Abstract
Let w be a Muckenhoupt A2(ℝn) weight and Ω a bounded Reifenberg flat domain in ℝn. Assume that p (·):Ω → (1, ∞) is a variable exponent satisfying the log-Hölder continuous condition. In this article, the authors investigate the weighted W1, p (·)(Ω, w)-regularity of the weak solutions of second order degenerate elliptic equations in divergence form with Dirichlet boundary condition, under the assumption that the degenerate coefficients belong to weighted BMO spaces with small norms.
1 Introduction
In this article, we consider the following second order degenerate elliptic equation in divergence form with Dirichlet boundary condition:
where Ω is a bounded domain of ℝn and ∂Ω denotes its boundary,
In this article, we always let Λ be as in (1.2) and A a symmetric matrix. Recall that a non-negative locally integrable function w on ℝn is said to belong to the Muckenhoupt class Ap(ℝn), denoted by w ∈ Ap(ℝ,n), if, when p ∈ (1, ∞),
and, when p=1,
where the suprema are taken over all balls B of ℝn. We also let
Let Ω be a measurable subset of ℝn. For any k ∈ ℕ, p ∈ [1, ∞), and w ∈ Ap(ℝn), the weighted Lebesgue space Lp(Ω, w) and the weighted Sobolev space W1, p(Ω, w) are defined, respectively, to be the sets of all measurable functions f on Ω such that
and
where
Definition. 1.1
Let p ∈ (1, ∞), w ∈ Ap(ℝn), A satisfy (1.2), and
When w ∈ A2( ℝn) and
This W1, p-regularity was achieved via the theory of Calderón–Zygmund operators in [17]. By a counterexample constructed by Meyer [42], it is known that, if A is only assumed to be uniformly elliptic, it is not sufficient to guarantee that (1.5) holds true. Di Fazio [22] showed that (1.5) holds true under the assumptions that A ∈ VMO (the vanishing mean oscillation space) and ∂Ω ∈ C1, 1 which was further weakened to ∂Ω ∈ C1 by Auscher and Qafsaoui [5]. Moreover, Byun and Wang [15] generalized (1.5) to the case that A ∈ BMO (the bounded mean oscillation space) with a small norm and Ω is a Reifenberg flat domain. Dong and Kim [26,27,28] established (1.5) under the assumptions that A has partial small BMO coefficients and Ω is a bounded Lipschitz domain with small Lipschitz constant or a bounded Reifenberg flat domain. Furthermore, Shen [56] proved that (1.5) holds true for any given
where C0 is a positive constant depending only on n, [w]A2( ℝn)∩ Ap( ℝn), p, Λ, R, and diam(Ω). Here and thereafter, diam(Ω)≔ supx, y∈Ω∣x − y∣ denotes the diameter of Ω.
On the other hand, the study of variable Lebesgue spaces originated from Orlicz [49] in 1931, which was further developed by Nakano [47,48]. The next major step in the investigation of variable function spaces was made in the article of Kováčik and Rákosník [40] in 1991. In [40], many basic properties of variable Lebesgue and Sobolev spaces were established. It was shown, in [51,52,54,55], that the theory of variable function spaces has a close connection with modelling of eletrorheological fluids, the study of which is an important issue in physics and engineering [23,55]. Moreover, the variable function spaces have many applications in, for instance, elastic mechanics [51,66], fluid mechanics [1,55], and image processing [19]. Inspired by [2,3], where Acerbi and Mingione established the Calderón–Zygmund type estimates for a non-homogeneous p [or p (·)] Laplacian system (see also [6,7,20,29,43]), Byun et al. [14] extended the Sobolev type regularity (1.5) to the setting of variable Sobolev spaces. Moreover, Bui et al. [8,10,11,12,13] did a lot of nice works on the variable Sobolev type estimates for the solutions of elliptic or parabolic equations. Very recently, we [65] established the variable Lorentz regularity of the weak solution of (1.1) in Reifenberg domains. Let Ω be a measurable subset of ℝn and
Recall that a function
equipped with the Luxemburg (or called the Luxemburg–Nakano) (quasi-)norm
The weighted variable Sobolev space W1, p (·)(Ω, w) is defined to be the set of all measurable functions f on Ω such that
where ∇ f is as in (1.4).
Remark 1.2
Let Ω be a measurable subset of ℝn,
By (1.8), it is easy to see that, for any f ∈ Lp (·)(Ω, w),
By [24, Theorem 3.4.12], we find that
A Banach space X is said to be continuously imbedded into a Banach space Y, denoted by X↪ Y, if X ⊂ Y and there exists a positive constant C such that, for any x ∈ X, ∣x∣Y ⩽ C∣x∣X. If
Motivated by [8,11,12,13,14,18], in this article, we extend (1.6) to the case that p is a variable exponent function [see (1.17) below]. To state the main result of this article, we first introduce some notions and notation.
Notation
For any r ∈ (0, ∞) and x0 ∈ ℝn, Br(x0)≔ {x ∈ ℝn: ∣x − x0∣<r} always denotes a ball centered at x0 with radius r. Define
Let
In this article, let ℕ≔ {1, 2, … }. We always use C to denote a positive constant which may change from line to line, but is independent of the main parameters, and use C(α,β,…) to denote a positive constant depending on the indicated parameters α,β, … . The symbol f ≲ g means that f ⩽ Cg. If f ≲ g and g ≲ f, we then write f ∼ g. If f ⩽ Cg and g=h or g ⩽ h, we then write f ≲ g ∽ h or f ≲ g ≲ h, rather than f ≲ g=h or f ≲ g ≤ h. For any p ∈ [1, ∞], p′ denotes its conjugate index, namely, 1/p + 1/p′=1.
Definition 1.3
Let δ, R ∈ (0, ∞) and Ω be a bounded domain of ℝn. Then Ω is said to be a (δ, R)-Reifenberg flat domain if, for any x ∈ ∂Ω and r ∈ (0, R), there exists a Y-coordinate system
where
Remark 1.4
Reifenberg flat domains were first introduced by Reifenberg [53], which appear naturally in the theory of minimal surfaces and free boundary problems. Intuitively, a Reifenberg flat domain is a domain such that, at its any boundary point and at any scale locally, the boundary can be placed between two hyperplanes, but does not require any smoothness on the boundary. It is known (see [59]) that Lipschitz domains with sufficiently small Lipschitz constants are Reifenberg flat domains, but generally Lipschitz domains may not be Reifenberg flat domains. Moreover, Reifenberg flat domains are W1, p, with p ∈ [1,∞), extension domains (see, for instance, [36]), which are very important to the study on the W1, p-regularity (0. In recent years, boundary value problems of elliptic or parabolic equations on Reifenberg flat domains have been widely studied (see, for instance, [8,9,14,15,27,28,62,63,64]).
By [41, Remark 3.2], we know that, if Ω is a (δ, R)-Reifenberg flat domain for some δ ∈ (0, 1) and R ∈ (0, ∞), then, for any x ∈ ∂Ω and r ∈ (0, (1−δ)R), there exists a Z-coordinate system of basis
where
Let δ, R ∈ (0, ∞) and Ω be a (δ, R)-Reifenberg flat domain. It is easy to see that, for any x0 ∈ ℝn and τ ∈ (0, ∞),
Equation (1.1) is translation and scaling invariant in the following sense. Let A and
In what follows, we use
Definition 1.5
([18]). Let Ω be a measurable subset of ℝn, R ∈ (0, ∞), and w ∈ A∞( ℝn). The space of functions of weighted bounded mean oscillation, denoted by BMOR(Ω, w), is defined to be the set of all
Here and thereafter,
denotes the average of f over the ball Br(x), and w(Br(x)) is as in (1.9) with E therein replaced by Br(x).
Remark 1.6
The weighted bounded mean oscillation space BMOR(Ω, w) when Ω≔ ℝn and R≔ ∞ was first introduced by Muckenhoupt and Wheeden [46]. The study of BMOR(Ω, w) is motivated by the behavior of the Hilbert transform of any measurable function bounded by a multiple of the weighted function. Let p ∈ (1, ∞) and w ∈ Ap( ℝn). Then, for any given r ∈ [1, p′], by [46, Theorem 4], we know that f ∈ BMOR(Ω, w) if and only if
Moreover, ∥·∥BMOR(Ω, w) is equivalent to ∥·∥BMOR(Ω, w) with the positive equivalence constants depending only on n, r, and [w]Ap( ℝn). Recalling that the space BMOR(Ω, w) in [18, Definition 2.3] was defined via the norm ∥·∥BMOR(Ω, w), by the aforementioned fact, we know that Definition 1.5 of this article coincides with [18, Definition 2.3].
In what follows, for any given symmetric matrix A of measurable functions
and
In this article, let Ω be a bounded domain of ℝn. We always assume that
and, for any x, y ∈ Ω,
where ω:[0, ∞) → [0, ∞)is nondecreasing continuous function with ω (0)=0 and
By (1.13), it is easy to see that there exists a positive constant c0 such that, for any r ∈ (0, 1), ω(r)log(1/r) ⩽ c0, which is equivalent to
For any
and w ∈ Aσ( ℝn). Then, by Lemma 2.2(iii) below, we know that there exists some q ∈ (1, ∞) such that
The following theorem is the main result of this article.
Theorem 1.7
Let Ω be a bounded domain of ℝn,
where C is a positive constant depending only on n, Λ, δ, σ, q, M, R, w (Ω), diam(Ω), p−, and p+.
Remark 1.8
In particular, if p (·) ≡ p ∈ (1, ∞) is a constant exponent, then Theorem 1.7 in this case coincides with [18, Theorem 2.7], namely, (1.17) in this case is just (1.6). However, when dealing with the variable Lebesgue space, the method used in [18] is not feasible. The approach used in [18] is called the approximation method of Caffarelli and Peral [16]. The main idea is to locally consider the equation (1.1) as the perturbation of an equation whose regularity of the solution is well known. This method depends on the Vitali covering lemma and the Lp boundedness of Hardy-Littlewood maximal function. We prove Theorem 1.7 by using the so-called “maximal function free technique”, which is introduced by Acerbi and Mingione [2,3] to overcome the scaling deficiency for a class of p [or p (·)] Laplacian systems, and developed by Byun et al. in [14]. This technique is a powerful tool to deal with the variable Sobolev type estimate and was also used in [8,12]. Note that there have been some known results on the estimate like (1.17) with w (·) ≡ 1 for some PDEs of divergence form, which were mainly based on the theory of Calderón–Zygmund operators (see [24,25,60]). But, in [14], it was pointed out that the tools from harmonic analysis, such as Calderón–Zygmund operators and the maximal functions, might not be feasible for the estimate (1.17) with w (·) ≡ 1. This motivates us to use the maximal function free technique. Besides this, we establish the weighted Lp Caccioppoli estimate for any given p ∈ (1, ∞) and the weighted gradient approximation estimate of u (see Sections 3 and 4 below). Moreover, to obtain (1.17), we need to establish a higher integrability result for ∇ u (see Lemma 4.8 below). To achieve this, we take the advantage of the weighted estimate of level sets of ∇ u (see Lemma 4.7 below) and borrow some ideas used in the proof of [18, Theorem 2.7] (see also [15, Theorem 1.5]), which are essentially influenced by Caffarelli and Peral [16].
We make a comment on the weighted Lp Caccioppoli estimate for any given p ∈ (1, ∞) in Section 3 below. The proof of Theorem 1.7 strongly depends on the weighted gradient approximation estimate of u established in Lemmas 4.3 and 4.5 below. When p− ∈ (2, ∞), the conclusions of Lemmas 4.3 and 4.5 for the integration index p=2 therein are enough for (1.0) being true in this case, which has been established in [18, Propositions 4.4 and 5.5]. When p (·)=p ∈ (1, 2) is a constant exponent, (1.0 in this case was showed in [18, Theorem 2.7] via a dual argument of the case p ∈ (2, ∞). In general, when p− ∈ (1, 2] and p (·) is not a constant, such a dual argument does not work anymore. Thus, we need to extend the results of [18, Propositions 4.4 and 5.5] to p ∈ (1, 2). However, to the best of our knowledge, this is unknown. To achieve this, we need to establish the weighted Lp Caccioppoli estimate for any given p ∈ (1, 2) [see Lemmas 3.3 and 3.5 below, where we prove a more general result for any given p ∈ (1, ∞)]. To this end, we apply the method used in [4,38,39,50], which depends on an integration by parts combined with a covering and iteration argument.
Via constructing a counterexample, Cao et al. [18, pp. 2232-2233] showed the necessity of the assumption ∣A∣BMOR(Ω, w)<δ such that (1.6) holds true, and constructed a specific example of such a matrix A. Indeed, let n≥3,
where In denotes the n × n identity matrix and u(α)(x)≔ ∞ when
which contradicts (1.6) with
Let Ω be a bounded domain of ℝn. If
(1.18)Conversely, if p (·) ∈ Clog(Ω), then, by taking
The remainder of this article is organized as follows.
In Section 2, we recall some preliminary results on the Vitali covering lemma, Muckenhoupt weights, the truncated Fefferman–Stein sharp maximal function, and the weighted Sobolev and Poincaré inequalities; in Section 3, we establish the local weighted interior and boundary Caccioppoli type estimates of the weak solutions of degenerate elliptic equations for any given p ∈ (1, ∞); in Section 4, based on the results obtained in Section 3, we establish the weighted interior and boundary gradient approximation estimates of the weak solutions of degenerate elliptic equations; Section 5 is devoted to the proof of Theorem 1.7.
2 Preliminaries
In this section, we recall several auxiliary and necessary well-known conclusions which are needed for the proof of Theorem 1.7.
The following is the well-known Vitali covering lemma (see, for instance, [57, p. 9]s70).
Lemma 2.1
([57]). Let E be a measurable subset of ℝn. Suppose that there exists a class {Bα}α∈I of balls of uniform bounded radii such that E⊂∪α∈IBα, where I denotes an index set. Then there exists a subclass
Let q ∈ (1, ∞]. A nonnegative locally integrable function w is said to belong to the reverse Hölder class RHq( ℝn) if there exists a positive constant C such that, for any ball B of ℝn,
where we replace {∣B∣−1∫B [w(x)]q dx}1/q by ∣w∣L∞(B) when q=∞.
We recall some properties of Muckenhoupt weights and reverse Hölder classes in the following lemma (see, for instance, [34, Chapter 9]).
Lemma 2.2
([34])
For any 1 ⩽ p ⩽ q<∞, Ap(ℝn)⊂ Aq(ℝn).
For any 1 < p ⩽ q ⩽ ∞, RHq(ℝn)⊂ RHp(ℝn).
A∞(ℝn)=⋃p∈[1, ∞)Ap(ℝn)=⋃r∈(1, ∞]RHr(ℝn).
Let p ∈ [1, ∞), r ∈ (1, ∞], and w ∈ Ap(ℝn)∩ RHr( ℝn). Then there exists a positive constant C1≔ C([w]Ap( ℝn)) such that, for any ball B of ℝn and any subset E ⋃ B,
Let w ∈ A∞( ℝn). The weighted central Hardy–Littlewood maximal function
Mw is defined by setting, for any
In particular, if w ≡ 1, we denote Mw(f) simply by M(f) which is just the classical central Hardy–Littlewood maximal function.
Remark 2.3
Let w ∈ A∞( ℝn). By Lemma 2.2, we know that w is a doubling measure on ℝn. Then it is well known (see, for instance, [58, p. 13]) that, for any given p ∈ (1, ∞), Mw is strong type (p, p) and weak type (1, 1), namely, there exist positive constants C≔ C(w, p) and
and, for any f ∈ L1(ℝn, w) and λ ∈ (0, ∞),
Let p ∈ (1, ∞) and w ∈ Ap( ℝn). It is well known that M is bounded on Lp(ℝn, w), namely, there exists a positive constant
Let ρ ∈ (0, ∞). Recall that the truncated central Fefferman–Stein sharp maximal function
Mρ, ♯(f) is defined by setting, for any
where 〈f〉Br(x) is as in (1.10). If ρ=∞, Mρ, ♯ is just the usual central sharp maximal function and we denote it simply by M♯. Let p ∈ (1, ∞). It is well known (see, for instance, [35, Chapter 7]) that there exists a positive constant C≔ C(p) such that, for any f ∈ Lp( ℝn),
The following lemma is a truncated weighted version of (2.2), which was established in [50, Corollary 2.7].
Lemma 2.4
([50]) Let M ∈ (0, ∞), ρ ∈ (0, ∞), x0 ∈ ℝn, p ∈ (1, ∞), and w ∈ Ap( ℝn) with [w]Ap( ℝn) ⩽ M. Then there exist constants
The following lemma shows that the Reifenberg flat domain satisfies the measure density condition.
Lemma 2.5
Let p ∈ [1, ∞), w ∈ Ap( ℝn), δ ∈ (0, 1), and R ∈ (0, ∞). Assume that Ω⊂ℝn is a (δ, R)-Reifenberg flat domain. Then
where C1 is the positive constant same as in Lemma 2.2(iv).
Proof. Let p ∈ [1, ∞), w ∈ Ap( ℝn), δ ∈ (0, 1), and R ∈ (0, ∞). Fix any r ∈ (0, 2R) and x ∈ Ω. If dist(x, ∂Ω) ⩾ r/2, then Br/2(x)⊂Ω. By this and Lemma 2.2, we have
If dist(x, ∂Ω)<r/2, then there exists some y ∈ ∂Ω such that dist(x, ∂Ω)=dist(x, y) < r/2. By Remark 1.4(ii), we know that there exists a Z-coordinate system
Thus, we have
Moreover, it is easy to see that Br/2(y)∩{z ∈ ℝn: zn>δr/2}⊃ Br (1−δ)/4(y0), where
This, together with (2.4), then finishes the proof of Lemma 2.5. □
Let Ω be a measurable subset of ℝn, w ∈ A∞( ℝn), and p ∈ (0, ∞). It is well known that, for any f ∈ Lp(Ω, w),
From this, it is easy to deduce the following lemma, which is just [18, Lemma 3.1].
Lemma 2.6
([18]). Let w ∈ A∞(ℝn) and p ∈ [1, ∞). Assume that Ω is a bounded domain of ℝn and g ∈ Lp(Ω, w). Then, for any given θ ∈ (0, ∞) and γ ∈ (1, ∞), there exists a positive constant C≔ C(θ, γ, p) such that
where
The following technical lemma was established in [37, Lemma 4.3].
Lemma 2.7
([37]). Let 0 < a<b<∞ and ϕ be a nonnegative bounded function on [a, b]. Assume that there exist positive constants θ ∈ (0, 1), τ ∈ (0, ∞), and κ1, κ2 ∈ [0, ∞) such that, for any a ⩽ α ⩽ β ⩽ b,
Then there exists a positive constant C≔ C(θ, τ) such that, for any a ⩽ α ⩽ β ⩽ b,
To prove Theorem 1.7, we also need the following weighted imbedding inequality, which is just [30, Theorems (1.2) and (1.3)].
Lemma 2.8
([30]). Let Ω be a bounded domain of ℝn, p ∈ (1, ∞), and w ∈ Ap( ℝn). Then there exist positive constants C and δ such that, for any given
where the positive constant C depends only on n, p, [w]Ap( ℝn), and diam(Ω).
Remark 2.9
Let B≔ B(x0, R) be a ball ofℝn with x0 ∈ ℝn and R ∈ (0, ∞). By the classical Sobolev imbedding theorem, we know that there exists a positive constant C≔ C(n, p) such that, for any
where
The following weighted Poincaré inequality was established in [30, Theorem (1.5)].
Lemma 2.10
([30]). Let p ∈ (1, ∞) and w ∈ Ap( ℝn). Then there exist positive constants δ and C≔ C(n, p, [w]Ap( ℝn)) such that, for any given
where either
3 Weighted Caccioppoli type estimates
In this section, we establish the weighted Lp Caccioppoli estimate for any given p ∈ (1, ∞), which plays a key role in the proof of weighted gradient approximation estimates in Section 4.
Let p ∈ (1, ∞) and w ∈ Ap( ℝn). Then, by the reverse Hölder property and the self-improving property of Muckenhoupt weights (see, for instance,[35, Theorem 9.2.2 and Corollary 9.2.6]), we know that there exists some q ∈ (1, ∞)such that
Lemma 3.1
Let B be a ball of ℝn, M ∈ (0, ∞), p ∈ [1, ∞), and w ∈ Ap( ℝn) with [w]Ap( ℝn) ⩽ M.
Let
Let
By the Hölder inequality, the proof of Lemma 3.1 is quite direct and we omit the details.
Definition 3.2
Let R ∈ (0, ∞),
if, for any
The following lemma can be viewed as a local interior weighted Lp Caccioppoli estimate.
Lemma 3.3
Let R ∈ (0, ∞), M ∈ (0, ∞), p ∈ (1, ∞), w ∈ Ap( ℝn) with [w]Ap( ℝn) ⩽ M, q ∈ (1, ∞)be as in (3.1), and
We prove Lemma 3.3 at the end of this section. Before that, we first present its local boundary version. To this end, let Ω be a bounded domain of ℝn. Recall that, for any given R ∈ (0, ∞), ΩR≔ Ω∩{x ∈ ℝn: ∣x∣<R}.
Definition 3.4
Let R ∈ (0, ∞), p ∈ (1, ∞), w ∈ Ap( ℝn), A satisfy (1.2), and
if, for any
and the zero extension of u to BR is in W1, p(BR, w), namely, there exists a U ∈ W1, p(BR, w) such that U(x)=u(x) for any x ∈ ΩR, and U(x)=0 for any x ∈ BR∖Ω.
For the weak solution u of (3.5), we have the following Lemma 3.5, which is a local boundary version of Lemma 3.3. The proof of Lemma 3.5 is similar to that of Lemma 3.3 (see also the proof of [4, Theorem 6.1]). To limit the length of this paper, we omit the details here.
Lemma 3.5
Let R ∈ (0, ∞), M ∈ (0, ∞), p ∈ (1, ∞), w ∈ Ap( ℝn) with [w]Ap( ℝn) ⩽ M, q ∈ (1, ∞)be as in (3.1), and
In the remainder of this section, we prove Lemma 3.3 and always let R ∈ (0, ∞), p ∈ (1, ∞), u be a weak solution of (3.2), h ∈ (1, ∞), and
For any x ∈ BR, define
Let p0 ∈ (1, ∞). For any z ∈ Bκr and τ ∈ (0, hκ r), consider the following equation
To show Lemma 3.3, we need the following estimate on v − u* on Bτ(z).
Lemma 3.6
Let p ∈ (1, ∞), M ∈ (0, ∞), w ∈ Ap( ℝn), [w]Ap( ℝn) ⩽ M, p* and q be as in (3.1), and z, τ, u* as in (3.8). Let
For any given p0 ∈ (1, q*), assume that
Proof. Let z, τ, u* be as in (3.8) and w≔ v − u*. Then (3.8) is equivalent to
Let
From this and (3.11), we deduce that
Now, we consider the following two equations
and
Then
Let
Then, from (3.12) and (3.13), we deduce that
where Aτ, z satisfies the classical uniform elliptic condition, namely, (1.2) with w ≡ 1 therein. It is well known that
To estimate
From this and Remark 2.9, it follows that
where, for any given p ∈ (1, ∞)and subset U of ℝn, W−1, p′(U) denotes the dual space of
Combining (3.14) and (3.15), we find that
For I1, by p0 ∈ (1, q*) with q* as in (3.9), we easily conclude that
Thus, there exist positive constants 𝛼, 𝛽, 𝛾 ∈ (1, ∞)such that
Moreover, we have
where
where q0≔ 𝛽(p0−1)+1 and we used the fact that
For I2, let 𝛽 and 𝛾 be as in (3.18). Then we have 𝛽>𝛾′. By (1.2), the Hölder inequality, respectively, with exponents 𝛾 and 𝛾′, and
where q1≔ 𝛾′(p0−1)+1.
Next, we estimate I3, I4, and I5, respectively. For I3, from (3.16) and Remark 2.9, it follows that
Let
Moreover, from
By this and (3.22), we find that there exists some
For any
For I4, by (3.16) and Remark 2.9, we conclude that
For any
For K1, by (1.2), (3.6), the Hölder inequality, Remark 2.9, and 𝛕 є (0, hkr), we conclude that
For K2, from the Hölder inequality and an argument similar to that used in the estimation of (3.27), it follows that
Combining this, (3.27), (3.26), and (3.25), we obtain
Next, we estimate I5. By (3.16) and Remark 2.9, we know that
For any
From this and an argument similar to that used in the estimation of (3.21), we deduce that
where
The following lemma gives a C1, α estimate for the weak solution of (3.8), which was established in [4, Lemma 5.5].
Lemma 3.7
[4]. Let p ∈ (1, ∞), u be a weak solution of (3.2), p0 ∈ (1, p), and z, τ, u* be as in (3.8). Assume that
We are now ready to prove Lemma 3.3 by using Lemmas 3.6 and 3.7.
(Proof of Lemma 3.3). Let R ∈ (0, ∞), M ∈ (0, ∞), p ∈ (1, ∞), w ∈ Ap( ℝn) with [w]Ap( ℝn) ≤ M, q ∈ (1, ∞)be as in (3.1), and
Take p0 ∈ (1, p) sufficiently small such that
where
Observing that supp
From this, (3.29), Remark 2.3, and
Fix h ∈ (1, ∞) large enough such that
where c4≔ C(n, 𝛬, p, q, M, R, h, κ). Let δ0 ∈ (0, ∞) be such that
By this and u*(x)=u(x) for any x ∈ Brθ, we obtain
To complete the proof of this lemma, we use the covering and iteration method to absorb the first term on the right hand side. Indeed, by the proof of (3.31), we know that (3.31) holds true not only for balls Br centered at the origin
where N(n) is a positive constant depending only on n. By (3.31), we know that, for any i ∈ I,
From this and (3.32), via summing up over i ∈ I, we deduce that
where we used 2Bi ⊂ Bl2 for any i ∈ I. Let δ1 ∈ (0, ∞)be such that
Fixing δ ∈ (0, ∞)small enough such that ∥A∥BMOR(BR, w) ≤ δ<min{δ0, δ1}, where δ0 is as in (3.30), we find that, for any
where c5≔ 2c4(1−θ)pN(n). Using Lemma 2.7 with
4 Weighted gradient approximation estimates
In this section, we establish the weighted interior and boundary gradient approximation estimates of the weak solution u of degenerate elliptic equations, which play a key role in the proof of Theorem 1.7. Moreover, we obtain the higher integrability result for ∇ u (see Lemma 4.8 below).
Let w ∈ A2( ℝn), A satisfy (1.2), and
and the corresponding homogeneous elliptic equation
where
By [18, Lemma 4.1] and its proof (see also [37, Theorem 4.1]), we have the following well-known result on the regularity of the weak solution of (4.2).
Lemma 4.1
([18]). Let
Remark 4.2
Let p ∈ (1, ∞) and v ∈ W1, p(B4) be a weak solution of (4.2). Since
For a weak solution u of (4.1), we have the following lemma, which shows that the gradient ∇ u can be approximated via the gradient of a weak solution v of the associated homogeneous equation (4.2).
Lemma 4.3
Let p ∈ (1, ∞), M ∈ (0, ∞), w ∈ Ap( ℝn) with [w]Ap( ℝn) ≤ M, q ∈ (1, ∞) be as in (3.1), A satisfy (1.2), and
and
then there exist an n × n matrix
and
where
and C2 := C(n, 𝛬, p, q, M) is a positive constant. Moreover, it holds true that
Remark 4.4
In particular, if p=2, then Lemma 4.3 in this case is just [18 Proposition 4.4].
Now, we show Lemma 4.3.
Proof of Lemma 4.3. Let p ∈ (1, ∞), M ∈ (0, ∞), w ∈ Ap(ℝn)∩ RHq(ℝn) with [w]Ap(ℝn) ≤ M, q ∈ (1, ∞) be as in (3.1), A satisfy (1.2),
We first prove (4.4) by using a contradiction argument. Indeed, if (4.4) is not true, then there exists some ε0 ∈ (0, ∞) such that, for any k ∈ ℕ, if wk ∈ Ap(ℝn) with [wk]Ap(ℝn) ≤ M and
and
then, for any constant matrix
Now, we show that there exist a constant matrix
namely,
On the other hand, from Lemma 2.10 and (4.11), it follows that there exists a positive constant c1≔ C(n, p, M) such that
where
where
Moreover, by the fact that
We claim that v0 is a weak solution of
By (4.13) and
Moreover, from (4.9), (4.11), and the Hölder inequality, we deduce that
This, combined with (4.16), implies that
By an argument similar to that used in the estimation of (4.17), we know that
From this, (4.18), and (4.15), it follows that v0 is a weak solution of
Then, for any k ∈ ℕ, vk is also a weak solution of
Next, we prove that
If this is true, then it contradicts (4.12), which further shows that (4.4) holds true. To prove (4.19), fix
Let τ ∈ (0, p) be a small number which is determined later, and
From this, the Hölder inequality, respectively, with exponents τ/pθ and (τ/pθ)′, and (4.20), we deduce that
Moreover, by Lemma 3.1(ii), we know that there exists a positive constant c3≔ C(n, q0, q, M) such that
where we fixed some
Combining this, (4.22), and (4.21), we conclude that
This, together with (4.14), implies that (4.19) holds true. Thus, we complete the proof of (4.4).
Now, we prove (4.5). Let
This shows (4.5).
To complete the proof of this lemma, we still need to prove (4.7). Indeed, from (4.5) and Remark 4.2, we deduce that
On the other hand, by the assumption that v and u solve, respectively, (4.2) and (4.1), we conclude that
where
For I, by (4.3), we find that
For II, from (4.4) and ∣A∣BMO4(B4, w) ≤ δ, we deduce that
For III, by (4.4), we further have
From this, (4.25), (4.24), (4.23), and
where δ is chosen small enough such that δp ∈ (0, ε), and c8≔ C(n, Λ, p, q, M) same as above. Since ε ∈ (0, ∞) is arbitrary, we obtain (4.7). This finishes the proof of Lemma 4.3. □
Let w ∈ A2(ℝn), A satisfy (1.2), and
where
and the zero extension of v to B4 is in W1, q(B4), namely, there exists a function V ∈ W1, q(B4) such that V(x)=v(x) for any
By Lemma 3.5 and an argument similar to that used in the proof of Lemma 4.3, we have the following approximation estimate at the boundary for the weak solution of (3.5) and we omit the details here.
Lemma 4.5
Let p ∈ (1, ∞), M ∈ (0, ∞), w ∈ Ap(ℝn) with [w]Ap(ℝn) ≤ M, q ∈ (1, ∞) be as in (3.1), A satisfy (1.2), and
and
then there exist an n × n matrix
where V is the zero extension of v to B4 and C3≔ C(n, Λ, p, q, M) is a positive constant.
Remark 4.6
In particular, if p=2, then Lemma 4.5 in this case is just [18, Proposition 5.5].
By Lemma 4.5 and an argument similar to that used in the proof of [18, Lemma 5.10], we have the following conclusion and we omit the details here. Recall that Ω1≔ Ω∩{x ∈ ℝn: ∣x∣<1} and Ω6≔ Ω∩{x ∈ ℝn: ∣x∣<6}.
Lemma 4.7
Let R ∈ (0, ∞), p ∈ (1, ∞), M ∈ (0, ∞), w ∈ Ap(ℝn) with [w]Ap(ℝn) ≤ M, q ∈ (1, ∞) be as in (3.1), and
where Mw is the weighted Hardy-Littlewood maximal function as in (2.1), then, for any j ∈ ℕ and
To prove Theorem 1.7, we also need the following lemma which establishes the higher integrability result for the gradient of the weak solution of (1.1).
Lemma 4.8
Let R, p, M, w, q, and
where C4≔ C(n, Λ, p, q, r, M) is a positive constant.
Proof. We prove this lemma by borrowing some ideas from the proof of [18, Theorem 2.7]. Let R ∈ (0, ∞), p ∈ (1, ∞), r ∈ (p, ∞), M ∈ (0, ∞), w ∈ Ap(ℝn), q ∈ (1, ∞) be as in (3.1), [w]Ap(ℝn) ≤ M, and
Indeed, by Remark 2.3(i), we find that there exists a positive constant c0≔ C(M) such that
We fix N large enough such that
Moreover, from Lemma 2.2(iv), we deduce that, for any y ∈ Ω1,
where C1 ∈ (0, ∞) is as in Lemma 2.2(iv). This, combined with (4.28) and (4.30), implies that (4.27) holds true.
Let
Then, by Lemma 2.6, we conclude that there exists a positive constant c1≔ C(γ, p, r) such that
On the other hand, from (4.27) and Lemmas 4.7 and 2.6, it follows that
where c2≔ C(γ, δ, p, r) is a positive constant,
where c3≔ C(γ, δ, p, r, M) is a positive constant. Combining this, (4.31), and, for almost every x ∈ Ω1,
Multiplying the both sides of the above formula by Nr, we conclude that
Moreover, from (4.29), we deduce that
where c4≔ (c0C14pn/ε)1/pγ−1. By this and (4.33), we find that
where
where C1 is as in Lemma 2.2(iv). This, together with (4.34), implies that
where C≔ C(n, Λ, p, q, r, M) is a positive constant. This finishes the proof of Lemma 4.8. □
5 Proof of Theorem 1.7
In this section, we show Theorem 1.7. We begin with establishing the following lemma, which plays a key role in the proof of Theorem 1.7.
Lemma 5.1
Let Ω be a bounded domain of ℝn, p (·) ∈ Ƥ(Ω) satisfy (1.11), (1.12), and (1.13), σ be as in (1.15), q as in (1.16), M ∈ (0, ∞), w ∈ Aσ(ℝn) with [w]Aσ(ℝn) ≤ M, A satisfy (1.2), and
where c*≔ max{R[2+w (Ω)], 1}, K1 and K2 are positive constants, respectively, as in (5.6) and (5.33) below,
Proof. Assume that
where δ1 and δ2 are, respectively, as in Lemmas 4.3 and 4.5. For any x0 ∈ Ω̅, δ ∈ (0, δ4), and
where c*≔ max {R[2+w (Ω)], 1}. By the assumption that ∣∣∇ u∣∣Lp (·)(Ω, w) < ∞ and
For any given δ ∈ (0, δ4), 1 ≤ α < β ≤ 2, and λ ∈ (0, ∞), and for any y ∈ E (λ) and r ∈ (0, (β−α)R0), define
From
where C1 is as in Lemma 2.2. Let
Then, from (5.5), it follows that, for any given λ ∈ (K1λ0, ∞) and y ∈ E (λ), and for any
On the other hand, by (5.4), it is easy to see that, for any given λ ∈ (0, ∞) and y ∈ E (λ), Φy(·) is continuous on (0, [β−α]R0). Moreover, from the definition of E (λ) in (5.2) and the Lebesgue differential theorem, we deduce that, for any given λ ∈ (0, ∞) and almost every y ∈ E (λ), limr → 0Φy(r)>λ. This, combined with (5.7), implies that, for any given λ ∈ (K1λ0, ∞) and almost every y ∈ E (λ), there exists some
and
where E0 is a measurable subset of ℝn and ∣E0∣ = 0. Let K2 ∈ (1, ∞) be a constant fixed later. Then, for any λ ∈ (0, ∞), we have
where
Now, we consider two cases.
Case 1) i ∈ Λ1. In this case, for any i ∈ Λ1, let
Then, for any i ∈ Λ1, by
and
Next, we show that there exists a positive constant c6≔ C(n, σ, M, R, w (Ω), diam (Ω)) such that, for any i ∈ Λ1 and λ ∈ (K1λ0, ∞),
and
where p− and p+ are as in (1.11).
To this end, we first prove that there exists a positive constant C(n, σ, M, R, w (Ω), diam (Ω)) such that, for any i ∈ Λ1,
Indeed, fix any y* ∈ Ω. Then we have
where C1 ∈ (0, ∞) is as in Lemma 2.2. This, together with (1.12), (1.14), and
On the other hand, by the assumption that
Combining this and (1.6), we conclude that
where C0 is a positive constant as in (1.16). From this,
Combining this and (5.16), we obtain (5.15). Observe that
where we used λ>K1λ0 ≥ λ0 > 1. This shows that (5.13) holds ture. By (5.12)and an argument similar to that used in the estimation of (5.13), we obtain (5.14).
For any i ∈ Λ1 and x ∈ ℝn, let
where C3 is a positive constant as in Lemma 4.5. Then it is easy to see that [wi]Aσ(ℝn) = [w]Aσ(ℝn). From the assumption that u solves (1.1), it follows that u is a weak solution of
From ∣∣A∣∣BMOR(Ω, w) < δ, Remark 1.6,
Combining this, Remark 1.6, (5.18), and Lemma 4.3, we find that there exists a matrix
where C2 is as in Lemma 4.3. By this and changing of variables, we know that, for any i ∈ Λ1,
where
From ∣∇ u(x)∣σ ≤ 2σ−1∣∇ u(x)−∇ v(x)∣σ+2σ−1∣∇ v(x)∣σ for any x ∈ ℝn, it follows that, for any i ∈ Λ1,
where
Case 2) i ∈ Λ2, namely,
By this and δ ∈ (0, 1/5), we find that, in this Z-coordinate system, for any i ∈ Λ2,
which further implies that
In what follows, we denote u (Tz),
By (5.21), we find that
where λ ∈ (K1λ0, ∞). For any i ∈ Λ, let
Next, we show that there exists a positive constant c8≔ C(n, σ, M, R, w (Ω), diam (Ω)) such that, for any i ∈ Λ2,
and
where
Indeed, by
where
On the other hand, by an argument similar to that used in the estimation of (5.17), we know that, for any i ∈ Λ2,
Combining this and (5.27), we obtain (5.26). By
Thus, we obtain (5.24). Similarly, we can show that (5.25) holds true and we omit the details.
For any i ∈ Λ2 and z ∈ ℝn, let
By the assumption that u solves (1.1) and Remark 1.4(ii), we find that ui is a weak solution of
where
and
here and thereafter, for any r ∈ (0, ∞),
Combining this, (5.28), (5.29), (5.30), and Lemma 4.5, we further conclude that there exist a matrix
where C3 is as in Lemma 4.5 and Vi the zero extension of vi to B4. By changing of variables, we know that, for any i ∈ Λ2,
and
where
Fix
From this,
By
This, together with (5.34) and δ ∈ (0, 1/5), implies that, for any i ∈ Λ2,
From this, (5.19), and (5.10), we deduce that, for any λ ∈ (K1λ0, ∞),
where
On the other hand, by (5.8), we know that, for any i ∈ ℕ and λ ∈ (K1λ0, ∞),
which further implies that, for any i ∈ ℕ and λ ∈ (K1λ0, ∞),
From this,
Combining this and (5.35), we obtain (5.1), which completes the proof of Lemma 5.1. □
We now prove Theorem 1.7.
Proof of Theorem 1.7. Let R ∈ (0, ∞), Ω be a (δ, R)-Reifenberg domain of ℝn, p (·) ∈ Ƥ(Ω) satisfy (1.11), (1.12), and (1.13), M ∈ (0, ∞), σ be as in (1.15), w ∈ Aσ(ℝn) with [w]Aσ(ℝn) ≤ M, and q ∈ (1, ∞) as in (1.16), where δ is a positive constant determined later. Suppose that A is a matrix satisfying (1.2) and ∣A∣BMOR(Ω, w) < δ, and
Setp 1) Assume that
Under this prior assumption, we now prove (1.17). To this end, by an argument of scaling, we only need to show that there exists a positive constant C6≔ C(n, δ, σ, q, M, R, w (Ω), diam (Ω), p−, p+, Λ) such that
under the assumption that
where p− and p+ are as in (1.7). For any given ε ∈ (0, ∞), let δ3 and δ4 be the positive constants, respectively, as in Lemmas 4.8 and 5.1 and fix some δ ∈ (0, min{δ3, δ4}). Let 1 ≤ α < β ≤ 2 and
where
For I, from (5.3) and (5.6), we deduce that, for any given
where
By
On the other hand, from
By this, Lemma 4.8, (5.38), and Remark 1.2(i), we conclude that there exists a positive constant
where C0 is a positive constant as in (1.6) and c13≔ C(n, σ, q, M, p−, p+, Λ, R, diam(Ω)). From this,
where c14≔ C(n, δ, σ, q, M, R, w (Ω), diam (Ω), p−, p+, Λ) is a positive constant.
For II, by Lemma 5.1, we know that
where C5 is as in (5.1), λ0 as in (5.3), and D1, D2 are as in (5.2). Notice that, for any given subset U of ℝn, any 0 < r < p < ∞ and f ∈ Lp(U, w),
To estimate II1, using (5.42) with
where
Similarly, from (5.38) and Remark 1.2(i), we deduce that, for any given
Combining this and (5.43), we find that, for any given
This, together with (5.41), implies that, for any given
By fixing ε ∈ (0, ∞) small enough such that 0 < c15ε < 1/2, we further know that, for any given
where
where C7≔ C(n, δ, σ, q, M, R, w (Ω), diam (Ω), p−, p+, Λ) and we used
Thus, we have
which further implies (5.37).
Step 2) In this step, we show the existence and the uniqueness of weak solution
Obviously, for any k ∈ ℕ,
Moreover, by Remark 1.2(iii), we find that, for any k ∈ ℕ, ∣∇ uk∣Lp (·)(Ω, w) < ∞. Thus, for any k ∈ ℕ, ∫Ω∣∇ uk(x)∣p(x)w(x) dx < ∞. From this, Step 1), and (5.44), it follows that, for any k ∈ ℕ large enough,
where C6 is a positive constant as in (5.37). By this, Lemma 2.8, and Remark 1.2(iii), we know that there exist positive constants C(n, σ, M, diam (Ω)) and C(p (·), n, σ, M, diam (Ω)) such that, for any k, l ∈ ℕ,
This, combined with (5.44), further shows that {uk}k∈ℕ is a Cauchy sequence in
From this, we deduce that, for almost every x ∈ Ω, limk → ∞∣∇ uk(x)∣ = ∣∇ u(x)∣. By the Fatou lemma for Lp (·)(Ω, w) (see, for instance, [21, Theorem 2.61] or [24, Lemma 3.2.8]), we conclude that
Now, we prove that u is a weak solution of (1.1). Indeed, from the fact that uk is a weak solution of (5.45), it follows that, for any k ∈ ℕ and
By this, (5.46), the Hölder inequality, (1.2),
This shows that
To complete the proof of Theorem 1.7, we need to prove that the above weak solution u is unique. Indeed, if
From (5.47), we deduce that ∣∇(u − v)∣Lp (·)(Ω, w) = 0. By this, Remark 1.2(iii), and Lemma 2.8, we conclude that
Acknowledgements
The authors want to express their sincere thanks to the referees for their valuable remarks and suggestions, which made this paper more readable.
Junqiang Zhang is supported by the National Natural Science Foundation of China (Grant Nos. 11801555, 11971058 and 12071431) and the Fundamental Research Funds for the Central Universities (Grant No. 2021YQLX10). Dachun Yang is supported by the National Natural Science Foundation of China (Grant Nos. 11971058 and 12071197) and the National Key Research and Development Program of China (Grant No. 2020YFA0712900). Sibei Yang is supported by the National Natural Science Foundation of China (Grant Nos. 11871254 and 12071431) and the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2021-ey18).
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Conflict of interest: Authors state no conflict of interest.
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