Global gradient estimates for Dirichlet problems of elliptic operators with a BMO antisymmetric part

Sibei Yang; Dachun Yang; Wen Yuan

doi:10.1515/anona-2022-0247

Open Access Published by De Gruyter May 17, 2022

Global gradient estimates for Dirichlet problems of elliptic operators with a BMO antisymmetric part

Sibei Yang , Dachun Yang and Wen Yuan

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2022-0247

Abstract

Let n ≥ 2 and Ω ⊂ R n be a bounded nontangentially accessible domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second-order elliptic equations of divergence form with an elliptic symmetric part and a BMO antisymmetric part in Ω . More precisely, for any given p ∈ ( 2 , ∞ ) , the authors prove that a weak reverse Hölder inequality with exponent p implies the global W 1 , p estimate and the global weighted W 1 , q estimate, with q ∈ [ 2 , p ] and some Muckenhoupt weights, of solutions to Dirichlet boundary value problems. As applications, the authors establish some global gradient estimates for solutions to Dirichlet boundary value problems of second-order elliptic equations of divergence form with small BMO symmetric part and small BMO antisymmetric part, respectively, on bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, C 1 domains, or (semi-)convex domains, in weighted Lebesgue spaces. Furthermore, as further applications, the authors obtain the global gradient estimate, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (Musielak–)Orlicz spaces, and variable Lebesgue spaces. Even on global gradient estimates in Lebesgue spaces, the results obtained in this article improve the known results via weakening the assumption on the coefficient matrix.

Keywords: elliptic operator; Dirichlet problem; NTA domain; quasi-convex domain; weak reverse Hölder inequality; gradient estimate; Muckenhoupt weight

MSC 2010: Primary 35J25; Secondary 35J15; 42B35; 42B37

1 Introduction

It is well known that the research of global regularity estimates in various function spaces for (non)linear elliptic equations (or systems) in nonsmooth domains is one of the most interesting and important topics in partial differential equations (see, for instance, [6,22,24,36,44] for the linear case, and [10,19,46,68,71] for the nonlinear case). Moreover, the global regularity estimates for solutions to elliptic boundary problems depend not only on the structure of equations, the integrability of the right-hand side datum, and the properties of coefficients appearing in equations but also on the smooth property or the geometric property of the boundary of domains (see, for instance, [1,17,25,29,65]).

Motivated by [29,43,54,58,65], in this article, we study global gradient estimates in various function spaces for Dirichlet boundary value problems of second-order elliptic equations of divergence form with an elliptic symmetric part and a BMO antisymmetric part on nonsmooth domains of R n . More precisely, let n ≥ 2 and Ω ⊂ R n be a bounded nontangentially accessible domain. For any given p ∈ ( 2 , ∞ ) , using a real-variable argument, we show that a weak reverse Hölder inequality with exponent p implies the global W 1 , p estimate and the global weighted W 1 , q estimate, with q ∈ [ 2 , p ] and some Muckenhoupt weights, of solutions to Dirichlet boundary value problems. As applications, we obtain some global gradient estimates for solutions to Dirichlet boundary value problems of second-order elliptic equations of divergence form with small BMO symmetric part and small BMO antisymmetric part, respectively, on bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, C 1 domains, or (semi-)convex domains, in the scale of weighted Lebesgue spaces. Applying these weighted global estimates and some technique from harmonic analysis, such as properties of Muckenhoupt weights, the interpolation theorem of operators, and the extrapolation theorem, we further establish the global gradient estimate, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (Musielak–)Orlicz spaces, and variable Lebesgue spaces. Even on global gradient estimates in Lebesgue spaces, the results obtained in this article improve the corresponding results in [1,17,37,58] via weakening the assumption on the coefficient matrix.

To describe the main results of this article, we first recall the concepts of the Muckenhoupt weight class and the reverse Hölder class (see, for instance, [5,33,59]).

Definition 1.1

Let q ∈ [ 1 , ∞ ) . A nonnegative and locally integrable function ω on R n is said to belong to the Muckenhoupt weight class A q ( R n ) , denoted by ω ∈ A q ( R n ) , if, when q ∈ ( 1 , ∞ ) ,

[ ω ] A q ( R n ) ≔ sup B ⊂ R n 1 ∣ B ∣ ∫ B ω ( x ) d x 1 ∣ B ∣ ∫ B [ ω ( x ) ] − 1 q − 1 d x q − 1 < ∞

[ ω ] A 1 ( R n ) ≔ sup B ⊂ R n 1 ∣ B ∣ ∫ B ω ( x ) d x { ess sup y ∈ B [ ω ( y ) ] − 1 } < ∞ ,

where the suprema are taken over all balls B of R n .

Let r ∈ ( 1 , ∞ ] . A nonnegative and locally integrable function ω on R n is said to belong to the reverse Hölder class RH r ( R n ) , denoted by ω ∈ RH r ( R n ) , if, when r ∈ ( 1 , ∞ ) ,

[ ω ] RH r ( R n ) ≔ sup B ⊂ R n 1 ∣ B ∣ ∫ B [ ω ( x ) ] r d x 1 r 1 ∣ B ∣ ∫ B ω ( x ) d x − 1 < ∞

[ ω ] RH ∞ ( R n ) ≔ sup B ⊂ R n [ ess sup y ∈ B ω ( y ) ] 1 ∣ B ∣ ∫ B ω ( x ) d x − 1 < ∞ ,

where the suprema are taken over all balls B of R n .

Furthermore, we recall the definition of the so-called nontangentially accessible domain introduced by Jerison and Kenig [35] (see also [41,61]) as follows. We begin with recalling several concepts. For any given x ∈ R n and any given measurable subset E ⊂ R n , let

dist ( x , E ) ≔ inf { ∣ x − y ∣ : y ∈ E } .

Meanwhile, for any measurable subsets E , F ⊂ R n , let

dist ( E , F ) ≔ inf { ∣ x − y ∣ : x ∈ E , y ∈ F }

and

diam ( E ) ≔ sup { ∣ x − y ∣ : x , y ∈ E } .

Moreover, for any given x ∈ R n and r ∈ ( 0 , ∞ ) , let

B ( x , r ) ≔ { y ∈ R n : ∣ y − x ∣ < r } .

Definition 1.2

Let n ≥ 2 , Ω ⊂ R n be a domain, which means that Ω is a connected open set, and Ω ∁ ≔ R n \ Ω . Denote by ∂ Ω the boundary of Ω .

Then, the domain Ω is said to satisfy the interior [resp., the exterior] corkscrew condition if there exist positive constants R ∈ ( 0 , ∞ ) and σ ∈ ( 0 , 1 ) such that, for any x ∈ ∂ Ω and r ∈ ( 0 , R ) , there exists a point x 0 ∈ Ω [resp., x 0 ∈ Ω ∁ ], depending on x , such that B ( x 0 , σ r ) ⊂ Ω ∩ B ( x , r ) [resp., B ( x 0 , σ r ) ⊂ Ω ∁ ∩ B ( x , r ) ].
The domain Ω is said to satisfy the Harnack chain condition if there exist constants m 1 ∈ ( 1 , ∞ ) and m 2 ∈ ( 0 , ∞ ) such that, for any x 1 , x 2 ∈ Ω satisfying
M ≔ ∣ x 1 − x 2 ∣ min { dist ( x 1 , ∂ Ω ) , dist ( x 2 , ∂ Ω ) } > 1 ,
there exists a chain { B i } i = 1 N of open Harnack balls, B i ⊂ Ω for any i ∈ { 1 , … , N } , that connects x 1 to x 2 ; namely, x 1 ∈ B 1 , x 2 ∈ B N , B i ∩ B i + 1 ≠ ∅ for any i ∈ { 1 , … , N − 1 } , and, for any i ∈ { 1 , … , N } ,
m 1 − 1 diam ( B i ) ≤ dist ( B i , ∂ Ω ) ≤ m 1 diam ( B i ) ,
where the integer N satisfies N ≤ m 2 log 2 M .
The domain Ω is called a nontangentially accessible domain (NTA domain) if Ω satisfies the interior and the exterior corkscrew conditions, and the Harnack chain condition.

We point out that NTA domains include Lipschitz domains, Zygmund domains, quasi-spheres, and some Reifenberg flat domains as special examples (see, for instance, [35,41,61]).

Let n ≥ 2 and Ω be a bounded NTA domain of R n . Assume that p ∈ [ 1 , ∞ ) and ω ∈ A q ( R n ) with some q ∈ [ 1 , ∞ ) . Recall that the weighted Lebesgue space L ω p ( Ω ) is defined by setting

(1.1) L ω p ( Ω ) ≔ f is measurable on Ω : ‖ f ‖ L ω p ( Ω ) ≔ ∫ Ω ∣ f ( x ) ∣ p ω ( x ) d x 1 p < ∞ .

Moreover, let

(1.2) L ω p ( Ω ; R n ) ≔ { f ≔ ( f 1 , … , f n ) : for any i ∈ { 1 , … , n } , f i ∈ L ω p ( Ω ) }

and

‖ f ‖ L ω p ( Ω ; R n ) ≔ ∑ i = 1 n ‖ f i ‖ L ω p ( Ω ) .

Denote by W ω 1 , p ( Ω ) the weighted Sobolev space on Ω equipped with the norm

‖ f ‖ W ω 1 , p ( Ω ) ≔ ‖ f ‖ L ω p ( Ω ) + ‖ ∇ f ‖ L ω p ( Ω ; R n ) ,

where ∇ f denotes the distributional gradient of f . Furthermore, W 0 , ω 1 , p ( Ω ) is defined to be the closure of C c ∞ ( Ω ) in W ω 1 , p ( Ω ) , where C c ∞ ( Ω ) denotes the set of all infinitely differentiable functions on Ω with compact support contained in Ω . In particular, when ω ≡ 1 , the weighted spaces L ω p ( Ω ) , W ω 1 , p ( Ω ) , and W 0 , ω 1 , p ( Ω ) are denoted simply, respectively, by L p ( Ω ) , W 1 , p ( Ω ) , and W 0 1 , p ( Ω ) , which are, respectively, the classical Lebesgue space and the classical Sobolev spaces.

Let Ω be a domain of R n . Denote by L loc 1 ( Ω ) the set of all locally integrable functions on Ω .

Definition 1.3

Let Ω ⊂ R n be a domain and f ∈ L loc 1 ( Ω ) . Then, f is said to belong to the space BMO ( Ω ) if

‖ f ‖ BMO ( Ω ) ≔ sup B ⊂ Ω 1 ∣ B ∣ ∫ B ∣ f ( x ) − f B ∣ d x < ∞ ,

where the supremum is taken over all balls B ⊂ Ω and f B ≔ 1 ∣ B ∣ ∫ B f ( y ) d y .

For any given x ∈ Ω , let a ( x ) ≔ { a i j ( x ) } i , j = 1 n denote an n × n matrix with real-valued, bounded, and measurable entries. Then, a is said to satisfy the uniform ellipticity condition if there exists a positive constant μ 0 ∈ ( 0 , 1 ] such that, for any x ∈ Ω and ξ ≔ ( ξ 1 , … , ξ n ) ∈ R n ,

(1.3) μ 0 ∣ ξ ∣ 2 ≤ ∑ i , j = 1 n a i j ( x ) ξ i ξ j ≤ μ 0 − 1 ∣ ξ ∣ 2 .

Recall that the matrix b ≔ { b i j } i , j = 1 n is said to be antisymmetric if b i j = − b j i for any i , j ∈ { 1 , … , n } . Throughout this article, we always assume that the matrix A satisfies the following assumption.

Assumption 1.4

Let Ω ⊂ R n be a domain. Assume that, for any given x ∈ Ω , A ( x ) is an n × n matrix satisfying that A ( x ) = a ( x ) + b ( x ) , where the matrix a ( x ) ≔ { a i j ( x ) } i , j = 1 n is real-valued, symmetric, and measurable, and satisfies the uniform ellipticity condition (1.3), and the matrix b ( x ) ≔ { b i j ( x ) } i , j = 1 n is real-valued, antisymmetric, and measurable, and satisfies b i j ∈ BMO ( Ω ) for any i , j ∈ { 1 , … , n } .

Remark 1.5

Let Ω ⊂ R n be a domain and A ≔ a + b satisfy Assumption 1.4.

By the assumption that a satisfies (1.3), we conclude that a ∈ L ∞ ( Ω ; R n 2 ) , which, together with the facts that L ∞ ( Ω ) ⊂ BMO ( Ω ) and b ∈ BMO ( Ω ; R n 2 ) , further implies that A ∈ BMO ( Ω ; R n 2 ) .
Via replacing Ω by R n in Definition 1.3, we obtain the definition of the space BMO ( R n ) . Jones [39] proved that any given function f ∈ BMO ( Ω ) admits an extension to some f ˜ ∈ BMO ( R n ) if and only if the domain Ω is a uniform domain (namely, the domain satisfying the interior corkscrew condition and the Harnack chain condition). Thus, if Ω is an NTA domain, then, for any given f ∈ BMO ( Ω ) , there exists an f ˜ ∈ BMO ( R n ) such that
f ˜ ∣ Ω = f and ∥ f ˜ ∥ BMO ( R n ) ≤ C ‖ f ‖ BMO ( Ω ) ,
where C is a positive constant depending only on Ω and n .
By the assumptions that a satisfies (1.3) and b is antisymmetric, we conclude that, for any x ∈ Ω and ξ ∈ R n ,
( A ( x ) ξ ) ⋅ ξ = ( a ( x ) ξ ) ⋅ ξ ≥ μ 0 ∣ ξ ∣ 2 .

Let Ω ⊂ R n be a bounded domain and the matrix A satisfy Assumption 1.4. Assume that p ∈ ( 1 , ∞ ) , ω ∈ A q ( R n ) with some q ∈ [ 1 , ∞ ) , and f ∈ L ω p ( Ω ; R n ) . Then, a function u is called a weak solution of the following weighted Dirichlet boundary value problem

(1.4) − div ( A ∇ u ) = div ( f ) in Ω , u = 0 on ∂ Ω ( D ) p , ω

if u ∈ W 0 , ω 1 , p ( Ω ) and, for any φ ∈ C c ∞ ( Ω ) ,

(1.5) ∫ Ω A ( x ) ∇ u ( x ) ⋅ ∇ φ ( x ) d x = − ∫ Ω f ( x ) ⋅ ∇ φ ( x ) d x .

In particular, when ω ≡ 1 , the weighted Dirichlet problem ( D ) p , ω is just the Dirichlet problem ( D ) p . The weighted Dirichlet problem ( D ) p , ω is said to be uniquely solvable if, for any given f ∈ L ω p ( Ω ; R n ) , there exists a unique u ∈ W 0 , ω 1 , p ( Ω ) such that (1.5) holds true for any φ ∈ C c ∞ ( Ω ) .

Let L ≔ − div ( A ∇ ) with the matrix A satisfying Assumption 1.4. The elliptic operator L naturally arises in the study of the elliptic equation of the form

(1.6) − Δ u + c ⋅ ∇ u = f

(see, for instance, [45,54]) and the parabolic equation

∂ u ∂ t − Δ u + c ⋅ ∇ u = f ,

where the drift term c satisfies div ( c ) = 0 . By div ( c ) = 0 , we know that c = div ( b ) for some antisymmetric tensor b ≔ { b i j } i , j = 1 n . Therefore, equation (1.6) becomes

− div ( ( I − b ) ∇ u ) = f ,

where I denotes the unit matrix on R n . In particular, Seregin et al. [54] discovered that the well-known Moser iteration works for such an elliptic operator L . Via the Moser iteration, Seregin et al. [54] proved the Liouville theorem and the Harnack inequality for solutions to equation (1.6) or its parabolic case. Moreover, Li and Pipher [43] studied the boundary behavior of solutions of the equation L u = 0 in NTA domains. Furthermore, Dong and Phan [27] investigated the mixed-norm Sobolev estimate for solutions to nonstationary Stokes systems with coefficients having unbounded antisymmetric part in cylindrical domains.

Remark 1.6

Let Ω ⊂ R n be a bounded NTA domain and the matrix A ≔ a + b satisfy Assumption 1.4. For any u , v ∈ W 0 1 , 2 ( Ω ) , let

B [ u , v ] ≔ ∫ Ω A ( x ) ∇ u ( x ) ⋅ ∇ v ( x ) d x .

From Remark 1.5(iii), it follows that, for any u ∈ W 0 1 , 2 ( Ω ) ,

B [ u , u ] ≥ μ 0 ‖ ∇ u ‖ L 2 ( Ω ; R n ) 2 .

Moreover, it was showed in [43, (2.11)], via using the div-curl lemma, that, for any u , v ∈ W 0 1 , 2 ( Ω ) ,

∣ B [ u , v ] ∣ ≤ C ‖ ∇ u ‖ L 2 ( Ω ; R n ) ‖ ∇ v ‖ L 2 ( Ω ; R n ) ,

where C is a positive constant depending only on A and Ω . Thus, by the Lax–Milgram theorem (see, for instance, [32, Theorem 5.8]), we conclude that the Dirichlet problem ( D ) 2 is uniquely solvable and, for any given f ∈ L 2 ( Ω ; R n ) , the weak solution u ∈ W 0 1 , 2 ( Ω ) of ( D ) 2 satisfies that

‖ ∇ u ‖ L 2 ( Ω ; R n ) ≤ μ 0 − 1 ‖ f ‖ L 2 ( Ω ; R n ) ,

where μ 0 is as in (1.3).

Moreover, via an example given by Meyers [48, Section 5] (see also [17, p. 1285]), we find that, for any p ∈ ( 1 , ∞ ) with p ≠ 2 , the Dirichlet problem ( D ) p may not be uniquely solvable, even when the domain Ω is smooth.

Let n ≥ 2 , Ω ⊂ R n be a bounded NTA domain, and the matrix A satisfy Assumption 1.4. Assume further that A satisfies the ( δ , R ) -BMO condition (see Definition 2.1) or A belongs to the space VMO ( Ω ) (see, for instance, [53]). In this article, our aim is to establish the weighted Calderón–Zygmund type estimates

(1.7) ‖ ∇ u ‖ L ω p ( Ω ; R n ) ≤ C ‖ f ‖ L ω p ( Ω ; R n )

for the Dirichlet problem (1.4), with p ∈ ( 1 , ∞ ) and ω ∈ A q ( R n ) for some q ∈ [ 1 , ∞ ) , and then give their applications, where C is a positive constant independent of both u and f .

Let n ≥ 2 , Ω ⊂ R n be a bounded domain, and the matrix A ≔ a + b satisfy Assumption 1.4. For the Dirichlet problem ( D ) p , the estimate (1.7) with p ∈ ( 1 , ∞ ) and ω ≡ 1 was established by Di Fazio [22], under the assumptions that a ∈ VMO ( R n ) , b ≡ 0 , and ∂ Ω ∈ C 1 , 1 , which was weakened to ∂ Ω ∈ C 1 by Auscher and Qafsaoui [6]. Moreover, for the Dirichlet problem ( D ) p , estimate (1.7) with p ∈ ( 1 , ∞ ) and ω ≡ 1 was obtained by Byun and Wang in [13,17], under the assumptions that a satisfies the ( δ , R ) -BMO condition for sufficiently small δ ∈ ( 0 , ∞ ) , b ≡ 0 , and Ω is a bounded Lipschitz domain with small Lipschitz constant or a bounded Reifenberg flat domain (see, for instance, [52,61]). Furthermore, for the Dirichlet problem ( D ) p with a having partial small BMO coefficients and b ≡ 0 , estimate (1.7) with p ∈ ( 1 , ∞ ) and ω ≡ 1 was studied, respectively, by Dong and Kim [24] and Krylov [42], under the assumption that Ω is a bounded Lipschitz domain with small Lipschitz constant. Moreover, if Ω is a bounded quasi-convex domain, a satisfies the ( δ , R ) -BMO condition for sufficiently small δ ∈ ( 0 , ∞ ) , and b ≡ 0 , estimate (1.7) with p ∈ ( 1 , ∞ ) and ω ≡ 1 was established by Jia et al. [37] for the Dirichlet problem ( D ) p . For the Dirichlet problem ( D ) p in a general Lipschitz domain Ω , it was proved by Shen [58] that, if a ∈ VMO ( R n ) and b ≡ 0 , then (1.7) with ω ≡ 1 holds true for any given p ∈ 3 2 − ε , 3 + ε when n ≥ 3 , or p ∈ 4 3 − ε , 4 + ε when n = 2 , where ε ∈ ( 0 , ∞ ) is a positive constant depending on Ω . It is worth pointing out that, when A ≔ I (the identity matrix) in (1.7), the range of p obtained in [58] is even sharp for general Lipschitz domains (see, for instance, [36]).

For the weighted Dirichlet problem ( D ) p , ω with a having partial small BMO coefficients and b ≡ 0 , (1.7) with p ∈ ( 2 , ∞ ) and ω ∈ A p / 2 ( R n ) was obtained by Byun and Palagachev [15] under the assumption that Ω is a bounded Reifenberg flat domain. Furthermore, for the problem ( D ) p , ω with a having partial small BMO coefficients and b ≡ 0 , the estimate (1.7) with p ∈ ( 1 , ∞ ) and ω ∈ A p ( R n ) was established by Dong and Kim [26] under the assumption that Ω is a bounded Reifenberg flat domain. For the problem ( D ) p , ω with a having small BMO coefficients and b ≡ 0 , (1.7) with p ∈ ( 1 , ∞ ) and ω ∈ A p ( R n ) was obtained by Adimurthi et al. [1] under the assumption that Ω is a bounded Lipschitz domain with small Lipschitz constant; see also [60] for another short proof of this result. Recently, the weighted global gradient estimates for the Neumann problem of second-order elliptic equations of divergence form with the coefficient matrix A in NTA domains were studied in [66].

Now, we state the main results of this article as follows.

Theorem 1.7

Let n ≥ 2 , Ω ⊂ R n be a bounded NTA domain, the matrix A satisfy Assumption 1.4, and p ∈ ( 2 , ∞ ) . Assume that there exist positive constants C 0 ∈ ( 0 , ∞ ) and r 0 ∈ ( 0 , diam ( Ω ) ) such that, for any ball B ( x 0 , r ) having the property that r ∈ ( 0 , r 0 / 4 ) and either x 0 ∈ ∂ Ω or B ( x 0 , 2 r ) ⊂ Ω , the weak reverse Hölder inequality

(1.8) 1 ∣ B Ω ( x 0 , r ) ∣ ∫ B Ω ( x 0 , r ) ∣ ∇ v ( x ) ∣ p d x 1 p ≤ C 0 1 ∣ B Ω ( x 0 , 2 r ) ∣ ∫ B Ω ( x 0 , 2 r ) ∣ ∇ v ( x ) ∣ 2 d x 1 2

holds true for any function v ∈ W 1 , 2 ( B Ω ( x 0 , 2 r ) ) satisfying div ( A ∇ v ) = 0 in B Ω ( x 0 , 2 r ) , and v = 0 on B ( x 0 , 2 r ) ∩ ∂ Ω when x 0 ∈ ∂ Ω , where B Ω ( x 0 , r ) ≔ B ( x 0 , r ) ∩ Ω .

Then, the weak solution u ∈ W 0 1 , 2 ( Ω ) of the Dirichlet problem ( D ) p with f ∈ L p ( Ω ; R n ) exists and, moreover, u ∈ W 0 1 , p ( Ω ) and there exists a positive constant C , depending only on n , p , and Ω , such that
(1.9) ‖ ∇ u ‖ L p ( Ω ; R n ) ≤ C ‖ f ‖ L p ( Ω ; R n ) .
Let q ∈ [ 2 , p ] , q 0 ∈ 1 , q p ′ , r 0 ∈ p q ′ , ∞ , and ω ∈ A q 0 ( R n ) ∩ RH r 0 ( R n ) . Then, a weak solution u of the weighted Dirichlet problem ( D ) q , ω with f ∈ L ω q ( Ω ; R n ) exists and, moreover, u ∈ W 0 , ω 1 , q ( Ω ) and there exists a positive constant C , depending only on n , p , q , [ ω ] A q 0 ( R n ) , [ ω ] RH r 0 ( R n ) , and Ω , such that
(1.10) ‖ ∇ u ‖ L ω q ( Ω ; R n ) ≤ C ‖ f ‖ L ω q ( Ω ; R n ) .
Here and thereafter, for any s ∈ [ 1 , ∞ ] , s ′ denotes its conjugate exponent, namely, 1 / s + 1 / s ′ = 1 .

We prove Theorem 1.7 via using a (weighted) real-variable argument (see Theorem 3.1), which was essentially established in [57, Theorem 3.4] (see also [29,30,56,58,65]) and inspired by [19,62]. It is worth pointing out that a similar real-variable argument with the different motivation was used in [4,5]. Moreover, a different weighted real-variable argument was obtained by Shen [55, Theorem 2.1]. Furthermore, the linear structure of second-order elliptic operators of divergence form and the properties of Muckenhoupt weights are subtly used in the proof of Theorem 1.7.

Let the ( δ , R ) -BMO condition and the space VMO ( Ω ) be as in Definition 2.1. As an application of Theorem 1.7, we obtain the (weighted) global gradient estimates for solutions to Dirichlet boundary problems on bounded Lipschitz domains as follows.

Theorem 1.8

Let n ≥ 2 , Ω ⊂ R n be a bounded Lipschitz domain, and the matrix A satisfy Assumption 1.4.

Then, there exist positive constants ε 0 , δ 0 ∈ ( 0 , ∞ ) , depending only on n and the Lipschitz constant of Ω , such that, for any given p ∈ ( ( 3 + ε 0 ) ′ , 3 + ε 0 ) when n ≥ 3 , or p ∈ ( ( 4 + ε 0 ) ′ , 4 + ε 0 ) when n = 2 , if A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then, the Dirichlet problem ( D ) p with f ∈ L p ( Ω ; R n ) is uniquely solvable and there exists a positive constant C , depending only on n , p , and the Lipschitz constant of Ω , such that, for any weak solution u , u ∈ W 0 1 , p ( Ω ) and
(1.11) ‖ ∇ u ‖ L p ( Ω ; R n ) ≤ C ‖ f ‖ L p ( Ω ; R n ) .
Let ε 0 be as in (i) and p 0 ≔ 3 + ε 0 when n ≥ 3 , and p 0 ≔ 4 + ε 0 when n = 2 . Then, for any given p ∈ ( p 0 ′ , p 0 ) and any ω ∈ A p p 0 ′ ( R n ) ∩ RH p 0 p ′ ( R n ) , there exists a positive constant δ 0 ∈ ( 0 , ∞ ) , depending only on n , p , the Lipschitz constant of Ω , [ ω ] A p p 0 ′ ( R n ) , and [ ω ] RH p 0 p ′ ( R n ) , such that, if A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then, the weighted Dirichlet problem ( D ) p , ω with f ∈ L ω p ( Ω ; R n ) is uniquely solvable and there exists a positive constant C , depending only on n , p , [ ω ] A p p 0 ′ ( R n ) , [ ω ] RH p 0 p ′ ( R n ) , and the Lipschitz constant of Ω , such that, for any weak solution u , u ∈ W 0 , ω 1 , p ( Ω ) and
(1.12) ‖ ∇ u ‖ L ω p ( Ω ; R n ) ≤ C ‖ f ‖ L ω p ( Ω ; R n ) .

Let A ≔ a + b satisfy Assumption 1.4 and p 0 be as in Theorem 1.8(ii). The key to proving Theorem 1.8 is to show that the weak reverse Hölder inequality (1.8) is valid for any p ∈ ( 2 , p 0 ) . To do this, we flexibly apply the real-variable argument established in Theorem 3.1, the method of perturbation, the assumptions that b ∈ BMO ( Ω ; R n 2 ) and b is antisymmetric, and the properties of Muckenhoupt weights.

Remark 1.9

Let n ≥ 2 , Ω ⊂ R n be a bounded Lipschitz domain, and A ≔ a + b satisfy Assumption 1.4. If a ∈ VMO ( Ω ) and b ≡ 0 , then Theorem 1.8 in this case was essentially established by Shen [58, Theorem C]. Thus, Theorem 1.8 improves [58, Theorem C] via weakening the condition for the matrix A .

Let the quasi-convex domain be as in Definition 2.3. We further obtain the following (weighted) global gradient estimates for the Dirichlet problems on quasi-convex domains by using Theorem 1.7.

Theorem 1.10

Let n ≥ 2 , Ω ⊂ R n be a bounded NTA domain, and p ∈ ( 1 , ∞ ) . Assume further that the matrix A satisfies Assumption 1.4, and Ω is a ( δ , σ , R ) quasi-convex domain with some δ , σ ∈ ( 0 , 1 ) and R ∈ ( 0 , ∞ ) .

Then, there exists a positive constant δ 0 ∈ ( 0 , 1 ) , depending only on n , p , and Ω , such that, if Ω is a ( δ , σ , R ) quasi-convex domain and A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) , σ ∈ ( 0 , 1 ) , and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then the Dirichlet problem ( D ) p with f ∈ L p ( Ω ; R n ) is uniquely solvable and, for any weak solution u of the problem ( D ) p , u ∈ W 0 1 , p ( Ω ) and
‖ ∇ u ‖ L p ( Ω ; R n ) ≤ C ‖ f ‖ L p ( Ω ; R n ) ,
where C is a positive constant depending only on n , p , and Ω .
Let ω ∈ A p ( R n ) . Then, there exists a positive constant δ 0 ∈ ( 0 , 1 ) , depending only on n , p , Ω , and [ ω ] A p ( R n ) , such that, if Ω is a ( δ , σ , R ) quasi-convex domain and A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) , σ ∈ ( 0 , 1 ) , and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then the weighted Dirichlet problem ( D ) p , ω with f ∈ L ω p ( Ω ; R n ) is uniquely solvable and there exists a positive constant C , depending only on n , p , [ ω ] A p ( R n ) , and Ω , such that, for any weak solution u , u ∈ W 0 , ω 1 , p ( Ω ) and
(1.13) ‖ ∇ u ‖ L ω p ( Ω ; R n ) ≤ C ‖ f ‖ L ω p ( Ω ; R n ) .

Assume that p ∈ [ 2 , ∞ ) . To prove Theorem 1.10 via using Theorem 1.7, we need to prove that there exists a δ 0 ∈ ( 0 , 1 ) , depending on p and Ω , such that, if Ω is a bounded ( δ , σ , R ) quasi-convex domain with some δ ∈ ( 0 , δ 0 ) , σ ∈ ( 0 , 1 ) , and R ∈ ( 0 , ∞ ) , then the weak reverse Hölder inequality (1.8) is valid for the exponent p . To this end, we adequately use the real-variable argument obtained in Theorem 3.1, the method of perturbation, and the geometric properties of quasi-convex domains (see Lemma 5.3).

Let the (semi-)convex domain be as in Remark 2.5(i). As a corollary of Theorem 1.10, we have the following conclusion.

Corollary 1.11

Let n ≥ 2 , Ω ⊂ R n be a bounded C 1 or (semi-)convex domain, p ∈ ( 1 , ∞ ) , and ω ∈ A p ( R n ) . Assume that the matrix A satisfies Assumption 1.4. Then, there exists a positive constant δ 0 ∈ ( 0 , ∞ ) depending only on n , p , Ω , and [ ω ] A p ( R n ) , such that, if A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then the weighted Dirichlet problem ( D ) p , ω with f ∈ L ω p ( Ω ; R n ) is uniquely solvable and there exists a positive constant C , depending only on n , p , [ ω ] A p ( R n ) , and Ω , such that, for any weak solution u , u ∈ W 0 , ω 1 , p ( Ω ) and

‖ ∇ u ‖ L ω p ( Ω ; R n ) ≤ C ‖ f ‖ L ω p ( Ω ; R n ) .

Moreover, let the Reifenberg flat domain be as in Remark 2.5(ii). By Theorem 1.10 and Remark 2.5(ii), we have the following corollary.

Corollary 1.12

Let n ≥ 2 , Ω ⊂ R n be a bounded ( δ , R ) -Reifenberg flat domain with some δ ∈ ( 0 , 1 ) and R ∈ ( 0 , ∞ ) , p ∈ ( 1 , ∞ ) , and ω ∈ A p ( R n ) . Assume that the matrix A satisfies Assumption 1.4. Then there exists a positive constant δ 0 ∈ ( 0 , 1 ) , depending only on n , p , Ω , and [ ω ] A p ( R n ) , such that, if Ω is a bounded ( δ , R ) -Reifenberg flat domain and A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then the weighted Dirichlet problem ( D ) p , ω with f ∈ L ω p ( Ω ; R n ) is uniquely solvable and there exists a positive constant C , depending only on n , p , [ ω ] A p ( R n ) , and Ω , such that, for any weak solution u , u ∈ W 0 , ω 1 , p ( Ω ) and

‖ ∇ u ‖ L ω p ( Ω ; R n ) ≤ C ‖ f ‖ L ω p ( Ω ; R n ) .

Remark 1.13

Let n ≥ 2 and Ω ⊂ R n be a bounded ( δ , σ , R ) quasi-convex domain with some δ , σ ∈ ( 0 , 1 ) and R ∈ ( 0 , ∞ ) . Assume that the matrix A ≔ a + b satisfies Assumption 1.4.

If a satisfies the ( δ , R ) - BMO condition for some sufficiently small δ ∈ ( 0 , ∞ ) and b ≡ 0 , then Theorem 1.10(i) in this case was established by Jia et al. in [37, Theorem 1.1]. Thus, Theorem 1.10(i) improves [37, Theorem 1.1] via weakening the condition for the matrix A . Moreover, even when b ≡ 0 , the conclusion of Theorem 1.10(ii) in this case is also new.
When a satisfies the ( δ , R ) - BMO condition for some sufficiently small δ ∈ ( 0 , ∞ ) , b ≡ 0 , and ω ≡ 1 , Corollary 1.12 in this case was obtained by Byun and Wang in [17, Theorem 1.5]. Therefore, even when ω ≡ 1 , Corollary 1.12 in this case also improves [17, Theorem 1.5] via weakening the assumption on A . Moreover, a similar parabolic version of Corollary 1.12 was given by Nguyen in [51, Theorem 1.2]. Furthermore, we point out that the approach used in this article to establish the global gradient estimates is different from that used in [13,17, 37]. Indeed, the global estimates were obtained in [13, 17,37] by using an approximation argument, the modified Vitali covering lemma, and a compactness method. However, in this article, we establish the (weighted) global estimates via using a (weighted) real-variable argument (see Theorem 3.1), the method of perturbation, and the geometric properties of quasi-convex domains.
We point out that, even when Ω is a bounded convex domain and ω ≡ 1 , the conclusion of Corollary 1.11 in this case is also new.

Applying the weighted global regularity estimates obtained in Theorems 1.8(ii) and 1.10(ii), and some tools from harmonic analysis, such as the properties of Muckenhoupt weights, the interpolation theorem of operators, and the Rubio de Francia extrapolation theorem, we obtain the global gradient estimates for the Dirichlet problem (1.4), respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (Musielak–)Orlicz spaces, and variable Lebesgue spaces, which have independent interests and are presented in Section 6. It is worth pointing out that the approach used in this article to establish the global estimates in both Orlicz spaces and variable Lebesgue spaces is quite different from that used in [14,18]. In [14,18], the global estimates in variable Lebesgue spaces or in Orlicz spaces were established via the so-called maximum function free technique. However, in this article, we obtain the global gradient estimates in both Orlicz spaces and variable Lebesgue spaces by simply using weighted global estimates in Theorems 1.8(ii) and 1.10(ii), and the Rubio de Francia extrapolation theorem. Furthermore, we point out that the extrapolation theorem used in this article is also valid for the boundary value problem studied in [14,18] and independent of the boundary value condition and the considered equation.

Moreover, the global estimates in Orlicz spaces were obtained in [38] through an approximation argument, the modified Vitali covering lemma, and a compactness method. However, in this article, the global gradient estimates in Orlicz spaces are obtained as corollaries of both the weighted norm inequality in Theorem 1.10(ii) and the extrapolation theorem.

The remainder of this article is organized as follows.

In Section 2, we present several concepts on the ( δ , R ) - BMO condition, the space VMO ( Ω ) , and several domains, and clarify their relations. We then prove Theorems 1.7, 1.8, and 1.10, and Corollary 1.11, respectively, in Sections 3, 4, and 5. Several applications of Theorems 1.8 and 1.10 are given in Section 6.

Finally, we make some conventions on notation. Throughout this article, we always denote by C a positive constant, which is independent of the 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 ≤ C g . If f ≲ g and g ≲ f , then we write f ∼ g . If f ≤ C g 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 ( x B , r B ) of R n , with some x B ∈ R n and r B ∈ ( 0 , ∞ ) , and α ∈ ( 0 , ∞ ) , let α B ≔ B ( x B , α r B ) ; furthermore, denote set B ( x , r ) ∩ Ω by B Ω ( x , r ) , and the set ( α B ) ∩ Ω by α B Ω . For any subset E of R n , we denote the set R n ⧹ E by E ∁ , and its characteristic function by 1 E . For any ω ∈ A p ( R n ) with p ∈ [ 1 , ∞ ) , and any measurable set E ⊂ R n , let

ω ( E ) ≔ ∫ E ω ( x ) d x .

For any given q ∈ [ 1 , ∞ ] , we denote by q ′ its conjugate exponent, namely, 1 / q + 1 / q ′ = 1 . Finally, for any measurable set E ⊂ R n , ω ∈ A q ( R n ) with some q ∈ [ 1 , ∞ ) , and f ∈ L 1 ( E ) , we denote the integral ∫ E ∣ f ( x ) ∣ ω ( x ) d x simply by ∫ E ∣ f ∣ ω d x and, when ∣ E ∣ < ∞ , we use the notation

⨏ E f d x ≔ 1 ∣ E ∣ ∫ E f ( x ) d x .

2 Several concepts

In this section, we present the definitions of the ( δ , R ) - BMO condition, the space VMO ( Ω ) , and several domains including quasi-convex domains, (semi-)convex domains, and Reifenberg flat domains. Furthermore, we also clarify the relations between NTA domains, Lipschitz domains, quasi-convex domains, Reifenberg flat domains, and semi-convex domains.

First, we recall the concepts of the ( δ , R ) - BMO condition and the space VMO ( Ω ) as follows (see, for instance, [16,17,53]).

Definition 2.1

Let Ω ⊂ R n be a domain and R , δ ∈ ( 0 , ∞ ) .

A function f ∈ L loc 1 ( Ω ) is said to satisfy the ( δ , R ) - BMO condition if
‖ f ‖ ∗ , R ≔ sup B ( x , r ) ⊂ Ω , r ∈ ( 0 , R ) 1 ∣ B ( x , r ) ∣ ∫ B ( x , r ) ∣ f ( y ) − f B ( x , r ) ∣ d y ≤ δ ,
where the supremum is taken over all balls B ( x , r ) ⊂ Ω with r ∈ ( 0 , R ) . Furthermore, f is said to belong to the space VMO ( Ω ) if f satisfies the ( δ , R ) - BMO condition for some δ , R ∈ ( 0 , ∞ ) , and
lim r → 0 + sup B ( x , r ) ⊂ Ω 1 ∣ B ( x , r ) ∣ ∫ B ( x , r ) ∣ f ( y ) − f B ( x , r ) ∣ d y = 0 ,
where r → 0 + means r ∈ ( 0 , ∞ ) and r → 0 .
A matrix A ≔ { a i j } i , j = 1 n is said to satisfy the ( δ , R ) - BMO condition [resp., A ∈ VMO ( Ω ) ] if, for any i , j ∈ { 1 , … , n } , a i j satisfies the ( δ , R ) - BMO condition [resp., a i j ∈ VMO ( Ω ) ].

Remark 2.2

Let Ω ⊂ R n be a domain and δ ∈ ( 0 , ∞ ) . If f ∈ BMO ( Ω ) and ‖ f ‖ BMO ( Ω ) ≤ δ , then f satisfies the ( δ , R ) - BMO condition for any R ∈ ( 0 , ∞ ) . Moreover, if f ∈ VMO ( Ω ) , then f satisfies the ( γ , R ) - BMO condition for any γ ∈ ( 0 , ∞ ) and some R ∈ ( 0 , ∞ ) .

Now, we recall the concept of quasi-convex domains as follows. Let E 1 , E 2 ⊂ R n be nonempty measurable subsets. Then the Hausdorff distance between E 1 and E 2 is defined by setting

d H ( E 1 , E 2 ) ≔ max { sup x ∈ E 1 inf y ∈ E 2 ∣ x − y ∣ , sup y ∈ E 2 inf x ∈ E 1 ∣ x − y ∣ } .

Definition 2.3

Let Ω ⊂ R n be a domain, δ , σ ∈ ( 0 , 1 ) , and R ∈ ( 0 , ∞ ) . Then Ω is called a ( δ , σ , R ) quasi-convex domain if, for any x ∈ ∂ Ω and r ∈ ( 0 , R ] ,

There exists an x 0 ∈ Ω , depending on x , such that B ( x 0 , σ r ) ⊂ Ω ∩ B ( x , r ) .
There exists a convex domain V ≔ V ( x , r ) , depending on x and r , such that B ( x , r ) ∩ Ω ⊂ V and d H ( ∂ ( B ( x , r ) ∩ Ω ) , ∂ V ) ≤ δ r .

Remark 2.4

The concept of quasi-convex domains was introduced by Jia et al. [37] to study the global regularity of second-order elliptic equations. Roughly speaking, a quasi-convex domain is a domain satisfying that the local boundary is close to be convex at small scales. It is easy to see that, if Ω is a convex domain, then Ω is a ( δ , σ , R ) quasi-convex domain for any δ ∈ ( 0 , 1 ) , some σ ∈ ( 0 , 1 ) , and some R ∈ ( 0 , ∞ ) .
We may always assume that the convex domain V in Definition 2.3(b) is the convex hull of B ( x , r ) ∩ Ω , which is the smallest convex domain containing B ( x , r ) ∩ Ω (see [37, Theorem 3.1] and [73, Remark 1.2] for more details).
It was showed in [37, Theorem 3.10] that Definition 2.3(b) can be replaced by the following condition:
1. For any x ∈ ∂ Ω and r ∈ ( 0 , R ] , there exist an ( n − 1 ) -dimensional plane L ( x , r ) containing x , a choice of the unit normal vector to L ( x , r ) , denoted by ν x , r , and the half space
  H ( x , r ) ≔ { y + t ν x , t : y ∈ L ( x , t ) , t ∈ [ − δ r , ∞ ) }
  such that
  Ω ∩ B ( x , r ) ⊂ H ( x , r ) ∩ B ( x , r ) .
More precisely, it was proved in [37, Theorem 3.10] that, if Ω is a domain satisfying the assumptions (a) and (c) with some δ , σ ∈ ( 0 , 1 ) and R ∈ ( 0 , ∞ ) , then Ω is a ( δ 1 , σ , R ) quasi-convex domain with δ 1 ≔ 8 δ / σ .

In the following remark, we recall the concepts of semi-convex domains and Reifenberg flat domains and then clarify the relations between NTA domains, Lipschitz domains, quasi-convex domains, Reifenberg flat domains, and semi-convex domains.

Remark 2.5

A set E ⊂ R n is said to satisfy an exterior ball condition at x ∈ ∂ E if there exist a v ∈ S n − 1 and an r ∈ ( 0 , ∞ ) such that
(2.1) B ( x + r v , r ) ⊂ ( R n ⧹ E ) ,
where S n − 1 denotes the unit sphere of R n . For such an x ∈ ∂ E , let
r ( x ) ≔ sup { r ∈ ( 0 , ∞ ) : ( 2.1 ) holds true for some v ∈ S n − 1 } .
A set E is said to satisfy a uniform exterior ball condition (UEBC) with radius r ∈ ( 0 , ∞ ] if
(2.2) inf x ∈ ∂ E r ( x ) ≥ r ,
and the value r in (2.2) is referred to the UEBC constant. A set E is said to satisfy a UEBC if there exists an r ∈ ( 0 , ∞ ] such that E satisfies the uniform exterior ball condition with radius r . Moreover, the largest positive constant r as mentioned earlier is called the uniform ball constant of E .

It is known that, for any open set Ω ⊂ R n with compact boundary, Ω is a Lipschitz domain satisfying a UEBC if and only if Ω is a semi-convex domain in R n (see, for instance, [49, Theorem 2.5] or [50, Theorem 3.9]). Moreover, more equivalent characterizations of semi-convex domains are given in [49,50].

It is worth pointing out that, if Ω ⊂ R n is convex, then Ω satisfies a UEBC with the uniform ball constant ∞ (see, for instance, [28]). Thus, convex domains of R n are semi-convex domains (see, for instance, [49,50, 64,67]). Moreover, (semi-)convex domains are special cases of Lipschitz domains. More precisely,
class of convex domains ⊊ class of semi-convex domains ⊊ class of Lipschitz domains .
Let n ≥ 2 , δ ∈ ( 0 , 1 ) , and R ∈ ( 0 , ∞ ) . A domain Ω ⊂ R n is called a ( δ , R ) -Reifenberg flat domain if, for any x 0 ∈ ∂ Ω and r ∈ ( 0 , R ] , there exists a system of coordinates, { y 1 , … , y n } , which may depend on x 0 and r , such that, in this coordinate system, x 0 = 0 and
B ( 0 , r ) ∩ { y ∈ R n : y n > δ r } ⊂ B ( 0 , r ) ∩ Ω ⊂ B ( 0 , r ) ∩ { y ∈ R n : y n > − δ r } ,
where 0 denotes the origin of R n . The Reifenberg flat domain was introduced by Reifenberg in [52], which naturally appears in the theory of minimal surfaces and free boundary problems. A typical example of Reifenberg flat domains is the well-known Van Koch snowflake (see, for instance, [61]). In recent years, boundary value problems of elliptic or parabolic equations on Reifenberg flat domains have been widely concerned and studied (see, for instance, [9,11,12, 14,17,18, 26,46,69, 70,72]).

Moreover, for any given δ ∈ ( 0 , 1 ) and R ∈ ( 0 , ∞ ) , a ( δ , R ) -Reifenberg flat domain is a ( δ , σ , R ) quasi-convex domain with σ ≔ 1 − δ 2 (see, for instance, [37]). However, a quasi-convex domain may not be a Reifenberg flat domain. Indeed, let
Ω ≔ { ( x 1 , x 2 ) ∈ R 2 : 1 > x 2 > ∣ x 1 ∣ } .
Then Ω is convex and hence a quasi-convex domain, but Ω is not a Reifenberg flat domain. Thus,
class of Reifenberg flat domains ⊊ class of quasi-convex domains .
Furthermore, it was showed by Kenig and Toro [41] that, if δ is sufficiently small, then ( δ , R ) -Reifenberg flat domains are also NTA domains.
By the facts that Lipschitz domains with small Lipschitz constants are Reifenberg flat domains and that C 1 domains are Lipschitz domains with small Lipschitz constants (see, for instance, [61]), we conclude that C 1 domains are ( δ , σ , R ) quasi-convex domains with any δ ∈ ( 0 , δ 0 ) , some σ ∈ ( 0 , 1 ) , and some R ∈ ( 0 , ∞ ) , where δ 0 ∈ ( 0 , 1 ) is a positive constant depending only on Ω .

Let Ω ⊂ R n be a bounded semi-convex domain. Then Ω is a ( δ , σ , R ) quasi-convex domain for any δ ∈ ( 0 , δ 0 ) , some σ ∈ ( 0 , 1 ) , and some R ∈ ( 0 , ∞ ) , where δ 0 ∈ ( 0 , 1 ) is a positive constant depending only on Ω (see Lemma 5.4).
On NTA domains, quasi-convex domains, Reifenberg flat domains, Lipschitz domains, C 1 domains, and (semi-)convex domains, we have the following relations.
1. class of C 1 domains ⊊ class of Lipschitz domains with small Lipschitz constants ⊊ class of Lipschitz domains ⊊ class of NTA domains ; class of (semi-)convex domains ⊊ class of Lipschitz domains .
2. class of C 1 domains ⊊ class of Lipschitz domains with small Lipschitz constants ⊊ class of Reifenberg flat domains ⊊ class of quasi-convex domains ; class of (semi-)convex domains ⊊ class of quasi-convex domains .
3. Reifenberg flat domains or quasi-convex domains may not be Lipschitz domains, and generally Lipschitz domains may also not be Reifenberg flat domains (see, for instance, [61]). Moreover, (semi-)convex domains may not be Lipschitz domains with small Lipschitz constants, or Reifenberg flat domains.

3 Proof of Theorem 1.7

In this section, we prove Theorem 1.7 via using a real-variable argument for (weighted) L p ( Ω ) estimates, which is inspired by the work of Caffarelli and Peral [19] (see also [62]). When Ω is a bounded Lipschitz domain of R n , the conclusion of Theorem 3.1 was essentially established in [57, Theorem 3.4] (see also [29, Theorems 2.1 and 2.2], [56, Theorem 4.2.6], [58, Theorem 3.3], and [65, Theorem 3.1]). It is worth pointing out that the proofs of [57, Theorem 3.4], [56, Theorem 4.2.6], and [65, Theorem 3.1] are also valid in the case of bounded NTA domains. Thus, we omit the proof of Theorem 3.1 here. Furthermore, we also mention that a similar argument with a different motivation was established in [4,5]. Moreover, a different weighted real-variable argument was obtained by Shen [55, Theorem 2.1].

Theorem 3.1

Let n ≥ 2 , Ω ⊂ R n be a bounded NTA domain, p 1 , p 2 ∈ [ 1 , ∞ ) satisfy p 2 > p 1 , F ∈ L p 1 ( Ω ) , and f ∈ L q ( Ω ) with some q ∈ ( p 1 , p 2 ) . Assume that, for any ball B ≔ B ( x B , r B ) ⊂ R n having the property that ∣ B ∣ ≤ β 1 ∣ Ω ∣ and either 2 B ⊂ Ω or x B ∈ ∂ Ω , there exist two measurable functions F B and R B on 2 B such that ∣ F ∣ ≤ ∣ F B ∣ + ∣ R B ∣ on 2 B ∩ Ω ,

(3.1) ⨏ 2 B Ω ∣ R B ∣ p 2 d x 1 p 2 ≤ C 1 ⨏ β 2 B Ω ∣ F ∣ p 1 d x 1 p 1 + sup B ˜ ⊃ B ⨏ B ˜ Ω ∣ f ∣ p 1 d x 1 p 1

and

(3.2) ⨏ 2 B Ω ∣ F B ∣ p 1 d x 1 p 1 ≤ ε ⨏ β 2 B Ω ∣ F ∣ p 1 d x 1 p 1 + C 2 sup B ˜ ⊃ B ⨏ B ˜ Ω ∣ f ∣ p 1 d x 1 p 1 ,

where C 1 , C 2 , ε , and β 1 < 1 < β 2 are positive constants independent of F , f , R B , F B , and B , and the suprema are taken over all balls B ˜ ⊃ B .

Then, for any ω ∈ A q / p 1 ( R n ) ∩ RH s ( R n ) with s ∈ p 2 q ′ , ∞ , there exists a positive constant ε 0 , depending only on C 1 , C 2 , n , p 1 , p 2 , q , β 1 , β 2 , [ ω ] A q / p 1 ( R n ) , and [ ω ] RH s ( R n ) , such that, if ε ∈ [ 0 , ε 0 ) , then

1 ω ( Ω ) ∫ Ω ∣ F ∣ q ω d x 1 q ≤ C 1 ∣ Ω ∣ ∫ Ω ∣ F ∣ p 1 d x 1 p 1 + 1 ω ( Ω ) ∫ Ω ∣ f ∣ q ω d x 1 q ,

where C is a positive constant depending only on C 1 , C 2 , n , p 1 , p 2 , q , β 1 , β 2 , [ ω ] A q / p 1 ( R n ) , and [ ω ] RH s ( R n ) .

To show Theorem 1.7 via using Theorem 3.1, we need the following auxiliary conclusion.

Lemma 3.2

Let n ≥ 2 , Ω ⊂ R n be a bounded NTA domain, p ∈ ( 1 , ∞ ) , and p ′ ∈ ( 1 , ∞ ) be given by 1 / p + 1 / p ′ = 1 . Assume that, if the matrix A satisfies Assumption 1.4 and f ∈ L p ( Ω ; R n ) , then the weak solution u ∈ W 0 1 , p ( Ω ) of the Dirichlet problem

(3.3) − div ( A ∇ u ) = div ( f ) in Ω , u = 0 on ∂ Ω

satisfies the estimate

(3.4) ‖ ∇ u ‖ L p ( Ω ; R n ) ≤ C ‖ f ‖ L p ( Ω ; R n ) ,

where C is a positive constant independent of u and f . Let g ∈ L p ′ ( Ω ; R n ) and v ∈ W 0 1 , p ′ ( Ω ) be the weak solution of the Dirichlet problem (3.3) with f replaced by g . Then

‖ ∇ v ‖ L p ′ ( Ω ; R n ) ≤ C ‖ g ‖ L p ′ ( Ω ; R n ) ,

where C is a positive constant independent of both v and g .

Proof

Let f ∈ L p ( Ω ; R n ) and w ∈ W 0 1 , p ( Ω ) be the weak solution of the Dirichlet problem

(3.5) − div ( A ∗ ∇ w ) = div ( f ) in Ω , w = 0 on ∂ Ω ,

where A ∗ denotes the transpose of A . By the assumption that A satisfies Assumption 1.4, we find that A ∗ also satisfies Assumption 1.4, which, combined with the assumption (3.4), further implies that (3.4) also holds true for the weak solution w of (3.5). Moreover, from the fact that v is the weak solution of (3.3) with f replaced by g , it follows that

∫ Ω g ⋅ ∇ w d x = − ∫ Ω A ∇ v ⋅ ∇ w d x = − ∫ Ω A ∗ ∇ w ⋅ ∇ v d x = ∫ Ω f ⋅ ∇ v d x ,

which, together with (3.4) and the Hölder inequality, further implies that

‖ ∇ v ‖ L p ′ ( Ω ; R n ) = sup ‖ f ‖ L p ( Ω ; R n ) ≤ 1 ∫ Ω f ⋅ ∇ v d x = sup ‖ f ‖ L p ( Ω ; R n ) ≤ 1 ∫ Ω g ⋅ ∇ w d x ≤ sup ‖ f ‖ L p ( Ω ; R n ) ≤ 1 ‖ g ‖ L p ′ ( Ω ; R n ) ‖ ∇ w ‖ L p ( Ω ; R n ) ≲ sup ‖ f ‖ L p ( Ω ; R n ) ≤ 1 ‖ g ‖ L p ′ ( Ω ; R n ) ‖ f ‖ L p ( Ω ; R n ) ≲ ‖ g ‖ L p ′ ( Ω ; R n ) .

This finishes the proof of Lemma 3.2.□

Moreover, to prove Theorem 1.7, we need the following properties of A p ( R n ) weights, which are well known (see, for instance, [33, Chapter 7] and [5, Proposition 2.1 and Lemma 4.4]).

Lemma 3.3

Let q ∈ ( 1 , ∞ ) , ω ∈ A q ( R n ) , and Ω ⊂ R n be a bounded domain.

Then there exists a q 1 ∈ ( 1 , q ) , depending only on n , q , and [ ω ] A q ( R n ) , such that ω ∈ A q 1 ( R n ) .
Then there exists a γ ∈ ( 0 , ∞ ) , depending only on n , q , and [ ω ] A q ( R n ) , such that ω ∈ RH 1 + γ ( R n ) .
If q ′ denotes the conjugate number of q , namely, 1 / q + 1 / q ′ = 1 , then ω − q ′ / q ∈ A q ′ ( R n ) and [ ω − q ′ / q ] A q ′ ( R n ) = [ ω ] A q ( R n ) q ′ / q .
If ω ∈ RH p ( R n ) with p ∈ ( 1 , ∞ ) , then there exists a p 1 ∈ ( p , ∞ ] , depending only on n , p , and [ ω ] RH p ( R n ) , such that ω ∈ RH p 1 ( R n ) .
Let p 0 , q 0 ∈ ( 1 , ∞ ) , p ∈ ( p 0 , q 0 ) , and v ∈ L loc 1 ( R n ) . Then v ∈ A p p 0 ( R n ) ∩ RH q 0 p ′ ( R n ) if and only if v 1 − p ′ ∈ A p ′ ( q 0 ) ′ ( R n ) ∩ RH ( p 0 ) ′ p ′ ′ ( R n ) .
Let ω ∈ A p ( R n ) with some p ∈ [ 1 , ∞ ) and γ ∈ ( 0 , 1 ) . Then ω γ ∈ RH γ − 1 ( R n ) , and there exists a positive constant C , depending only on both [ ω ] A p ( R n ) and γ , such that [ ω γ ] RH γ − 1 ( R n ) ≤ C .
Let q 2 ≔ q 1 + 1 γ with γ as in (ii), and let q 1 be as in (i). Then L q 2 ( Ω ) ⊂ L ω q ( Ω ) ⊂ L q q 1 ( Ω ) .

Furthermore, we also need Lemma 3.4, whose proof is similar to that of [8, Lemma 4.38]; we omit the details here.

Lemma 3.4

Let n ≥ 2 , Ω ⊂ R n be a bounded NTA domain, 0 < p 0 < q ≤ ∞ , and r 0 ∈ ( 0 , diam ( Ω ) ) . Assume that x ∈ Ω ¯ and the weak reverse Hölder inequality

⨏ B Ω ( x , r ) ∣ g ∣ q d x 1 q ≤ C 3 ⨏ B Ω ( x , 2 r ) ∣ g ∣ p 0 d x 1 p 0

holds true for a given measurable function g on Ω and any r ∈ ( 0 , r 0 ) , where C 3 is a positive constant, independent of x and r , which may depend on g . Then, for any given p ∈ ( 0 , ∞ ] , there exists a positive constant C , depending only on p , p 0 , q , and C 3 , such that

⨏ B Ω ( x , r ) ∣ g ∣ q d x 1 q ≤ C ⨏ B Ω ( x , 2 r ) ∣ g ∣ p d x 1 p

holds true for any r ∈ ( 0 , r 0 ) .

Now, we prove Theorem 1.7 by using Theorem 3.1 and Lemmas 3.2, 3.3, and 3.4.

Proof of Theorem 1.7

We first show (i). Let B ≔ B ( x B , r B ) ⊂ R n be a ball satisfying r B ∈ ( 0 , r 0 / 4 ) and either 2 B ⊂ Ω or x B ∈ ∂ Ω . Take ϕ ∈ C c ∞ ( R n ) such that ϕ ≡ 1 on 2 B , 0 ≤ ϕ ≤ 1 , and supp ( ϕ ) ⊂ 4 B . Let w , v ∈ W 0 1 , 2 ( Ω ) be, respectively, the weak solutions of the Dirichlet problems

(3.6) − div ( A ∇ w ) = div ( ϕ f ) in Ω , w = 0 on ∂ Ω

and

(3.7) − div ( A ∇ v ) = div ( ( 1 − ϕ ) f ) in Ω , v = 0 on ∂ Ω .

Then u = w + v and ∇ u = ∇ w + ∇ v . Let F ≔ ∣ ∇ u ∣ , f ≔ ∣ f ∣ , F B ≔ ∣ ∇ w ∣ , and R B ≔ ∣ ∇ v ∣ . It is easy to see that 0 ≤ F ≤ F B + R B . By f ∈ L 2 ( Ω ; R n ) , (3.6), and the fact that (1.9) holds true for p = 2 (see Remark 1.6), we conclude that

(3.8) ⨏ 2 B Ω F B 2 d x = ⨏ 2 B Ω ∣ ∇ w ∣ 2 d x ≲ 1 ∣ 2 B Ω ∣ ∫ Ω ∣ f ϕ ∣ 2 d x ≲ ⨏ 4 B Ω ∣ f ∣ 2 d x ∼ ⨏ 4 B Ω f 2 d x .

Moreover, from (3.7) and the assumption (1.8) of this theorem, it follows that (1.8) holds true for the above v , which, together with the self-improvement property of the weak reverse Hölder inequality (see, for instance, [31, pp. 122–123]), further implies that there exists an ε 0 ∈ ( 0 , ∞ ) such that the inequality (1.8) holds true with p replaced by p + ε 0 . By this and Lemma 3.4, we conclude that, for any q ∈ ( 0 , 2 ] , the weak reverse Hölder inequality

(3.9) ⨏ B Ω ∣ ∇ v ∣ p + ε 0 d x 1 / ( p + ε 0 ) ≲ ⨏ 2 B Ω ∣ ∇ v ∣ q d x 1 / q

holds true, which, combined with (3.8), further implies that

(3.10) ⨏ B Ω R B p + ε 0 d x 1 / ( p + ε 0 ) ≲ ⨏ 2 B Ω ∣ ∇ v ∣ 2 d x 1 / 2 ≲ ⨏ 4 B Ω ∣ ∇ u ∣ 2 d x 1 / 2 + ⨏ 4 B Ω f 2 d x 1 / 2 ∼ ⨏ 4 B Ω F 2 d x 1 / 2 + ⨏ 4 B Ω f 2 d x 1 / 2 .

From (3.8) and (3.10), we deduce that (3.1) and (3.2) hold true with p 2 ≔ p + ε 0 and p 1 ≔ 2 . Thus, by Theorem 3.1 with ω ≡ 1 and q ≔ p , and the Hölder inequality, we conclude that

⨏ Ω ∣ ∇ u ∣ p d x 1 / p ≲ ⨏ Ω ∣ ∇ u ∣ 2 d x 1 / 2 + ⨏ Ω ∣ f ∣ p d x 1 / p ≲ ⨏ Ω ∣ f ∣ 2 d x 1 / 2 + ⨏ Ω ∣ f ∣ p d x 1 / p ≲ ⨏ Ω ∣ f ∣ p d x 1 / p ,

which implies that (1.9) holds true. Thus, (i) holds true.

Next, we prove (ii). Let q ∈ [ 2 , p ] , q 0 ∈ 1 , q p ′ , r 0 ∈ p q ′ , ∞ , and ω ∈ A q 0 ( R n ) ∩ RH r 0 ( R n ) . Assume that ε 0 ∈ ( 0 , ∞ ) is as in (3.9). Then, ω ∈ A q ( p + ε 0 ) ′ ( R n ) ∩ RH s ( R n ) with s ∈ p + ε 0 q ′ , ∞ . Let u be the weak solution of the Dirichlet problem ( D ) q , ω with f ∈ L ω q ( Ω ; R n ) . Then, from Lemma 3.3(vii), it follows that

(3.11) L ω q ( Ω ) ⊂ L q / q 0 ( Ω ) .

By the fact that q 0 ≤ q / p ′ < q / ( p + ε 0 ) ′ , we find that ( p + ε 0 ) ′ < q / q 0 , which, together with the Hölder inequality and the assumption that Ω is bounded, implies that L q / q 0 ( Ω ) ⊂ L ( p + ε 0 ) ′ ( Ω ) . From this and (3.11), we deduce that f ∈ L q / q 0 ( Ω ; R n ) ⊂ L ( p + ε 0 ) ′ ( Ω ; R n ) .

Let B ≔ B ( x B , r B ) ⊂ R n be a ball satisfying ∣ B ∣ ≤ β 1 ∣ Ω ∣ and either 2 B ⊂ Ω or x B ∈ ∂ Ω , where β 1 ∈ ( 0 , 1 ) is as shown in Theorem 3.1. Take ϕ ∈ C c ∞ ( R n ) such that ϕ ≡ 1 on 2 B , 0 ≤ ϕ ≤ 1 , and supp ( ϕ ) ⊂ 4 B . Let w and v be, respectively, as in (3.6) and (3.7). Then, u = w + v and ∇ u = ∇ w + ∇ v . Recall that F ≔ ∣ ∇ u ∣ , f ≔ ∣ f ∣ , F B ≔ ∣ ∇ w ∣ , and R B ≔ ∣ ∇ v ∣ . By the proof of (i), we know that (1.9) holds true with p replaced by p + ε 0 , which, combined with Lemma 3.2 and (3.6), further implies that

‖ ∇ w ‖ L ( p + ε 0 ) ′ ( Ω ; R n ) ≲ ∥ ϕ f ∥ L ( p + ε 0 ) ′ ( Ω ; R n ) ≲ ∥ f ∥ L ( p + ε 0 ) ′ ( 4 B Ω ; R n ) .

From this, it follows that

(3.12) ⨏ 2 B Ω F B ( p + ε 0 ) ′ d x ≲ ⨏ 4 B Ω f ( p + ε 0 ) ′ d x .

Furthermore, by assumption (1.8) of this theorem and (3.7), we conclude that (1.8) holds true for the above v . Thus, (3.10) holds true for the above v , which, together with (3.12), further implies that

(3.13) ⨏ B Ω R B p + ε 0 d x 1 / ( p + ε 0 ) = ⨏ B Ω ∣ ∇ v ∣ p + ε 0 d x 1 / ( p + ε 0 ) ≲ ⨏ 2 B Ω ∣ ∇ v ∣ ( p + ε 0 ) ′ d x 1 / ( p + ε 0 ) ′ ≲ ⨏ 2 B Ω ∣ ∇ u ∣ ( p + ε 0 ) ′ d x 1 / ( p + ε 0 ) ′ + ⨏ 4 B Ω ∣ f ∣ ( p + ε 0 ) ′ d x 1 / ( p + ε 0 ) ′ ≲ ⨏ 4 B Ω F ( p + ε 0 ) ′ d x 1 / ( p + ε 0 ) ′ + ⨏ 4 B Ω f ( p + ε 0 ) ′ d x 1 / ( p + ε 0 ) ′ .

From (3.12) and (3.13), we deduce that (3.1) and (3.2) hold true with p 2 ≔ p + ε 0 and p 1 ≔ ( p + ε 0 ) ′ , which, combined with q < p + ε 0 , Theorem 3.1, and (3.11), further implies that

1 ω ( Ω ) ∫ Ω ∣ ∇ u ∣ q ω d x 1 / q = 1 ω ( Ω ) ∫ Ω F q ω d x 1 / q ≲ 1 ∣ Ω ∣ ∫ Ω ∣ F ∣ ( p + ε 0 ) ′ d x 1 / ( p + ε 0 ) ′ + 1 ω ( Ω ) ∫ Ω ∣ f ∣ q ω d x 1 / q ∼ 1 ∣ Ω ∣ ∫ Ω ∣ ∇ u ∣ ( p + ε 0 ) ′ d x 1 / ( p + ε 0 ) ′ + 1 ω ( Ω ) ∫ Ω ∣ f ∣ q ω d x 1 / q ≲ 1 ∣ Ω ∣ ∫ Ω ∣ f ∣ ( p + ε 0 ) ′ d x 1 / ( p + ε 0 ) ′ + 1 ω ( Ω ) ∫ Ω ∣ f ∣ q ω d x 1 / q ≲ 1 ω ( Ω ) ∫ Ω ∣ f ∣ q ω d x 1 / q .

Therefore, (1.10) holds true, which shows (ii). This finishes the proof of Theorem 1.7.□

4 Proof of Theorem 1.8

In this section, we prove Theorem 1.8 by using Theorem 1.7 and the method of perturbation. We begin with the following reverse Hölder inequality established in [43, Lemma 3.2 and Corollary 4.1].

Lemma 4.1

Let Ω ⊂ R n be a bounded NTA domain, B ( x 0 , r ) a ball such that r ∈ ( 0 , r 0 / 4 ) and either x 0 ∈ ∂ Ω or B ( x 0 , 2 r ) ⊂ Ω , where r 0 ∈ ( 0 , diam ( Ω ) ) is a constant. Assume that the matrix A ≔ a + b satisfies Assumption 1.4 and u ∈ W 1 , 2 ( B Ω ( x 0 , 2 r ) ) is the weak solution of the following Dirichlet problem

− div ( A ∇ u ) = 0 in B Ω ( x 0 , 2 r ) , u = 0 on B ( x 0 , 2 r ) ∩ ∂ Ω .

Then, there exists a positive constant p ∈ ( 2 , ∞ ) , depending on Ω , n , and μ 0 in (1.3), such that

⨏ B Ω ( x 0 , r ) ∣ ∇ u ∣ p d x 1 / p ≤ C [ 1 + ‖ b ‖ BMO ( Ω ) ] ⨏ B Ω ( x 0 , 2 r ) ∣ ∇ u ∣ 2 d x 1 / 2 ,

where C is a positive constant depending only on n , Ω , and p .

To show Theorem 1.8, we need the following perturbation argument, which is motivated by [19].

Lemma 4.2

Let n ≥ 2 , Ω ⊂ R n be a bounded NTA domain, and r 0 ∈ ( 0 , diam ( Ω ) ) be a constant. Assume that the matrix A satisfies Assumption 1.4. Let u ∈ W 1 , 2 ( B Ω ( x 0 , 4 r ) ) be a solution of equation div ( A ∇ u ) = 0 in B Ω ( x 0 , 4 r ) with u = 0 on B ( x 0 , 4 r ) ∩ ∂ Ω , where x 0 ∈ Ω ¯ and r ∈ ( 0 , r 0 / 4 ) . Then, there exists a function θ ≔ θ ( r ) , a p ∈ ( 2 , ∞ ) , and a function v ∈ W 1 , p ( B Ω ( x 0 , r ) ) such that

(4.1) ⨏ B Ω ( x 0 , r ) ∣ ∇ v ∣ p d x 1 / p ≤ C ⨏ B Ω ( x 0 , 4 r ) ∣ ∇ u ∣ 2 d x 1 / 2

and

(4.2) ⨏ B Ω ( x 0 , r ) ∣ ∇ ( u − v ) ∣ 2 d x 1 / 2 ≤ θ ( r ) ⨏ B Ω ( x 0 , 4 r ) ∣ ∇ u ∣ 2 d x 1 / 2 ,

where C is a positive constant independent of u , v , x 0 , and r .

Proof

By the assumption that A satisfies Assumption 1.4, we find that A = a + b , where a ≔ { a i j } i , j = 1 n is real valued, symmetric, and measurable, and a satisfies (1.3), and b ≔ { b i j } i , j = 1 n is real valued, antisymmetric, and measurable, and b satisfies b i j ∈ BMO ( Ω ) for any i , j ∈ { 1 , … , n } .

Let a 0 ≔ { c i j } i , j = 1 n , where, for any i , j ∈ { 1 , … , n } ,

c i j ≔ ⨏ B Ω ( x 0 , 2 r ) a i j d x .

Assume that v ∈ W 1 , 2 ( B Ω ( x 0 , 2 r ) ) is the solution of the following boundary value problem

(4.3) div ( a 0 ∇ v ) = 0 in B Ω ( x 0 , 2 r ) , v = u on ∂ B Ω ( x 0 , 2 r ) .

Then, from (4.3) and the fact that u − v ∈ W 0 1 , 2 ( B Ω ( x 0 , 2 r ) ) , we deduce that

(4.4) ∫ B Ω ( x 0 , 2 r ) a 0 ∇ ( u − v ) ⋅ ∇ ( u − v ) d x = ∫ B Ω ( x 0 , 2 r ) ( a 0 − A ) ∇ u ⋅ ∇ ( u − v ) d x .

Denote by u ˜ the W 1 , 2 extension of u to R n ; namely, u ˜ ∈ W 1 , 2 ( R n ) and u ˜ ∣ B Ω ( x 0 , 2 r ) = u . Furthermore, denote by u − v ˜ the zero extension of ( u − v ) ∣ B Ω ( x 0 , 2 r ) to R n . Obviously, u − v ˜ ∈ W 1 , 2 ( R n ) and supp ( u − v ˜ ) ⊂ B Ω ( x 0 , 2 r ) , which, combined with the divergence theorem and the assumption that

b B Ω ( x 0 , 2 r ) ≔ ⨏ B Ω ( x 0 , 2 r ) b i j d x i , j = 1 n

is antisymmetric, further implies that

∫ B Ω ( x 0 , 2 r ) b B Ω ( x 0 , 2 r ) ∇ u ⋅ ∇ ( u − v ) d x = ∫ B ( x 0 , 2 r ) b B Ω ( x 0 , 2 r ) ∇ u ˜ ⋅ ∇ ( u − v ˜ ) d x = ∫ B ( x 0 , 2 r ) div ( b B Ω ( x 0 , 2 r ) ∇ u ˜ ) ( u − v ˜ ) d x − ∫ ∂ B ( x 0 , 2 r ) b B Ω ( x 0 , 2 r ) ∇ u ˜ ⋅ ν ( u − v ˜ ) d σ ( x ) = 0 ,

where ν denotes the outward unit normal to ∂ B ( x 0 , 2 r ) . By this, (4.4), and the definitions of a 0 and b B Ω ( x 0 , 2 r ) , we find that

(4.5) ∫ B Ω ( x 0 , 2 r ) a 0 ∇ ( u − v ) ⋅ ∇ ( u − v ) d x = ∫ B Ω ( x 0 , 2 r ) ( a 0 − A + b B Ω ( x 0 , 2 r ) ) ∇ u ⋅ ∇ ( u − v ) d x = ∫ B Ω ( x 0 , 2 r ) ( A B Ω ( x 0 , 2 r ) − A ) ∇ u ⋅ ∇ ( u − v ) d x ,

where

A B Ω ( x 0 , 2 r ) ≔ ⨏ B Ω ( x 0 , 2 r ) ( a i j + b i j ) d x i , j = 1 n .

Therefore, from (4.5), (1.3), and the Hölder inequality, it follows that, for any given ε ∈ ( 0 , ∞ ) , there exists a positive constant C ( ε ) , depending on ε , such that

(4.6) μ 0 ∫ B Ω ( x 0 , 2 r ) ∣ ∇ ( u − v ) ∣ 2 d x ≤ ∫ B Ω ( x 0 , 2 r ) ∣ A − A B Ω ( x 0 , 2 r ) ∣ ∣ ∇ u ∣ ∣ ∇ ( u − v ) ∣ d x ≤ ε ∫ B Ω ( x 0 , 2 r ) ∣ ∇ ( u − v ) ∣ 2 d x + C ( ε ) ∫ B Ω ( x 0 , 2 r ) ∣ A − A B Ω ( x 0 , 2 r ) ∣ 2 ∣ ∇ u ∣ 2 d x ,

where μ 0 ∈ ( 0 , 1 ) is as in (1.3). Take ε ≔ μ 0 / 2 in (4.6). Then, by (4.6), we conclude that

(4.7) ∫ B Ω ( x 0 , 2 r ) ∣ ∇ ( u − v ) ∣ 2 d x 1 / 2 ≤ C 4 ∫ B Ω ( x 0 , 2 r ) ∣ A − A B Ω ( x 0 , 2 r ) ∣ 2 ∣ ∇ u ∣ 2 d x 1 / 2 .

Moreover, from Lemma 4.1, we deduce that there exist positive constants p ˜ ∈ ( 1 , ∞ ) and C 5 ∈ ( 0 , ∞ ) , independent of u , x 0 , and r , such that

(4.8) ⨏ B Ω ( x 0 , 2 r ) ∣ ∇ u ∣ 2 p ˜ d x 1 / ( 2 p ˜ ) ≤ C 5 [ 1 + ‖ b ‖ BMO ( Ω ) ] ⨏ B Ω ( x 0 , 4 r ) ∣ ∇ u ∣ 2 d x 1 / 2 .

Furthermore, by [39, Theorem 1], we find that there exist an A ˜ ∈ BMO ( R n ; R n 2 ) and a positive constant C , independent of both x 0 and r , such that

(4.9) A ˜ ∣ B Ω ( x 0 , 2 r ) = A and ‖ A ˜ ‖ BMO ( R n ; R n 2 ) ≤ C ‖ A ‖ BMO ( B Ω ( x 0 , 2 r ) ; R n 2 ) .

From the Hölder inequality and the well-known John–Nirenberg inequality on BMO ( R n ) (see, for instance, [34,59]), it follows that

⨏ B Ω ( x 0 , 2 r ) ∣ A − A B Ω ( x 0 , 2 r ) ∣ 2 p ˜ ′ d x 1 / ( 2 p ˜ ′ ) ≲ ⨏ B ( x 0 , 2 r ) ∣ A ˜ − A ˜ B ( x 0 , 2 r ) ∣ 2 p ˜ ′ d x 1 / ( 2 p ˜ ′ ) + ⨏ B ( x 0 , 2 r ) ∣ A ˜ B ( x 0 , 2 r ) − A B Ω ( x 0 , 2 r ) ∣ 2 p ˜ ′ d x 1 / ( 2 p ˜ ′ ) ≲ ‖ A ˜ ‖ BMO ( R n ; R n 2 ) + ∣ A ˜ B ( x 0 , 2 r ) − A B Ω ( x 0 , 2 r ) ∣ ≲ ‖ A ˜ ‖ BMO ( R n ; R n 2 ) + ⨏ B Ω ( x 0 , 2 r ) ∣ A − A ˜ B ( x 0 , 2 r ) ∣ d x ≲ ‖ A ˜ ‖ BMO ( R n ; R n 2 ) + ⨏ B ( x 0 , 2 r ) ∣ A ˜ − A ˜ B ( x 0 , 2 r ) ∣ d x ≲ ‖ A ˜ ‖ BMO ( R n ; R n 2 ) ,

where p ˜ ′ ∈ ( 1 , ∞ ) is given by 1 / p ˜ + 1 / p ˜ ′ = 1 , which, together with (4.9), further implies that there exists a positive constant C 6 , independent of x 0 , r , and A , such that

⨏ B Ω ( x 0 , 2 r ) ∣ A − A B Ω ( x 0 , 2 r ) ∣ 2 p ˜ ′ d x 1 / ( 2 p ˜ ′ ) ≤ C 6 ‖ A ‖ BMO ( B Ω ( x 0 , 2 r ) ; R n 2 ) .

By this, (4.7) and (4.8), we conclude that

(4.10) ⨏ B Ω ( x 0 , 2 r ) ∣ ∇ ( u − v ) ∣ 2 d x 1 / 2 ≤ C 4 ⨏ B Ω ( x 0 , 2 r ) ∣ A − A B Ω ( x 0 , 2 r ) ∣ 2 p ˜ ′ d x 1 / ( 2 p ˜ ′ ) ⨏ B Ω ( x 0 , 2 r ) ∣ ∇ u ∣ 2 p ˜ d x 1 / ( 2 p ˜ ) ≤ C 4 C 5 C 6 [ 1 + ‖ b ‖ BMO ( Ω ) ] ‖ A ‖ BMO ( B Ω ( x 0 , 2 r ) ; R n 2 ) ⨏ B Ω ( x 0 , 4 r ) ∣ ∇ u ∣ 2 d x 1 / 2 ≤ θ ( r ) ⨏ B Ω ( x 0 , 4 r ) ∣ ∇ u ∣ 2 d x 1 / 2 ,

where

(4.11) θ ( r ) ≔ C 4 C 5 C 6 [ 1 + ‖ b ‖ BMO ( Ω ) ] sup B ( x , 2 t ) ⊂ Ω , t ∈ ( 0 , r ] ⨏ B ( x , 2 t ) A − ⨏ B ( x , 2 t ) A d z d y .

Thus, (4.2) holds true. Furthermore, from the known regularity theory of the second-order elliptic equations (see, for instance, [40, Chapter 1] and [31, Chapter V]), we deduce that there exists a p ∈ ( 2 , ∞ ) , such that

⨏ B Ω ( x 0 , 2 r ) ∣ ∇ v ∣ p d x 1 / p ≲ ⨏ B Ω ( x 0 , 4 r ) ∣ ∇ v ∣ 2 d x 1 / 2 ,

which, combined with (4.10), (4.11), and the fact that A ∈ BMO ( Ω ; R n 2 ) , further implies that (4.1) holds true. This finishes the proof of Lemma 4.2.□

Furthermore, to prove Theorem 1.8, we need the following conclusion for the constant coefficient boundary value problems, which was established in [58, Lemma 4.1] (see also [29, Lemma 4.1]).

Lemma 4.3

Let n ≥ 2 , Ω ⊂ R n be a bounded Lipschitz domain, and a 0 ≔ { a i j } i , j = 1 n a symmetric constant coefficient matrix that satisfies (1.3). Assume that v ∈ W 1 , 2 ( B Ω ( x 0 , 2 r ) ) is a weak solution of the equation div ( a 0 ∇ v ) = 0 in B Ω ( x 0 , 2 r ) with v = 0 on B ( x 0 , 2 r ) ∩ ∂ Ω , where B ( x 0 , r ) is a ball such that r ∈ ( 0 , r 0 / 4 ) and either x 0 ∈ ∂ Ω or B ( x 0 , 2 r ) ⊂ Ω , and r 0 ∈ ( 0 , diam ( Ω ) ) is a constant. Then, the weak reverse Hölder inequality

⨏ B Ω ( x 0 , r ) ∣ ∇ v ∣ p d x 1 p ≤ C ⨏ B Ω ( x 0 , 2 r ) ∣ ∇ v ∣ 2 d x 1 2

holds true for p ≔ 3 + ε when n ≥ 3 , or p ≔ 4 + ε when n = 2 , where C and ε are positive constants depending only on n , the Lipschitz constant of Ω , and μ 0 in (1.3).

Now, we prove Theorem 1.8 via using Theorem 3.1 and Lemmas 4.2 and 4.3.

Proof of Theorem 1.8

We first show (i). Let A satisfy Assumption 1.4 and u be the weak solution of the following Dirichlet problem:

(4.12) − div ( A ∇ u ) = div ( f ) in Ω , u = 0 on ∂ Ω .

Based on Lemma 3.2, to prove (1.11), it suffices to show that there exists a positive constant ε 0 ∈ ( 0 , ∞ ) , depending only on n and the Lipschitz constant of Ω , such that, for any given p ∈ [ 2 , 3 + ε 0 ) when n ≥ 3 , or p ∈ [ 2 , 4 + ε 0 ) when n = 2 , there exists a δ 0 ∈ ( 0 , ∞ ) , depending only on n , p , and the Lipschitz constant of Ω , such that, if A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then (1.11) holds true.

By the definition of VMO ( Ω ) , we conclude that if A ∈ VMO ( Ω ) , then there exists an r 1 ∈ ( 0 , ∞ ) such that, for any r ∈ ( 0 , r 1 ) , θ ( r ) < ε 0 / 2 , where θ ( r ) and ε 0 are, respectively, as in (4.11) and Theorem 3.1. Let B ( x 0 , r ) ⊂ R n be such that r ∈ ( 0 , min { r 0 , r 1 } / 4 ) and either x 0 ∈ ∂ Ω or B ( x 0 , 2 r ) ⊂ Ω , where r 0 ∈ ( 0 , diam ( Ω ) ) is as in Lemma 4.3. Assume that v ∈ W 1 , 2 ( B Ω ( x 0 , 2 r ) ) is a weak solution of the equation div ( A ∇ v ) = 0 in B Ω ( x 0 , 2 r ) with v = 0 on B ( x 0 , 2 r ) ∩ ∂ Ω . Let w ∈ W 1 , 2 ( B Ω ( x 0 , 2 r ) ) be a weak solution of the equation div ( a 0 ∇ w ) = 0 in B Ω ( x 0 , 2 r ) with w = v on ∂ B Ω ( x 0 , 2 r ) , where a 0 ≔ { c i j } i , j = 1 n with c i j ≔ ⨏ B Ω ( x 0 , 2 r ) a i j d x for any i , j ∈ { 1 , … , n } . Then, from Lemmas 4.2 and 4.3, it follows that

(4.13) ⨏ B Ω ( x 0 , r ) ∣ ∇ w ∣ p d x 1 p ≲ ⨏ B Ω ( x 0 , 2 r ) ∣ ∇ v ∣ 2 d x 1 2

and

(4.14) ⨏ B Ω ( x 0 , r ) ∣ ∇ ( v − w ) ∣ 2 d x 1 2 ≤ θ ( r ) ⨏ B Ω ( x 0 , 2 r ) ∣ ∇ v ∣ 2 d x 1 2 ,

where θ ( r ) is as in (4.11). By the well-known John–Nirenberg inequality on BMO ( Ω ) (see, for instance, [34,59]), we conclude that there exists a δ 0 ∈ ( 0 , ∞ ) sufficiently small such that, if A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) and R ∈ ( 0 , ∞ ) , then, for any r ∈ ( 0 , r 0 ) , θ ( r ) < ε 0 / 2 , where ε 0 is as in Theorem 3.1. From this, (4.13), (4.14), r ∈ ( 0 , min { r 0 , r 1 } / 4 ) , and Theorem 3.1 with ω ≡ 1 , we deduce that, if A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then ∣ ∇ v ∣ ∈ L p ( B Ω ( x 0 , 2 r ) ) and

(4.15) ⨏ B Ω ( x 0 , r ) ∣ ∇ v ∣ p d x 1 p ≲ ⨏ B Ω ( x 0 , 2 r ) ∣ ∇ v ∣ 2 d x 1 2 ,

which, together with Theorem 1.7 and the fact that the Dirichlet problem ( D ) 2 is uniquely solvable, further implies that (1.11) holds true in the case p ∈ [ 2 , 3 + ε 0 ) when n ≥ 3 , or p ∈ [ 2 , 4 + ε 0 ) when n = 2 . This finishes the proof of (i).

Now, we prove (ii). Let ε 0 be as in (i) and u the weak solution of the Dirichlet problem (4.12) with f ∈ L ω p ( Ω ; R n ) . We first assume that p ∈ [ 2 , p 0 ) and ω ∈ A p p 0 ′ ( R n ) ∩ RH p 0 p ′ ( R n ) , where p 0 ≔ 3 + ε 0 when n ≥ 3 , or p 0 ≔ 4 + ε 0 when n = 2 . Then, by (4.15) and Theorem 1.7, we conclude that there exists a δ 0 ∈ ( 0 , ∞ ) , depending only on n , p , [ ω ] A p p 0 ′ ( R n ) , [ ω ] RH p 0 p ′ ( R n ) , and the Lipschitz constant of Ω , such that, if A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then (1.12) holds true.

Next, we assume that p ∈ ( p 0 ′ , 2 ) and ω ∈ A p p 0 ′ ( R n ) ∩ RH p 0 p ′ ( R n ) . Then, from Lemma 3.3(v), we deduce that p ′ ∈ ( 2 , p 0 ) and ω 1 − p ′ ∈ A p ′ p 0 ′ ( R n ) ∩ RH ( p 0 p ′ ) ′ ( R n ) . Let g ∈ L ω 1 − p ′ p ′ ( Ω ; R n ) and v be the weak solution of the following Dirichlet problem:

− div ( A ∗ ∇ v ) = div ( g ) in Ω , v = 0 on ∂ Ω ,

where A ∗ denotes the transpose of A . By the assumption that A satisfies Assumption 1.4, we find that A ∗ also satisfies Assumption 1.4. Thus, we have

(4.16) ‖ ∇ v ‖ L ω 1 − p ′ p ′ ( Ω ; R n ) ≲ ‖ g ‖ L ω 1 − p ′ p ′ ( Ω ; R n ) .

Moreover,

∫ Ω g ⋅ ∇ u d x = − ∫ Ω A ∗ ∇ v ⋅ ∇ u d x = − ∫ Ω A ∇ u ⋅ ∇ v d x = ∫ Ω f ⋅ ∇ v d x ,

which, combined with (4.16) and the Hölder inequality, further implies that

(4.17) ‖ ∇ u ‖ L ω p ( Ω ; R n ) = sup ‖ g ‖ L ω 1 − p ′ p ′ ( Ω ; R n ) ≤ 1 ∫ Ω g ⋅ ∇ u d x = sup ‖ g ‖ L ω 1 − p ′ p ′ ( Ω ; R n ) ≤ 1 ∫ Ω f ⋅ ∇ v d x ≤ sup ‖ g ‖ L ω 1 − p ′ p ′ ( Ω ; R n ) ≤ 1 ‖ f ‖ L ω p ( Ω ; R n ) ‖ ∇ v ‖ L ω 1 − p ′ p ′ ( Ω ; R n ) ≲ sup ‖ g ‖ L ω 1 − p ′ p ′ ( Ω ; R n ) ≤ 1 ‖ f ‖ L ω p ( Ω ; R n ) ‖ g ‖ L ω 1 − p ′ p ′ ( Ω ; R n ) ≲ ‖ f ‖ L ω p ( Ω ; R n ) .

From this, it follows that (1.12) holds true when p ∈ ( p 0 ′ , 2 ) . This finishes the proof of (ii) and hence of Theorem 1.8.□

5 Proof of Theorem 1.10

In this section, we show both Theorem 1.10 and Corollary 1.11 by using Theorem 1.7 and some properties of quasi-convex domains. We begin with Lemma 5.1.

Lemma 5.1

Let n ≥ 2 and Ω ⊂ R n be a bounded NTA domain. Assume that p ∈ ( 2 , ∞ ) , Ω is a ( δ , σ , R ) quasi-convex domain for some δ , σ ∈ ( 0 , 1 ) and R ∈ ( 0 , ∞ ) , a 0 ≔ { a i j } i , j = 1 n a symmetric constant coefficient matrix that satisfies (1.3). Let v ∈ W 1 , 2 ( B Ω ( x 0 , 2 r ) ) be a weak solution of the equation div ( a 0 ∇ v ) = 0 in B Ω ( x 0 , 2 r ) with v = 0 on B ( x 0 , 2 r ) ∩ ∂ Ω , where B ( x 0 , r ) is a ball such that r ∈ ( 0 , r 0 / 4 ) and either x 0 ∈ ∂ Ω or B ( x 0 , 2 r ) ⊂ Ω , and r 0 ∈ ( 0 , R ) is a constant. Then, there exists a positive constant δ 0 ∈ ( 0 , 1 ) , depending only on n , p , and Ω , such that, if Ω is a ( δ , σ , R ) quasi-convex domain for some δ ∈ ( 0 , δ 0 ) , σ ∈ ( 0 , 1 ) , and R ∈ ( 0 , ∞ ) , then the weak reverse Hölder inequality

(5.1) ⨏ B Ω ( x 0 , r ) ∣ ∇ v ∣ p d x 1 p ≤ C ⨏ B Ω ( x 0 , 2 r ) ∣ ∇ v ∣ 2 d x 1 2

holds true, where C is a positive constant depending only on n , δ , σ , R , and diam ( Ω ) .

To show Lemma 5.1, we need Lemma 5.2, which is a special case of Theorem 3.1.

Lemma 5.2

Let n ≥ 2 , Ω ⊂ R n be a bounded NTA domain, p 1 , p 2 ∈ [ 1 , ∞ ) with p 2 > p 1 , F ∈ L p 1 ( Ω ) , and q ∈ ( p 1 , p 2 ) . Suppose that, for any ball B ≔ B ( x B , r B ) ⊂ R n having the property that ∣ B ∣ ≤ β 1 ∣ Ω ∣ and either 2 B ⊂ Ω or x B ∈ ∂ Ω , there exist two measurable functions F B and R B on 2 B , such that ∣ F ∣ ≤ ∣ F B ∣ + ∣ R B ∣ on 2 B ∩ Ω ,

⨏ 2 B Ω ∣ R B ∣ p 2 d x 1 p 2 ≤ C 7 ⨏ β 2 B Ω ∣ F ∣ p 1 d x 1 p 1

and

⨏ 2 B Ω ∣ F B ∣ p 1 d x 1 p 1 ≤ ε ⨏ β 2 B Ω ∣ F ∣ p 1 d x 1 p 1 ,

where C 7 , ε , and β 1 < 1 < β 2 are positive constants independent of F , R B , F B , and B , respectively. Then, there exists a positive constant ε 0 , depending only on C 7 , n , p 1 , p 2 , q , β 1 , and β 2 , such that, if ε ∈ [ 0 , ε 0 ) , then

⨏ Ω ∣ F ∣ q d x 1 q ≤ C ⨏ Ω ∣ F ∣ p 1 d x 1 p 1 ,

where C is a positive constant depending only on C 7 , ε 0 , n , p 1 , p 2 , q , β 1 , and β 2 .

Moreover, we also need Lemma 5.3, which was established in [73, Lemmas 3.4 and 3.5].

Lemma 5.3

Let n ≥ 2 and Ω ⊂ R n be a bounded ( δ , σ , R ) quasi-convex domain for some δ , σ ∈ ( 0 , 1 ) and R ∈ ( 0 , ∞ ) . Assume that x 0 ∈ ∂ Ω , r ∈ ( 0 , R / 4 ) , and V 4 r is the convex hull of B Ω ( x 0 , 4 r ) . For any t ∈ ( 0 , 1 ) , let

(5.2) Ω t r ≔ { x ∈ Ω : dist ( x , ∂ Ω ) < t r }

and

(5.3) W r , t ≔ { x ∈ V 4 r : dist ( x , ∂ V 4 r ∩ B ( x 0 , 3 r ) ) ≤ ( t + δ ) r } .

Then, for any t ∈ ( 0 , 1 ) , Ω t r ∩ B ( x 0 , r ) ⊂ W r , t . Moreover, there exists a positive constant C , depending only on n and σ , such that ∣ W r , t ∣ ≤ C ( t + δ ) r n .
Let u ∈ W 1 , 2 ( B ( x 0 , 4 r ) ) satisfy u = 0 on B ( x 0 , 4 r ) \ V 4 r . Then, for any t ∈ ( 0 , 1 − δ ) , there exists a positive constant C , depending only on n and σ , such that
∫ B ( x 0 , r ) ∩ Ω t r u 2 d x ≤ C ( t + δ ) 2 r 2 ∫ W r , t ∣ ∇ u ∣ 2 d x .

Now, we prove Lemma 5.1 via using Lemmas 4.1, 5.2, and 5.3.

Proof of Lemma 5.1

We borrow some ideas from [73]. Observing that the matrix a 0 is symmetric and elliptic and has constant coefficients, without loss of generality, by a change of the coordinate system, we may assume that a 0 = I (the unit matrix), namely, Δ v = 0 in B ( x 0 , 2 r ) ∩ Ω , and v = 0 on B ( x 0 , 2 r ) ∩ ∂ Ω . If B ( x 0 , 2 r ) ⊂ Ω , from the interior gradient estimates of harmonic functions (see, for instance, [32, Theorem 2.10]), it follows that (5.1) holds true for any given p ∈ ( 2 , ∞ ) .

Next, we assume that x 0 ∈ ∂ Ω . By the assumption that v = 0 on B ( x 0 , 2 r ) ∩ ∂ Ω , we know that v can be extended to a function v ˜ ∈ W 1 , 2 ( B ( x 0 , 2 r ) ) by zero-extension. Let s ∈ ( 0 , r / 16 ) . Since Ω is a bounded ( δ , σ , R ) quasi-convex domain, from Definition 2.3 and Remark 2.4(ii), it follows that the convex hull of B Ω ( x 0 , s ) , denoted by V s , satisfies that

V s ∩ Ω = B ( x 0 , s ) ∩ Ω and d H ( ∂ V s , ∂ ( B ( x 0 , s ) ∩ Ω ) ) ≤ δ s .

Let w ∈ W 1 , 2 ( V s ) be the weak solution of the following Dirichlet problem

− Δ w = 0 in V s , w = v ˜ on ∂ V s .

By the assumptions that v ˜ = 0 on ∂ V s ∩ B ( x 0 , s ) and that V s is convex, and the well-known result for the boundary gradient estimates of harmonic functions in convex domains (see, for instance, [7, Theorem 1.1]), we find that

(5.4) ‖ ∇ w ‖ L ∞ ( B Ω ( x 0 , s ) ) ≲ ⨏ B Ω ( x 0 , 2 s ) ∣ ∇ w ∣ 2 d x 1 / 2 .

We now claim that there exists a positive constant ε , such that

(5.5) ⨏ B Ω ( x 0 , s ) ∣ ∇ ( v − w ) ∣ 2 d x 1 / 2 ≤ C δ ε ⨏ B Ω ( x 0 , 2 s ) ∣ ∇ v ∣ 2 d x 1 / 2 ,

where C is a positive constant independent of x 0 , s , δ , and v . If (5.5) holds true, then, from (5.5), (5.4), and Lemma 5.2, we deduce that there exists a δ 0 ∈ ( 0 , 1 ) such that, if Ω is a ( δ , σ , R ) quasi-convex domain for some δ ∈ ( 0 , δ 0 ) , σ ∈ ( 0 , 1 ) , and R ∈ ( 0 , ∞ ) , then (5.1) holds true.

Next, we prove (5.5). For any given t ∈ ( 0 , 1 ) , let Ω t s be as in (5.2). Meanwhile, take θ δ s ∈ C c ∞ ( R n ) satisfying that 0 ≤ θ δ s ≤ 1 , θ δ s ≡ 1 on Ω \ Ω 2 δ s , θ δ s = 0 in Ω δ s , and ∣ ∇ θ δ s ∣ ≲ ( δ s ) − 1 . Then,

(5.6) ∫ V s ∣ ∇ ( w − v ˜ ) ∣ 2 d x = ∫ V s ∇ w ⋅ ∇ ( w − v ˜ ) d x − ∫ V s ∇ v ˜ ⋅ ∇ ( w − v ˜ ) d x = − ∫ V s ∇ v ˜ ⋅ ∇ ( w − v ˜ ) d x = − ∫ B Ω ( x 0 , s ) ∇ v ⋅ ∇ ( w − v ) d x = − ∫ B Ω ( x 0 , s ) ∇ ( θ δ s v ) ⋅ ∇ ( w − v ) d x − ∫ B Ω ( x 0 , s ) ∇ ( ( 1 − θ δ s ) v ) ⋅ ∇ ( w − v ) d x .

By the assumptions that Δ v = 0 in B Ω ( x 0 , s ) , and that v = 0 on B ( x 0 , 2 s ) ∩ ∂ Ω , and Lemma 5.3, we conclude that

∫ B Ω ( x 0 , s ) ∩ Ω 2 δ s ∣ v ∣ 2 d x 1 / 2 ≲ δ s ∫ W s , 2 δ ∣ ∇ v ∣ 2 d x 1 / 2 ≲ δ s ( δ s n ) ε ∫ B Ω ( x 0 , 2 s ) ∣ ∇ v ∣ p d x 1 / p ,

where W s , 2 δ is as in (5.3), p ∈ ( 2 , ∞ ) as in Lemma 4.1, and ε ≔ 1 2 − 1 p , which, combined with ∣ ∇ θ δ s ∣ ≲ ( δ s ) − 1 and the Hölder inequality, further implies that

(5.7) ∫ B Ω ( x 0 , s ) ∇ ( θ δ s v ) ⋅ ∇ ( w − v ) d x ≤ ∫ B Ω ( x 0 , s ) v ∇ θ δ s ⋅ ∇ ( w − v ) d x + ∫ B Ω ( x 0 , s ) θ δ s ∇ v ⋅ ∇ ( w − v ) d x ≲ [ ( δ s ) − 1 ‖ v ‖ L 2 ( B Ω ( x 0 , s ) ∩ Ω 2 δ s ) + ‖ ∇ v ‖ L 2 ( B Ω ( x 0 , s ) ∩ Ω 2 δ s ; R n ) ] ∥ ∇ ( w − v ) ∥ L 2 ( B Ω ( x 0 , s ) ; R n ) ≲ ( δ s n ) ε ‖ ∇ v ‖ L p ( B Ω ( x 0 , 2 s ) ; R n ) ‖ ∇ ( w − v ) ‖ L 2 ( B Ω ( x 0 , s ) ; R n ) .

Furthermore, similar to (5.7), we have

(5.8) ∫ B Ω ( x 0 , s ) ∇ ( ( 1 − θ δ s ) v ) ⋅ ∇ ( w − v ) d x ≲ ( δ s n ) ε ‖ ∇ v ‖ L p ( B Ω ( x 0 , 2 s ) ; R n ) ‖ ∇ ( w − v ) ‖ L 2 ( B Ω ( x 0 , s ) ; R n ) .

Thus, from (5.6), (5.7), and (5.8), it follows that

‖ ∇ ( w − v ˜ ) ‖ L 2 ( V s ; R n ) 2 ≲ ( δ s n ) ε ‖ ∇ v ‖ L p ( B Ω ( x 0 , 2 s ) ; R n ) ‖ ∇ ( w − v ) ‖ L 2 ( B Ω ( x 0 , s ) ; R n ) ,

which further implies that

(5.9) ‖ ∇ ( w − v ) ‖ L 2 ( B Ω ( x 0 , s ) ; R n ) ≲ ( δ s n ) ε ‖ ∇ v ‖ L p ( B Ω ( x 0 , 2 s ) ; R n ) .

Then, by (5.9), Lemma 4.1, and ε ≔ 1 2 − 1 p , we find that

⨏ B Ω ( x 0 , s ) ∣ ∇ ( v − w ) ∣ 2 d x 1 / 2 ≲ δ ε ⨏ B Ω ( x 0 , 2 s ) ∣ ∇ v ∣ p d x 1 / p ≲ δ ε ⨏ B Ω ( x 0 , 4 s ) ∣ ∇ v ∣ 2 d x 1 / 2 .

This finishes the proof of (5.5) and hence of Lemma 5.1.□

Now, we prove Theorem 1.10 via using Lemma 5.1 and Theorem 1.7.

Proof of Theorem 1.10

We first show (i). Via replacing Lemma 4.3 by Lemma 5.1 and repeating the proof of Theorem 1.8(i), we can prove (i). We omit the details here.

Next, we show (ii). Let p ∈ ( 1 , ∞ ) and ω ∈ A p ( R n ) . Assume that u is the weak solution of the Dirichlet problem (4.12) with f ∈ L ω p ( Ω ; R n ) .

We first assume that p ∈ [ 2 , ∞ ) . From ω ∈ A p ( R n ) and both (i) and (ii) of Lemma 3.3, it follows that there exists a sufficiently large p 0 ∈ ( p , ∞ ) , such that

(5.10) ω ∈ A p p ′ 0 ( R n ) ∩ RH p 0 p ′ ( R n ) .

Let v be as in (4.14). Using Lemmas 5.1 and 3.4, and repeating the proof of (4.15), we find that there exists a positive constant δ 0 ∈ ( 0 , 1 ) , depending only on n , p , and Ω , such that, if Ω is a ( δ , σ , R ) quasi-convex domain and A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) , σ ∈ ( 0 , 1 ) , and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , the inequality (4.15) holds true with p replaced by p 0 , namely,

(5.11) ⨏ B Ω ( x 0 , r ) ∣ ∇ v ∣ p 0 d x 1 p 0 ≲ ⨏ B Ω ( x 0 , 2 r ) ∣ ∇ v ∣ 2 d x 1 2 .

Then, by (5.10), (5.11), and Theorem 1.7(ii), we conclude that there exists a positive constant δ 0 ∈ ( 0 , 1 ) , depending only on n , p , and Ω , such that, if Ω is a ( δ , σ , R ) quasi-convex domain and A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) , σ ∈ ( 0 , 1 ) , and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then the weighted Dirichlet problem ( D ) p , ω is uniquely solvable and (1.13) holds true in this case.

Now, let p ∈ ( 1 , 2 ) . From this, ω ∈ A p ( R n ) , and Lemma 3.3(iii), we deduce that p ′ ∈ ( 2 , ∞ ) and ω 1 − p ′ ∈ A p ′ ( R n ) . Let g ∈ L ω 1 − p ′ p ′ ( Ω ; R n ) and w be the weak solution of the following Dirichlet problem

− div ( A ∗ ∇ w ) = div ( g ) in Ω , w = 0 on ∂ Ω ,

where A ∗ denotes the transpose of A . By the assumption that A satisfies Assumption 1.4, we find that A ∗ also satisfies Assumption 1.4. Thus, we know that there exists a positive constant δ 0 ∈ ( 0 , 1 ) , depending only on n , p , and Ω , such that, if Ω is a ( δ , σ , R ) quasi-convex domain and A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) , σ ∈ ( 0 , 1 ) , and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then

(5.12) ‖ ∇ w ‖ L ω 1 − p ′ p ′ ( Ω ; R n ) ≲ ‖ g ‖ L ω 1 − p ′ p ′ ( Ω ; R n ) .

Using (5.12) and repeating the proof of (4.17), we conclude that, when p ∈ ( 1 , 2 ) and ω ∈ A p ( R n ) , the weighted Dirichlet problem ( D ) p , ω is uniquely solvable and (1.13) holds true. This finishes the proof of (ii) and hence of Theorem 1.10.□

To prove Corollary 1.11 by using Theorem 1.10, we need Lemma 5.4.

Lemma 5.4

Let n ≥ 2 and Ω ⊂ R n be a bounded semi-convex domain. Then, there exists a δ 0 ∈ ( 0 , 1 ) such that Ω is a ( δ , σ , R ) quasi-convex domain for any δ ∈ ( 0 , δ 0 ) , some σ ∈ ( 0 , 1 ) , and some R ∈ ( 0 , ∞ ) .

Proof

Assume that x ∈ ∂ Ω and Ω has the UEBC constant R 0 ∈ ( 0 , ∞ ) . Then, there exists a v x ∈ S n − 1 , depending on x , such that B ( x + R 0 v x , R 0 ) ⊂ Ω ∁ , which further implies that, for any r ∈ ( 0 , R 0 ) ,

(5.13) B ( x + r v x , r ) ⊂ B ( x + R 0 v x , R 0 ) ⊂ Ω ∁ .

Denote by L x the ( n − 1 ) -dimensional plane such that L x contains x and has a unit normal direction v x . For any given δ ∈ ( 0 , 1 ) and any r ∈ ( 0 , 2 R 0 δ ) , let

H ( x , δ , r ) ≔ { y + t ( − v x ) : y ∈ L x , t ∈ [ − δ r , ∞ ) } .

Then, by (5.13) and a simple geometric observation, we conclude that, for any given δ ∈ ( 0 , 1 ) and any r ∈ ( 0 , 2 R 0 δ ) ,

(5.14) Ω ∩ B ( x , r ) ⊂ H ( x , δ , r ) ∩ B ( x , r ) .

Moreover, from Remark 2.5(i), it follows that Ω is a Lipschitz domain, which, together with the fact that Lipschitz domains are NTA domains, implies that there exist a σ ∈ ( 0 , 1 ) and an R 1 ∈ ( 0 , ∞ ) such that, for any x ∈ ∂ Ω and r ∈ ( 0 , R 1 ) , there exists an x 0 ∈ Ω , depending on x , such that B ( x 0 , σ r ) ⊂ Ω ∩ B ( x , r ) . By this, (5.14), and Remark 2.4(iii), we conclude that there exists a δ 0 ∈ ( 0 , 1 ) such that Ω is a ( δ , σ , R ) quasi-convex domain for any δ ∈ ( 0 , δ 0 ) , some σ ∈ ( 0 , 1 ) , and some R ∈ ( 0 , ∞ ) . This finishes the proof of Lemma 5.4.□

Next, we show Corollary 1.11 via using Theorem 1.10(ii) and Lemma 5.4.

Proof of Corollary 1.11

By Remark 2.5(iii), we find that, for any given bounded C 1 domain Ω , Ω is a ( δ , σ , R ) quasi-convex domain for any δ ∈ ( 0 , δ 0 ) , some σ ∈ ( 0 , 1 ) , and some R ∈ ( 0 , ∞ ) , where δ 0 ∈ ( 0 , 1 ) is a constant depending on Ω . Furthermore, from Lemma 5.4, it follows that, for any bounded semi-convex domain Ω , Ω is a ( δ , σ , R ) quasi-convex domain for any δ ∈ ( 0 , δ 0 ) , some σ ∈ ( 0 , 1 ) , and some R ∈ ( 0 , ∞ ) , where δ 0 ∈ ( 0 , 1 ) is a constant depending on Ω . Thus, for any given bounded C 1 domain or semi-convex domain Ω , Ω is a ( δ , σ , R ) quasi-convex domain for any δ ∈ ( 0 , δ 0 ) , some σ ∈ ( 0 , 1 ) , and some R ∈ ( 0 , ∞ ) , where δ 0 ∈ ( 0 , 1 ) is a constant depending on Ω . By this and Theorem 1.10(ii), we conclude that the conclusion of this corollary holds true, which completes the proof of Corollary 1.11.□

6 Several applications of Theorems 1.8 and 1.10

In this section, we give several applications of the weighted global estimates obtained in Theorems 1.8 and 1.10. More precisely, using both Theorems 1.8(ii) and 1.10(ii), we obtain the global gradient estimates, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (Musielak–)Orlicz spaces (also called generalized Orlicz spaces), and variable Lebesgue spaces. We begin with recalling the following concept of the weighted Lorentz space L ω q , r ( Ω ) on the domain Ω .

Definition 6.1

Let n ≥ 2 and Ω be a bounded NTA domain of R n . Assume that q ∈ [ 1 , ∞ ) , r ∈ ( 0 , ∞ ] , and ω ∈ A p ( R n ) with some p ∈ [ 1 , ∞ ) . The weighted Lorentz space L ω q , r ( Ω ) is defined by setting

L ω q , r ( Ω ) ≔ { f is measurable on Ω : ‖ f ‖ L ω q , r ( Ω ) < ∞ } ,

where, when r ∈ ( 0 , ∞ ) ,

‖ f ‖ L ω q , r ( Ω ) ≔ q ∫ 0 ∞ [ t q ω ( { x ∈ Ω : ∣ f ( x ) ∣ > t } ) ] r / q d t t 1 / r

and

‖ f ‖ L ω q , ∞ ( Ω ) ≔ sup t ∈ ( 0 , ∞ ) { t [ ω ( { x ∈ Ω : ∣ f ( x ) ∣ > t } ) ] 1 / q } .

Moreover, the space L ω q , r ( Ω ; R n ) is defined via replacing L ω p ( Ω ) in (1.1) by the aforementioned L ω q , r ( Ω ) in the definition of L ω p ( Ω ; R n ) in (1.2).

It is easy to see that, when q , r ∈ [ 1 , ∞ ) and q = r , L ω q , r ( Ω ) = L ω q ( Ω ) and L ω q , r ( Ω ; R n ) = L ω q ( Ω ; R n ) .

As applications of both Theorems 1.8(ii) and 1.10(ii) and the interpolation theorem of operators in the scale of (weighted) Lorentz spaces, we have the following global gradient estimates for the Dirichlet problem ( D ) p in (weighted) Lorentz spaces.

Theorem 6.2

Let n ≥ 2 , Ω ⊂ R n be a bounded Lipschitz domain, and the matrix A satisfy Assumption 1.4. Assume that r ∈ ( 0 , ∞ ] , ε 0 is as in Theorem 1.8(i), p 0 ≔ 3 + ε 0 when n ≥ 3 , or p 0 ≔ 4 + ε 0 when n = 2 . Then, for any given p ∈ ( p 0 ′ , p 0 ) and any ω ∈ A p p 0 ′ ( R n ) ∩ RH p 0 p ′ ( R n ) , there exists a positive constant δ 0 ∈ ( 0 , ∞ ) , depending only on n , p , the Lipschitz constant of Ω , [ ω ] A p p 0 ′ ( R n ) , and [ ω ] RH p 0 p ′ ( R n ) , such that, if A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then, for any weak solution u ∈ W 0 1 , 2 ( Ω ) of the problem ( D ) 2 with f ∈ L ω p , r ( Ω ; R n ) , ∇ u ∈ L ω p , r ( Ω ; R n ) and

(6.1) ‖ ∇ u ‖ L ω p , r ( Ω ; R n ) ≤ C ‖ f ‖ L ω p , r ( Ω ; R n ) ,

where C is a positive constant depending only on n , p , r , [ ω ] A p p 0 ′ ( R n ) , [ ω ] RH p 0 p ′ ( R n ) , and the Lipschitz constant of Ω .

Proof

Let p 0 be as in Theorem 6.2, p ∈ ( p 0 ′ , p 0 ) , r ∈ ( 0 , ∞ ] , and ω ∈ A p p 0 ′ ( R n ) ∩ RH p 0 p ′ ( R n ) . By both (i) and (iv) of Lemma 3.2, we find that there exists an ε ∈ ( 0 , min { p − p 0 ′ , p 0 − p } ) such that ω ∈ A p + ε p 0 ′ ( R n ) ∩ RH ( p 0 p + ε ) ′ ( R n ) and ω ∈ A p − ε p 0 ′ ( R n ) ∩ RH ( p 0 p − ε ) ′ ( R n ) .

For any f ∈ L 2 ( Ω ; R n ) , let T : f ↦ ∇ u f , where u f is the weak solution of the following Dirichlet problem:

(6.2) − div ( A ∇ u f ) = div ( f ) in Ω , u f = 0 on ∂ Ω .

From (1.12), it follows that T is a well-defined linear operator on both the spaces L ω p + ε ( Ω ; R n ) and L ω p − ε ( Ω ; R n ) . Let θ 0 ≔ p + ε 2 p . Then, θ 0 ∈ ( 0 , 1 ) and 1 p = 1 − θ 0 p − ε + θ 0 p + ε . By 1 p = 1 − θ 0 p − ε + θ 0 p + ε , (1.12), and the interpolation theorem of operators on Lorentz spaces (see, for instance, [33, Theorem 1.4.19]), we further conclude that, for any f ∈ L ω p , r ( Ω ; R n ) ,

(6.3) ∥ ∇ u f ∥ L ω p , r ( Ω ; R n ) = ∥ T ( f ) ∥ L ω p , r ( Ω ; R n ) ≲ ∥ f ∥ L ω p , r ( Ω ; R n ) ,

where u f is as in (6.2). This finishes the proof of (6.1) and hence of Theorem 6.2.□

Remark 6.3

We were told by Professor Q.-H. Nguyen that the method using the interpolation theorem of operators on Lorentz spaces to obtain (6.3) was first appeared in the proof of [51, Theorem 1.3].

Theorem 6.4

Let n ≥ 2 , Ω ⊂ R n be a bounded NTA domain, p ∈ ( 1 , ∞ ) , r ∈ ( 0 , ∞ ] , and ω ∈ A p ( R n ) . Assume that the matrix A satisfies Assumption 1.4 and Ω is a ( δ , σ , R ) quasi-convex domain with some δ , σ ∈ ( 0 , 1 ) and R ∈ ( 0 , ∞ ) . Then, there exists a positive constant δ 0 ∈ ( 0 , 1 ) , depending only on n , p , Ω , and [ ω ] A p ( R n ) , such that, if Ω is a ( δ , σ , R ) quasi-convex domain and A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) , σ ∈ ( 0 , 1 ) , and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then, for any weak solution u ∈ W 0 1 , 2 ( Ω ) of the problem ( D ) 2 with f ∈ L ω p , r ( Ω ; R n ) , ∇ u ∈ L ω p , r ( Ω ; R n ) and

(6.4) ‖ ∇ u ‖ L ω p , r ( Ω ; R n ) ≤ C ‖ f ‖ L ω p , r ( Ω ; R n ) ,

where C is a positive constant depending only on n , p , r , [ ω ] A p ( R n ) , and Ω .

The proof of Theorem 6.4 is similar to that of Theorem 6.2. We omit the details here.

Next, we recall the definition of the (Lorentz–)Morrey space on the domain Ω as follows.

Definition 6.5

Assume that n ≥ 2 and Ω is a bounded NTA domain of R n . Let p ∈ ( 1 , ∞ ) , r ∈ ( 0 , ∞ ] , and θ ∈ [ 0 , n ] . The Lorentz–Morrey space L p , r ; θ ( Ω ) is defined by setting

L p , r ; θ ( Ω ) ≔ { f is measurable on Ω : ‖ f ‖ L p , r ; θ ( Ω ) < ∞ } ,

where

‖ f ‖ L p , r ; θ ( Ω ) ≔ sup s ∈ ( 0 , diam ( Ω ) ] sup x ∈ Ω s θ − n p ‖ f ‖ L p , r ( B ( x , s ) ∩ Ω ) .

Moreover, the space L p , r ; θ ( Ω ; R n ) is defined via replacing L ω p ( Ω ) in (1.1) by the aforementioned L p , r ; θ ( Ω ) in the definition of L ω p ( Ω ; R n ) in (1.2).

It is worth pointing out that, when θ = n , the Lorentz–Morrey space L p , r ; θ ( Ω ) is just the Lorentz space; in this case, we denote the spaces L p , r ; θ ( Ω ) and L p , r ; θ ( Ω ; R n ) simply, respectively, by L p , r ( Ω ) and L p , r ( Ω ; R n ) . Moreover, when p = r , the space L p , r ; θ ( Ω ) is just the Morrey space; in this case, we denote the spaces L p , r ; θ ( Ω ) and L p , r ; θ ( Ω ; R n ) simply by ℳ p θ ( Ω ) and ℳ p θ ( Ω ; R n ) , respectively.

Applying Theorems 6.2 and 6.4, we further obtain the global gradient estimates for the Dirichlet problem ( D ) p in Lorentz–Morrey spaces as follows.

Theorem 6.6

Let A and Ω be as in Theorem 6.2, ε 0 as in Theorem 1.8(i), p 0 ≔ 3 + ε 0 when n ≥ 3 , or p 0 ≔ 4 + ε 0 when n = 2 , p ∈ ( p 0 ′ , p 0 ) , r ∈ ( 0 , ∞ ] , and θ ∈ ( p n / p 0 , n ] . Then, there exists a positive constant δ 0 ∈ ( 0 , ∞ ) , depending on n , p , r , θ , and Ω , such that, if A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then there exists a positive constant C , depending only on n , p , r , θ , and the Lipschitz constant of Ω , such that, for any weak solution u ∈ W 0 1 , 2 ( Ω ) of the problem ( D ) 2 with f ∈ L p , r ; θ ( Ω ; R n ) , ∇ u ∈ L p , r ; θ ( Ω ; R n ) and

(6.5) ‖ ∇ u ‖ L p , r ; θ ( Ω ; R n ) ≤ C ‖ f ‖ L p , r ; θ ( Ω ; R n ) .

Theorem 6.7

Let n ≥ 2 , Ω ⊂ R n be a bounded NTA domain, p ∈ ( 1 , ∞ ) , r ∈ ( 0 , ∞ ] , and θ ∈ ( 0 , n ] . Assume that the matrix A satisfies Assumption 1.4 and that Ω is a ( δ , σ , R ) quasi-convex domain with some δ , σ ∈ ( 0 , 1 ) and R ∈ ( 0 , ∞ ) . Then, there exists a positive constant δ 0 ∈ ( 0 , 1 ) , depending only on n , p , r , θ , and Ω , such that, if Ω is a ( δ , σ , R ) quasi-convex domain and A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) , σ ∈ ( 0 , 1 ) , and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then, for any weak solution u ∈ W 0 1 , 2 ( Ω ) of the problem ( D ) 2 with f ∈ L p , r ; θ ( Ω ; R n ) , ∇ u ∈ L p , r ; θ ( Ω ; R n ) , and

(6.6) ‖ ∇ u ‖ L p , r ; θ ( Ω ; R n ) ≤ C ‖ f ‖ L p , r ; θ ( Ω ; R n ) ,

where C is a positive constant depending only on n , p , r , θ , and Ω .

As corollaries of Theorems 6.6 and 6.7, we have the following global gradient estimates in Morrey spaces.

Corollary 6.8

Let Ω ⊂ R n be a bounded NTA domain and A satisfy Assumption 1.4.

Assume further that Ω is a bounded Lipschitz domain, ε 0 is as shown in Theorem 1.8(i), p 0 ≔ 3 + ε 0 when n ≥ 3 , or p 0 ≔ 4 + ε 0 when n = 2 , p ∈ ( p 0 ′ , p 0 ) , and θ ∈ ( p n / p 0 , n ] . Then, there exists a positive constant δ 0 ∈ ( 0 , ∞ ) , depending on n , p , θ , and the Lipschitz constant of Ω , such that, if A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then, there exists a positive constant C , depending only on n , p , θ , and the Lipschitz constant of Ω , such that, for any weak solution u ∈ W 0 1 , 2 ( Ω ) of the problem ( D ) 2 with f ∈ ℳ p θ ( Ω ; R n ) , ∇ u ∈ ℳ p θ ( Ω ; R n ) and
‖ ∇ u ‖ ℳ p θ ( Ω ; R n ) ≤ C ‖ f ‖ ℳ p θ ( Ω ; R n ) .
Let p ∈ ( 1 , ∞ ) and θ ∈ ( 0 , n ] . Then, there exists a positive constant δ 0 ∈ ( 0 , 1 ) , depending only on n , p , θ , and Ω , such that, if Ω is a ( δ , σ , R ) quasi-convex domain and A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) , σ ∈ ( 0 , 1 ) , and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then, for any weak solution u ∈ W 0 1 , 2 ( Ω ) of the problem ( D ) 2 with f ∈ ℳ p θ ( Ω ; R n ) , ∇ u ∈ ℳ p θ ( Ω ; R n ) and
‖ ∇ u ‖ ℳ p θ ( Ω ; R n ) ≤ C ‖ f ‖ ℳ p θ ( Ω ; R n ) ,
where C is a positive constant depending only on n , p , θ , and Ω .

Remark 6.9

Let Ω ⊂ R n be a bounded NTA domain and A ≔ a + b satisfy Assumption 1.4. For the Dirichlet problem (1.4), estimates (6.4) and (6.6) were established in [1, Corollary 2.2 and Theorem 2.3] under the assumptions that a satisfies the ( δ , R ) -BMO condition for some small δ ∈ ( 0 , ∞ ) and some R ∈ ( 0 , ∞ ) , b ≡ 0 , and Ω is a bounded Lipschitz domain with small Lipschitz constants. Thus, estimates (6.4) and (6.6) improve, respectively, [1, Corollary 2.2 and Theorem 2.3] via weakening the assumptions on the matrix A and the domain Ω .

Moreover, some estimates similar to (6.4) and (6.6) for the Dirichlet problem of some nonlinear elliptic or parabolic equations on Reifenberg flat domains were obtained in [2,11,12,46,47].

To show Theorem 6.6 via using Theorem 6.2, we need the following lemma, which is well known (see, for instance, [33, Section 7.1.2] and [46, Lemma 3.4]).

Lemma 6.10

Let s ∈ [ 1 , ∞ ) , ω ∈ A s ( R n ) , z ∈ R n , and k ∈ ( 0 , ∞ ) be a constant. Assume that τ z ( ω ) ( ⋅ ) ≔ ω ( ⋅ − z ) and ω k ≔ min { ω , k } . Then, τ z ( ω ) ∈ A s ( R n ) and [ τ z ( ω ) ] A s ( R n ) = [ ω ] A s ( R n ) , and ω k ∈ A s ( R n ) and [ ω k ] A s ( R n ) ≤ c ( s ) [ ω ] A s ( R n ) , where c ( s ) ≔ 1 when s ∈ [ 1 , 2 ] , and c ( s ) ≔ 2 s − 1 when s ∈ ( 2 , ∞ ) .
For any x ∈ R n , let ω γ ( x ) ≔ ∣ x ∣ γ , where γ ∈ R is a constant. Then, for any given s ∈ ( 1 , ∞ ) , ω γ ∈ A s ( R n ) if and only if γ ∈ ( − n , n [ s − 1 ] ) . Moreover, [ ω γ ] A s ( R n ) ≤ C ( n , s , γ ) , where C ( n , s , γ ) is a positive constant depending only on n , s , and γ .

Now, we show Theorem 6.6 by using both Theorem 6.2 and Lemma 6.10.

Proof of Theorem 6.6

We prove this theorem via borrowing some ideas from [46,47]. Let p ∈ ( p 0 ′ , p 0 ) , r ∈ ( 0 , ∞ ] , and θ ∈ ( p n / p 0 , n ] , where p 0 is as in Theorem 6.6. Assume that u is the weak solution of the Dirichlet problem (1.4) with f ∈ L q , r ; θ ( Ω ; R n ) . For any x , z ∈ Ω , ρ ∈ ( 0 , diam ( Ω ) ] , and ε ∈ ( 0 , θ − p n p 0 ) , let

ω z ( x ) ≔ min { ∣ x − z ∣ − n + θ − ε , ρ − n + θ − ε } .

Then, by Lemma 6.10, we conclude that, for any given z ∈ Ω , ω z ∈ A s ( R n ) for any given s ∈ ( 1 , ∞ ) , and there exists a positive constant C ( n , s , θ ) , depending only on n , s , and θ , such that [ ω z ] A s ( R n ) ≤ C ( n , s , θ ) . Moreover, from the assumptions θ > n p / p 0 and ε ∈ ( 0 , θ − p n p 0 ) , it follows that θ − n − ε > − n / p 0 p ′ . By this and Lemmas 3.3(vi) and 6.10, we conclude that, for any given z ∈ Ω , ω z ∈ RH p 0 p ′ ( R n ) and [ ω z ] RH p 0 p ′ ( R n ) ≲ 1 , which, combined with Theorem 6.2 and the assumption that, for any x ∈ B ( z , ρ ) , ω z ( x ) = ρ − n + θ − ε , further implies that, for any z ∈ Ω and ρ ∈ ( 0 , diam ( Ω ) ] ,

(6.7) ‖ ∇ u ‖ L p , r ( B ( z , ρ ) ∩ Ω ; R n ) = ρ n − θ + ε p ‖ ∇ u ‖ L ω z p , r ( B ( z , ρ ) ∩ Ω ; R n ) ≲ ρ n − θ + ε p ‖ f ‖ L ω z p , r ( Ω ; R n ) .

Moreover, similarly to the proofs of [46, (5.12) and (5.14)], we find that, for any z ∈ Ω and ρ ∈ ( 0 , diam ( Ω ) ] ,

‖ f ‖ L ω z p , r ( Ω ; R n ) ≲ ‖ f ‖ L p , r ; θ ( Ω ; R n ) ρ − ε p ,

which, together with (6.7), implies that, for any z ∈ Ω and ρ ∈ ( 0 , diam ( Ω ) ] ,

‖ ∇ u ‖ L p , r ( B ( z , ρ ) ∩ Ω ; R n ) ≲ ρ n − θ p ‖ f ‖ L p , r ; θ ( Ω ; R n ) .

From this and the definition of L p , r ; θ ( Ω ; R n ) , we deduce that (6.5) holds true, which completes the proof of Theorem 6.6.□

Proof of Theorem 6.7

The proof of this theorem is similar to that of Theorem 6.6. We omit the details here.

In what follows, a function f : [ 0 , ∞ ) → [ 0 , ∞ ] is said to be almost increasing (resp., almost decreasing) if there exists a positive constant L ∈ [ 1 , ∞ ) such that, for any s , t ∈ [ 0 , ∞ ) satisfying s ≤ t , f ( s ) ≤ L f ( t ) [resp., f ( s ) ≥ L f ( t ) ]; in particular, if L ≔ 1 , then f is said to be increasing (resp., decreasing). Now, we recall the definitions of weak Φ -functions and Musielak–Orlicz spaces (also called generalized Orlicz spaces) as follows (see, for instance, [21,63]). Recall that the symbol t → 0 + means t ∈ ( 0 , ∞ ) and t → 0 .

Definition 6.11

Let φ : [ 0 , ∞ ) → [ 0 , ∞ ] be an increasing function satisfying that

φ ( 0 ) = lim t → 0 + φ ( t ) = 0 and lim t → ∞ φ ( t ) = ∞ .

Then, φ is called a weak Φ -function, denoted by φ ∈ Φ w , if t → φ ( t ) t is almost increasing on ( 0 , ∞ ) .
The left-continuous generalized inverse of φ , denoted by φ − 1 , is defined by setting, for any s ∈ [ 0 , ∞ ] ,
φ − 1 ( s ) ≔ inf { t ∈ [ 0 , ∞ ) : φ ( t ) ≥ s } .
The conjugate Φ -function of φ , denoted by φ ∗ , is defined by setting, for any t ∈ [ 0 , ∞ ) ,
φ ∗ ( t ) ≔ sup s ∈ [ 0 , ∞ ) { s t − φ ( s ) } .
Let E ⊂ R n be a measurable set. A function φ : E × [ 0 , ∞ ) → [ 0 , ∞ ] is called a Musielak–Orlicz function (or a generalized Φ -function) on E if it satisfies
1. For any t ∈ [ 0 , ∞ ) , φ ( ⋅ , t ) is measurable.
2. For almost every x ∈ E , φ ( x , ⋅ ) ∈ Φ w .
Then, the set Φ w ( E ) is defined to be the collection of all Musielak–Orlicz functions on E .

Definition 6.12

Let E ⊂ R n be a measurable set and φ ∈ Φ w ( E ) . For any given f ∈ L loc 1 ( E ) , the Musielak–Orlicz modular of f is defined by setting

ρ φ ( f ) ≔ ∫ E φ ( x , ∣ f ( x ) ∣ ) d x .

Then, the Musielak–Orlicz space (also called generalized Orlicz space) L φ ( E ) is defined by setting

L φ ( E ) ≔ { u is measurable on E : there exists a λ ∈ ( 0 , ∞ ) such that ρ φ ( λ f ) < ∞ }

equipped with the Luxemburg (also called the Luxemburg–Nakano) norm

‖ u ‖ L φ ( E ) ≔ inf λ ∈ ( 0 , ∞ ) : ρ φ u λ ≤ 1 .

To obtain the global gradient estimates for the Dirichlet problem in the scale of Musielak–Orlicz spaces, we need several additional assumptions for the Musielak–Orlicz function φ . Let E ⊂ R n be a measurable set, φ ∈ Φ w ( E ) , and p ∈ ( 0 , ∞ ) .

Assumption (A0). There exist positive constants β ∈ ( 0 , 1 ) and γ ∈ ( 0 , ∞ ) such that, for any x ∈ E ,

φ ( x , β γ ) ≤ 1 ≤ φ ( x , γ ) .

Assumption (A1). There exists a β ∈ ( 0 , 1 ) such that, for any x , y ∈ E satisfying ∣ x − y ∣ ≤ 1 , and any t ∈ [ 1 , ∣ x − y ∣ − n ] ,

β φ − 1 ( x , t ) ≤ φ − 1 ( y , t ) .

Assumption (A2). There exist β , σ ∈ ( 0 , ∞ ) and h ∈ L 1 ( E ) ∩ L ∞ ( E ) such that, for any t ∈ [ 0 , σ ] and x , y ∈ E ,

φ ( x , β t ) ≤ φ ( y , t ) + h ( x ) + h ( y ) .

Assumption ( aInc ) p . The function s → φ ( x , s ) s p is almost increasing uniformly in x ∈ E .

Assumption ( aDec ) p . The function s → φ ( x , s ) s p is almost decreasing uniformly in x ∈ E .

By using the weighted global gradient estimates obtained in Theorems 1.8(ii) and 1.10(ii), and the limited range extrapolation theorem established in [21, Theorem 4.18 and Corollary 4.21] in the scale of Musielak–Orlicz spaces, we obtain the following global gradient estimates in Musielak–Orlicz spaces for the Dirichlet problem ( D ) p on bounded Lipschitz domains and ( δ , σ , R ) quasi-convex domains.

Theorem 6.13

Let A and Ω be as in Theorem 6.2, ε 0 as in Theorem 1.8(i), p 0 ≔ 3 + ε 0 when n ≥ 3 , or p 0 ≔ 4 + ε 0 when n = 2 , and p 1 , p 2 ∈ ( p 0 ′ , p 0 ) with p 1 ≤ p 2 . Assume that φ ∈ Φ w ( Ω ) satisfies Assumptions ( A 0 ) – ( A 2 ) , ( aInc ) p 1 , and ( aDec ) p 2 . Then, there exists a positive constant δ 0 ∈ ( 0 , ∞ ) , depending only on n , φ , and the Lipschitz constant of Ω , such that, if A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then, for any weak solution u ∈ W 0 1 , 2 ( Ω ) of the Dirichlet problem ( D ) 2 with f ∈ L φ ( Ω ; R n ) , ∇ u ∈ L φ ( Ω ; R n ) and

‖ ∇ u ‖ L φ ( Ω ; R n ) ≤ C ‖ f ‖ L φ ( Ω ; R n ) ,

where C is a positive constant depending only on n , φ , diam ( Ω ) , and the Lipschitz constant of Ω .

Theorem 6.14

Let n ≥ 2 , Ω ⊂ R n be a bounded NTA domain, and p 1 , p 2 ∈ ( 1 , ∞ ) with p 1 ≤ p 2 . Assume that the matrix A satisfies Assumption 1.4 and that φ ∈ Φ w ( Ω ) satisfies Assumptions ( A 0 ) – ( A 2 ) , ( aInc ) p 1 , and ( aDec ) p 2 . Then, there exists a positive constant δ 0 ∈ ( 0 , 1 ) , depending only on n , φ , and Ω , such that, if Ω is a ( δ , σ , R ) quasi-convex domain and A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) , σ ∈ ( 0 , 1 ) , and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then, for any weak solution u ∈ W 0 1 , 2 ( Ω ) of the problem ( D ) 2 with f ∈ L φ ( Ω ; R n ) , ∇ u ∈ L φ ( Ω ; R n ) and

(6.8) ‖ ∇ u ‖ L φ ( Ω ; R n ) ≤ C ‖ f ‖ L φ ( Ω ; R n ) ,

where C is a positive constant depending only on n , φ , and Ω .

To prove Theorem 6.13 via using the Rubio de Francia extrapolation theorem in the scale of Musielak–Orlicz spaces, we need Lemma 6.15, which is just [21, Corollary 4.21]. Furthermore, it is worth pointing out that the extrapolation theorem is a powerful tool to establish the boundedness of some operators on various function spaces (see, for instance, [3,21]).

Lemma 6.15

Let n ≥ 2 , Ω ⊂ R n be a bounded NTA domain, f and h be two given nonnegative measurable functions on Ω , and 1 < p 1 < p < p 2 < ∞ . Assume that, for any given ω ∈ A p / p 1 ( R n ) ∩ RH ( p 2 / p ) ′ ( R n ) ,

‖ f ‖ L ω p ( Ω ) ≤ C ‖ h ‖ L ω p ( Ω ) ,

where C is a positive constant depending only on n , p , Ω , [ ω ] A p / p 1 ( R n ) , and [ ω ] RH ( p 2 / p ) ′ ( R n ) . If φ ∈ Φ w ( Ω ) satisfies Assumptions ( A 0 ) – ( A 2 ) , ( aInc ) q 1 , and ( aDec ) q 2 for some p 1 < q 1 ≤ q 2 < p 2 , then there exists a positive constant C , depending only on n , Ω , and φ , such that ‖ f ‖ L φ ( Ω ) ≤ C ‖ h ‖ L φ ( Ω ) .

Now, we show Theorem 6.13 via using both Theorem 1.8 and Lemma 6.15.

Proof of Theorem 6.13

Let u be the weak solution of the Dirichlet problem

− div ( A ∇ u ) = div ( f ) in Ω , u = 0 on ∂ Ω .

Assume that p ∈ ( p 0 ′ , p 0 ) , where p 0 is as in Theorem 6.2. Then, by Theorem 1.8, we conclude that, for any given ω ∈ A p p 0 ′ ( R n ) ∩ RH p 0 p ′ ( R n ) , there exists a positive constant δ 0 ∈ ( 0 , ∞ ) , depending only on n , p , the Lipschitz constant of Ω , [ ω ] A p p 0 ′ ( R n ) , and [ ω ] RH p 0 p ′ ( R n ) , such that, if A satisfies the ( δ , R ) - BMO condition for some δ ∈ ( 0 , δ 0 ) and R ∈ ( 0 , ∞ ) , or A ∈ VMO ( Ω ) , then

‖ ∇ u ‖ L ω p ( Ω ; R n ) ≲ ‖ f ‖ L ω p ( Ω ; R n ) .

From this and Lemma 6.15 with f ≔ ∣ ∇ u ∣ , h ≔ ∣ f ∣ , p 1 ≔ p 0 ′ , and p 2 ≔ p 0 , it follows that

∥ ∇ u ∥ L φ ( Ω ; R n ) ∼ ‖ f ‖ L φ ( Ω ) ≲ ‖ h ‖ L φ ( Ω ) ∼ ∥ f ∥ L φ ( Ω ; R n ) ,

which completes the proof of Theorem 6.13.□

Proof of Theorem 6.14

The proof of this theorem is similar to that of Theorem 6.13. We omit the details here.□

To give more corollaries of both Theorems 6.13 and 6.14, we recall some necessary concepts for variable exponent functions p ( ⋅ ) as follows (see, for instance, [20,23]). Let P ( R n ) be the set of all measurable functions p : R n → [ 1 , ∞ ) . For any p ∈ P ( R n ) , let

(6.9) p + ≔ ess sup x ∈ R n p ( x ) and p − ≔ ess inf x ∈ R n p ( x ) .

Recall that a function p : R n → R is said to satisfy the local log-Hölder continuity condition if there exists a positive constant C loc such that, for any x , y ∈ R n with ∣ x − y ∣ ≤ 1 / 2 ,

∣ p ( x ) − p ( y ) ∣ ≤ C loc − log ( ∣ x − y ∣ ) ;

a function p : R n → R is said to satisfy the log-Hölder decay condition (at infinity) if there exist positive constants C ∞ ∈ ( 0 , ∞ ) and p ∞ ∈ [ 1 , ∞ ) , such that, for any x ∈ R n ,

∣ p ( x ) − p ∞ ∣ ≤ C ∞ log ( e + ∣ x ∣ ) .

If a function p satisfies both the local log-Hölder continuity condition and the log-Hölder decay condition, then the function p is said to satisfy the log-Hölder continuity condition.

Moreover, recall that, for any α ∈ ( 0 , 1 ] , the Hölder space C 0 , α ( Ω ) is defined by setting

C 0 , α ( Ω ) ≔ g is continuous on Ω : [ g ] C 0 , α ( Ω ) ≔ sup x , y ∈ Ω , x ≠ y ∣ g ( x ) − g ( y ) ∣ ∣ x − y ∣ α < ∞ .

Then, we have the following two corollaries of both Theorems 6.13 and 6.14.

Corollary 6.16

Assume that p ∈ P ( R n ) satisfies the log-Hölder continuity condition, ε 0 is as in Theorem 1.8(i), p 0 ≔ 3 + ε 0 when n ≥ 3 , or p 0 ≔ 4 + ε 0 when n = 2 , and p 0 ′ < p − ≤ p + < p 0 , where p − and p + are as in (6.9). Then, the conclusion of Theorem 6.13 holds true if φ satisfies one of the following cases:

For any x ∈ Ω and t ∈ [ 0 , ∞ ) , φ ( x , t ) ≔ ϕ ( t ) , where ϕ ∈ Φ w satisfies Assumptions ( aInc ) p 1 and ( aDec ) p 2 with p 0 ′ < p 1 ≤ p 2 < p 0 .
For any x ∈ Ω and t ∈ [ 0 , ∞ ) , φ ( x , t ) ≔ a ( x ) t p ( x ) , where C − 1 ≤ a ≤ C with C being a positive constant.
For any x ∈ Ω and t ∈ [ 0 , ∞ ) , φ ( x , t ) ≔ t p ( x ) log ( e + t ) .
For any x ∈ Ω and t ∈ [ 0 , ∞ ) , φ ( x , t ) ≔ t p + a ( x ) t q , where p 0 ′ < p < q < p 0 satisfy q p < 1 + 1 n , and 0 ≤ a ∈ L ∞ ( Ω ) ∩ C 0 , n p ( q − p ) ( Ω ) .
For any x ∈ Ω and t ∈ [ 0 , ∞ ) , φ ( x , t ) ≔ t p + a ( x ) t p log ( e + t ) , where p 0 ′ < p < p 0 , and 0 ≤ a ∈ L ∞ ( Ω ) satisfies the local log-Hölder continuity condition.

Corollary 6.17

Assume that p ∈ P ( R n ) satisfies the log-Hölder continuity condition and 1 < p − ≤ p + < ∞ , where p − and p + are as in (6.9). Then, the conclusion of Theorem 6.14 holds true if φ satisfies one of the following cases:

For any x ∈ Ω and t ∈ [ 0 , ∞ ) , φ ( x , t ) ≔ ϕ ( t ) , where ϕ ∈ Φ w satisfies Assumptions ( aInc ) p 1 and ( aDec ) p 2 with 1 < p 1 ≤ p 2 < ∞ .
For any x ∈ Ω and t ∈ [ 0 , ∞ ) , φ ( x , t ) ≔ a ( x ) t p ( x ) , where C − 1 ≤ a ≤ C with C being a positive constant.
For any x ∈ Ω and t ∈ [ 0 , ∞ ) , φ ( x , t ) ≔ t p ( x ) log ( e + t ) .
For any x ∈ Ω and t ∈ [ 0 , ∞ ) , φ ( x , t ) ≔ t p + a ( x ) t q , where 1 < p < q < ∞ satisfy q p < 1 + 1 n and 0 ≤ a ∈ L ∞ ( Ω ) ∩ C 0 , n p ( q − p ) ( Ω ) .
For any x ∈ Ω and t ∈ [ 0 , ∞ ) , φ ( x , t ) ≔ t p + a ( x ) t p log ( e + t ) , where p ∈ ( 1 , ∞ ) and 0 ≤ a ∈ L ∞ ( Ω ) satisfies the local log-Hölder continuity condition.

Remark 6.18

Let n ≥ 2 , Ω ⊂ R n be a bounded NTA domain, and A satisfy Assumption 1.4.

Assume that, for any x ∈ Ω and t ∈ [ 0 , ∞ ) , φ ( x , t ) ≔ ϕ ( t ) , where ϕ is as in Corollary 6.17(i). For the Dirichlet problem (1.4), the estimate (6.8) in this case was obtained in [38, Theorem 3.1] under the assumptions that a satisfies the ( δ , R ) -BMO condition for some small δ ∈ ( 0 , ∞ ) and some R ∈ ( 0 , ∞ ) , b ≡ 0 , and Ω is a bounded Reifenberg flat domain. Thus, Corollary 6.17(i) improves [38, Theorem 3.1] via weakening the assumptions on the matrix A and the domain Ω .
Assume that, for any x ∈ Ω and t ∈ [ 0 , ∞ ) , φ ( x , t ) ≔ t p ( x ) , where p ( ⋅ ) is as in Corollary 6.17. For the Dirichlet problem (1.4), the estimate (6.8) in this case was established in [14, Theorem 2.5] under the assumptions that a has partial small BMO coefficients, b ≡ 0 , and Ω is a bounded Reifenberg flat domain. Moreover, a variable exponent type estimate similar to (6.8) for the Dirichlet problem of some p -Laplace type elliptic equations on Reifenberg flat domains was also obtained in [9, Theorem 1.4].

Acknowledgments

The authors would like to thank both referees for their very careful reading and several valuable comments that indeed improve the presentation of this article. The authors would also like to thank Professor Quoc Hung Nguyen to bring our attentions to references [51,60].

Funding information: This work was supported by the National Natural Science Foundation of China (Grant Nos. 11871254, 12071431, 11971058, 12071197, 12122102, and 11871100), the National Key Research and Development Program of China (Grant No. 2020YFA0712900), and the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2021-ey18).
Conflict of interest: The authors state no conflict of interest.

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Received: 2022-01-13

Revised: 2022-02-18

Accepted: 2022-02-21

Published Online: 2022-05-17

This work is licensed under the Creative Commons Attribution 4.0 International License.

Global gradient estimates for Dirichlet problems of elliptic operators with a BMO antisymmetric part

Abstract

1 Introduction

Definition 1.1

Definition 1.2

Definition 1.3

Assumption 1.4

Remark 1.5

Remark 1.6

Theorem 1.7

Theorem 1.8

Remark 1.9

Theorem 1.10

Corollary 1.11

Corollary 1.12

Remark 1.13

2 Several concepts

Definition 2.1

Remark 2.2

Definition 2.3

Remark 2.4

Remark 2.5

3 Proof of Theorem 1.7

Theorem 3.1

Lemma 3.2

Proof

Lemma 3.3

Lemma 3.4

Proof of Theorem 1.7

4 Proof of Theorem 1.8

Lemma 4.1

Lemma 4.2

Proof

Lemma 4.3

Proof of Theorem 1.8

5 Proof of Theorem 1.10

Lemma 5.1

Lemma 5.2

Lemma 5.3

Proof of Lemma 5.1

Proof of Theorem 1.10

Lemma 5.4

Proof

Proof of Corollary 1.11

6 Several applications of Theorems 1.8 and 1.10

Definition 6.1

Theorem 6.2

Proof

Remark 6.3

Theorem 6.4

Definition 6.5

Theorem 6.6

Theorem 6.7

Corollary 6.8

Remark 6.9

Lemma 6.10

Proof of Theorem 6.6

Proof of Theorem 6.7

Definition 6.11

Definition 6.12

Theorem 6.13

Theorem 6.14

Lemma 6.15

Proof of Theorem 6.13

Proof of Theorem 6.14

Corollary 6.16

Corollary 6.17

Remark 6.18

Acknowledgments

References

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