Bất đẳng thức Cacciopoli có trọng cho nghiệm của phương trình p-Laplace

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Bài viết trình bày 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.

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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. 1
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. 2
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 3
HCMUE Journal of Science Vol. 18, No. 9 (2021): 1359-1367 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). 4
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). 5
HCMUE Journal of Science Vol. 18, No. 9 (2021): 1359-1367 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). 6
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  7
HCMUE Journal of Science Vol. 18, No. 9 (2021): 1359-1367 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.  8
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 9
HCMUE Journal of Science Vol. 18, No. 9 (2021): 1359-1367 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 10
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 11
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 12
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 13
HCMUE Journal of Science Vol. 18, No. 9 (2021): 1359-1367 ∞ 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 14
HCMUE Journal of Science Tran Quang Vinh 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).  15
HCMUE Journal of Science Vol. 18, No. 9 (2021): 1359-1367  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. 16
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 17