Weighted gradient estimates for elliptic problems with Neumann boundary conditions in Lipschitz and (semi-)convex domains

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Weighted gradient estimates for elliptic problems with Neumann boundary conditions in Lipschitz and (semi-)convex domains ✩ Sibei Yang a , Der-Chen Chang b,c , Dachun Yang d,∗ , Wen Yuan d a School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou

University, Lanzhou 730000, People’s Republic of China b Department of Mathematics, Georgetown University, Washington D.C. 20057, USA c Graduate Institute of Business Administration, College of Management, Fu Jen Catholic University, New Taipei City

242, Taiwan, ROC d Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences,

Beijing Normal University, Beijing 100875, People’s Republic of China Received 19 April 2019; revised 8 August 2019; accepted 12 September 2019

Abstract Let n ≥ 2 and  be a bounded Lipschitz domain in Rn . In this article, the authors investigate global (weighted) norm estimates for the gradient of solutions to Neumann boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in . More precisely, for any given p ∈ (2, ∞), two necessary and sufficient conditions for W 1,p estimates of solutions to Neumann boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted W 1,q estimates of solutions with q ∈ [2, p] and some Muckenhoupt weights, are obtained. As applications, for any given p ∈ (1, ∞) and ω ∈ Ap (Rn ) (the class of Muckenhoupt weights), 1,p

the authors establish weighted Wω estimates for solutions to Neumann boundary value problems of second order elliptic equations of divergence form with small BMO coefficients on bounded (semi-)convex domains. As further applications, the global gradient estimates are obtained, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (weighted) Orlicz spaces, and variable Lebesgue spaces.

✩ This work is supported by the National Natural Science Foundation of China (Grant Nos. 11871254, 11571289, 11971058, 11761131002, 11671185 and 11871100) and the China Scholarship Council (No. 201806185039). The second author is partially supported by an NSF grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University. * Corresponding author. E-mail addresses: [email protected] (S. Yang), [email protected] (D.-C. Chang), [email protected] (D. Yang), [email protected] (W. Yuan).

https://doi.org/10.1016/j.jde.2019.09.036 0022-0396/© 2019 Elsevier Inc. All rights reserved.

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MSC: primary 35J25; secondary 35J15, 42B35, 42B37 Keywords: Elliptic equation; Neumann boundary problem; Lipschitz domain; (Semi-)convex domain; Muckenhoupt weight; Weak reverse Hölder inequality

1. Introduction It is well known that the gradient estimates for linear or non-linear elliptic equations (or systems) in non-smooth domains are important in the theory of partial differential equations (see, for instance, [6,11,12,17,19,27,28,32,38,41] for the linear case and [4,15,18,22,44,43,49] for the non-linear case). Moreover, the global gradient estimates of solutions to elliptic boundary problems depend not only on the properties of the right-hand side datum and the coefficients appeared in equations, but also on the smoothness or geometric properties of the boundary of the considered underlying manifold (see, for instance, [2,6,11,16,21,23,38,41,42]). Motivated by [16,32,33,43], in this article we are going to study global weighted gradient estimates for Neumann boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in bounded Lipschitz and (semi-)convex domains. More precisely, let  be a bounded Lipschitz domain in Rn with n ≥ 2. For any given p ∈ (2, ∞), two necessary and sufficient conditions for W 1,p estimates of solutions to Neumann boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted W 1,q estimates of solutions with q ∈ [2, p] and some Muckenhoupt weights, are established. As applications, for any given p ∈ (1, ∞) and ω ∈ Ap (Rn ) 1,p (the class of Muckenhoupt weights), we obtain the weighted Wω estimates for solutions to Neumann boundary value problems of second order elliptic equations of divergence form with small BMO coefficients on bounded (semi-)convex domains. Moreover, as further applications of those global weighted estimates, the global gradient estimates are given in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (weighted) Orlicz spaces, and variable Lebesgue spaces, respectively. To state the main results of this article, we first recall the notions of the Muckenhoupt weight class and the reverse Hölder class (see, for instance, [24,37,57]). Definition 1.1. Let q ∈ [1, ∞). A non-negative and locally integrable function ω on Rn is called an Aq (Rn ) weight, denoted by ω ∈ Aq (Rn ), if, when q ∈ (1, ∞), ⎫⎧ ⎫q−1 ⎧ ⎬⎨ 1  ⎬ ⎨ 1  1 − [ω]Aq (Rn ) := sup ω(x) dx [ω(x)] q−1 dx <∞ ⎭ ⎩ |B| ⎭ B⊂Rn ⎩ |B| B

B

or ⎫ ⎧

⎬ ⎨ 1  −1 [ω]A1 (Rn ) := sup ω(x) dx ess sup [ω(y)] < ∞, ⎭ y∈B B⊂Rn ⎩ |B| B

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where the suprema are taken over all balls B ⊂ Rn . Let r ∈ (1, ∞]. A non-negative and locally integrable function ω on Rn is said to belong to the reverse Hölder class RHr (Rn ), denoted by ω ∈ RHr (Rn ), if, when r ∈ (1, ∞), ⎫1/r ⎧ ⎫−1 ⎧ ⎬ ⎨ 1  ⎬ ⎨ 1  [ω]RHr (Rn ) := sup [ω(x)]r dx ω(x) dx <∞ ⎭ ⎩ |B| ⎭ B⊂Rn ⎩ |B| B

B

or [ω]RH∞ (Rn ) := sup

B⊂Rn

⎫−1

⎧ ⎨ 1  ⎬ ω(x) dx < ∞, ess sup ω(y) ⎩ |B| ⎭ y∈B B

where the suprema are taken over all balls B ⊂ Rn . Let n ≥ 2 and  be a bounded Lipschitz domain in Rn . Denote by ν := (ν1 , . . . , νn ) the outward unit normal to ∂, the boundary of . Let p ∈ [1, ∞) and ω ∈ Aq (Rn ) with some p q ∈ [1, ∞). Recall that the weighted Lebesgue space Lω () is defined by setting  Lpω () := f is measurable on  : f Lpω () < ∞ , where f Lpω () :=

⎧ ⎨ ⎩

|f (x)|p ω(x) dx

⎫1/p ⎬ ⎭

.

(1.1)



Furthermore, let

 Lpω (; Rn ) := f := (f1 , . . . , fn ) : for any i ∈ {1, . . . , n}, fi ∈ Lpω ()

(1.2)

and fLpω (;Rn ) :=

n 

fi Lpω () .

i=1 1,p

Denote by Wω () the weighted Sobolev space on  equipped with the norm f W 1,p () := f Lpω () + ∇f Lpω (;Rn ) , ω

1,p

where ∇f denotes the distributional gradient of f . Moreover, W0, ω () stands for the closure

of Cc∞ () in Wω (), where Cc∞ () denotes the set of all infinitely differentiable functions on  with compact supports contained in . It is easy to see that, when ω ≡ 1, the weighted spaces p 1,p Lω () and Wω () are just the classical Lebesgue space Lp () and the classical Sobolev space 1,p W (), respectively. 1,p

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For any given x ∈ Rn , let A(x) := {aij (x)}ni,j =1 denote an n × n matrix with real-valued, bounded, measurable entries. Then A is said to satisfy the uniform ellipticity condition if there exists a positive constant μ0 ∈ (0, 1] such that, for any x := (x1 , . . . , xn ), ξ := (ξ1 , . . . , ξn ) ∈ Rn , μ0 |ξ |2 ≤

n 

2 aij (x)ξi ξj ≤ μ−1 0 |ξ | .

(1.3)

i,j =1

Throughout this article, we always assume that the matrix A is real-valued, bounded, and measurable, and satisfies the uniform ellipticity condition (1.3). p p Let p ∈ (1, ∞), ω ∈ Aq (Rn ) with some q ∈ [1, ∞), F ∈ Lω (), and g ∈ Lω (; Rn ). A function u is called a weak solution of the following weighted Neumann boundary value problem ⎧ ⎨−div(A∇u) = F + div(g) ∂u ⎩ =g·ν ∂ν

(N )p, ω

in , (1.4)

on ∂, 1,p

where ∂u ∂ν := (A∇u) · ν denotes the conormal derivative of u on ∂, if u ∈ Wω () and, for any ϕ ∈ C ∞ (Rn ) (the set of all infinitely differentiable functions on Rn ), 

 A(x)∇u(x) · ∇ϕ(x) dx =



 F (x)ϕ(x) dx −



g(x) · ∇ϕ(x) dx.

(1.5)



Moreover, the weighted Neumann boundary  value problem (N )p, ω ispsaid to be uniquely solvp able if, for any F ∈ Lω () satisfying  F (x) dx = 0 and g ∈ Lω (; Rn ), there exists a 1,p u ∈ Wω (), unique up to constants, such that (1.5) holds true. Furthermore, the function u is called a weak solution of the following weighted Dirichlet boundary value problem (D)p, ω

−div(A∇u) = F + div(g) u=0

in , on ∂

(1.6)

if u ∈ W0, ω () and (1.5) holds true for any ϕ ∈ Cc∞ (). The weighted Dirichlet problem (D)p, ω p p is said to be uniquely solvable if, for any F ∈ Lω () and g ∈ Lω (; Rn ), there exists a unique 1,p u ∈ W0, ω () such that (1.5) holds true. In particular, in (1.4) and (1.6), if ω ≡ 1, then the weighted Neumann problem (N )p, ω and the weighted Dirichlet problem (D)p, ω are just, respectively, the classical Neumann problem and the classical Dirichlet problem. In this case, we denote (N )p, ω and (D)p, ω simply, respectively, by (N )p and (D)p . Furthermore, by the Lax–Milgram theorem (see, for instance, [36, Theorem 5.8]), we know that the Neumann problem (N )2 and the Dirichlet problem (D)2 are uniquely solvable. Let n ≥ 2 and  ⊂ Rn be a bounded Lipschitz domain or a bounded (semi-)convex domain. Assume that the matrix A satisfies the (δ, R)-BMO condition (see Definition 1.4 below) or A belongs to the space VMO(Rn ) (see, for instance, [52]). The main purpose of this article is to obtain the weighted Calderón–Zygmund type estimates 1,p

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 ∇uLpω (;Rn ) ≤ C gLpω (;Rn ) + F Lpω () ,

(1.7)

for the Neumann problem (1.4), with p ∈ (1, ∞), ω ∈ Aq (Rn ) and some q ∈ [1, ∞), and then give their applications, where C is a positive constant independent of u, g and F . For the Dirichlet problem (D)p , the estimate (1.7) with F ≡ 0, p ∈ (1, ∞) and ω ≡ 1 was established in [25], under the assumptions that A ∈ VMO(Rn ) and ∂ ∈ C 1,1 , which was weakened to ∂ ∈ C 1 in [6]. Moreover, for the Dirichlet problem (D)p , the estimate (1.7) with F ≡ 0, p ∈ (1, ∞) and ω ≡ 1 was obtained in [11,17], under the assumptions that A satisfies the (δ, R)-BMO condition for sufficiently small δ ∈ (0, ∞) and  is a bounded Lipschitz domain with a small Lipschitz constant or a bounded Reifenberg flat domain (see, for instance, [51,58]). Furthermore, for the Dirichlet problem (D)p with partial small BMO coefficients, the estimate (1.7) with F ≡ 0, p ∈ (1, ∞), and ω ≡ 1 was studied in [27,41], under the assumption that  is a bounded Lipschitz domain with small Lipschitz constant. For the Dirichlet problem (D)p in a general Lipschitz domain , it was proved in [56] that, if A is symmetric and A ∈ VMO(Rn ), then (1.7) with F ≡ 0 and ω ≡ 1 holds true for any p ∈ ( 32 − ε, 3 + ε) when n ≥ 3 and p ∈ ( 43 − ε, 4 + ε) when n = 2, where ε ∈ (0, ∞) is a positive constant depending on . It is worth pointing out that, when A := I (the identity matrix) in (1.6), the range of p obtained in [56] is even sharp for general Lipschitz domains (see, for instance, [38]). Moreover, for the weighted Dirichlet problem (D)p, ω with partial small BMO coefficients, (1.7) with F ≡ 0, p ∈ (2, ∞), and ω ∈ Ap/2 (Rn ) was obtained in [14] under the assumption that  is a bounded Reifenberg flat domain. For the problem (D)p, ω with symmetric and small BMO coefficients, (1.7) with F ≡ 0, p ∈ (1, ∞) and ω ∈ Ap (Rn ) was established in [2] under the assumption that  is a bounded Lipschitz domain with small Lipschitz constant. For C 1 domains, Lipschitz domains, Reifenberg flat domains and (semi-)convex domains, we have the following relations. We know that C 1 domains are Lipschitz domains with small Lipschitz constants, Lipschitz domains with small Lipschitz constants are Reifenberg flat domains, but general Lipschitz domains may not be Reifenberg flat domains. Moreover, (semi-)convex domains are Lipschitz domains, but may not be C 1 domains, Lipschitz domains with small Lipschitz constants or Reifenberg flat domains. Furthermore, convex domains are semi-convex domains (see Remarks 1.10 and 1.11 below for the details). For the Neumann problem (N )p , it was proved in [6] that, if A ∈ VMO(Rn ) and ∂ ∈ C 1 , then (1.7) with F ≡ 0, p ∈ (1, ∞), and ω ≡ 1 holds true. Moreover, for the problem (N )p , if A has the small BMO coefficients and  is a bounded Reifenberg flat domain, or A has partial small BMO coefficients and  is a bounded Lipschitz domain with small Lipschitz constant, (1.7) with F ≡ 0, p ∈ (1, ∞), and ω ≡ 1 was established, respectively, in [12,16] and [27]. For the Neumann problem (N )p on a general Lipschitz domain, it was proved in [32] that, if A is symmetric and A ∈ VMO(Rn ), then (1.7) with F ≡ 0 and ω ≡ 1 holds true for any p ∈ ( 32 − ε, 3 + ε) when n ≥ 3 and p ∈ ( 43 − ε, 4 + ε) when n = 2, where ε ∈ (0, ∞) is a positive constant depending on . We point out that, when A := I in (1.4), the range of p obtained in [32] is even sharp for general Lipschitz domains (see, for instance, [31]). In particular, if A is symmetric, A ∈ VMO(Rn ) and  is a bounded convex domain in the Neumann problem (N )p , it was showed in [33] that (1.7) with F ≡ 0 and ω ≡ 1 holds true for any given p ∈ (1, ∞). Now we state the main results of this article as follows. Theorem 1.2. Let n ≥ 2,  ⊂ Rn be a bounded Lipschitz domain and p ∈ (2, ∞). Assume that the matrix A is real-valued, symmetric, bounded and measurable, and satisfies (1.3). Then the following three statements are mutually equivalent.

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(i) Let u ∈ W 1,2 () be the weak solution of the Neumann problem (N )2 with g ∈ Lp (; Rn ) and F ≡ 0. Then u ∈ W 1,p () and there exists a positive constant C, depending only on n, p, μ0 , and the Lipschitz constant of , such that ∇uLp (;Rn ) ≤ CgLp (;Rn ) .

(1.8)

(ii) There exist positive constants C0 ∈ (0, ∞) and r0 ∈ (0, diam()) such that, for any ball B(x0 , r) with r ∈ (0, r0 /4) and either x0 ∈ ∂ or B(x0 , 2r) ⊂ , the weak reverse Hölder inequality ⎧ ⎪ ⎨1 n ⎪ ⎩r

 |∇v(x)|p dx B(x0 ,r)∩

⎫1/p ⎪ ⎬ ⎪ ⎭

⎧ ⎪ ⎨1 ≤ C0 n ⎪ ⎩r

 |∇v(x)|2 dx B(x0 ,2r)∩

⎫1/2 ⎪ ⎬ ⎪ ⎭

(1.9)

holds true for any function v ∈ W 1,2 (B(x0 , 2r) ∩ ) satisfying div(A∇v) = 0 in B(x0 , 2r) ∩  and ∂v ∂ν = 0 on B(x0 , 2r) ∩ ∂ when x0 ∈ ∂, where ν denotes the outward unit normal to ∂ and diam() := sup{|y − z| : y, z ∈ }. (iii) Let q ∈ [2, p], q0 ∈ [1, q/p ], r0 ∈ [p/(p − q), ∞] and ω ∈ Aq0 (Rn ) ∩ RHr0 (Rn ), where p := p/(p − 1). Assume that u is a weak solution of the weighted Neumann problem q 1,q (N )q, ω with g ∈ Lω (; Rn ) and F ≡ 0. Then u ∈ Wω () and there exists a positive constant C, depending only on n, μ0 , p, q, [ω]Aq0 (Rn ) , [ω]RHr0 (Rn ) and the Lipschitz constant of , such that ∇uLqω (;Rn ) ≤ CgLqω (;Rn ) .

(1.10)

The proof of Theorem 1.2 is based on a weighted real variable argument (see Theorem 3.1 below), which was essentially established in [54, Theorem 3.4] (see also [32,33,53,56]) and inspired by [19,59]. We also point out that a similar real variable argument with the different motivation was used in [5]. Moreover, the linear structure of the Neumann problem (1.4), the properties of Muckenhoupt weights and the (weak) reverse Hölder inequality are subtly used in the proof of Theorem 1.2. We point out that, in the case of bounded Reifenberg flat domains, the global gradient estimate (1.8) was obtained in [16,17] via using the Vitali covering lemma, the Hardy–Littlewood maximal function, the compactness method, the geometric property of Reifenberg flat domains and the fact that Reifenberg flat domains are W 1,p (1 ≤ p ≤ ∞) extension domains. The approach used in this article to establish (1.8) is different from that used in [16,17]. Indeed, to obtain (1.8) [or (1.10)], we first prove the weak reverse Hölder inequality (1.9), and then show (1.8) [or (1.10)] by using Theorem 1.2. Furthermore, the method used in this article to prove (1.9) is closely dependent on a real variable argument (see Theorem 3.1 below). Remark 1.3. (i) The conclusion of Theorem 1.2 also holds true for the Dirichlet problem (1.6). More pre∂v =0 cisely, replacing the Neumann problem (N )2 in (i), the Neumann boundary condition ∂ν on B(x0 , 2r) ∩ ∂ in (ii) and the weighted Neumann problem (N )q, ω in (iii), respectively, by the Dirichlet problem (D)2 , the Dirichlet boundary condition u = 0 on B(x0 , 2r) ∩ ∂,

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and the weighted Dirichlet problem (D)q, ω , and then repeating the proof of Theorem 1.2, we find that Theorem 1.2 also holds true in this case. We point out that, for the Dirichlet problem (1.6), the equivalence of (i) and (ii) in Theorem 1.2 was essentially established in [56]. (ii) For the Neumann problem (1.4), the conclusion that (ii) implies (i) in Theorem 1.2 was obtained in [32, Theorem 1.1]. Thus, Theorem 1.2 improves [32, Theorem 1.1] via proving the equivalence of (i) and (ii) of Theorem 1.2. Moreover, by the proof of [32, Theorem 1.2] (see also [33, Theorem 1.2]), we conclude that, for the bounded Lipschitz domain  ⊂ Rn , there exists a positive constant δ0 ∈ (0, ∞), depending on n and , such that, if the symmetric matrix A satisfies the (δ, R)-BMO condition for some δ ∈ (0, δ0 ) and R ∈ (0, ∞) or A ∈ VMO(Rn ), then (1.8) holds true for any given p ∈ ( 32 − ε, 3 + ε) when n ≥ 3 or p ∈ ( 43 − ε, 4 + ε) when n = 2, where ε ∈ (0, ∞) is a positive constant depending on . From this and the equivalence of (i) and (iii) of Theorem 1.2, we deduce that, for the weighted Neumann problem (N )2, ω with F ≡ 0, (1.10) holds true for any ω ∈ A4/3 (Rn ) ∩ RH3 (Rn ) when n ≥ 3 or any ω ∈ A3/2 (R2 ) ∩ RH2 (R2 ). Moreover, even when A := I in (1.4), the range p ∈ ( 32 − ε, 3 + ε) (when n ≥ 3) or p ∈ ( 43 − ε, 4 + ε) (when n = 2) of p such that (1.8) holds true is sharp for general Lipschitz domains (see, for instance, [31,32]). By this and the equivalence of (i) and (iii) of Theorem 1.2, we find that the condition ω ∈ A4/3 (Rn ) ∩ RH3 (Rn ) (when n ≥ 3) or ω ∈ A3/2 (R2 ) ∩ RH2 (R2 ) such that (1.10) holds true for (N )2, ω with F ≡ 0 is sharp for general Lipschitz domains. Denote by L1loc (Rn ) the set of all locally integrable functions on Rn . To establish the global weighted gradient estimates for solutions to Neumann boundary problems on bounded (semi)convex domains via using Theorem 1.2, we need to recall notions of the (δ, R)-BMO condition and the space VMO(Rn ) as follows (see, for instance, [17,52]). Definition 1.4. Let R, δ ∈ (0, ∞). A function f ∈ L1loc (Rn ) is said to satisfy the (δ, R)-BMO condition if  1 f ∗, R := sup sup |f (y) − fB(x,r) | dy ≤ δ, (1.11) r∈(0,R) x∈Rn |B(x, r)| B(x,r)

where the suprema are taken over all x ∈ Rn and r ∈ (0, R) and fB(x,r) :=

1 |B(x, r)|

 f (y) dy. B(x,r)

Furthermore, f is said to belong to VMO(Rn ) if f satisfies the (δ, R)-BMO condition for some δ, R ∈ (0, ∞) and lim sup

r→0+ x∈Rn

1 |B(x, r)|

 |f (y) − fB(x,r) | dy = 0, B(x,r)

where r → 0+ means r ∈ (0, ∞) and r → 0.

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Moreover, a matrix A := {aij }ni,j =1 is said to satisfy the (δ, R)-BMO condition [or A ∈ VMO(Rn )] if, for any i, j ∈ {1, . . . , n}, aij satisfies the (δ, R)-BMO condition [or aij ∈ VMO(Rn )]. Remark 1.5. A function f ∈ L1loc (Rn ) is said to belong to the space BMO(Rn ) (the space of bounded mean oscillation), denoted by f ∈ BMO(Rn ), if f BMO(Rn ) := f ∗, ∞ < ∞, where f ∗, ∞ is as in (1.11). Similarly, a matrix A := {aij }ni,j =1 is said to belong to the space BMO(Rn ), denoted by A ∈ BMO(Rn ), if, for any i, j ∈ {1, . . . , n}, aij ∈ BMO(Rn ). Let δ ∈ (0, ∞). If f ∈ BMO(Rn ) and f BMO(Rn ) ≤ δ, then f satisfies the (δ, R)-BMO condition for any R ∈ (0, ∞). Moreover, if f ∈ VMO(Rn ), then f satisfies the (γ , R)-BMO condition for any γ ∈ (0, ∞) and some R ∈ (0, ∞). Using Theorem 1.2 and the geometric property of (semi-)convex domains, we obtain the following weighted gradient estimates for the Neumann problem (1.4) on bounded (semi-)convex domains. Theorem 1.6. Let n ≥ 2,  ⊂ Rn be a bounded (semi-)convex domain, p ∈ (1, ∞) and ω ∈ Ap (Rn ). Assume that the matrix A is real-valued, symmetric, bounded and measurable, and satisfies (1.3). Then there exists a positive constant δ0 ∈ (0, ∞), depending on n, p,  and [ω]Ap (Rn ) , such that, if A satisfies the (δ, R)-BMO condition for some δ ∈ (0, δ0 ) and p p R ∈ (0, ∞), then the Neumann problem (N )p, ω with g ∈ Lω (; Rn ) and F ∈ Lω () is uniquely solvable and there exists a positive constant C, depending only on n, p, [ω]Ap (Rn ) , diam() and the Lipschitz constant and the uniform ball constant of , such that, for any weak solution u, p ∇u ∈ Lω (; Rn ) and  ∇uLpω (;Rn ) ≤ C gLpω (;Rn ) + F Lpω () . (1.12) The key to proving Theorem 1.6 is to show that the weak reverse Hölder inequality (1.9) is valid for any p ∈ (2, ∞) in the case where  is bounded and (semi-)convex. To do this, we apply the real variable argument established in Theorem 3.1 below and some nice geometric properties of (semi-)convex domains. More precisely, via the uniform exterior ball property of semi-convex domains, a new Bernstein type identity established in [45, Theorem 3.2] and expressed by the Weingarten matrix of ∂, the boundedness from below of the Weingarten matrix of ∂ in the case that  is a bounded semi-convex domain, and some ideas from [34], it was proved in [60, Theorem 2.4] that (1.9) holds true for any p ∈ (2, ∞) if the coefficient matrix A = I (see Lemma 4.2 below). By this, a comparison principle (see, for instance, Lemma 4.1 below), the (δ, R)-BMO condition for A, and Theorem 3.1, we prove that (1.9) holds true for any p ∈ (2, ∞), which, combined with Theorem 1.2, then completes the proof of Theorem 1.6. Compared with convex domains, in the case of semi-convex domains, the new ingredients appeared in the proof of Theorem 1.6 are a new Bernstein type identity which is expressed by the Weingarten matrix of ∂ and the boundedness from below of the Weingarten matrix of ∂ (see, for instance, [45,60]). Moreover, we point out that the method used in this article to prove Theorem 1.6 is different from that used in [2] and [11,12,16,17]. The approach used in [2] employs a local sharp maximal function of Fefferman and Stein to obtain the global gradient estimate. Motivated by [19,59], the approach used in [11,12,16,17] depends on the Vitali covering lemma, the Hardy– Littlewood maximal function and the compactness method. However, the approach used in this

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article strongly depends on the weighted real variable argument established in Theorem 3.1, which is motivated by [32,33,54,56]. In particular, if the matrix A ∈ VMO(Rn ), from Definition 1.4 and Remark 1.5, we deduce that, for sufficiently small δ ∈ (0, ∞), there exists R ∈ (0, ∞) such that A satisfies the (δ, R)-BMO condition. By this and the proof of Theorem 1.6, we know that, if A ∈ VMO(Rn ), then the conclusion of Theorem 1.6 also holds true, which is stated as follows with the details omitted. Corollary 1.7. Let n ≥ 2,  ⊂ Rn be a bounded (semi-)convex domain, p ∈ (1, ∞) and ω ∈ Ap (Rn ). Assume that the matrix A is real-valued, symmetric, bounded and measurable, and p satisfies (1.3). If A ∈ VMO(Rn ), then the Neumann problem (N )p, ω with g ∈ Lω (; Rn ) and p F ∈ Lω () is uniquely solvable and there exists a positive constant C, depending only on n, p, [ω]Ap (Rn ) , diam(), and the Lipschitz constant and the uniform ball constant of , such that, p for any weak solution u, ∇u ∈ Lω (; Rn ) and  ∇uLpω (;Rn ) ≤ C gLpω (;Rn ) + F Lpω () .

(1.13)

Remark 1.8. (i) Let A and  be as in Theorem 1.6 or Corollary 1.7. Then, for any p ∈ (1, ∞) and ω ∈ Ap (Rn ), (1.12) and (1.13) hold true for the Dirichlet problem (D)p, ω . Indeed, from the proofs of Lemmas 4.1 and 4.2 below, it follows that the conclusions of Lemmas 4.1 and 4.2 are also valid if replacing the Neumann boundary condition by the Dirichlet boundary condition. By this and the fact that Theorem 1.2 holds true for the Dirichlet problem, we know that the proof of Theorem 1.6 is also valid for the Dirichlet problem (D)p, ω and hence (1.12) and (1.13) hold true for the Dirichlet problem (1.6). Even when ω ≡ 1 and  is convex, this improves the corresponding results established in [56] via extending the range p ∈ ( 32 − ε, 3 + ε) when n ≥ 3 and p ∈ ( 43 − ε, 4 + ε) when n = 2 of p into p ∈ (1, ∞), where ε ∈ (0, ∞) is a positive constant depending on . (ii) We point out that the estimate (1.12) with ω ≡ 1 and F ≡ 0 was established in [16, Theorem 2] under the assumptions that A satisfies the (δ, R)-BMO condition for some small δ ∈ (0, ∞) and some R ∈ (0, ∞), and that  is a bounded Reifenberg flat domain. Observing that bounded Reifenberg flat domains and bounded (semi-)convex domains cannot cover each other, in this sense, Theorem 1.6 is a supplement of [16, Theorem 2]. Moreover, when  is a bounded convex domain, F ≡ 0 and ω ≡ 1, the estimate (1.13) was obtained in [33, Theorem 1.3]. Thus, even when ω ≡ 1, Corollary 1.7 improves [33, Theorem 1.3] by weakening the assumption of convex domains for  into semi-convex domains. By applying the weighted norm inequality obtained in Theorem 1.6 and Corollary 1.7 and some tools from harmonic analysis, such as the interpolation of operators and the Rubio de Francia extrapolation theorem, we obtain the global gradient estimates for the Neumann problem (1.4), respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (weighted) Orlicz spaces, and variable Lebesgue spaces, which have independent interests and are presented in Section 2 below. We point out that the approach used in this article to establish the global gradient estimates in both (weighted) Orlicz spaces and variable Lebesgue spaces is quite different from that used in [13,18,39]. In [13,18], the global gradient estimates in variable Lebesgue spaces

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or in Orlicz spaces were established via the so-called “maximum function free technique” developed in [1]. Furthermore, the global gradient estimates in Orlicz spaces were obtained in [39] through an approximation argument, the modified Vitali covering lemma, and a compactness method. However, in this article, as corollaries of weighted norm inequalities in Theorem 1.6, the interpolation of operators and the extrapolation theorem, the global gradient estimates in both (weighted) Orlicz spaces, and variable Lebesgue spaces are obtained. Now, we recall the notion of semi-convex domains as follows. Definition 1.9. (i) Let O be an open set in Rn . The collection of semi-convex functions on O is defined to be the set of all continuous functions u : O → R having the property that there exists a positive constant C such that, for any x, h ∈ Rn with the ball B(x, |h|) ⊂ O, 2u(x) − u(x + h) − u(x − h) ≤ C|h|2 . The minimal positive constant C as above is referred as the semi-convexity constant of u. (ii) A non-empty, proper open subset  of Rn is said to be semi-convex provided there exist constants c, d ∈ (0, ∞) such that, for every x0 ∈ ∂, there exist an (n − 1)-dimensional affine variety H ⊂ Rn passing through x0 , a choice N of the unit normal to H , and an open set C := { x + tN :  x ∈ H, | x − x0 | < c, |t| < d} (which is called a coordinate cylinder near x0 with axis along N ) satisfying, for some semiconvex function ϕ : H → R, C ∩  = C ∩ { x + tN :  x ∈ H, t > ϕ( x )}, C ∩ ∂ = C ∩ { x + tN :  x ∈ H, t = ϕ( x )}, 

x + tN :  x ∈ H, t < ϕ( x )}, C ∩  = C ∩ { x )| < d/2 if | x − x0 | ≤ c, ϕ(x0 ) = 0 and |ϕ( 

where  and  respectively denote the closure of  in Rn and the complementary set of  in Rn . Remark 1.10. (i) Recall that a set  ⊂ Rn is called a domain if  is a connected open subset of Rn . Let n ≥ 2, δ ∈ (0, 1), and R0 ∈ (0, ∞). A domain  ⊂ Rn is called a Reifenberg flat domain if, for any x0 ∈ ∂ and any r ∈ (0, R0 ], there exists a system of coordinates {y1 , . . . , yn }, which may depend on x0 and r, such that, in this coordinate system, x0 = 0n and B(0n , r) ∩ {y ∈ Rn : yn > δr} ⊂ B(0n , r) ∩  ⊂ B(0n , r) ∩ {y ∈ Rn : yn > −δr}, (1.14)

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where 0n denotes the origin of Rn . The Reifenberg flat domain was introduced by Reifenberg [51], which naturally appears in the theory of minimal surfaces and free boundary problems. In recent years, boundary value problems of elliptic or parabolic equations on Reifenberg flat domains have been widely concerned and studied (see, for instance, [3,8–10,15–17,42]). (ii) On Reifenberg flat domains, Lipschitz domains, and (semi-)convex domains, we have the following relations. Lipschitz domains with small Lipschitz constants are Reifenberg flat domains, but generally Lipschitz domains may not be Reifenberg flat domains (see, for instance, [58]). (Semi-)convex domains are Lipschitz domains, but may not be Lipschitz domains with small Lipschitz constants or Reifenberg flat domains. For instance, let α0 , β0 ∈ (0, ∞) be constants and  := {(x1 , x2 ) ∈ R2 : β0 > x2 > α0 |x1 |}. It is easy to see that  ⊂ R2 is a bounded convex domain. However, at the points on ∂ near the vertexes of , (1.14) does not hold true. Thus,  is not a Reifenberg flat domain and hence not a Lipschitz domain with small Lipschitz constant. Remark 1.11. (i) A set E ⊂ Rn is said to satisfy an exterior ball condition at x ∈ ∂E if there exist v ∈ S n−1 and r ∈ (0, ∞) such that B(x + rv, r) ⊂ (Rn \ E),

(1.15)

where S n−1 denotes the unit sphere of Rn . For such an x ∈ ∂E, let  r(x) := sup r ∈ (0, ∞) : (1.15) holds true for some v ∈ S n−1 . A set E is said to satisfy a uniform exterior ball condition (for short, UEBC) with radius r ∈ (0, ∞] if inf r(x) ≥ r,

x∈∂E

(1.16)

and the value r in (1.16) is referred to the UEBC constant. A set E is said to satisfy a UEBC if there exists r ∈ (0, ∞] such that E satisfies the uniform exterior ball condition with radius r. Moreover, the largest constant r as above is called the uniform ball constant of E. It is well known that, for any open set  ⊂ Rn with compact boundary,  is a Lipschitz domain satisfying a UEBC if and only if  is a semi-convex domain in Rn (see, for instance, [45, Theorem 2.5] or [46, Theorem 3.9]). (ii) It is worth pointing out that, if  ⊂ Rn is convex, then  satisfies a UEBC with the uniform ball constant ∞ (see, for instance, [29]). Thus, convex domains in Rn are Lipschitz domains satisfying a UEBC and hence convex domains in Rn are semi-convex domains (see, for instance, [45,46,60,61]). This article is organized as follows. In Section 2, several applications of the global weighted gradient estimates in Theorem 1.6 are given (see Theorems 2.2, 2.4, 2.10 and 2.13 below), but their proofs are given in Section 5. In Section 3, we give the proof of Theorem 1.2 via a weighted real variable argument and, in Section 4, we prove Theorem 1.6 by using Theorem 1.2.

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Finally, we make some conventions on notation. Throughout the whole article, we always denote by C a positive constant which is independent of main parameters, but it may vary from line to line. We also use C(γ , β, ...) or c(γ , β, ...) to denote a positive constant depending on the indicated parameters γ , β, . . .. The symbol f  g means that f ≤ Cg. If f  g and g  f , then we write f ∼ g. We also use the following convention: If f ≤ Cg and g = h or g ≤ h, we then write f  g ∼ h or f  g  h, rather than f  g = h or f  g ≤ h. For each ball B := B(xB , rB ) in Rn , with some xB ∈ Rn , rB ∈ (0, ∞), and α ∈ (0, ∞), let αB := B(xB , αrB ); furthermore, denote the set B(x, r) ∩  by B (x, r) and the set (αB) ∩  by αB . For any subset E of Rn , we denote the set Rn \ E by E  and its characteristic function by 1E . For any  ω ∈ Ap (Rn ) with p ∈ [1, ∞) and any measurable set E ⊂ Rn , let ω(E) := E ω(x) dx. We also let N := {1, 2, . . .}. Finally, for any given q ∈ [1, ∞], we denote by q its conjugate exponent, namely, 1/q + 1/q = 1. 2. Several applications of Theorem 1.6 In this section, we give several applications of the global weighted gradient estimates obtained in Theorem 1.6 (see Section 5 for their proofs). More precisely, using Theorem 1.6 and Corollary 1.7, we obtain the global gradient estimates, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (weighted) Orlicz spaces, and variable Lebesgue spaces. We begin q,r with recalling the notion of the weighted Lorentz space Lω () on the domain . Definition 2.1. Assume that n ≥ 2 and  is a bounded Lipschitz domain in Rn . Let r ∈ (0, ∞], q,r q ∈ [1, ∞) and ω ∈ Ap (Rn ) with some p ∈ [1, ∞). The weighted Lorentz space Lω () is defined by setting  Lq,r < ∞ , ω () := f is measurable on  : f Lq,r () ω where, when r ∈ (0, ∞), ⎧ ∞ ⎫1/r ⎨   r/q dt ⎬ q t f Lq,r := ω ∈  : |f (x)| > t}) q ({x () ω ⎩ t ⎭ 0

and f Lq,∞ := sup t[ω ({x ∈  : |f (x)| > t})]1/q . ω () t∈(0,∞)

q,r

p

Moreover, the space Lω (; Rn ) is defined via replacing Lω () in (1.1) by the above p in the definition of Lω (; Rn ) in (1.2).

q,r Lω ()

q,r

q

q,r

It is easy to see that, when q, r ∈ [1, ∞) and q = r, Lω () = Lω () and Lω (; Rn ) =

q Lω (; Rn ).

Then we have the following global gradient estimates in weighted Lorentz spaces for the Neumann problem (1.4).

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Theorem 2.2. Let n ≥ 2,  ⊂ Rn be a bounded (semi-)convex domain, r ∈ (0, ∞], q ∈ (1, ∞) and ω ∈ Aq (Rn ). Assume that the matrix A is real-valued, symmetric, bounded and measurable, and satisfies (1.3). Then there exists a positive constant δ0 ∈ (0, ∞), depending on n, q, , and [ω]Aq (Rn ) , such that, if A satisfies the (δ, R)-BMO condition for some δ ∈ (0, δ0 ) and R ∈ (0, ∞) or A ∈ VMO(Rn ), then there exists a positive constant C, depending only on n, q, r, [ω]Aq (Rn ) diam(), the Lipschitz constant, and the uniform ball constant of , such that, for any weak q,r q,r solution u ∈ W 1,2 () of the Neumann problem (N )2 with g ∈ Lω (; Rn ) and F ∈ Lω (), q,r n ∇u ∈ Lω (; R ) and  ∇uLq,r (2.1) n ) ≤ C gLq,r (;Rn ) + F Lq,r () . (;R ω ω ω Definition 2.3. Assume that n ≥ 2 and  is a bounded Lipschitz domain in Rn . Let q ∈ (1, ∞), r ∈ (0, ∞] and θ ∈ [0, n]. The Lorentz–Morrey space Lq,r;θ () is defined by setting

 Lq,r;θ () := f is measurable on  : f Lq,r;θ () < ∞ , where f Lq,r;θ () :=

sup

 θ−n sup ρ q f Lq,r (B(x,ρ)∩) .

ρ∈(0,diam()] x∈ p

Moreover, the space Lq,r;θ (; Rn ) is defined via replacing Lω () in (1.1) by the above p in the definition of Lω (; Rn ) in (1.2).

Lq,r;θ ()

We point out that, when θ = n, the Lorentz–Morrey space Lq,r;θ () is just the Lorentz space; in this case, we denote the spaces Lq,r;θ () and Lq,r;θ (; Rn ) simply, respectively, by Lq,r () and Lq,r (; Rn ). Moreover, when q = r, the space Lq,r;θ () is just the Morrey space; in this case, we denote the spaces Lq,r;θ () and Lq,r;θ (; Rn ) simply by Mθq () and Mθq (; Rn ), respectively. Motivated by [2,4,42,43], using Theorem 2.2 and properties of Muckenhoupt weights, we obtain the following global gradient estimates in Lorentz–Morrey spaces for the Neumann problem (1.4). Theorem 2.4. Let A and  be as in Theorem 2.2, r ∈ (0, ∞], q ∈ (1, ∞) and θ ∈ (0, n]. Then there exists a positive constant δ0 ∈ (0, ∞), depending on n, q, r, θ , and , such that, if A satisfies the (δ, R)-BMO condition for some δ ∈ (0, δ0 ) and R ∈ (0, ∞) or A ∈ VMO(Rn ), then there exists a positive constant C, depending only on n, q, r, θ , diam(), the Lipschitz constant, and the uniform ball constant of , such that, for any weak solution u ∈ W 1,2 () of the Neumann problem (N )2 with g ∈ Lq,r;θ (; Rn ) and F ∈ Lq,r;θ (), ∇u ∈ Lq,r;θ (; Rn ) and 

∇uLq,r;θ (;Rn ) ≤ C gLq,r;θ (;Rn ) + F Lq,r;θ () .

(2.2)

As a corollary of Theorem 2.4, we have the following global gradient estimates in the Morrey space. Corollary 2.5. Let A and  be as in Theorem 2.2, q ∈ (1, ∞) and θ ∈ (0, n]. Then there exists a positive constant δ0 ∈ (0, ∞), depending on n, q, θ , and , such that, if A satisfies the

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(δ, R)-BMO condition for some δ ∈ (0, δ0 ) and R ∈ (0, ∞) or A ∈ VMO(Rn ), then there exists a positive constant C, depending only on n, q, θ , diam(), the Lipschitz constant, and the uniform ball constant of , such that, for any weak solution u ∈ W 1,2 () of the Neumann problem (N )2 with g ∈ Mθq (; Rn ) and F ∈ Mθq (), ∇u ∈ Mθq (; Rn ) and  ∇uMθq (;Rn ) ≤ C gMθq (;Rn ) + F Mθq () . Recall that, for any α ∈ (0, 1], the Hölder space C 0,α () on  is defined by setting C 0,α () := g is continuous on  : [g]C 0,α () :=

sup

x, y∈, x=y

|g(x) − g(y)| <∞ . |x − y|α

Then, by Corollary 2.5 and the Sobolev–Morrey embedding theorem (see, for instance, [36, Theorem 7.19]), we obtain the following conclusion. Corollary 2.6. Let A and  be as in Theorem 2.2, θ ∈ (0, n] and q ∈ (max{1, θ}, ∞). Then there exists a positive constant δ0 ∈ (0, ∞), depending on n, q, θ , and , such that, if A satisfies the (δ, R)-BMO condition for some δ ∈ (0, δ0 ) and R ∈ (0, ∞) or A ∈ VMO(Rn ), then there exists a positive constant C, depending only on n, q, θ , diam(), the Lipschitz constant, and the uniform ball constant of , such that, for any weak solution u ∈ W 1,2 () of the Neumann problem (N )2 with g ∈ Mθq (; Rn ) and F ∈ Mθq (), u ∈ C [u]

0,1− qθ C

0,1− qθ

() and

 ≤ C gMθq (;Rn ) + F Mθq () . ()

Remark 2.7. We point out that, for the Dirichlet problem (1.6) with F ≡ 0, the estimates (2.1) and (2.2) were established in [2, Corollary 2.2 and Theorem 2.3] under the assumptions that A is symmetric and satisfies the (δ, R)-BMO condition for some small δ ∈ (0, ∞) and some R ∈ (0, ∞), and that  is a bounded Lipschitz domain with small Lipschitz constants. Moreover, estimates similar to (2.1) and (2.2) for the Dirichlet problem of some nonlinear elliptic or parabolic equations on Reifenberg flat domains were obtained in [3,4,9,10,42, 43]. Let A and  be as in Theorem 2.2. By Remark 1.8 and the proofs of Theorems 2.2 and 2.4, we find that (2.1) and (2.2) also hold true for the Dirichlet problem (1.6). Furthermore, from Remark 1.10(ii), it follows that (semi-)convex domains may not be Reifenberg flat domains or Lipschitz domains with small Lipschitz constants. Thus, even in the case of the Dirichlet problem (1.6), the estimates (2.1) and (2.2) are extensions of [2, Corollary 2.2 and Theorem 2.3] along the direction of (semi-)convex domains. Now we recall the definitions of Young functions and Orlicz spaces as follows (see, for instance, [50]). Recall that the symbol t → 0+ means t ∈ (0, ∞) and t → 0. Definition 2.8. (i) A function  : [0, ∞) → [0, ∞) is called a Young function if it is continuous, convex and increasing, and satisfies limt→0+ (t)/t = 0 and limt→∞ (t)/t = ∞.

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(ii) The Young function  is said to satisfy the 2 -condition, denoted by  ∈ 2 , if there exists a positive constant κ1 ∈ (1, ∞) such that, for any r ∈ [0, ∞), (2r) ≤ κ1 (r). (iii) The Young function  is said to satisfy the ∇2 -condition, denoted by  ∈ ∇2 , if there exists a positive constant κ2 ∈ (1, ∞) such that, for any r ∈ [0, ∞), (κ2 r) ≥ 2κ2 (r). (iv) Let  be a Young function and, for any t ∈ (0, ∞), h (t) := sups∈(0,∞) (st) (s) . Then the upper and lower dilation indices, I and i , of  are defined, respectively, by setting I := lim

t→∞

log h (t) log h (t) log h (t) log h (t) = inf and i := lim = sup . + t∈(1,∞) log t log t log t log t t→0 t∈(0,1)

(v) Assume that  is a Young function and ω ∈ Ap (Rn ) with some p ∈ [1, ∞). Let  ⊂ Rn be a domain. For any locally integrable function u on , the weighted modular ρ, ω (u) of u is defined by setting  ρ, ω (u) :=

(|u(x)|)ω(x) dx. 

Then the weighted Orlicz space L ω () is defined by setting

 L ω () := u is measurable on  : there exists λ ∈ (0, ∞) such that ρ, ω (λu) < ∞ equipped with the Luxembourg norm (which is also called the Luxembourg–Nakano norm) ⎧ ⎨





uLω () := inf λ ∈ (0, ∞) : ⎩





⎫ ⎬

|u(x)| ω(x) dx ≤ 1 . ⎭ λ

 p

n  Moreover, the space L ω (; R ) is defined via replacing Lω () by Lω () in the definition p n of Lω (; R ).  n  In particular, when ω ≡ 1, we denote the spaces L ω () and Lω (; R ) simply by L ()  n and L (; R ), respectively. For Young functions, we have the following observation (see, for instance, [50, Chapter II]).

Remark 2.9. Let  : [0, ∞) → [0, ∞) be a Young function. (i) Then  ∈ 2 ∩ ∇2 if and only if 1 < i ≤ I < ∞. (ii) The simplest example for the function  ∈ 2 ∩ ∇2 is that, for any t ∈ [0, ∞), (t) := t p with p ∈ (1, ∞). Another typical example of  ∈ 2 ∩ ∇2 is that, for any t ∈ [0, ∞), (t) := t p [log(e + t)]α with p ∈ (1, ∞) and α ∈ R. By the global weighted gradient estimates established in Theorem 1.6 and the interpolation of operators in the scale of Orlicz spaces (see, for instance, [20,48] or Lemma 5.2 below), we obtain the following global gradient estimates in weighted Orlicz spaces for the Neumann problem (1.4).

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Theorem 2.10. Let A and  be as in Theorem 2.2,  ∈ 2 ∩ ∇2 and ω ∈ Ai (Rn ), where i is as in Definition 2.8(iv). Then there exists a positive constant δ0 ∈ (0, ∞), depending on n, , [ω]Ai (Rn ) , and , such that, if A satisfies the (δ, R)-BMO condition for some δ ∈ (0, δ0 ) and R ∈ (0, ∞) or A ∈ VMO(Rn ), then there exists a positive constant C, depending only on n, , [ω]Ai (Rn ) , diam(), the Lipschitz constant, and the uniform ball constant of , such that, n for any weak solution u ∈ W 1,2 () of the Neumann problem (N )2 with g ∈ L ω (; R ) and   n F ∈ Lω (), ∇u ∈ Lω (; R ) and  ∇uLω (;Rn ) ≤ C gLω (;Rn ) + F Lω () .

(2.3)

Remark 2.11. For the Dirichlet problem (1.6) with F ≡ 0, the estimate (2.3) with ω ≡ 1 was obtained in [39, Theorem 3.1] under the assumptions that A satisfies the (δ, R)-BMO condition for some small δ ∈ (0, ∞) and some R ∈ (0, ∞), and that  is a bounded Reifenberg flat domain. Moreover, unweighed Orlicz type estimates similar to (2.3) for the Dirichlet problem of some p-Laplace type elliptic equations on Reifenberg flat domains were established in [18]. Let A and  be as in Theorem 2.2. From Remark 1.8 and the proof of Theorem 2.10, we deduce that (2.3) also holds true for the Dirichlet problem (1.6), which is a supplement and extension of [39, Theorem 3.1]. Now we recall the notion of variable Lebesgue spaces as follows (see, for instance, [24,26]). Definition 2.12. Let  be a domain in Rn and P() the set of all measurable functions p :  → [1, ∞). For any given p ∈ P(), the variable exponent modular ρp(·) is defined by setting, for any locally integrable function f on ,  ρp(·) (f ) :=

|f (x)|p(x) dx. 

The variable Lebesgue space Lp(·) () is defined by setting

 Lp(·) () = f is measurable on  : there exists λ ∈ (0, ∞) such that ρp(·) (λf ) < ∞ equipped with the Luxembourg norm (which is also called the Luxembourg–Nakano norm)

 f Lp(·) () := inf λ ∈ (0, ∞) : ρp(·) (f/λ) ≤ 1 . Moreover, for any p ∈ P(), let p+ := ess sup p(x) x∈

and

p− := ess inf p(x). x∈

By Theorem 1.6 and the extrapolation theorem in the scale of variable Lebesgue spaces, we obtain the global gradient estimates in variable Lebesgue spaces as follows.

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Theorem 2.13. Let A and  be as in Theorem 2.2 and p ∈ P(). Assume that 1 < p− ≤ p+ < ∞ and the Hardy–Littlewood maximal operator M is bounded on Lp(·) (). Then there exists a positive constant δ0 ∈ (0, ∞), depending on n, p(·), and , such that, if A satisfies the (δ, R)-BMO condition for some δ ∈ (0, δ0 ) and R ∈ (0, ∞) or A ∈ VMO(Rn ), then there exists a positive constant C, depending only on n, p− , p+ , diam(), the Lipschitz constant, and the uniform ball constant of , such that, for any weak solution u ∈ W 1,2 () of the Neumann problem (N )2 with g ∈ Lp(·) (; Rn ) and F ∈ Lp(·) (), ∇u ∈ Lp(·) (; Rn ) and

 ∇uLp(·) (;Rn ) ≤ C gLp(·) (;Rn ) + F Lp(·) () .

(2.4)

Remark 2.14. For the Dirichlet problem (1.6) with F ≡ 0, the estimate (2.4) was established in [13, Theorem 2.5] (see also [26]) under the assumptions that A is symmetric and has partial small BMO coefficients,  is a bounded Reifenberg flat domain, and p(·) satisfies the so-called log-Hölder continuity (see, for instance, [24,26]). Moreover, the variable exponent type estimate similar to (2.4) for the Dirichlet problem of some p-Laplace type elliptic equations on Reifenberg flat domains was also obtained in [8, Theorem 1.4]. Let A and  be as in Theorem 2.2. By Remark 1.8 and the proof of Theorem 2.13, we know that the variable exponent type estimate (2.4) also holds true for the Dirichlet problem (1.6), which is an extension of [13, Theorem 2.5] along the direction of (semi-)convex domains. 3. Proof of Theorem 1.2 In this section, we prove Theorem 1.2 via a weighted real variable argument given in Theorem 3.1 below, which was essentially established in [54, Theorem 3.4] (see also [32, Theorems 2.1 and 2.2], [33, Theorem 2.1], [53, Theorem 4.2.6], [55, Theorem 2.32] and [56, Theorem 3.3]) and inspired by the work of Caffarelli and Peral [19] (see also [59]). This weighted real variable argument may be seen as a duality argument of the well-known Calderón–Zygmund decomposition. We point out that the following Theorem 3.1 is a slight generalization of [54, Theorem 3.4] and [33, Theorem 2.1]. More precisely, if ε ≡ 0 in (3.2) below, Theorem 3.1 is just [54, Theorem 3.4]. Moreover, if ω ≡ 1 and we replace F and f , respectively, by |F |2 and |f |2 in Theorem 3.1 below, then Theorem 3.1 is just [33, Theorem 2.1] (see also [53, Theorem 4.2.6]). For the sake of completeness, we give the proof of Theorem 3.1 here. Theorem 3.1. Let n ≥ 2,  ⊂ Rn be a bounded Lipschitz domain, p0 ∈ (1, ∞), F ∈ L1 () and f ∈ Lq () with some q ∈ (1, p0 ). Suppose that, for any ball B := B(xB , rB ) ⊂ Rn having the property that |B| ≤ β1 || and either 2B ⊂  or xB ∈ ∂, there exist two measurable functions FB and RB on 2B such that |F | ≤ |FB | + |RB | on 2B ∩ , ⎧ ⎪ ⎨

1 ⎪ ⎩ |2B |

≤ C1

 |RB (x)|p0 dx 2B

⎧ ⎪ ⎨

1 ⎪ ⎩ |β2 B |



⎫1/p0 ⎪ ⎬ ⎪ ⎭ 1  |  |B B⊂B



|F (x)| dx + sup β2 B

|f (x)| dx  B

⎫ ⎪ ⎬ ⎪ ⎭

(3.1)

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18

and 1 |2B |

 2B

ε |FB (x)| dx ≤ |β2 B |

 β2 B

1 |F (x)| dx + C2 sup   |B | B⊂B

 |f (x)| dx,

(3.2)

 B

where C1 , C2 , ε ∈ (0, ∞) and 0 < β1 < 1 < β2 < ∞ are positive constants independent of  ⊃ B. Then, for any ω ∈ F, f, RB , FB , and B, and the suprema are taken over all balls B Aq (Rn ) ∩ RHs (Rn ) with s ∈ (p0 /(p0 − q), ∞], there exists a positive constant ε0 , depending on C1 , C2 , n, p0 , q, β1 , β2 , [ω]Aq (Rn ) , and [ω]RHs (Rn ) , such that, if ε ∈ [0, ε0 ), then ⎫1/q ⎧ ⎬ ⎨ 1  |F (x)|q ω(x) dx ⎭ ⎩ ω() 

⎡

⎢ 1 ≤C⎣ ||



⎧ ⎫1/q ⎤ ⎨ 1  ⎬ ⎥ |F (x)| dx + |f (x)|q ω(x) dx ⎦, ⎩ ω() ⎭



(3.3)



where C is a positive constant depending only on C1 , C2 , n, p0 , q, β1 , β2 , [ω]Aq (Rn ) and [ω]RHs (Rn ) . Recall that the Hardy–Littlewood maximal operator M on Rn is defined by setting, for any f ∈ L1loc (Rn ) and x ∈ Rn , M(f )(x) := sup Bx

1 |B|

 |f (y)| dy, B

where the supremum is taken over all balls B ⊂ Rn satisfying B  x. Moreover, let B0 ⊂ Rn be a ball. For any f ∈ L1loc (Rn ), the localized Hardy–Littlewood maximal function MB0 (f ) is defined by setting, for any x ∈ B0 , 1 MB0 (f )(x) := sup |B| Bx

 |f (y)| dy, B

where the supremum is taken over all balls B ⊂ B0 satisfying B  x. It is easy to see that, for any f ∈ L1loc (Rn ) and x ∈ B0 , MB0 (f )(x) ≤ M(f 1B0 )(x).

(3.4)

To show Theorem 3.1, we need the following properties of Ap (Rn ) weights, which are well known (see, for instance, [37, Chapter 7]). Lemma 3.2. Let q ∈ (1, ∞), ω ∈ Aq (Rn ), and  ⊂ Rn be a bounded Lipschitz domain. (i) There exists q0 ∈ (1, q), depending on n, q, and [ω]Aq (Rn ) , such that ω ∈ Aq0 (Rn ).

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(ii) There exists γ ∈ (0, ∞), depending on n, q, and [ω]Aq (Rn ) , such that ω ∈ RH1+γ (Rn ). q /q



(iii) ω−q /q ∈ Aq (Rn ) and [ω]Aq (Rn ) = [ω]Aq (Rn ) , where 1/q + 1/q = 1.  depending on n, s, (iv) If ω ∈ RHs (Rn ) with s ∈ (1, ∞], then there exists a positive constant C, and [ω]RHs (Rn ) , such that, for any ball B ⊂ Rn and any measurable set E ⊂ B,   ω(E)  |E| (s−1)/s . ≤C ω(B) |B| q () ⊂ L () ⊂ Lq/q0 (). (v) Let  q := q(1 + γ1 ) with γ as in (ii) and let q0 be as in (i). Then L ω q

Moreover, we also need the boundedness of the Hardy–Littlewood maximal operator M on (weak) weighted Lebesgue spaces (see, for instance, [37, Theorem 7.1.9]). Lemma 3.3. Let M be the Hardy–Littlewood maximal operator on Rn , p ∈ [1, ∞), and ω ∈ Ap (Rn ). Then there exists a positive constant C, depending on n, p, and [ω]Ap (Rn ) , such that, p for any f ∈ Lω (Rn ),    1/p sup λ ω x ∈ Rn : |M(f )(x)| > λ ≤ Cf Lpω (Rn ) .

λ∈(0,∞)

p

Moreover, if p ∈ (1, ∞), then M is bounded on Lω (Rn ). Now we prove Theorem 3.1 by using Lemmas 3.2 and 3.3 and by borrowing some ideas from the proofs of [54, Theorem 3.4] and [53, Theorem 4.2.3].  and Proof of Theorem 3.1. Take a ball B0 ⊂ Rn such that  ⊂ B0 and rB0 = diam(). Let F n  f be, respectively, the zero extensions of F and f to R . To show (3.3), it suffices to prove that ⎧ ⎪ ⎨

1 ⎪ ⎩ ω(B0 )





(x)|q ω(x) dx |F

B0

1 |B0 |

 B0

(x)| dx + |F

⎫1/q ⎪ ⎬ ⎪ ⎭

⎧ ⎪ ⎨

1 ⎪ ω(B 0) ⎩

 B0

|f(x)|q ω(x) dx

⎫1/q ⎪ ⎬ ⎪ ⎭

.

(3.5)

Let Q0 ⊂ Rn be a cube such that 2Q0 ⊂ 2B0 and |Q0 | ∼ |B0 |. Then we only need to show that ⎧ ⎪ ⎨

1 ⎪ ⎩ ω(Q0 )

 Q0

(x)|q ω(x) dx |F

⎫1/q ⎪ ⎬ ⎪ ⎭

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20



1 |2B0 |



(x)| dx + |F

2B0

⎧ ⎪ ⎨

1 ⎪ ⎩ ω(2B0 )



|f(x)|q ω(x) dx

2B0

⎫1/q ⎪ ⎬ ⎪ ⎭

.

(3.6)

Indeed, if (3.6) holds true, then (3.5) can be obtained by using (3.6) and covering B0 with a finite number of non-overlapping cubes Q0 of the same size such that 2Q0 ⊂ 2B0 . Now we prove (3.6). For any λ ∈ (0, ∞), let

 )(x) > λ . E(λ) := x ∈ Q0 : M2B0 (F To show (3.6), similarly to the proofs of [54, Theorem 3.4], [32, Theorem 2.1] and [53, Theorem 4.2.3], we only need to prove that there exists a positive constant ε0 , depending on p0 , q, β1 , β2 , C1 , C2 , [ω]Aq (Rn ) , and [ω]RHs (Rn ) , such that, if ε ∈ [0, ε0 ), then there exist positive constants δ ∈ (0, 1/3), γ ∈ (0, 1) and C3 ∈ (0, ∞), depending on p0 , q, β1 , β2 , C1 , C2 , [ω]Aq (Rn ) , and [ω]RHs (Rn ) , such that, for any λ ∈ (λ0 , ∞), ω(E(αλ)) ≤ δ (s−1)/s ω(E(λ)) + ω

  x ∈ Q0 : M2B0 (f)(x) > γ λ ,

(3.7)

where C3 λ0 := |2B0 |



(x)| dx |F

(3.8)

2B0

and α := [2δ (s−1)/s ]−1/q . Now we show (3.7). From (3.4) and Lemma 3.3, we deduce that there exists a positive constant C4 , depending only on n, such that, for any λ ∈ (0, ∞), |E(λ)| ≤

C4 λ



(x)| dx. |F

(3.9)

2B0

Let C3 := C4 |2B0 ||Q0 |−1 δ −1 . Then, by (3.8) and (3.9), we conclude that, for any λ ∈ (λ0 , ∞), |E(λ)| < δ|Q0 |. From now on, we fix λ ∈ (λ0 , ∞). From the fact that E(λ) is open in Q0 and the Calderón– Zygmund decomposition (see, for instance, [57, p. 17] and [53, Lemma 4.2.2]), it follows that there exist an index set I and a sequence {Qk }k∈I of disjoint and maximal dyadic subcubes of Q0 such that (a) for any k ∈ I , Qk ⊂ E(λ); k ⊂ Q0 , but Q k  E(λ), where Q k denotes the dyadic parent cube of Qk ; (b) for any k!∈ I , Q (c) |E(λ)\( k∈I Qk )| = 0. For any k ∈ I , denote by Bk the ball with the same center and diameter as Qk ; namely, rBk := √ n(Qk )/2, where (Qk ) denotes the side length of Qk . By the definition of E(λ) and the above property (b) of {Qk }k∈I , we find that there exists a positive constant C5 , depending only on n,  ⊂ 2B0 satisfying B  ∩ Bk = ∅ and rB ≥ rBk , such that, for any ball B

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1  |B|



(x)| dx ≤ C5 λ, |F

21

(3.10)

 B

which, combined with the definition of M2B0 , further implies that, for any x ∈ Qk with k ∈ I ,

 )(x) ≤ max M2Bk (F )(x), C5 λ . M2B0 (F

(3.11)

Let B := B(x0 , r) be a ball satisfying B ⊂ 2B0 and |B| ≤ β1 ||/5n , where x0 ∈ Rn and r ∈ B := FB and R B := RB ; if 2B ∩  = ∅, let F B := 0 and R B := 0; if (0, ∞). If 2B ⊂ , let F    2B ∩  = ∅ and 2B ∩  = ∅, let FB := FB1 1 and RB := RB1 1 , where B1 := B(y0 , 4r) and y0 ∈ ∂ satisfy 2B ⊂ B(y0 , 4r). Then it is easy to see that, for any ball B satisfying 2B ⊂ 2B0 | ≤ |F B | + |R B | on 2B, (3.1) and (3.2) also hold true if replacing F , f , and |B| ≤ β1 ||/5n , |F  B , R B , and 2B0 (see, for instance, [32, pp. 2435-2436] FB , RB , and , respectively, by F , f, F for more details). Take δ ∈ (0, 1/3) to be sufficiently small such that α > C5 . Then, from (3.11), the fact that | ≤ |F Bk | + |F Bk | on 2Bk , (3.4), and Lemma 3.3, it follows that, for any k ∈ I , |F " " )(x) > αλ " |E(αλ) ∩ Qk | ≤ " x ∈ Qk : M2Bk (F "# $" "# $" " " " " Bk )(x) > αλ " + " x ∈ Qk : M2Bk (R Bk )(x) > αλ " ≤ "" x ∈ Qk : M2Bk (F " " 2 2 "   C 2C4 Bk (x)| dx + (n, p0 ) Bk (x)|p0 dx. |F |R (3.12) ≤ αλ (αλ)p0 2Bk

2Bk

Now we claim that, for any k ∈ I , if Qk ∩ {x ∈ Q0 : M2B0 (f)(x) ≤ γ λ} = ∅,

(3.13)

−s/(s−1) δ|Qk |, |E(αλ) ∩ Qk | ≤ C

(3.14)

then

 is as in Lemma 3.2(iv). Indeed, for such a k, by (3.2), (3.10), and (3.13), we know that where C 1 |2Bk |



Bk (x)| dx ≤ (C2 γ + εC5 )λ. |F

(3.15)

2Bk

Moreover, from (3.1), (3.10), and (3.13), we deduce that 1 |2Bk |



Bk (x)|p0 dx ≤ C 0 {C5 λ + γ λ}p0 = [C1 (C5 + γ )]p0 λp0 . |R 1 p

(3.16)

2Bk

By (3.12), (3.15), (3.16), and α = [2δ (s−1)/s ]−1/q , we conclude that, for any k ∈ I satisfying (3.13),

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|E(αλ) ∩ Qk | # $ 2C4 (C2 γ + εC5 ) C(n,p0 ) [C1 (C5 + γ )]p0 |2Bk | + |Qk | ≤ α α p0 |Qk | # s−1 1+ 1 −1 n/2 ≤ n c(n) 2 q C4 (C2 γ + εC5 )δ sq +2p0 /q C(n,p0 ) [C1 (C5 + γ )]p0 δ

(s−1)p0 −1 sq

$ δ|Qk |,

(3.17)

where c(n) denotes the volume of the unit ball in Rn . From the assumption s > p0 /(p0 − q), we 0 deduce that (s−1)p > 1. Via this inequality, we choose δ ∈ (0, 1/3) sufficiently small such that sq nn/2 c(n) 2p0 /q C(n,p0 ) [C1 (C5 + γ )]p0 δ

(s−1)p0 −1 sq

1 −s/(s−1) ≤ C , 2

(3.18)

 is as in Lemma 3.2(iv). Furthermore, for such a fixed δ, we take γ ∈ (0, 1) and ε0 ∈ where C (0, 1) small enough such that, for any ε ∈ [0, ε0 ), nn/2 c(n) 2

1+ q1

C4 (C2 γ + εC5 )δ

s−1 sq −1

1 −s/(s−1) ≤ C , 2

which, together with (3.17) and (3.18), further implies that, for any k ∈ I such that (3.13) holds true, −s/(s−1) δ|Qk |. |E(αλ) ∩ Qk | ≤ C Thus, for any k ∈ I satisfying (3.13), (3.14) holds true. Now we show (3.7) by using (3.14). By ω ∈ RHs (Rn ), Lemma 3.2(iv), and (3.14), we conclude that, for any k satisfying (3.13), ω(E(αλ) ∩ Qk ) ≤ δ (s−1)/s ω(Qk ).

(3.19)

Let I1 := {k ∈ I : (3.13) holds true for k}. Then, from (3.19) and the above property (c), it follows that, for any λ ∈ (λ0 , ∞), % ω(E(αλ)) = ω E(αλ) ≤



% & '

(( Qk

k∈I

ω (E(αλ) ∩ Qk ) + ω

k∈I1

≤ δ (s−1)/s



ω (Qk ) + ω

k∈I1

≤δ

(s−1)/s

ω(E(λ)) + ω

  x ∈ Q0 : M2B0 (f)(x) > γ λ



 x ∈ Q0 : M2B0 (f)(x) > γ λ

  x ∈ Q0 : M2B0 (f)(x) > γ λ ,

which completes the proof of (3.7) and hence of Theorem 3.1.

2

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23

To prove Theorem 1.2, we need the following auxiliary conclusions. Lemma 3.4. Assume that n ≥ 2,  ⊂ Rn is a bounded Lipschitz domain, p ∈ (1, ∞) and f ∈ Lp (; Rn ), where 1/p + 1/p = 1. Let v ∈ W 1,p () be the weak solution of the Neumann problem −div(A∇v) = div(f) in . Assume further that the weak solution u ∈ W 1,p () of the Neumann problem (N )p with g ∈ Lp (; Rn ) and F ≡ 0 satisfies the estimate ∇uLp (;Rn ) ≤ CgLp (;Rn ) ,

(3.20)

where C is a positive constant independent of u and g. Then there exists a positive constant C, independent of v and f, such that ∇vLp (;Rn ) ≤ C fLp (;Rn ) . Proof. By the assumptions that −div(A∇u) = div(g) and −div(A∇v) = div(f) in , we conclude that    A(x)∇u(x) · ∇v(x) dx = − g(x) · ∇v(x) dx = − f(x) · ∇u(x) dx, 





which, combined with (3.20) and the Hölder inequality, further implies that " " " " " " " " " " " " " " " " ∇vLp (;Rn ) = sup sup " ∇v(x) · f(x) dx " = " ∇u(x) · g(x) dx " fLp (;Rn ) ≤1 " " fLp (;Rn ) ≤1 " " 

≤ 



sup

∇uLp (;Rn ) gLp (;Rn )

sup

fLp (;Rn ) gLp (;Rn )  gLp (;Rn ) .

fLp (;Rn ) ≤1 fLp (;Rn ) ≤1

This finishes the proof Lemma 3.4.

2

Lemma 3.5. Let n ≥ 2,  ⊂ Rn be a bounded Lipschitz domain, p ∈ (1, ∞) and g ∈ Lq (), where q ∈ (1, ∞) is given by q1 := p1 + n1 . Assume that v is the weak solution of the Neumann problem −div(A∇v) = g in . If the weak solution u ∈ W 1,p () of the Neumann problem (N )p with g ∈ Lp (; Rn ) and F ≡ 0 satisfies (3.20), then ∇vLp (;Rn ) ≤ CgLq () , where C is a positive constant independent of v and g.

Proof. Let w ∈ W 1,p () be  the weak solution of the Neumann problem −div(A∇w) = div(h) with h ∈ Lp (; Rn ) and  w(x) dx = 0, where 1/p + 1/p = 1. Then, by Lemma 3.4, we know that ∇wLp (;Rn )  hLp (;Rn ) .

(3.21)

Moreover, from the facts that w and v are, respectively, the weak solutions of the Neumann problems −div(A∇w) = div(h) and −div(A∇v) = g in , it follows that

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24







g(x)w(x) dx = 

A(x)∇v(x) · ∇w(x) dx = − 

h(x) · ∇v(x) dx, 

which, together with (3.21), the Hölder inequality and the Sobolev embedding theorem (see, for instance, [36, Theorem 7.26]), implies that " " " " " " " " " " " " " h(x) · ∇v(x) dx " = " g(x)w(x) dx " ∇vLp (;Rn ) = sup sup " " " " h p ≤1 " " hLp (;Rn ) ≤1 " " L (;Rn ) 

≤ 

h

sup

gLq () wLq () 

sup

gLq () hLp (;Rn )  gLq () .

≤1 Lp (;Rn )

h



≤1 Lp (;Rn )

This finishes the proof Lemma 3.5.

h

sup

≤1 Lp (;Rn )

gLq () ∇wLp (;Rn )

2

By Lemmas 3.4 and 3.5, we have the following corollary. Corollary 3.6. Let n ≥ 2,  ⊂ Rn be a bounded Lipschitz domain, p ∈ (1, ∞), f ∈ Lp (; Rn ) and g ∈ Lq (), where q ∈ (1, ∞) is given by q1 := p1 + n1 . Assume that v ∈ W 1,p () is the weak

solution of the Neumann problem −div(A∇v) = div(f) + g. If the weak solution u ∈ W 1,p () of the Neumann problem (N )p with g ∈ Lp (; Rn ) and F ≡ 0 satisfies (3.20) with p replaced by p , then 

∇vLp (;Rn ) ≤ C fLp (;Rn ) + gLq () , where C is a positive constant independent of v, f and g and 1/p + 1/p = 1. Lemma 3.7. Let n ≥ 2,  ⊂ Rn be a Lipschitz domain and 0 < p0 < q ≤ ∞. Assume that x0 ∈ , r0 ∈ (0, diam()) and the weak reverse Hölder inequality ⎧ ⎪ ⎨1 n ⎪ ⎩r

 |g(x)|q dx B (x0 ,r0 )

⎫1/q ⎪ ⎬ ⎪ ⎭

⎧ ⎪ ⎨1 ≤ C6 n ⎪ ⎩ r0

 |g(x)|p0 dx B (x0 ,2r0 )

⎫1/p0 ⎪ ⎬ ⎪ ⎭

holds true for a given measurable function g on , where C6 is a positive constant, independent of x0 and r0 , which may depend on g. Then, for any given p ∈ (0, ∞], there exists a positive constant C, depending on p, p0 , q, and C6 , such that ⎧ ⎪ ⎨1 n ⎪ ⎩r

 |g(x)|q dx B (x0 ,r0 )

⎫1/q ⎪ ⎬ ⎪ ⎭

⎧ ⎪ ⎨1 ≤C n ⎪ ⎩ r0

 |g(x)|p dx B (x0 ,2r0 )

⎫1/p ⎪ ⎬ ⎪ ⎭

.

The proof of Lemma 3.7 is similar to that of [7, Lemma 4.38] and the details are omitted here. Now we prove Theorem 1.2 by using Theorem 3.1 and Lemmas 3.4, 3.5, and 3.7.

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25

Proof of Theorem 1.2. We first show that (i) and (ii) are equivalent. Assume that Theorem 1.2(i) holds true. Let r0 ∈ (0, diam()) be a constant and B := B(x0 , r) as in Theorem 1.2(ii); namely, r ∈ (0, r0 /4) and either x0 ∈ ∂ or B(x0 , 2r) ⊂ . Let η ∈ Cc∞ (Rn ) satisfy that 0 ≤ η ≤ 1, η ≡ 1 on B, supp (η) ⊂ 2B, and |∇η|  r −1 . Moreover, assume that −div(A∇v) = 0 in 2B ∩  and ∂v ∂ν = 0 on B(x0 , 2r) ∩ ∂ when x0 ∈ ∂. Then −div(A∇(vη)) = −div(vA∇η) − A∇v · ∇η

(3.22)

in the sense of (1.5). Indeed, by the assumption that −div(A∇v) = 0 in 2B ∩ , we find that, for any φ ∈ C ∞ (Rn ),  A(x)∇(vη)(x) · ∇φ(x) dx 



 A(x)η(x)∇v(x) · ∇φ(x) dx +

= 





A(x)∇v(x) · ∇(ηφ)(x) dx −

= 

A(x)v(x)∇η(x) · ∇φ(x) dx 

A(x)∇v(x) · ∇η(x)φ(x) dx 



A(x)v(x)∇η(x) · ∇φ(x) dx

+ 





A(x)∇v(x) · ∇η(x)φ(x) dx +

=− 

A(x)v(x)∇η(x) · ∇φ(x) dx, 

 which implies that (3.22) holds true. Assume further that 2B∩ v(x) dx = 0 and q ∈ (1, ∞) is given by q1 := p1 + n1 . From Theorem 1.2(i), (3.22), Corollary 3.6, (1.3), |∇η|  r −1 , and the Sobolev embedding theorem (see, for instance, [36, Theorem 7.26]), it follows that ∇(vη)Lp (;Rn )  A∇v · ∇ηLq () + vA∇ηLp (;Rn )    r −1 ∇vLq (2B ;Rn ) + vLp (2B )  r −1 ∇vLq (2B ;Rn ) , which, together with the fact that η ≡ 1 on B, implies that ∇vLp (B ;Rn )  r −1 ∇vLq (2B ;Rn ) . By this and

1 q

=

1 p

+ n1 , we conclude that ⎫1/p ⎧ ⎫1/q ⎧ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨1  ⎬ ⎨1  p q |∇v(x)| dx  |∇v(x)| dx . n n ⎪ ⎪ ⎪ ⎪ ⎭ ⎩r ⎭ ⎩r B

(3.23)

2B

From (3.23) and Lemma 3.7, we deduce that (1.9) holds true, which completes the proof that (i) implies (ii).

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26

Now we prove that (ii) implies (i). Assume that (ii) holds true. Let B := B(xB , rB ) ⊂ Rn be a ball satisfying rB ∈ (0, r0 /4) and either 2B ⊂  or xB ∈ ∂. Take φ ∈ Cc∞ (Rn ) such that φ ≡ 1 on 2B, 0 ≤ φ ≤ 1, and supp (φ) ⊂ 4B. Let w, v ∈ W 1,2 () be, respectively, the weak solutions of the Neumann problems ⎧ ⎨−div(A∇w) = div(φg) in , (3.24) ∂w ⎩ = φg · ν on ∂ ∂ν and ⎧ ⎨−div(A∇v) = div((1 − φ)g) ∂v ⎩ = (1 − φ)g · ν ∂ν

in , (3.25)

on ∂.

Then u = w + v up to constants, and hence ∇u = ∇w + ∇v. Let F := |∇u|2 , f := |g|2 , FB := 2|∇w|2 , and RB := 2|∇v|2 . It is easy to see that 0 ≤ F ≤ FB + RB . By g ∈ L2 (; Rn ), (3.24) and the fact that (1.8) holds true for p = 2, we conclude that    1 1 1 2 FB (x) dx = |∇w(x)| dx  |g(x)φ(x)|2 dx |2B | |2B | |2B | 2B

2B



1 |4B |



|g(x)|2 dx ∼ 4B

1 |4B |





(3.26)

f (x) dx. 4B

Moreover, from (3.25) and Theorem 1.2(ii), it follows that (1.9) holds true for v, which, combined with the self-improvement property of the weak reverse Hölder inequality (see, for instance, [35, pp. 122-123]), further implies that there exists an ε0 ∈ (0, ∞) such that the inequality (1.9) holds true with p replaced by p + ε0 . By this and Lemma 3.7, we find that, for any q ∈ (0, 2], the weak reverse Hölder inequality ⎧ ⎪ ⎨1 n ⎪ ⎩r

 |∇v(x)|p+ε0 dx B (x0 ,r)

⎫1/(p+ε0 ) ⎪ ⎬ ⎪ ⎭

⎧ ⎪ ⎨1  n ⎪ ⎩r

 |∇v(x)|q dx B (x0 ,2r)

⎫1/q ⎪ ⎬ ⎪ ⎭

(3.27)

holds true. This, together with (3.26), further implies that ⎧ ⎪ ⎨

1 ⎪ ⎩ |2B |

 [RB (x)](p+ε0 )/2 dx 2B

1  |4B | ∼

1 |4B |



⎫2/(p+ε0 ) ⎪ ⎬ ⎪ ⎭

1 |∇v(x)| dx  |4B |



2

4B



F (x) dx + 4B

1 |4B |



1 |∇u(x)| dx + |4B |

4B

f (x) dx. 4B



2

f (x) dx 4B

(3.28)

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27

From (3.26) and (3.28), we deduce that (3.1) and (3.2) hold true with p0 := (p + ε0 )/2. Then, by Theorem 3.1 with ω ≡ 1 and q := p/2 and the Hölder inequality, we conclude that ⎫1/p ⎧ ⎬ ⎨ 1  |∇u(x)|p dx ⎭ ⎩ || 

⎧ ⎫1/2 ⎧ ⎫1/p ⎨ 1  ⎬ ⎨ 1  ⎬  |∇u(x)|2 dx + |g(x)|p dx ⎩ || ⎭ ⎩ || ⎭ 



⎧ ⎫1/2 ⎧ ⎫1/p ⎧ ⎫1/p ⎨ 1  ⎬ ⎨ 1  ⎬ ⎨ 1  ⎬  |g(x)|2 dx + |g(x)|p dx  |g(x)|p dx , ⎩ || ⎭ ⎩ || ⎭ ⎩ || ⎭ 





which implies that (1.8) holds true. This shows that (ii) implies (i) and hence that (i) and (ii) of Theorem 1.2 are equivalent. Now we prove that (ii) and (iii) are equivalent. To this end, we first show that (ii) implies (iii). We assume that Theorem 1.2(ii) holds true. Let q ∈ [2, p], q0 ∈ [1, q/p ], r0 ∈ [p/(p − q), ∞], and ω ∈ Aq0 (Rn ) ∩ RHr0 (Rn ). Assume that ε0 ∈ (0, ∞) is as in (3.27). Then ω ∈ Aq (Rn ) ∩ RHs (Rn ) with s ∈ ((p + ε0 )/(p + ε0 − q), ∞]. Assume that u is the weak solution q of the Neumann problem (N )q, ω with g ∈ Lω (; Rn ) and F ≡ 0. Then, from Lemma 3.2(v), it follows that Lqω () ⊂ Lq/q0 ().

(3.29)

By the fact that q0 ≤ q/p < q/(p + ε0 ) , we find that (p + ε0 ) < q/q0 , which, together with the Hölder inequality and the fact that  is bounded, implies Lq/q0 () ⊂ L(p+ε0 ) (). From this and (3.29), we deduce that g ∈ Lq/q0 (; Rn ) ⊂ L(p+ε0 ) (; Rn ). Let B := B(xB , rB ) ⊂ Rn be a ball satisfying |B| ≤ β1 || and either 2B ⊂  or xB ∈ ∂, where β1 ∈ (0, 1) is as in Theorem 3.1. Take φ ∈ Cc∞ (Rn ) such that φ ≡ 1 on 2B, 0 ≤ φ ≤ 1, and supp (φ) ⊂ 4B. Let w and v be, respectively, as in (3.24) and (3.25). Then ∇u = ∇w + ∇v. Let F := |∇u|(p+ε0 ) , f := |g|(p+ε0 ) , FB := 2(p+ε0 ) |∇w|(p+ε0 ) , and RB := 2(p+ε0 ) |∇v|(p+ε0 ) . It is easy to see that 0 ≤ F ≤ FB + RB . Furthermore, by the equivalence of (i) and (ii), we know that (1.8) holds true with p replaced by p + ε0 , which, combined with Lemma 3.4 and (3.24), implies that ∇wL(p+ε0 ) (;Rn )  φgL(p+ε0 ) (;Rn )  gL(p+ε0 ) (4B

 ;R

n)

.

From this, it follows that 1 |2B |

 2B

1 FB (x) dx  |4B |

 f (x) dx.

(3.30)

4B

Furthermore, by Theorem 1.2(ii) and (3.25), we conclude that (1.9) holds true for v. Thus, (3.27) holds true for v, which, together with (3.30), further implies that

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28

⎧ ⎪ ⎨

1 ⎪ ⎩ |2B |

=



∼





[RB (x)](p+ε0 )/(p+ε0 ) dx 2B

⎧ ⎪ ⎨

1 ⎪ ⎩ |2B | 1 |4B | 1 |4B |

 |∇v(x)|p+ε0 dx 2B



⎪ ⎭





F (x) dx + 4B

1 |4B |

⎪ ⎭

⎫(p+ε0 ) /(p+ε0 ) ⎪ ⎬

|∇u(x)|(p+ε0 ) dx + 4B

⎫(p+ε0 ) /(p+ε0 ) ⎪ ⎬

1 |4B |





1 |4B |







|∇v(x)|(p+ε0 ) dx 4B

|g(x)|(p+ε0 ) dx 4B

(3.31)

f (x) dx. 4B

From (3.30) and (3.31), we deduce that (3.1) and (3.2) hold true with p0 := (p + ε0 )/(p + ε0 ) . This, combined with q < p + ε0 , (3.3) and (3.29), further implies that ⎫1/q ⎧ ⎬ ⎨ 1  |∇u(x)|q ω(x) dx ⎭ ⎩ ω() 

⎧ ⎫1/q ⎨ 1  ⎬ = [F (x)]q/(p+ε0 ) ω(x) dx ⎩ ω() ⎭ 

⎧ ⎫1/(p+ε0 ) ⎧ ⎫1/q ⎨ 1  ⎬ ⎨ 1  ⎬  |F (x)| dx + |g(x)|q ω(x) dx ⎩ || ⎭ ⎩ ω() ⎭ 



⎧ ⎫1/(p+ε0 ) ⎧ ⎫1/q  ⎨ 1  ⎬ ⎨ ⎬ 1 ∼ |∇u(x)|(p+ε0 ) dx + |g(x)|q ω(x) dx ⎩ || ⎭ ⎩ ω() ⎭ 



⎧ ⎫1/(p+ε0 ) ⎧ ⎫1/q ⎨ 1  ⎬ ⎨ 1  ⎬  |g(x)|(p+ε0 ) dx + |g(x)|q ω(x) dx ⎩ || ⎭ ⎩ ω() ⎭ 

⎧ ⎫1/q ⎨ 1  ⎬  |g(x)|q ω(x) dx . ⎩ ω() ⎭





Therefore, (1.10) holds true, which shows that Theorem 1.2(iii) holds true. Finally, we prove that (iii) impels (ii). By taking q := p and ω ≡ 1 in Theorem 1.2(iii), we know that (i) holds true, which, combined with the equivalence of (i) and (ii), implies that Theorem 1.2(ii) holds true. This finishes the proof of Theorem 1.2. 2

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29

4. Proof of Theorem 1.6 In this section, we prove Theorem 1.6 by using Theorem 1.2. We begin with the following Lemma 4.1, which was essentially obtained in [32, Lemma 4.2] (see also [33, Lemma 3.2]) and inspired by [19]. Indeed, under the additional assumption that the matrix A ∈ VMO(Rn ), Lemma 4.1 was obtained in [32, Lemma 4.2] (see also [33, Lemma 3.2]). However, from the proof of Lemma 4.1 below, we deduce that the condition A ∈ VMO(Rn ) is not essential. For the sake of completeness, we give the proof of Lemma 4.1 here. Lemma 4.1. Let n ≥ 2,  ⊂ Rn be a bounded Lipschitz domain and r0 ∈ (0, diam()) be a constant. Assume that the matrix A is real-valued, symmetric, bounded and measurable, and satisfies (1.3). Let u ∈ W 1,2 (B(x0 , 4r) ∩ ) be a solution of the equation div(A∇u) = 0 in B(x0 , 4r) ∩  with ∂u ∂ν = 0 on B(x0 , 4r) ∩ ∂, where x0 ∈  and r ∈ (0, r0 /4). Then there exist a function θ := θ (r), p ∈ (2, ∞), and a function v ∈ W 1,p (B(x0 , r) ∩ ) such that ⎧ ⎪ ⎨

1 ⎪ ⎩ |B (x0 , r)|

 |∇v(x)|p dx B (x0 ,r)

⎫1/p ⎪ ⎬ ⎪ ⎭

≤C

⎧ ⎪ ⎨

1 ⎪ ⎩ |B (x0 , 4r)|

 |∇u(x)|2 dx B (x0 ,4r)

⎫1/2 ⎪ ⎬ ⎪ ⎭

(4.1)

and ⎧ ⎪ ⎨

1 ⎪ |B (x ⎩  0 , r)|

≤ θ (r)

⎧ ⎪ ⎨

 |∇(u − v)(x)|2 dx B (x0 ,r)

1 ⎪ ⎩ |B (x0 , 4r)|

⎫1/2 ⎪ ⎬ ⎪ ⎭

 |∇u(x)|2 dx B (x0 ,4r)

⎫1/2 ⎪ ⎬ ⎪ ⎭

,

(4.2)

where C is a positive constant independent of u, v, x0 and r. Proof. Let A0 := {cij }ni,j =1 , where, for any i, j ∈ {1, . . . , n}, 1 cij := |B(x0 , 2r)|

 aij (x) dx. B(x0 ,2r)

Assume that v ∈ W 1,2 (B (x0 , 2r)) is the solution of the following boundary value problem in B (x0 , 2r), div(A0 ∇v) = 0 (4.3) on ∂B (x0 , 2r). A0 ∇v · ν = A∇u · ν Using u − v as a test function, we find that  A0 ∇(u − v)(x) · ∇(u − v)(x) dx = B (x0 ,2r)



B (x0 ,2r)

(A0 − A(x))∇u(x) · ∇(u − v)(x) dx,

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30

which, together with (1.3) and the Hölder inequality, implies that, for any ε ∈ (0, ∞), there exists a positive constant C(ε) , depending on ε, such that  |∇(u − v)(x)|2 dx

μ0 B (x0 ,2r)



≤

|A0 − A(x)||∇u(x)||∇(u − v)(x)| dx

B (x0 ,2r)





≤ε

|∇(u − v)(x)| dx + C(ε)

|A0 − A(x)|2 |∇u(x)|2 dx,

2

B (x0 ,2r)

(4.4)

B (x0 ,2r)

where μ0 ∈ (0, 1) is as in (1.3). Take ε := μ0 /2 in (4.4). Then, by (4.4), we conclude that 

 |∇(u − v)(x)| dx ≤ C7

|A0 − A(x)|2 |∇u(x)|2 dx.

2

B (x0 ,2r)

(4.5)

B (x0 ,2r)

Moreover, it is well known that there exist positive constants p1 ∈ (1, ∞) and C8 ∈ (0, ∞), independent of u, x0 , and r, such that ⎧ ⎪ ⎨

1 ⎪ ⎩ |B (x0 , 2r)|

 |∇u(x)|2p1 dx B (x0 ,2r)

⎫1/(2p1 ) ⎪ ⎬ ⎪ ⎭

≤ C8

⎧ ⎪ ⎨

1 ⎪ ⎩ |B (x0 , 4r)|

 |∇u(x)|2 dx B (x0 ,4r)

⎫1/2 ⎪ ⎬ ⎪ ⎭

(see, for instance, [40, Chapter 1] and [35, Chapter V]), which, combined with (4.5) and the Hölder inequality, further implies that ⎧ ⎪ ⎨

1 ⎪ |B (x ⎩  0 , 2r)|

≤ C7

×

⎧ ⎪ ⎨

 |∇(u − v)(x)|2 dx B (x0 ,2r)

1 ⎪ |B(x 0 , 2r)| ⎩

⎧ ⎪ ⎨



≤ θ (r) where 1/p1 + 1/p1 = 1 and

⎪ ⎭



|A0 − A(x)|2p1 dx B(x0 ,2r)

1 ⎪ ⎩ |B (x0 , 2r)| ⎧ ⎪ ⎨

⎫1/2 ⎪ ⎬

 |∇u(x)|2p1 dx B (x0 ,2r)

1 ⎪ |B (x ⎩  0 , 4r)|

⎪ ⎭

⎫1/(2p1 ) ⎪ ⎬ ⎪ ⎭

 |∇u(x)|2 dx B (x0 ,4r)

⎫1/(2p ) 1 ⎪ ⎬

⎫1/2 ⎪ ⎬ ⎪ ⎭

,

(4.6)

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# θ (r) :=C7 C8

sup x∈, t∈(0,r]

 × B(x,2t)

31

1 |B(x, 2t)|

⎫1/(2p ) "2p " 1 " 1 ⎪ " ⎪  " ⎬ " 1 " " A(z) dz" dy . "A(y) − ⎪ " " |B(x, 2t)| ⎪ ⎭ " " B(x,2t)

(4.7)

Thus, (4.2) holds true. Furthermore, from the known regularity theory of second order elliptic equations (see, for instance, [40, Chapter 1] and [35, Chapter V]), it follows that there exists a p ∈ (2, ∞) such that ⎧ ⎪ ⎨

1 ⎪ ⎩ |B (x0 , 2r)|

 |∇v(x)|p dx B (x0 ,2r)

⎫1/p ⎪ ⎬ ⎪ ⎭



⎧ ⎪ ⎨

1 ⎪ ⎩ |B (x0 , 4r)|

 |∇v(x)|2 dx B (x0 ,4r)

⎫1/2 ⎪ ⎬ ⎪ ⎭

,

which, together with (4.6) and the fact that A = {aij }ni,j =1 ∈ L∞ (Rn ; Rn ), further implies that (4.1) holds true. This finishes the proof of Lemma 4.1. 2 2

Additionally, to show Theorem 1.6, we need the following lemma. Lemma 4.2. Let n ≥ 2 and  ⊂ Rn be a bounded (semi-)convex domain with ∂ ∈ C 2 . Assume that p ∈ (2, ∞) and A0 := {aij }ni,j =1 is a symmetric matrix with constant coefficients that satisfies (1.3). Let v ∈ W 1,2 (B (x0 , 2r)) be a weak solution of the equation div(A0 ∇v) = 0 in ∂v B (x0 , 2r) with ∂ν = 0 on B(x0 , 2r) ∩ ∂, where B(x0 , r) is a ball such that r ∈ (0, r0 /4) and either x0 ∈ ∂ or B(x0 , 2r) ⊂ , and r0 ∈ (0, diam()) is a constant. Then there exists a positive constant C, depending on n, p, the Lipschitz constant, and the uniform ball constant of , such that ⎫1/p ⎧ ⎫1/2 ⎧ ⎪ ⎪ ⎪ ⎪   ⎬ ⎨ ⎬ ⎨ 1 1 p 2 |∇v(x)| dx ≤C |∇v(x)| dx . (4.8) ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ |B (x0 , 2r)| ⎭ ⎩ |B (x0 , r)| B (x0 ,r)

B (x0 ,2r)

Proof. Via the fact that the matrix A0 is symmetric and elliptic and has constant coefficients, by a change of the coordinate system, we can assume that A0 = I , namely, v = 0 in B(x0 , 2r) ∩  and ∂v ∂ν = 0 on B(x0 , 2r) ∩ ∂. If B(x0 , 2r) ⊂ , then, by the interior estimate for harmonic functions (see, for instance, [36, Section 2.7]), we find that

sup y∈B(x0 ,r)

|∇v(y)| 

⎧ ⎪ ⎨

1 ⎪ ⎩ |B(x0 , 2r)|

 |∇v(x)|2 dx B(x0 ,2r)

⎫1/2 ⎪ ⎬ ⎪ ⎭

,

which implies that (4.8) holds true in this case. Moreover, if x0 ∈ ∂, (4.8) can be obtained as in [60, Theorem 2.4] via the geometric property of the semi-convex domain and borrowing some ideas from [34]. The details are omitted here. This finishes the proof of Lemma 4.2. 2

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32

Lemma 4.3. Let n ≥ 2 and  be a bounded (semi-)convex domain in Rn . Then there exists a sequence {j }j ∈N of bounded (semi-)convex domains such that  ⊂ j and ∂j ∈ C ∞ for any j ∈ N, and ∩j ∈N j = . Moreover, for any j ∈ N, j has the uniform Lipschitz constant and the uniform ball constant with . The proof of Lemma 4.3 is similar to that of [47, Lemma 6.4], and we omit the details here. Now we show Theorem 1.6 by using Theorem 1.2 and Lemmas 4.1 through 4.3. Proof of Theorem 1.6. We split the proof of Theorem 1.6 into the following five steps. Step 1. We assume that ∂ ∈ C 2 and u1 is the weak solution of the following Neumann problem ⎧ ⎨−div(A∇u1 ) = div(g1 ) ∂u ⎩ 1 = g1 · ν ∂ν

in , (4.9)

on ∂.

In this step we show that, for any given p ∈ (1, ∞), there exists δ0 ∈ (0, ∞), depending on n, p, and , such that, if A satisfies the (δ, R)-BMO condition for some δ ∈ (0, δ0 ) and R ∈ (0, ∞), then ∇u1 Lp (;Rn )  g1 Lp (;Rn ) .

(4.10)

We first assume that p ∈ (2, ∞). Let B(x0 , r) ⊂ Rn be a ball such that r ∈ (0, r0 /4) and either x0 ∈ ∂ or B(x0 , 2r) ⊂ , where r0 ∈ (0, diam()) is a positive constant as in Theorem 1.2(ii). Assume that v1 ∈ W 1,2 (B (x0 , 2r)) is a weak solution of the equation div(A∇v1 ) = 0 in 1,2 (B (x , 2r)) be a weak solution 1 B (x0 , 2r) with ∂v  0 ∂ν = 0 on B(x0 , 2r) ∩ ∂. Let w1 ∈ W of (4.3). Namely, div(A0 ∇w1 ) = 0 in B (x0 , 2r) and A0 ∇w1 · ν = A∇v · ν on ∂B (x0 , 2r), where A0 := {cij }ni,j =1 with cij := |B(x01,2r)| B(x0 ,2r) aij (x) dx for any i, j ∈ {1, . . . , n}. Then, from Lemmas 4.1 and 4.2, it follows that ⎧ ⎪ ⎨

1 ⎪ ⎩ |B (x0 , r)|



 |∇w1 (x)|p+1 dx B (x0 ,r)

⎧ ⎪ ⎨

1 ⎪ |B (x ⎩  0 , 2r)|

⎫1/(p+1) ⎪ ⎬ ⎪ ⎭

 |∇v1 (x)|2 dx B (x0 ,2r)

⎫1/2 ⎪ ⎬ ⎪ ⎭

and ⎧ ⎪ ⎨

1 ⎪ ⎩ |B (x0 , r)|

 |∇(v1 − w1 )(x)|2 dx B (x0 ,r)

⎫1/2 ⎪ ⎬ ⎪ ⎭

(4.11)

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≤ θ (r)

⎧ ⎪ ⎨

1 ⎪ ⎩ |B (x0 , 2r)|

 |∇v1 (x)|2 dx B (x0 ,2r)

33

⎫1/2 ⎪ ⎬ ⎪ ⎭

,

(4.12)

where θ (r) is as in (4.7). By the well-known John-Nirenberg inequality on BMO(Rn ) (see, for instance, [32,57]), we conclude that there exists a δ0 ∈ (0, ∞) sufficiently small such that, if A satisfies the (δ, R)-BMO condition with some δ ∈ (0, δ0 ) and R ∈ (r0 , ∞), then, for any r ∈ (0, r0 ), θ (r) < ε0 /2, where ε0 is as in Theorem 3.1. From this, (4.11), (4.12), and Theorem 3.1 with ω ≡ 1, we deduce that |∇v1 | ∈ Lp (B (x0 , 2r)) and ⎧ ⎪ ⎨

1 ⎪ ⎩ |B (x0 , r)|



⎧ ⎪ ⎨

 |∇v1 (x)|p dx B (x0 ,r)

1 ⎪ |B (x ⎩  0 , 2r)|

⎫1/p ⎪ ⎬ ⎪ ⎭

 |∇v1 (x)|2 dx B (x0 ,2r)

⎫1/2 ⎪ ⎬ ⎪ ⎭

,

(4.13)

which, combined with Theorem 1.2, further implies that (4.10) holds true in the case p ∈ (2, ∞). Assume now that p ∈ (1, 2). In this case, p ∈ (2, ∞), where 1/p + 1/p = 1. By this, Lemma 3.4, and the fact that (4.10) holds true for any given p ∈ (2, ∞), we conclude that (4.10) also holds true for any given p ∈ (1, 2). Furthermore, when p = 2, (4.10) is a simple conclusion of (1.5) and the Hölder inequality. Thus, (4.10) holds true for any given p ∈ (1, ∞). Step 2. In this step, we remove the additional assumption that ∂ ∈ C 2 in Step 1. Indeed, by Lemma 4.3 and the standard approximation argument (see, for instance, [21, pp. 170-171]), we know that (4.10) holds true when  ⊂ Rn is (semi-)convex without the assumption ∂ ∈ C 2 . The details are omitted here. Step 3. Let u1 be the weak solution of the Neumann problem (4.9). In this step, we show that, for any given p ∈ (1, ∞) and ω ∈ Ap (Rn ), there exists a δ0 ∈ (0, ∞), depending on n, p, [ω]Ap (Rn ) , and , such that, if A satisfies the (δ, R)-BMO condition for some δ ∈ (0, δ0 ), and R ∈ (0, ∞), then ∇u1 Lpω (;Rn )  g1 Lpω (;Rn ) .

(4.14)

To this end, we first assume that p ∈ [2, ∞). From ω ∈ Ap (Rn ) and Lemma 3.2(i), it follows that there exists a p0 ∈ (1, p) such that ω ∈ Ap/p0 (Rn ). Moreover, by Lemma 3.2(ii), we find that there exists a sufficiently large p2 ∈ (p, ∞) such that ω ∈ RHp2 /(p2 −p) (Rn ). n n Let p  := max{p0 , p2 } + 1. Then ω ∈ Ap/p0 (Rn ) ⊂ Ap/ p (R ) and ω ∈ RHp2 /(p2 −p) (R ) ⊂ n RHp/( p−p) (R ) and hence n n ω ∈ Ap/ /( p −p) (R ). p (R ) ∩ RHp

From the proof of Step 1, we deduce that there exists δ0 ∈ (0, ∞), depending on n, p, , and [ω]Ap (Rn ) , such that, if A satisfies the (δ, R)-BMO condition with some δ ∈ (0, δ0 ) and R ∈

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(r0 , ∞), then (4.13) holds true for the exponent p , which, together with Theorem 1.2, implies that (4.14) holds true in this case. Let p ∈ (1, 2). Then p ∈ (2, ∞) and, by Lemma 3.2(iii), we know that ω1 := ω−p /p ∈ p Ap (Rn ). Assume that g2 ∈ Lω1 (; Rn ) and u2 is the weak solution of the following Neumann problem ⎧ ⎨−div(A∇u2 ) = div(g2 ) ∂u ⎩ 2 = g2 · ν ∂ν

in , on ∂.

Then 

 g1 (x) · ∇u2 (x) dx = −



 A(x)∇u1 (x) · ∇u2 (x) dx =



g2 (x) · ∇u1 (x) dx 

and, from the fact that (4.14) holds true for any given p ∈ (2, ∞), it follows that ∇u2 

p

Lω1 (;Rn )

 g2 

p

Lω1 (;Rn )

.

By this and the Hölder inequality, we conclude that

∇u1 Lpω (;Rn ) =

=

g2 

sup

p Lω (;Rn ) 1

g2 

sup

p Lω (;Rn ) 1

 

g2 

" " " " " " " g2 (x) · ∇u1 (x) dx " " " ≤1 " "



sup

g1 Lpω (;Rn ) ∇u2 

sup

g1 Lpω (;Rn ) g2 

≤1 p Lω (;Rn ) 1

g2 



" " " " " " " g1 (x) · ∇u2 (x) dx " " " ≤1 " "

≤1 p Lω (;Rn ) 1

p

Lω1 (;Rn )

p

Lω1 (;Rn )

 g1 Lpω (;Rn ) ,

which implies that (4.14) holds true in this case p ∈ (1, 2). Thus, (4.14) holds true for any given p ∈ (1, ∞) and ω ∈ Ap (Rn ). Step 4. Let u3 be the weak solution of the following Neumann problem ⎧ ⎨−div(A∇u3 ) = F1 ∂u ⎩ 3 =0 ∂ν

in , on ∂.

(4.15)

In this step, we prove that, for any given p ∈ (1, ∞) and ω ∈ Ap (Rn ), there exists a δ0 ∈ (0, ∞), depending on n, p, , and [ω]Ap (Rn ) , such that, if A satisfies the (δ, R)-BMO condition with some δ ∈ (0, δ0 ) and R ∈ (0, ∞), then

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∇u3 Lpω (;Rn )  F1 Lpω () .

(4.16)

Let p ∈ (1, ∞) and ω ∈ Ap (Rn ). Then p ∈ (1, ∞) and, by Lemma 3.2(iii), we find that ω1 :=  ω−p /p ∈ Ap (Rn ). Let u1 and g1 be as in (4.9). Assume further that  u1 (x) dx = 0. From the weighted Sobolev inequality (see, for instance, [30, Theorem 1.5]), it follows that there exists a positive constant C, depending on n, p , and [ω1 ]Ap (Rn ) , such that u1 

p

Lω1 ()

≤ Cdiam()∇u1 

p

Lω1 (;Rn )

(4.17)

.

Moreover, by (4.9) and (4.15), we conclude that  −

 g1 (x) · ∇u3 (x) dx =



 A(x)∇u3 (x) · ∇u1 (x) dx =



F1 (x)u1 (x) dx, 

which, combined with the Hölder inequality, (4.17) and (4.14), further implies that

∇u3 Lpω (;Rn ) =   

g1 

sup

p Lω (;Rn ) 1

g1 

" " " " " " " g1 (x) · ∇u3 (x) dx " = sup " " ≤1 " " g1  p

sup

F1 Lpω () u1 

sup

F1 Lpω () ∇u1 

sup

F1 Lpω () g1 

≤1 p Lω (;Rn ) 1

g1 

≤1 p Lω (;Rn ) 1

g1 

Lω (;Rn ) 1



≤1 p Lω (;Rn ) 1

" " " " " " " F1 (x)u1 (x) dx " " " ≤1 " " 

p

Lω1 () p

Lω1 (;Rn )

p

Lω1 (;Rn )

 F1 Lpω () .

Thus, (4.16) holds true. Step 5. Assume that p ∈ (1, ∞) and ω ∈ Ap (Rn ). In this step, we show (1.12) by using (4.14) and (4.16). p Let u be the weak solution of the Neumann problem (1.4) with g ∈ Lω (; Rn ) and F ∈ p Lω (). Then u = ug + uF + C, where ug and uF are, respectively, the weak solutions of the Neumann problems (4.9) and (4.15) with g1 and F1 replaced, respectively, by g and F , and C is a constant. From ∇u = ∇ug + ∇uF , (4.14) and (4.16), we deduce that ∇uLpω (;Rn ) ≤ ∇ug Lpω (;Rn ) + ∇uF Lpω (;Rn )  gLpω (;Rn ) + F Lpω () , which implies that (1.12) holds true. This finishes the proof of Theorem 1.6.

2

5. Proofs of Theorems 2.2, 2.4, 2.10 and 2.13 In this section, we prove Theorems 2.2, 2.4, 2.10, and 2.13 by applying Theorem 1.6 and some technical tools from harmonic analysis.

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Proof of Theorem 2.2. Let q ∈ (1, ∞), r ∈ (0, ∞], and ω ∈ Aq (Rn ). By Lemma 3.2(i), we know that there exists an 0 ∈ (0, q − 1) such that ω ∈ Aq−0 (Rn ) ⊂ Aq (Rn ) ⊂ Aq+0 (Rn ). For any g ∈ L2 (; Rn ), let T1 : g → ∇ug , where ug is the weak solution of the following Neumann problem ⎧ ⎨−div(A∇ug ) = div(g) in , (5.1) ∂ug ⎩ =g·ν on ∂. ∂ν q+

From (1.12), it follows that T1 is a well-defined linear operator on both the spaces Lω 0 (; Rn ) q− 1−θ0 θ0 1 0 and Lω 0 (; Rn ). Let θ0 := q+ 2q . Then θ0 ∈ (0, 1) and q = q−0 + q+0 , which, together with (1.12) and the interpolation theorem of operators on Lorentz spaces (see, for instance, [37, q,r Theorem 1.4.19]), further implies that, for any g ∈ Lω (; Rn ), ) ) )∇ug )

q,r

Lω (;Rn )

= T1 (g)Lq,r n  gLq,r (;Rn ) , ω (;R ) ω

(5.2)

 where ug is as in (5.1). Moreover, for any F ∈ L2 () with  F (x) dx = 0, let T2 : F → ∇uF , where uF is the weak solution of the following Neumann problem ⎧ ⎨−div(A∇uF ) = F in , (5.3) ∂uF ⎩ =0 on ∂. ∂ν Then, similarly to (5.2), we find that ∇uF Lq,r n = T2 (F )Lq,r (;Rn )  F Lq,r () . ω (;R ) ω ω

(5.4) q,r

Let u be the weak solution of the Neumann problem (1.4) with g ∈ Lω (; Rn ) and F ∈ q,r Lω (). Then u = ug + uF + C, where ug and uF are, respectively, as in (5.1) and (5.3), and C is a constant. By this, (5.2) and (5.4), we conclude that ) ) ) ) q,r ∇uLq,r + ∇uF Lq,r n )  ∇ug n  gLq,r (;Rn ) + F Lq,r () . (;R L (;Rn ) ω ω (;R ) ω ω ω

This finishes the proof of (2.1) and hence of Theorem 2.2.

2

To prove Theorem 2.4 via using Theorem 2.2, we need the following lemma, which is well known (see, for instance, [37, Section 7.1.2] and [42, Lemma 3.4]). Lemma 5.1. Let s ∈ [1, ∞), ω ∈ As (Rn ), z ∈ Rn , and k ∈ (0, ∞) be a constant. (i) For any x ∈ Rn , let τ z (ω)(x) := ω(x − z). Then τ z (ω) ∈ As (Rn ) and [τ z (ω)]As (Rn ) = [ω]As (Rn ) . (ii) Let ωk := min{ω, k}. Then ωk ∈ As (Rn ) and [ωk ]As (Rn ) ≤ c(s) [ω]As (Rn ) , where c(s) := 1 when s ∈ [1, 2], and c(s) := 2s−1 when s ∈ (2, ∞). (iii) For any x ∈ Rn , let ωa (x) := |x|a , where a ∈ R is a constant. Then ωa ∈ As (Rn ), with s ∈ (1, ∞), if and only if a ∈ (−n, n[s − 1]). Moreover, [ωa ]As (Rn ) ≤ C(n, s, a) , where C(n, s, a) is a positive constant depending only on n, s and a.

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Now we show Theorem 2.4 by using Theorem 2.2 and Lemma 5.1. Proof of Theorem 2.4. We show this theorem by borrowing some ideas from [42,43]. Let q ∈ (1, ∞), r ∈ (0, ∞], and θ ∈ (0, n]. Assume that u is the weak solution of the Neumann problem (1.4) with g ∈ Lq,r;θ (; Rn ) and F ∈ Lq,r;θ (). For any x, z ∈ , ρ ∈ (0, diam()], and  ∈ (0, θ ), let 

ωz (x) := min |x − z|−n+θ− , ρ −n+θ− . Then, by Lemma 5.1, we find that, for any given z ∈ , ωz ∈ Ap (Rn ) with any given p ∈ (1, ∞), and there exists a positive constant C(n,p,θ) , depending only on n, p, and θ , such that [ωz ]Ap (Rn ) ≤ C(n,p,θ) . From this, Theorem 2.2 and the fact that, for any x ∈ B(z, ρ), ωz (x) = ρ −n+θ− , it follows that, for any z ∈  and ρ ∈ (0, diam()], ∇uLq,r (B(z,ρ)∩;Rn ) = ρ

n−θ+ q

∇uLq,r n ωz (B(z,ρ)∩;R ) n−θ+  q,r ρ q + F  gLq,r n Lω () . ω (;R ) z

(5.5)

z

Moreover, similarly to the proofs of [42, (5.12) and (5.14)], we know that, for any z ∈  and ρ ∈ (0, diam()], gLq,r n  gLq,r;θ (;Rn ) ρ ω (;R )

− q

z

and F Lq,r  F Lq,r;θ () ρ ω () z

− q

,

which, together with (5.5), implies that, for any z ∈  and ρ ∈ (0, diam()], ∇uLq,r (B(z,ρ)∩;Rn )  ρ

n−θ q

 gLq,r;θ (;Rn ) + F Lq,r;θ () .

From this, we deduce that (2.2) holds true, which completes the proof of Theorem 2.4.

2

To prove Theorem 2.10 by using Theorem 1.6, we need the following interpolation theorem for weighted Orlicz spaces, which is a special case of interpolation theorems in the scale of generalized modular spaces established in [48, Theorem 2] (see also [20]). Lemma 5.2. Let n ≥ 2,  ⊂ Rn be a bounded Lipschitz domain,  a Young function satisfying  ∈ 2 , p0 , q0 ∈ (1, ∞), p0 < q0 , and ω ∈ Ap0 (Rn ). Assume that the linear operator T is well p q defined and bounded on both Lω0 () and Lω0 (). If the function  satisfies the condition that there exists a positive constant C such that, for any t ∈ (0, ∞), ⎧ ⎨

t

max t p0 ⎩ 0

(s) ds, t q0 s p0 +1

∞ t

⎫ (s) ⎬ ds ≤ C(t) s q0 +1 ⎭

and, for any f ∈ L ω (), 

 (|Tf (x)|)ω(x) dx ≤ C



(|f (x)|)ω(x) dx, 

(5.6)

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where C is a positive constant depending only on n, , , and [ω]Ap0 (Rn ) . Now we prove Theorem 2.10 by using Theorem 1.6 and Lemma 5.2. Proof of Theorem 2.10. By  ∈ 2 ∩ ∇2 and Remark 2.9(i), we know that 1 < i ≤ I < ∞, which, combined with ω ∈ Ai (Rn ) and Lemma 3.2(i), implies that there exists a p0 ∈ (1, i ) such that ω ∈ Ap0 (Rn ). Let q0 ∈ (I , ∞). Then ω ∈ Ap0 (Rn ) ⊂ Ai (Rn ) ⊂ Aq0 (Rn ). From Theorem 1.6, we deduce that there exists a δ0 ∈ (0, ∞) such that, if A satisfies the (δ, R)-BMO condition for some δ ∈ (0, δ0 ) and R ∈ (0, ∞) or A ∈ VMO(Rn ), then the weak solution of the Neumann problem (1.4) satisfies the estimates 





|∇u(x)|p0 ω(x) dx  

p |g(x)| + |F (x)| 0 ω(x) dx

(5.7)

q |g(x)| + |F (x)| 0 ω(x) dx.

(5.8)



and 





|∇u(x)|q0 ω(x) dx  



Furthermore, by the definitions of i and I , we find that, for any given  ∈ (0, min{i − p0 , q0 − I }), there exists a positive constant C() , depending on , such that, for any s ∈ (0, ∞), a ∈ (0, 1], and b ∈ [1, ∞), (as) ≤ C() a i − (s) and (bs) ≤ C() bI + (s), which, combined with the facts that 1 < p0 < i ≤ I < q0 < ∞ and  ∈ (0, min{i − p0 , q0 − I }), further implies that, for any t ∈ (0, ∞), t t

p0

(s) ds  t p0 (t) s p0 +1

 t * +i − 1 s  ds  (t) t s p0 +1

0

0

and ∞ t

q0

(s) ds  t q0 (t) s q0 +1

t

∞* +I + 1 s  ds  (t). q t s 0 +1 t

From this, it follows that (5.6) holds true for , which, together with (5.7), (5.8), and Lemma 5.2, further implies that 

 (|∇u(x)|)ω(x) dx 



 (|g(x)|)ω(x) dx +



(|F (x)|)ω(x) dx. 

By this and the linear structure of the equation (1.4), we conclude that, for any λ ∈ (0, ∞),

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39

       |∇u(x)| |g(x)| |F (x)| ω(x) dx   ω(x) dx +  ω(x) dx, λ λ λ







which, combined with the definition of the norm  · Lω () , further implies that (2.3) holds true. This finishes the proof of Theorem 2.10. 2 To show Theorem 2.13, we need the following extrapolation theorem in the scale of variable Lebesgue spaces (see, for instance, [24, Theorem 5.24]). Lemma 5.3. Let n ≥ 2,  ⊂ Rn be a bounded Lipschitz domain, and let f, h be given nonnegative measurable functions on  and p ∈ P(). Assume that there exists a p0 ∈ [1, ∞) such that, for any ω0 ∈ A1 (Rn ), 

 [f (x)]

p0

ω0 (x) dx ≤ C



[h(x)]p0 ω0 (x) dx, 

where C is a positive constant depending on n, , p0 , and [ω0 ]A1 (Rn ) . If p0 < p− ≤ p+ < ∞ and the Hardy–Littlewood maximal operator M is bounded on Lp(·) (), then there exists a positive constant C, depending on n, , and p(·), such that f Lp(·) () ≤ ChLp(·) () . Now we prove Theorem 2.13 via using Theorem 1.6 and Lemma 5.3. Proof of Theorem 2.13. Let p0 ∈ (1, p− ). Then, by Theorem 1.6, we conclude that, for any given ω0 ∈ A1 (Rn ), 

 |∇u(x)| ω0 (x) dx  p0





p |g(x)| + |F (x)| 0 ω0 (x) dx.



From this and Lemma 5.3 with f := |∇u| and h := |g| + |F |, we deduce that ∇uLp(·) (;Rn ) = f Lp(·) ()  hLp(·) () ∼ |g| + |F |Lp(·) ()  gLp(·) (;Rn ) + F Lp(·) () , which completes the proof of (2.4) and hence of Theorem 2.13.

2

Acknowledgments The authors would like to thank the referee for her/his very careful reading and several constructive comments, which indeed improve the presentation of this article. References [1] E. Acerbi, G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2007) 285–320. [2] K. Adimurthi, T. Mengesha, N.C. Phuc, Gradient weighted norm inequalities for linear elliptic equations with discontinuous coefficients, Appl. Math. Optim. (2018), https://doi.org/10.1007/s00245-018-9542-5.

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