Some Properties of Solutions to a Class of Dirichlet Boundary Value Problems

Abstract

This paper deals with the following Dirichlet problem: 𝑑^∗𝐴(𝑥,𝑔+𝑑𝑢)=𝑑^∗ℎinΩ,𝑢_𝑇=0 on 𝜕Ω. Based on its solvability, we derive some properties of its solutions. In this paper, we mainly get three results. Firstly, we establish an integral estimate for the solutions of the above Dirichlet boundary value problem. Secondly, a stability result of solutions for varying differential forms 𝑔 and ℎ is obtained. Lastly, we present a weak reverse Hölder inequality for solutions.

1. Introduction

Let Ω be a bounded domain in ℝ^𝑛 with smooth boundary 𝜕Ω. Throughout this paper we assume that 1<𝑝, 𝑞<∞, is a Hölder conjugate pair, 𝑝+𝑞=𝑝𝑞. First, we consider the nonhomogeneous A-harmonic equation: 𝑑^∗𝐴𝑥,𝑔+𝑑𝑢=𝑑^∗ℎ,(1.1) where (𝑔,ℎ)∈𝐿^𝑝(Ω,∧^𝑙)×𝑊^1,𝑞(Ω,∧^𝑙) and 𝐴∶Ω×∧^𝑙(ℝ^𝑛)→∧^𝑙(ℝ^𝑛) is a Carathéodory mapping satisfying the following assumptions for fixed 0<𝛼≤𝛽<∞:(1)Lipschitz continuity 𝐴(𝑥,𝜉)−𝐴(𝑥,𝜁)≤𝛽𝜉−𝜁𝜉+𝜁^𝑝−2;(1.2)(2)uniform monotonicity 𝐴(𝑥,𝜉)−𝐴(𝑥,𝜁),𝜉−𝜁≥𝛼𝜉−𝜁²𝜉+𝜁^𝑝−2;(1.3)(3)homogeneity 𝐴(𝑥,𝜆𝜉)=𝜆|𝜆|^𝑝−2𝐴(𝑥,𝜉),(1.4) for almost every 𝑥∈Ω and all 𝜉,𝜁∈∧^𝑙(ℝ^𝑛), 𝜆∈ℝ.

In particular, for 𝐴(𝑥,𝜉)=|𝜉|^𝑝−2𝜉, then (1.1) is simplified to the nonhomogeneous 𝑝-harmonic equation: 𝑑^∗𝑔+𝑑𝑢^𝑝−2𝑔+𝑑𝑢=𝑑^∗ℎ.(1.5)

By Browder-Minty theory, see [1], the existence and uniqueness of a solution to the Dirichlet problem 𝑑^∗𝐴𝑥,𝑔+𝑑𝑢=𝑑^∗𝑢ℎinΩ,_𝑇=0on𝜕Ω,(1.6) in 𝑊^1,𝑝(Ω,∧^𝑙−1) has been obtained by Iwaniec et al., see [2]. For the solution 𝑢∈𝑊^1,𝑝(Ω,∧^𝑙−1) with vanishing tangential component on 𝜕Ω, we write it as 𝑢∈𝑊_𝑇^1,𝑝(Ω,∧^𝑙−1).

Definition 1.1. Given that (𝑔,ℎ)∈𝐿^𝑝(Ω,∧^𝑙)×𝑊^1,𝑞(Ω,∧^𝑙), a differential form 𝑢 is called a solution to the Dirichlet problem (1.6) if 𝑢∈𝑊_𝑇^1,𝑝(Ω,∧^𝑙−1) and it holds _Ω𝐴𝑥,𝑔+𝑑𝑢,𝑑𝜙d𝑥=_Ωℎ,𝑑𝜙d𝑥,(1.7) for all 𝜙∈𝑊_𝑇^1,𝑝(Ω,∧^𝑙−1).

A-harmonic equations for differential forms have been a very active field in recent years because they are an invaluable tool to describe various systems of partial differential equations and to express different geometrical structures on manifolds. Moreover, they can be used in many fields, such as physics, nonlinear elasticity theory, and the theory of quasiconformal mappings, see [3–10]. The purpose of this paper is to study properties of solutions of the Dirichlet boundary value problem (1.6) based on the existence of its solutions.

2. Notation and Preliminary Results

This section is devoted to the notation of the exterior calculus and a few necessary preliminaries. For more details the reader can refer to [2, 3].

We denote by ∧^𝑙=∧^𝑙(ℝ^𝑛) the space of 𝑙-covectors in ℝ^𝑛 and the direct sum ∧ℝ^𝑛=^𝑛_𝑙=0∧^𝑙ℝ^𝑛(2.1) is a graded algebra with respect to the wedge product ∧. We will make use of the exterior derivative: 𝑑∶𝐶^∞Ω,∧^𝑙→𝐶^∞Ω,∧^𝑙+1(2.2) and its formal adjoint operator 𝑑^∗=(−1)^𝑛𝑙+1∗𝑑∗∶𝐶^∞Ω,∧^𝑙+1→𝐶^∞Ω,∧^𝑙,(2.3) known as the Hodge codifferential, where the symbol ∗ denotes the Hodge star duality operator. Note that each of the operators 𝑑 and 𝑑^∗ applied twice gives zero.

Let 𝐶^∞(Ω,∧^𝑙) be the class of infinitely differentiable 𝑙-forms on Ω⊂ℝ^𝑛. Since Ω is a smooth domain, near each boundary point one can introduce a local coordinate system (𝑥₁,𝑥₂,…,𝑥_𝑛) such that 𝑥_𝑛=0 on 𝜕Ω and such that the 𝑥_𝑛-curve is orthogonal to 𝜕Ω. Near this boundary point, every differential form 𝜔∈𝐶^∞(Ω,∧^𝑙) can be decomposed as 𝜔(𝑥)=𝜔_𝑇(𝑥)+𝜔_𝑁(𝑥), where 𝜔_𝑇(𝑥)=_{1≤𝑖1<⋯<𝑖𝑙<𝑛}𝜔_{𝑖1,…,𝑖𝑙}(𝑥)𝑑𝑥_𝑖1∧⋯∧𝑑𝑥_𝑖𝑙,𝜔_𝑁(𝑥)=_{1≤𝑖1<⋯<𝑖𝑙=𝑛}𝜔_{𝑖1,…,𝑖𝑙}(𝑥)𝑑𝑥_𝑖1∧⋯∧𝑑𝑥_𝑖𝑙(2.4) are called the tangential and the normal parts of 𝜔, respectively. Now, the duality between 𝑑 and 𝑑^∗ is expressed by the integration by parts formula _Ω⟨𝑑𝑢,𝑣⟩=_Ω𝑢,𝑑^∗𝑣,(2.5) for all 𝑢∈𝐶^∞(Ω,∧^𝑙) and 𝑣∈𝐶^∞(Ω,∧^𝑙+1), provided 𝑢_𝑇=0 or 𝑣_𝑁=0. The symbol ⟨⋅,⋅⟩ denotes the inner product, that is, let 𝛼=∑_𝐼𝛼_𝐼(𝑥)d𝑥_𝐼 and 𝛽=∑_𝐼𝛽_𝐼(𝑥)d𝑥_𝐼, then ⟨𝛼,𝛽⟩=∑_𝐼𝛼_𝐼(𝑥)𝛽_𝐼(𝑥).

Due to (2.5), extended definitions for 𝑑 and 𝑑^∗ can be introduced as the introduction of weak derivatives.

Definition 2.1 (see [2]). Suppose that 𝜔∈𝐿¹_loc(Ω,∧^𝑙) and 𝑣∈𝐿¹_loc(Ω,∧^𝑙+1). If _Ω𝜔,𝑑^∗𝜂=_Ω𝑣,𝜂(2.6) for every test form 𝜂∈𝐶^∞₀(Ω,∧^𝑙+1), one says that 𝜔 has generalized exterior derivative 𝑣 and write 𝑣=𝑑𝜔.

The notion of the generalized exterior coderivative 𝑑^∗ can be defined analogously.

Definition 2.2 (see [2]). Suppose that 𝜔∈𝐿¹_loc(Ω,∧^𝑙) and 𝑣∈𝐿¹_loc(Ω,∧^𝑙−1). If 𝜔,𝑑𝜂=_ΩΩ𝑣,𝜂(2.7) for every test form 𝜂∈𝐶^∞₀(Ω,∧^𝑙−1), one says that 𝜔 has generalized exterior coderivative 𝑣 and write 𝑣=𝑑^∗𝜔.

Remark 2.3. (i) Observe that generalized exterior derivatives have many properties similar to those of weak derivatives. For example, (𝑖₁) if it exists, it is unique; (𝑖₂) if 𝜔 is differentiable in the conventional sense, then its generalized exterior derivative 𝑑𝜔 is identical to its the classical exterior differential 𝑑𝜔. Analogous results hold for generalized exterior coderivative.
(ii) If the generalized exterior derivative of 𝜔, 𝑑𝜔, exists, then 𝑑𝜔 also has its generalized exterior derivative 𝑑(𝑑𝜔). Moreover, 𝑑(𝑑𝜔)=0.
In fact, according to Definition 2.1, it holds _Ω𝜔,𝑑^∗𝜙=_Ω𝑑𝜔,𝜙(2.8) for every test form 𝜙∈𝐶^∞₀(Ω,∧^𝑙+1). Thus, for every 𝜂∈𝐶^∞₀(Ω,∧^𝑙+2), we have 𝑑^∗𝜂∈𝐶^∞₀(Ω,∧^𝑙+1) and by taking 𝜙=𝑑^∗𝜂 in the above integral equality implies _Ω𝑑𝜔,𝑑^∗𝜂=_Ω𝜔,𝑑^∗𝑑^∗𝜙=_Ω⟨𝜔,0⟩=0.(2.9) Therefore, Definition 2.1 yields that 𝑑(𝑑𝜔) exists and 𝑑(𝑑𝜔)=0. Similarly, we have 𝑑^∗(𝑑^∗𝜔)=0.
(iii) Together with the expression of differential forms, the definition of weak derivative and its uniqueness, we can prove that 𝑑 and 𝑑 have analogous expressions, that is, for 𝜔(𝑥)=∑_𝐼𝜔_𝐼(𝑥)d𝑥_𝐼, we have 𝑑𝜔(𝑥)=_𝐼^𝑛_𝑘=1𝜕𝜔_𝐼𝜕𝑥_𝑘d𝑥_𝑘∧d𝑥_𝐼,𝑑𝜔(𝑥)=_𝐼^𝑛_𝑘=1𝜕𝜔_𝐼𝜕𝑥_𝑘d𝑥_𝑘∧d𝑥_𝐼,(2.10) where 𝜕 denotes the ordinary derivative and 𝜕 the weak derivative. So in the next we use 𝑑 to represent the action instead of 𝑑, similar for 𝑑^∗ and 𝑑^∗.
(iv) Lastly, we refer to ker𝑑=𝜔∈𝐿¹_locΩ,∧^𝑙∶𝑑𝜔=0(2.11) as the closed 𝑙-forms and to ker𝑑^∗=𝜔∈𝐿¹_locΩ,∧^𝑙∶𝑑^∗𝜔=0(2.12) as the coclosed 𝑙-forms.

Definition 2.4 (see [2]). A 𝑙-form 𝜔 is said to have vanishing tangential component at 𝜕Ω in a generalized sense, if both 𝜔 and 𝑑𝜔 belong to 𝐿¹(Ω,∧^𝑙) and _Ω𝜔,𝑑^∗𝜂=_Ω𝑑𝜔,𝜂(2.13) holds for any 𝜂∈𝐶^∞(Ω,∧^𝑙+1). One writes 𝜔_𝑇=0.

The notion of vanishing normal part 𝜔_𝑁=0 can be defined analogously. Now, the following extension of the identity (2.5) can be introduced, by an approximation argument proved by Iwaniec and Lutoborski [3].

Proposition 2.5 (see [3]). For (𝑝,𝑞) a Hölder conjugate pair and 𝑢∈𝑊^1,𝑝(Ω,∧^𝑙), 𝑣∈𝑊^1,𝑞(Ω,∧^𝑙+1) one has _Ω⟨𝑑𝑢,𝑣⟩=_Ω𝑢,𝑑^∗𝑣(2.14) provided 𝑢_𝑇=0 or 𝑣_𝑁=0.

Finally, we present briefly some spaces of differential forms:  𝐿^𝑝(Ω,∧^𝑙)—the space of 𝑙-forms 𝜔 with coefficients in 𝐿^𝑝(Ω); 𝐿^𝑝₁(Ω,∧^𝑙)—the space of 𝑙-forms 𝜔 such that ∇𝜔 is a regular distribution of 𝐿^𝑝(Ω,∧^𝑙); 𝑊^1,𝑝(Ω,∧^𝑙)—the Sobolev space of 𝑙-forms 𝜔 defined by 𝐿^𝑝(Ω,∧^𝑙)∩𝐿^𝑝₁(Ω,∧^𝑙); 𝑊_𝑇^1,𝑝(Ω,∧^𝑙)—the space of 𝑙-forms 𝜔 in 𝑊^1,𝑝(Ω,∧^𝑙) with vanishing tangential component on 𝜕Ω. which are used throughout this paper.

3. Main Results

We study the properties of solutions of the Dirichlet boundary value problem (1.6) whose existence can be deduced by the following.

Lemma 3.1 (see [2]). For each data (𝑔,ℎ)∈𝐿^𝑝(Ω,∧^𝑙)×𝑊^1,𝑞(Ω,∧^𝑙), there exists a solution 𝑢∈𝑊_𝑇^1,𝑝(Ω,∧^𝑙−1) to the Dirichlet problem (1.6).

3.1. An Integral Estimate

We start with a proposition which gives an important estimate for solutions of (1.6).

Proposition 3.2. Given (𝑔,ℎ)∈𝐿^𝑝(Ω,∧^𝑙)×𝑊^1,𝑞(Ω,∧^𝑙), suppose that 𝑢 is a solution of (1.6) in 𝑊_𝑇^1,𝑝(Ω,∧^𝑙−1). Then one has _Ω|𝑑𝑢|^𝑝d𝑥≤𝑐𝑝,𝛼,𝛽_Ω𝑔^𝑝d𝑥+_Ω|ℎ|^𝑞d𝑥.(3.1)

Proof. Taking the solution 𝑢 as the test function in (1.7) yields _Ω𝐴𝑥,𝑔+𝑑𝑢,𝑑𝑢=_Ω⟨ℎ,𝑑𝑢⟩,(3.2) thus we have _Ω𝐴𝑥,𝑔+𝑑𝑢,𝑔+𝑑𝑢=_Ω𝐴𝑥,𝑔+𝑑𝑢,𝑔+_Ω⟨ℎ,𝑑𝑢⟩.(3.3) Next we apply Young’s inequalityas follows: 𝑎𝑏≤𝜀𝑎^𝑟+𝜀^{−1/(𝑟−1)}𝑏^{𝑟/(𝑟−1)},𝜀>0,𝑟>1.(3.4) It follows from the structural assumptions (1) and (2) that 𝛼_Ω𝑔+𝑑𝑢^𝑝≤_Ω𝐴=𝑥,𝑔+𝑑𝑢,𝑔+𝑑𝑢_Ω⟨ℎ,𝑑𝑢⟩+_Ω𝐴𝑥,𝑔+𝑑𝑢,𝑔≤_Ω|ℎ||𝑑𝑢|+𝛽_Ω𝑔+𝑑𝑢^𝑝−1𝑔≤_Ω|ℎ||𝑑𝑢|+𝛽_Ω𝜀𝑔+𝑑𝑢^𝑝+𝑐𝜀,𝑝𝑔^𝑝=𝜀𝛽_Ω𝑔+𝑑𝑢^𝑝+_Ω|ℎ||𝑑𝑢|+𝑐𝜀,𝑝𝛽_Ω𝑔^𝑝.(3.5) By choosing 𝜀=𝛼/(2𝛽) and using again Young’s inequality, we have _Ω𝑔+𝑑𝑢^𝑝≤2𝛼_Ω|ℎ||𝑑𝑢|+𝑐𝑝,𝛼,𝛽_Ω𝑔^𝑝≤2𝛼_Ω𝜀|𝑑𝑢|^𝑝+𝑐𝜀,𝑝|ℎ|^𝑞+𝑐𝑝,𝛼,𝛽_Ω𝑔^𝑝.(3.6) Therefore, we get that _Ω|𝑑𝑢|^𝑝≤_Ω𝑔+𝑑𝑢+𝑔^𝑝≤2^𝑝_Ω𝑔+𝑑𝑢^𝑝+2^𝑝_Ω𝑔^𝑝≤2^𝑝+1𝛼𝜀_Ω|𝑑𝑢|^𝑝+2^𝑝+1𝛼𝑐𝜀,𝑝_Ω|ℎ|^𝑞+𝑐𝑝,𝛼,𝛽_Ω𝑔^𝑝.(3.7) Finally, we obtain by choosing 𝜀=𝛼/2^𝑝+2 that _Ω|𝑑𝑢|^𝑝d𝑥≤𝑐𝑝,𝛼,𝛽_Ω𝑔^𝑝d𝑥+_Ω|ℎ|^𝑞d𝑥.(3.8) The theorem follows.

3.2. Stability of Solutions

In this section, we establish the weak convergence of solutions of (1.6) with varying differential forms 𝑔 and ℎ. Particularly, given a sequence (𝑔_𝑗,ℎ_𝑗)∈𝐿^𝑝(Ω,∧^𝑙)×𝑊^1,𝑞(Ω,∧^𝑙), Lemma 3.1 implies that there exists solution 𝑢_𝑗∈𝑊_𝑇^1,𝑝(Ω,∧^𝑙−1) to 𝑑^∗𝐴𝑥,𝑔_𝑗+𝑑𝑢_𝑗=𝑑^∗ℎ_𝑗.(3.9) Suppose that (𝑔_𝑗,ℎ_𝑗)→(𝑔,ℎ) in 𝐿^𝑝(Ω,∧^𝑙)×𝑊^1,𝑞(Ω,∧^𝑙+1), and 𝑢_𝑗⇀𝑢 in 𝑊^1,𝑝(Ω,∧^𝑙−1), we will show that 𝑢∈𝑊_𝑇^1,𝑝(Ω,∧^𝑙−1) is a solution of (1.6).

Theorem 3.3. Under the hypotheses above, 𝑢∈𝑊_𝑇^1,𝑝(Ω,∧^𝑙−1) is a solution of (1.6).

To prove Theorem 3.3, we need firstly to give the definition of the weak convergence for sequences in spaces of differential forms.

Definition 3.4. One says that 𝜑_𝑗 is weakly convergent to 𝜑 in 𝐿^𝑝(Ω,∧^𝑙) if _Ω𝜑_𝑗,ℎ→_Ω𝜑,ℎ,(3.10) whenever ℎ∈𝐿^𝑞(Ω,∧^𝑙) and write 𝜑_𝑗⇀𝜑 in 𝐿^𝑝(Ω,∧^𝑙).
It is easy to verify that it has the following equivalent definition.

Definition 3.5. One says that 𝜑_𝑗 weakly convergent to 𝜑 in 𝐿^𝑝(Ω,∧^𝑙) if _Ω𝜑_𝑗∧ℎ→_Ω𝜑∧ℎ(3.11) whenever ℎ∈𝐿^𝑞(Ω,∧^𝑛−𝑙) and write 𝜑_𝑗⇀𝜑 in 𝐿^𝑝(Ω,∧^𝑙).

According to the well-known results in Sobolev space in terms of functions, and together with the expression of differential forms and the diagonal rule we can easily obtain that

Proposition 3.6. For 1<𝑝<∞, 𝐿^𝑝(Ω,∧^𝑙) is reflexive.

Proposition 3.7. 𝜑_𝑗⇀𝜑 in 𝑊^1,𝑝(Ω,∧^𝑙) if and only if 𝜑_𝑗⇀𝜑 in 𝐿^𝑝(Ω,∧^𝑙) and ∇𝜑_𝑗⇀∇𝜑 in 𝐿^𝑝(Ω,∧^𝑙), where ∇𝜑=((𝜕𝜑/𝜕𝑥₁),…,(𝜕𝜑/𝜕𝑥_𝑛)) and the partial differentiation is applied to the coefficients of 𝜑.

Lemma 3.8. Suppose that a sequence of differential forms {𝜑_𝑗} converges to 𝜑 weakly in 𝐿^𝑝(Ω,∧^𝑙) while the generalized exterior derivatives of 𝜑_𝑗, 𝑑𝜑_𝑗, exist and stay bounded in 𝐿^𝑝(Ω,∧^𝑙+1). Then the generalized exterior derivative of 𝜑, 𝑑𝜑, exists and 𝑑𝜑_𝑗⇀𝑑𝜑 in 𝐿^𝑝(Ω,∧^𝑙+1).

Proof. On the one hand, since 𝜑_𝑗 has generalized exterior derivative 𝑑𝜑_𝑗, then according to Definition 2.1 we have _Ω𝑑𝜑_𝑗,𝜂=_Ω𝜑_𝑗,𝑑^∗𝜂,(3.12) for every 𝜂∈𝐶^∞₀(Ω,∧^𝑙+1). Notice that 𝜑_𝑗⇀𝜑 in 𝐿^𝑝(Ω,∧^𝑙) and 𝑑^∗𝜂∈𝐶^∞₀(Ω,∧^𝑙)⊂𝐿^𝑞(Ω,∧^𝑙), it follows from Definition 3.4 that _Ω𝜑_𝑗,𝑑^∗𝜂→_Ω𝜑,𝑑^∗𝜂.(3.13) Thus, we obtain _Ω𝑑𝜑_𝑗,𝜂→_Ω𝜑,𝑑^∗𝜂,(3.14) for every 𝜂∈𝐶^∞₀(Ω,∧^𝑙+1).
On the other hand, since 𝑑𝜑_𝑗 stays bounded in 𝐿^𝑝(Ω,∧^𝑙+1), then it follows from Proposition 3.6 that there exists weakly convergence subsequence of 𝑑𝜑_𝑗, we may assume that 𝑑𝜑_𝑗⇀𝑔(3.15) in 𝐿^𝑝(Ω,∧^𝑙+1). Hence, for 𝜂∈𝐶^∞₀(Ω,∧^𝑙+1)⊂𝐿^𝑞(Ω,∧^𝑙+1) we have _Ω𝑑𝜑_𝑗,𝜂→_Ω𝑔,𝜂.(3.16) Combining (3.14) and (3.16), the uniqueness of the limit yields _Ω𝜑,𝑑^∗𝜂=_Ω𝑔,𝜂,(3.17) for every 𝜂∈𝐶^∞₀(Ω,∧^𝑙+1). Thus, Definition 2.1 implies that the generalized exterior derivative of 𝜑 exists and 𝑔=𝑑𝜑. Furthermore, we have from (3.15) that 𝑑𝜑_𝑗⇀𝑑𝜑 in 𝐿^𝑝(Ω,∧^𝑙+1). The lemma follows.

Proof of Theorem 3.3. Taking (𝑢_𝑗−𝑢_𝑘) as test differential form in (1.7) for both 𝑑^∗𝐴𝑥,𝑔_𝑗+𝑑𝑢_𝑗=𝑑^∗ℎ_𝑗,𝑑^∗𝐴𝑥,𝑔_𝑘+𝑑𝑢_𝑘=𝑑^∗ℎ_𝑘,(3.18) then we obtain _Ω𝐴𝑥,𝑔_𝑗+𝑑𝑢_𝑗−𝐴𝑥,𝑔_𝑘+𝑑𝑢_𝑘,𝑑𝑢_𝑗−𝑑𝑢_𝑘=_Ωℎ_𝑗−ℎ_𝑘,𝑑𝑢_𝑗−𝑑𝑢_𝑘,(3.19) or, equivalently, we have _Ω𝐴𝑥,𝑔_𝑗+𝑑𝑢_𝑗−𝐴𝑥,𝑔_𝑘+𝑑𝑢_𝑘,𝑔_𝑗+𝑑𝑢_𝑗−𝑔_𝑘+𝑑𝑢_𝑘=_Ω𝐴𝑥,𝑔_𝑗+𝑑𝑢_𝑗−𝐴𝑥,𝑔_𝑘+𝑑𝑢_𝑘,𝑔_𝑗−𝑔_𝑘+_Ωℎ_𝑗−ℎ_𝑘,𝑑𝑢_𝑗−𝑑𝑢_𝑘.(3.20) Write 𝜉=𝑔_𝑗+𝑑𝑢_𝑗 and 𝜁=𝑔_𝑘+𝑑𝑢_𝑘, then the above identity can be simplified to _Ω𝐴()()=𝑥,𝜉−𝐴𝑥,𝜁,𝜉−𝜁_Ω𝐴(𝑥,𝜉)−𝐴(𝑥,𝜁),𝑔_𝑗−𝑔_𝑘+_Ωℎ_𝑗−ℎ_𝑘,𝑑𝑢_𝑗−𝑑𝑢_𝑘.(3.21) Then it follows from the Lipschitz condition (1), the monotonicity condition (2), and Hölder inequality that: 𝛼_Ω𝜉−𝜁^𝑝≤𝛽_Ω𝜉+𝜁^𝑝−1𝑔_𝑗−𝑔_𝑘+_Ωℎ_𝑗−ℎ_𝑘𝑑𝑢_𝑗−𝑑𝑢_𝑘≤𝛽_Ω𝜉+𝜁^𝑝1/𝑞_Ω𝑔_𝑗−𝑔_𝑘^𝑝1/𝑝+_Ω𝑑𝑢_𝑗−𝑑𝑢_𝑘^𝑝1/𝑝_Ωℎ_𝑗−ℎ_𝑘^𝑞1/𝑞≤𝑐𝑝,𝛽𝜉_𝑝^𝑝−1+𝜁_𝑝^𝑝−1𝑔_𝑗−𝑔_𝑘𝑝+𝑑𝑢_𝑗−𝑑𝑢_𝑘𝑝ℎ_𝑗−ℎ_𝑘𝑞.(3.22) Therefore, we obtain 𝜉−𝜁_𝑝≤𝑐𝑝,𝛼,𝛽𝜉_𝑝^1/𝑞+𝜁_𝑝^1/𝑞𝑔_𝑗−𝑔_𝑘𝑝^1/𝑝+𝑐𝑝,𝛼𝑑𝑢_𝑗−𝑑𝑢_𝑘𝑝^1/𝑝ℎ_𝑗−ℎ_𝑘𝑞^1/𝑝.(3.23) Since 𝑔_𝑗 and ℎ_𝑗 are Cauchy sequences in 𝐿^𝑝(Ω,∧^𝑙) and 𝐿^𝑞(Ω,∧^𝑙), respectively, we have ‖𝑔_𝑗‖_𝑝<∞ and ‖ℎ_𝑗‖_𝑞<∞ for all 𝑗∈ℕ, and further we get by applying Proposition 3.2 that 𝜉_𝑝^1/𝑞+𝜁_𝑝^1/𝑞≤𝑐𝑝𝑔_𝑗𝑝^1/𝑞+𝑑𝑢_𝑗𝑝^1/𝑞+𝑔_𝑘𝑝^1/𝑞+𝑑𝑢_𝑘𝑝^1/𝑞≤𝑐𝑝𝑔_𝑗𝑝^1/𝑞+𝑐𝑝,𝛼,𝛽𝑔_𝑗𝑝^1/𝑞+ℎ_𝑗𝑞^1/𝑝+𝑔_𝑘𝑝^1/𝑞+𝑐𝑝,𝛼,𝛽𝑔_𝑘𝑝^1/𝑞+ℎ_𝑘𝑞^1/𝑝≤𝑐𝑝,𝛼,𝛽𝑔_𝑗𝑝^1/𝑞+𝑔_𝑘𝑝^1/𝑞+ℎ_𝑗𝑞^1/𝑝+ℎ_𝑘𝑞^1/𝑝<∞,𝑑𝑢_𝑗−𝑑𝑢_𝑘𝑝^1/𝑝≤𝑐𝑝𝑑𝑢_𝑗𝑝^1/𝑝+𝑑𝑢_𝑘𝑝^1/𝑝≤𝑐𝑝,𝛼,𝛽𝑔_𝑗𝑝^1/𝑝+𝑔_𝑘𝑝^1/𝑝+ℎ_𝑗𝑞^{(𝑞−1)/𝑝}+ℎ_𝑘^(𝑞−1)/p_𝑞<∞.(3.24) Combining (3.23)-(3.24), we obtain 𝜉−𝜁_𝑝→0,(3.25) as 𝑗,𝑘→∞. Thus, the Minkowski inequality implies 𝑑𝑢_𝑗−𝑑𝑢_𝑘𝑝=(𝜉−𝜁)−𝑔_𝑗−𝑔_𝑘𝑝≤𝜉−𝜁_𝑝+𝑔_𝑗−𝑔_𝑘𝑝→0,(3.26) as 𝑗,𝑘→∞, that is, {𝑑𝑢_𝑗} is a Cauchy sequence in 𝐿^𝑝(Ω,∧^𝑙). Thus, we have 𝑑𝑢_𝑗 is bounded in 𝐿^𝑝(Ω,∧^𝑙). Then, it follows from Lemma 3.8 that 𝑑𝑢 exists and 𝑑𝑢_𝑗⇀𝑑𝑢 in 𝐿^𝑝(Ω,∧^𝑙). Note that {𝑑𝑢_𝑗} is a Cauchy sequence in 𝐿^𝑝(Ω,∧^𝑙), then we have by the uniqueness of weak limit that 𝑑𝑢_𝑗→𝑑𝑢(3.27) in 𝐿^𝑝(Ω,∧^𝑙). According to Definition 2.4, we know that _Ω𝑢_𝑗,𝑑^∗𝜂=_Ω𝑑𝑢_𝑗,𝜂,(3.28) for any 𝜂∈𝐶^∞(Ω,∧^𝑙). And it follows easily that _Ω𝑢,𝑑^∗𝜂=_Ω𝑑𝑢,𝜂,(3.29) since 𝑢_𝑗⇀𝑢 in 𝑊^1,𝑝(Ω,∧^𝑙−1) and Ω is bounded. Hence, we have 𝑢∈𝑊_𝑇^1,𝑝(Ω,∧^𝑙−1). The final step in our proof is to show that 𝑢 satisfies (1.7). Recall that _Ω𝐴𝑥,𝑔_𝑗+𝑑𝑢_𝑗,𝑑𝜙=_Ωℎ_𝑗,𝑑𝜙,(3.30) for all 𝜙∈𝑊_𝑇^1,𝑝(Ω,∧^𝑙−1). It is rather easy to obtain the following: _Ω𝐴𝑥,𝑔_𝑗+𝑑𝑢_𝑗−𝐴𝑥,𝑔+𝑑𝑢,𝑑𝜙≤𝛽_Ω𝑔_𝑗+𝑑𝑢_𝑗−𝑔+𝑑𝑢^𝑝−1𝑔𝑑𝜙≤𝛽_𝑗+𝑑𝑢_𝑗−𝑔+𝑑𝑢_𝑝^𝑝−1𝑑𝜙_𝑝≤𝛽𝑑𝜙_𝑝𝑔_𝑗−𝑔_𝑝^𝑝−1+𝑑𝑢_𝑗−𝑑𝑢_𝑝^𝑝−1→0,(3.31) as 𝑗→∞, that is, it holds _Ω𝐴𝑥,𝑔_𝑗+𝑑𝑢_𝑗,𝑑𝜙→_Ω𝐴𝑥,𝑔+𝑑𝑢,𝑑𝜙,(3.32) as 𝑗→∞. On the other hand, an easy computation gives that _Ωℎ_𝑗,𝑑𝜙→_Ωℎ,𝑑𝜙,(3.33) as 𝑗→∞. Therefore, we obtain _Ω𝐴𝑥,𝑔+𝑑𝑢,𝑑𝜙=_Ωℎ,𝑑𝜙,(3.34) for all 𝜙∈𝑊_𝑇^1,𝑝(Ω,∧^𝑙−1), which implies the desired result.

3.3. Weak Reverse Hölder Inequality

In virtue of the fact that 𝑑^∗(𝑑^∗)=0, we can write (1.1) as the following: 𝐴𝑥,𝑔+𝑑𝑢=ℎ+𝑑^∗𝑣,(3.35) for some 𝑣∈𝑊^1,𝑞(Ω,∧^𝑙+1). For 𝑣 in (3.35) we have 𝑑^∗𝑣=𝐴𝑥,𝑔+𝑑𝑢−ℎ≤𝛽𝑔+𝑑𝑢^𝑝−1+|ℎ|,(3.36) thus, by using the Minkowski inequality and Proposition 3.2, we have the estimate 𝑑^∗𝑣_𝑞≤𝑐𝑝,𝛼,𝛽𝑔_𝑝^𝑝/𝑞+‖ℎ‖_𝑞.(3.37)

For an arbitrary nonnegative test function 𝜂∈𝐶^∞₀(Ω), we multiply (3.35) for 𝜂^𝑝 and by the homogeneity assumption (3), and we obtain that 𝐴𝑥,𝜂^𝑞𝑔+𝜂^𝑞𝑑𝑢=𝜂^𝑝ℎ+𝜂^𝑝𝑑^∗𝑣.(3.38) Since 𝜂^𝑞𝑑𝑢=𝑑𝜂^𝑞𝑢−𝑑𝜂^𝑞𝜂∧𝑢,_𝑝𝑑^∗𝑣=𝑑^∗𝜂^𝑝𝑣−(−1)^𝑛𝑙+1⋆𝑑𝜂^𝑝∧⋆𝑣,(3.39) we have 𝐴𝑥,𝜂^𝑞𝑔−𝑑𝜂^𝑞∧𝑢+𝑑𝜂^𝑞𝑢=𝜂^𝑝ℎ−(−1)^𝑛𝑙+1⋆𝑑𝜂^𝑝∧⋆𝑣+𝑑^∗𝜂^𝑝𝑣.(3.40) Applying the operator 𝑑^∗ to the above equation leads to a new nonhomogeneous A-harmonic equation as follows: 𝑑^∗𝐴𝑥,𝜂^𝑞𝑔−𝑑𝜂^𝑞∧𝑢+𝑑𝜂^𝑞𝑢=𝑑^∗𝜂^𝑝ℎ−(−1)^𝑛𝑙+1⋆𝑑𝜂^𝑝∧⋆𝑣,(3.41) where 𝜂^𝑞𝑢∈𝑊₀^1,𝑝Ω,∧^𝑙−1,𝜂^𝑞𝑔−𝑑𝜂^𝑞∧𝑢∈𝐿^𝑝Ω,∧^𝑙,𝜂^𝑝ℎ−(−1)^𝑛𝑙+1⋆𝑑𝜂^𝑝∧⋆𝑣∈𝑊^1,𝑞Ω,∧^𝑙+1.(3.42) Then by applying Proposition 3.2 to (3.41) we have _Ω𝑑𝜂^𝑞𝑢^𝑝≤𝑐𝑝,𝛼,𝛽_Ω𝜂^𝑞𝑔−𝑑𝜂^𝑞∧𝑢^𝑝+_Ω𝜂^𝑝ℎ−(−1)^𝑛𝑙+1⋆𝑑𝜂^𝑝∧⋆𝑣^𝑞,(3.43) which implies _Ω𝜂^𝑞𝑑𝑢^𝑝=_Ω𝑑𝜂^𝑞𝑢−𝑑𝜂^𝑞∧𝑢^𝑝≤2^𝑝_Ω𝑑𝜂^𝑞∧𝑢^𝑝+2^𝑝_Ω𝑑𝜂^𝑞𝑢^𝑝≤2^𝑝_Ω𝑑𝜂^𝑞∧𝑢^𝑝+𝑐𝑝,𝛼,𝛽_Ω𝜂^𝑞𝑔^𝑝+_Ω𝑑𝜂^𝑞∧𝑢^𝑝+_Ω𝜂^𝑝ℎ^𝑞+_Ω𝑑𝜂^𝑝𝑞|𝑣|^𝑞.(3.44) Therefore, we obtain _Ω𝜂^𝑝+𝑞|𝑑𝑢|^𝑝≤𝑐𝑝,𝛼,𝛽_Ω|𝑢|^𝑝𝑑𝜂^𝑞𝑝+_Ω𝜂^𝑝+𝑞𝑔^𝑝+|ℎ|^𝑞+_Ω|𝑣|^𝑞𝑑𝜂^𝑝𝑞.(3.45) For an arbitrary cube 𝑄⊂ℝ^𝑛, we can choose a cut-off function 𝜂∈𝐶^∞₀(2𝑄) such that 0≤𝜂≤1, |∇𝜂|≤𝑐(𝑛)|𝑄|^−1/𝑛 and 𝜂≡1 on 𝑄, where |𝑄| denotes the Lebesgue measure of 𝑄. Now inequality (3.45) yields _𝑄|𝑑𝑢|^𝑝1≤𝑐𝑛,𝑝,𝛼,𝛽|𝑄|^{𝑝/𝑛2𝑄}|𝑢|^𝑝+1|𝑄|^{𝑞/𝑛2𝑄}|𝑣|^𝑞+_2𝑄𝑔^𝑝+|ℎ|^𝑞(3.46)

Note that (3.35) is not affected if a closed form 𝑢₀ is subtracted from 𝑢 and a coclosed form 𝑣₀ is subtracted from 𝑣. Therefore, the above calculation shows _𝑄|𝑑𝑢|^𝑝1≤𝑐𝑛,𝑝,𝛼,𝛽|𝑄|^{𝑝/𝑛2𝑄}𝑢−𝑢₀^𝑝+1|𝑄|^{𝑞/𝑛2𝑄}𝑣−𝑣₀^𝑞+_2𝑄𝑔^𝑝+|ℎ|^𝑞.(3.47)

Now, a Poincaré-Sobolev inequality for differential forms is needed.

Lemma 3.9 (see [2]). Let 𝑄 be a cube in ℝ^𝑛. Suppose that 𝜉∈𝑊^1,𝑟(𝑄,∧^𝑙) and 𝜁∈𝑊^1,𝑠(Ω,∧^𝑙), where 1<𝑟, 𝑠<𝑛. Then there exist a closed form 𝜉₀∈𝐿^𝑟(𝑄,∧^𝑙) and a coclosed form 𝜁^∗₀∈𝐿^𝑠(𝑄,∧^𝑙) such that 𝜉−𝜉_{0𝑛𝑟/(𝑛−𝑟)}≤𝑐(𝑛,𝑟)𝑑𝜉_𝑟,𝜁−𝜁^∗_{0𝑛𝑠/(𝑛−𝑠)}≤𝑐(𝑛,𝑠)𝑑^∗𝜁_𝑠.(3.48)

As a consequence of Lemma 3.9, we obtain the following by proceeding as in the proof of Corollary  3 in [5].

Corollary 3.10. Let 𝑄 be a cube in ℝ^𝑛. Suppose that 𝜉∈𝑊^1,𝑟(𝑄,∧^𝑙) and 𝜁∈𝑊^1,𝑠(Ω,∧^𝑙), where 1<𝑟, 𝑠<∞. Then there exist a closed form 𝜉₀∈𝐿^𝑟(𝑄,∧^𝑙) and a coclosed form 𝜁^∗₀∈𝐿^𝑠(𝑄,∧^𝑙) such that 1diam(𝑄)_𝑄𝜉−𝜉₀^𝑟1/𝑟≤𝑐(𝑛,𝑟)_𝑄𝑑𝜉^{𝑛𝑟/(𝑛+𝑟−1)(𝑛+𝑟−1)/𝑛𝑟},1(3.49)diam(𝑄)_𝑄𝜁−𝜁^∗₀^𝑠1/𝑠≤𝑐(𝑛,𝑠)_𝑄𝑑^∗𝜁^{𝑛𝑠/(𝑛+𝑠−1)(𝑛+𝑠−1)/𝑛𝑠},(3.50) where ⨍_𝑄 denotes the integral mean over 𝑄, that is, _𝑄=1|𝑄|_𝑄.(3.51)

It follows from (3.49) that _2𝑄𝑢−𝑢₀^𝑝d𝑥≤𝑐𝑛,𝑝|𝑄|^1/𝑛_2𝑄|𝑑𝑢|^{𝑛𝑝/(𝑛+𝑝−1)}d𝑥^{(𝑛+𝑝−1)/𝑛},(3.52) and we have by (3.50), Hölder inequality, and (3.37) that 𝑣−𝑣_0𝑞;2𝑄≤𝑐𝑛,𝑝|𝑄|^1/𝑛𝑞𝑑^∗𝑣_{𝑛𝑞/(𝑛+𝑞−1);2𝑄}≤𝑐𝑛,𝑝|𝑄|^1/𝑛𝑞|2𝑄|^{(𝑛+𝑞−1)/𝑛𝑞−1/𝑞}𝑑^∗𝑣_𝑞;2𝑄≤𝑐𝑛,𝑝,𝛼,𝛽|𝑄|^1/𝑛𝑔^𝑝/𝑞_𝑝;2𝑄+‖ℎ‖_𝑞;2𝑄.(3.53) Combining (3.52) and (3.53) with (3.47), we obtain that _𝑄|𝑑𝑢|^𝑝≤𝑐𝑛,𝑝,𝛼,𝛽|𝑄|^{(1−𝑝)/𝑛}_2𝑄|𝑑𝑢|^{𝑛𝑝/(𝑛+𝑝−1)(𝑛+𝑝−1)/𝑛}+_2𝑄𝑔^𝑝+|ℎ|^𝑞.(3.54) Therefore, we obtain that _𝑄|𝑑𝑢|^𝑝≤𝑐𝑛,𝑝,𝛼,𝛽_2𝑄|𝑑𝑢|^{𝑛𝑝/(𝑛+𝑝−1)(𝑛+𝑝−1)/𝑛}+_2𝑄𝑔^𝑝+|ℎ|^𝑞.(3.55) This is the desired estimate.

In conclusion, we summarize the above results in the following theorem.

Theorem 3.11. For (𝑝,𝑞) a Hölder conjugate pair. Given (𝑔,ℎ)∈𝐿^𝑝(Ω,∧^𝑙)×𝑊^1,𝑞(Ω,∧^𝑙), suppose that 𝑢∈𝑊_𝑇^1,𝑝(Ω,∧^𝑙−1) is a solution of (1.6). Then one has _𝑄|𝑑𝑢|^𝑝≤𝑐𝑛,𝑝,𝛼,𝛽_2𝑄|𝑑𝑢|^{𝑛𝑝/(𝑛+𝑝−1)(𝑛+𝑝−1)/𝑛}+_2𝑄𝑔^𝑝+|ℎ|^𝑞,(3.56) where ⨍_𝑄 denotes the integral mean over 𝑄, that is, _𝑄=1|𝑄|_𝑄.(3.57)

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11071048). The authors would like to express their sincere gratitude towards the reviewers for their efforts and valuable suggestions which greatly improved the paper.