
γ
d ( x) A(∇u ( x)) dx + ∫Ω j d γ
( y ) A(∇ u ( y )) d y
i − j ≥ 2 ( ri + r j ) (ri + r j )σ
≤ C.(( )11 + ( )12 ), (3.16)
where
riα −γ r jβ riα r jβ −γ
= ( )11 : ∑ σ ∫Ωi
d γ
( x ) A( ∇ u ( x )) d x + σ ∫Ω j
d γ
( y ) A( ∇ u ( y )) d y ,
i − j ≥2 i( r + r ) ( r + r )
j i j
and
riα −γ r jβ riα r jβ −γ
= ( )12 : ∑ σ ∫Ωi
d γ
( x ) A (∇ u ( x )) d x + σ ∫Ω j
d γ
( y ) A (∇ u ( y )) d y .
(
j −i ≥ 2 i r + r ) ( r + r )
j i j
From what have already been proved in (3.13), (3.14) and (3.16), ( )1 can be estimated as
∞
( )1 ≤ C ∫ d γ ( x) A(∇u ( x)) dx ∑ rαj + β −γ −σ ,
Ω0
j =1
which allows us to get (3.15).
Lemma 3.6. Assume that α , β > 0 , γ ≥ 0 ; α > γ , β > γ and α + β − γ > σ . Then, there
exists constant C > 0 such that
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- HCMUE Journal of Science Vol. 18, No. 9 (2021): 1359-1367
∞ ∞
∑ ∫Ωi ∫Ωi ( x, y)dxdy ≤ C ∫Ω0 d γ ( x) A(∇u ( x)) dx + µ (Ω0 ) ∑ riα + β −γ −σ . (3.17)
i 1 =i 1
Proof of Lemma 3.6. In this proof, let us denote
∞
()2 = ∑ ∫ ∫ ( x, y )dxdy.
Ωi Ωi
i =1
To continue estimates ( ) 2 , our idea is to decompose the Ωi into open balls with a radius
ri then applying the local inequality (1.5) in Theorem 1.1.
∂Ω
Notice that Ωi can be covered with Ni ~ ri
balls centered at zki ∈ Ωi with radius ri ,
k = 1, Ni . It means
Ni
Ωi ⊂ Bri ( zki ) = Bri ( zki ).
k =1 zki ∈Ωi
Let Pi be the set of all centers, i.e.
P=
i : {zk ∈ Ωi : k ∈ {1, 2, …, N i }}.
i
Now, we estimate ( ) 2 as follows
∞ ∞
=()2
Ωi Ωi ∑∫ ∫
B ( z i ) Bri ( zli )
( x, y )dxdy ≤ ∑ ∑ ∫ ∫ ( x, y )dxdy.
=i 1 = i 1 zki , zli∈Pi ri k
Let Pi , z i be the set of all centers that are closed to zki , which means
k
Pi , z i=:
k
{z ∈ P : B
i
l i 3ri /2 ( zl ) ∩ B3ri /2 ( zk )
i i
≠∅ . }
It is not difficult for us to check that
Bri ( zli ) ⊂ B3ri /2 ( zli ) ⊂ B4 ri ( zki ), ∀zli ∈ Pi , z i .
k
Moreover, the cardinality of Pi , z i is finite, means there exists a constant C such that
k
Pi , zi ≤ C . So, we can decompose the integral Ωi × Ωi as follows
k
∫Ω ∫Ω ( x, y)dxdy ≤ ∑ ∫B
i i
i
ri ( zk )
∫B i
ri ( zl )
( x, y )dxdy
zki , zli ∈Pi
≤ ∑ ∑ ∫B i
ri ( zk )
∫B i
ri ( zl )
( x, y )dxdy
zki ∈Pi zli∈P i
i,z
k
+ ∑ ∑ ∫B ri
∫
( zki ) Bri ( zli )
( x, y )dxdy. (3.18)
zki ∈Pi zli ∈Pi \ P
i , zki
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- HCMUE Journal of Science Tran Quang Vinh
Applying (3.8), (3.9) to (3.18), we reach that
∞
()2 = ∑ ∫ ∫ ( x, y )dxdy
Ωi Ωi
i =1
∞
≤ C. ∑ riα + β −γ −σ ∫ d γ ( x) A(∇u ( x)) dx + riα + β −γ −σ +1 µ (Ω0 )
Ω0
i =1
∞
≤ C. ∫ d γ ( x) A(∇u ( x)) dx + µ (Ω0 ) ∑ riα + β −γ −σ .
Ω0 i =1
(3.19)
which leads to the desired result.
Lemma 3.7. Assume that α , β > 0 , γ ≥ 0 ; α > γ , β > γ and α + β − γ > σ . Then, there
exists constant C > 0 such that
∞
∑ ∫Ω ∫Ω ( x, y )dxdy ≤ C ∫ d γ ( x) A(∇u ( x)) dx + µ (Ω0 ) ∑ riα + β −γ −σ . (3.20)
i− j 1 =
i j Ω0 i 1
Proof of Lemma 3.7. Let us establish
( )3 := ∑ ∫Ω ∫Ω ( x, y )dxdy.
i− j =
1 i j
We estimate ( )3 with note that
∞ ∞
=( )3
Ωi Ω j ∑ ∫ ∫ ( x, y)dxdy 2∑ ∫
=
Ω Ωi +1 A A ∫ ( x, y )dxdy ≤ 2∑ ∫ ∫ ( x, y )dxdy,
=i− j 1 = i 1 i= i 1 i i
where Ai is defined by
r
Ai := Ωi ∪ Ωi +1 = x ∈ Ω : i < d ( x) ≤ ri .
4
In a similar way, for ( )3 we may estimate by the same the way to ( ) 2 in (3.18) and reach
that
∞
( )3 ≤ C. ∫ d γ ( x) A(∇u ( x)) dx + µ (Ω0 ) ∑ riα + β −γ −σ , (3.21)
Ω0 i =1
which leads to the desired result.
Lemma 3.8. Assume that α , β > 0 , γ ≥ 0 ; α > γ , β > γ and α + β − γ > σ . Then, there
exists a constant C > 0 such that
( x, y )dxdy ≤ C ∫ d γ ( x) A(∇u ( x)) dx + µ (Ω0 ) .
∫Ω ∫Ω (3.22)
0 0 Ω0
Proof of Lemma 3.8. In this proof, let us set
( ) := ∫ ∫ ( x, y )dxdy.
Ω0 Ω0
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∞
Since Ω0 = Ωk , we can rewrite () as follows
k =1
∞
=() ∑
= ∫Ω ∫Ω ( x, y)dxdy ∑ ∫Ω ∫Ω ( x, y )dxdy + ∑ ∫Ω ∫Ω ( x, y )dxdy
=i, j 1 i j
i − j ≥2 i
=i− j 1
j i j
∞
+∑ ∫ ∫ ( x, y )dxdy
Ωi Ωi
i =1
= ( )1 + ( )3 + ( ) 2 , (3.23)
with
( )1 ∑ ∫ ∫ ( x, y)dxdy; ()3
=
Ωi Ω j ∑ ∫Ω ∫Ω ( x, y )dxdy
i − j ≥2 i− j =
1 i j
and
∞
()2 = ∑ ∫ ∫ ( x, y )dxdy.
Ωi Ωi
i =1
We can estimate each term on the right-hand side of (3.23) by applying Lemma 3.5, 3.6
and 3.7. Then, we can find a constant C > 0 such that
( ) ≤ C ∫ d γ ( x) A(∇u ( x)) dx + µ (Ω0 ) . (3.24)
Ω0
Notice that, the assumption α + β − γ > σ help us to find
∞ ∞ (α + β −γ −σ )i
1
∑ riα + β −γ −σ CR0α +=
= β −γ −σ
∑ 2
, with C < ∞,
i 1 =i 1
which completes the proof.
Lemma 3.9. For every α , β > 0 , γ ≥ 0 satisfying α > γ , β > γ and α + β − γ > σ ,
there exists a constant C > 0 such that
∫Ω ∫Ω \ Ω ( x, y )dxdy ≤ C ∫ d γ ( x ) A(∇u( x )) dx + µ (Ω \ Ω0 ) . (3.25)
0 0 Ω \ Ω 0
∂Ω
Proof of Lemma 3.9. Note that Ωi can be covered with Ni ~ ri
balls centered at zli with
radius ri , i.e.
Ni
Ωi ⊂ Bri ( zli ) = Bri ( zli ),
l =1 zli∈Pi
and Ω Ω0 can be covered by finite balls centered at zk with radius r1 , i.e
N
Ω \ Ω0 ⊂
Br1 ( zk ) = Br1 ( zk ),
= k 1 zk ∈P
where
P=
i : {zl ∈ Ωi : l ∈ {1, 2, …, N i }} and
i
P=: {zk ∈ Ω \ Ω0 : k ∈ {1, 2, …, N }}.
It is not difficult for us to check that
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Bri ( zli ) ⊂ B4 r1 ( zk ), ∀zli ∈ Pi .
Now, we estimate ( ) as follows
∞
=( ) ∫=
∫ ( x, y )dxdy
Ω0 Ω \ Ω0 ∑ ∫Ω ∫Ω \ Ω ( x, y )dxdy
i =1 i 0
∞ ∞
≤∑ ∑ ∑∫ ∫ ∑∑ ∑∫
( x, y )dxdy = ∫ i ( x, y)dxdy
B ( z i ) Br1 ( zk ) B ( z ) Bri ( zl )
i=
1 zli ∈Pi zk ∈P ri l i=
1 zk ∈P zli ∈Pi r1 k
≤C ∑ ∫B 4 r1 ( zk )
∫B 4 r1 ( zk )
( x, y )dxdy.
zk ∈P
(3.26)
Combining between (3.5) and (3.26), we reach that
( ) ≤ C (n, p, c A , σ , R0 )r1α + β −γ −σ ∑ ∫ d γ ( x) A(∇u ( x)) dx + r1 ∑ µ ( B8r1 ) .
z ∈P B8 r1 ( zk )
k zk ∈P
(3.27)
Substituting (3.7) to (3.27), we obtain that
( ) ≤ C ( n, p, c A , σ , R0 ) r1α + β −γ −σ ∫ d γ ( x ) A(∇u( x )) dx + r1[ µ (Ω \ Ω0 )]
Ω \ Ω0
≤ C ( n, p, c A , α , β , γ , σ , R0 ) ∫ d γ ( x ) A(∇u( x )) dx + µ (Ω \ Ω0 ) . (3.28)
Ω \ Ω 0
This achieves the proof of the desired result.
Thanks to some lemmas that have been proved and some important properties of
weighted fractional Sobolev's spaces discussed in Section 2, now we prove the main
theorem.
Proof of Theorem 3.1. The integral of over Ω × Ω can be rewritten as
∫ ∫ ( x, y )dxdy =
Ω Ω ∫ ∫ ( x, y )dxdy + 2∫ ∫ ( x, y )dxdy + ∫ ∫ ( x, y )dxdy
Ω0 Ω0 Ω0 Ω \ Ω0 Ω \ Ω0 Ω \ Ω0
=( ) + 2( ) + ( ), (3.29)
with
() ∫=
Ω ∫Ω
0
( x, y )dxdy;
0
( ) ∫ ∫
Ω Ω\Ω 0 0
( x, y )dxdy ,
and
( ) = ∫ ∫ ( x, y )dxdy.
Ω \ Ω0 Ω \ Ω0
We can estimate each term ( ) , ( ) and ( ) by using Lemmas 3.2, 3.8 and 3.9. Then,
there exists constant C C (n, p, c , α , β , γ , σ , R0 ) > 0 such that
=
∫Ω ∫Ω ( x, y)dxdy ≤ C ( ∫Ω d )
γ
( x) (∇u ( x)) dx + µ (Ω) , (3.30)
which leads to the desired result (3.1) from (3.30).
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Conflict of Interest: Authors have no conflict of interest to declare.
REFERENCES
Adams, D. R., & Hedberg, L.I. (1996). Function spaces and potential theory.
Berlin: Springer,
Avelin, B., T. Kuusi, T., & Mingione, G. (2018). Nonlinear Calderón-Zygmund theory in
the limiting case. Arch. Rational. Mech. Anal, 227, 663-714.
Balci, A. Kh., Diening, L., & Weimar, M. (2020). Higher order Calderón-Zygmund
estimates for the p-Laplace equation. J. Differential Equations, 268, 590-635.
Benilan, P., Boccardo, L., Gallouet, T., Gariepy, R., Pierre, M., & Vazquez, J. L. (1995).
An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations.
Ann. Scuola Norm. Sup. Pisa Cl. Sci., (IV), 22, 241-273.
Maso, G. D., Murat, F., Orsina, L., & Prignet, A. (1999). Renormalized solutions for
elliptic equations with general measure data. Ann. Sc. Norm. Super Pisa Cl. Sci., 28,
741-808.
Brezis, H. (2011). Functional analysis, Sobolev spaces and Partial Differential Equations.
Springer.
Di Nezza, E., Palatucci, G., & Valdinoci, E. (2012). Hitchhiker's guide to fractional
Sobolev spaces. Bulletin des Sciences Mathématique, 136(5), 521-573.
Grafakos, L. (2004). Classical and Modern Fourier Analysis. Pearson/Prentice Hall.
Mingione, G. (2007). The Calderón-Zygmund theory for elliptic problems with measure
data. Ann Scu. Norm. Sup. Pisa Cl. Sci., (V), 6, 195--261.
Tran, M.-P. (2019). Good-λ type bounds of quasilinear elliptic equations for the singular
case. Nonlinear Anal., 178, 266-281.
Tran, M. P., & Nguyen, T. N. (2019). Global gradient estimates for very singular
nonlinear elliptic equations with measure data, arXiv:1909.06991, 33pp.
Tran, M. P., & Nguyen, T. N. (2020a). Lorentz-Morrey global bounds for singular
quasilinear elliptic equations with measure data. Commun. Contemp. Math., 22(5),
19500330.
Tran, M. P., & Nguyen, T. N. (2020b). An endpoint case of the renormalization property
for the relativistic Vlasov–Maxwell system. Journal of Mathematical Physics, 61(7),
071512.
Tran, M. P., & Nguyen, T. N. (2021a). Pointwise gradient bounds for a class of very
singular quasilinear elliptic equations. Discrete and Continuous Dynamical Systems -
Series A, 41(9), 4461-4476.
Tran, M. P., & Nguyen, T. N. (2021b). A global fractional Caccioppoli-type estimate for
solutions to nonlinear elliptic problems with measure data. Studia Mathematica,
accepted, 18pp.
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- HCMUE Journal of Science Tran Quang Vinh
A WEIGHTED APPROACH FOR CACCIOPOLI INEQUALITY
FOR SOLUTIONS TO P-LAPLACE EQUATIONS
Trần Quang Vinh
Trường Đại học Sư phạm Thành phố Hồ Chí Minh, Việt Nam
Tác giả liên hệ: Trần Quang Vinh – Email: tranvinh3111@gmail.com
Ngày nhận bài: 01-6-2021; ngày nhận bài sửa: 16-8-2021; ngày duyệt đăng: -9-2021
ABSTRACT
Weighted fractional Sobolev spaces have many applications in partial differential equations.
In this paper, we study a class of weighted fractional Sobolev spaces, where the weights are the
distance functions to the boundary of the defined domain. This class has been used to obtain a
weighted Cacciopoli-type inequality for solutions to p-Laplace equations with measure data. Our
result expands to the Cacciopoli inequality in the recent paper (Tran & Nguyen, 2021b).
Keywords: Cacciopoli-type inequality; partial differential equations; p-Laplace equations;
weighted fractional Sobolev spaces
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nguon tai.lieu . vn