On the Regularity of Solutions to an Adjoint Elliptic Equation with Partially VMO Coefficients

Abstract

We establish, in dimension two, a regularity result for nonnegative solutions to an adjoint elliptic equation, generalizing a previous result of Escauriaza (1994). We consider elliptic equations with coefficients 𝑎_𝑖𝑗(𝑥₁,𝑥₂) which are measurable with respect to one variable and VMO with respect to the other.

1. Introduction

Let us consider a planar elliptic operator of nondivergence form:ℳ=_𝑖𝑗𝑎_𝑖𝑗(𝑥)𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗,(1.1) where 𝑎_𝑖𝑗=𝑎_𝑗𝑖 for 𝑖,𝑗=1,2 are measurable and the symmetric matrix𝐴(𝑥)=𝑎¹¹𝑎(𝑥)𝑎₁₂(𝑥)₁₂(𝑥)𝑎₂₂(𝑥)(1.2) is uniformly elliptic, that is, ||𝜉||²√𝐾≤⟨𝐴(𝑥)𝜉,𝜉⟩≤√𝐾||𝜉||²,(1.3) for all 𝜉∈ℝ² and a.e. 𝑥=(𝑥₁,𝑥₂)∈Ω, a bounded open subset of ℝ². Here the ratio √𝐾/1/√𝐾=𝐾 is the ellipticity constant.

The study of weak solutions 𝑣 to the adjoint equation (adjoint solutions, for short)ℳ^∗[𝑣]=_𝑖𝑗𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗𝑎_𝑖𝑗(𝑥)𝑣(𝑥)=0(1.4) often occurs in the literature (see Section 3, and for a very recent paper, see [1]).

We say that the function 𝑣∈𝐿^1loc(Ω) is a weak solution to (1.4) if _Ω𝑣ℳ[𝜑]=0∀𝜑∈𝐶^∞0(Ω).(1.5) In this paper we make the assumption that the coefficients 𝑎_𝑖𝑗(𝑥₁,𝑥₂) are VMO with respect to one of the two variables (see Section 2). This kind of assumption has been recently considered mainly for divergence (𝐿[𝑢]=div(𝐴(𝑥)∇𝑢)=0) or nondivergence (𝑀[𝑤]=Tr(𝐴(𝑥)𝐷²𝑤)=0) elliptic equations.

On the other hand, in [2] Escauriaza gave a regularity result for nonnegative solutions to adjoint equation with VMO coefficients.

Here, in case 𝑛=2, we give a generalized form of Theorem  1.2 in which he proves that, in particular,𝑣∈_𝑞>1𝐺_𝑞,(1.6) where 𝐺_𝑞 is the Gehring class, as defined in Section 2.

2. Definitions and Notations

In order to describe the results of the present paper, it is necessary to introduce some definitions. We start recalling basic definitions of the 𝐺_𝑞 classes, introduced by Gehring [3], in connection with local integrability properties of the gradient of quasiconformal mappings.

Let us assume that 𝑣 is a weight, that is, a nonnegative locally integrable function on ℝ² and consider cubes 𝑄⊂ℝ² with sides parallel to the coordinate axes. We will set𝑣_𝑄=_Q𝑣(𝑥)𝑑𝑥=1||𝑄||_𝑄𝑣(𝑥)𝑑𝑥(2.1) to denote the mean value of 𝑣 over 𝑄, where |𝑄| denotes the 2-dimensional Lebesgue measure of a subset 𝑄 of ℝ².

Definition 2.1. A weight 𝑣 satisfies the 𝐺_𝑞-condition if there exists a constant 𝐺≥1 such that, for all cubes 𝑄⊂ℝ² as above, one has ⨍^𝑄𝑣^𝑞(𝑥)𝑑𝑥^1/𝑞⨍_𝑄𝑣(𝑥)𝑑𝑥≤𝐺,(2.2) and one refers to (2.2) as a “reverse” Hölder inequality.

In the following, we will consider elliptic differential equations with coefficients 𝑎_𝑖𝑗(𝑥) of the matrix 𝐴 measurable with respect to one variable and vanishing mean oscillation (VMO) with respect to the other (we say partially-VMO, for short). We recall that the space VMO, introduced by Sarason [4], is a subspace of the functions in the John-Nirenberg space BMO. More precisely, VMO is defined as the closure in BMO of the subspace of uniformly continuous functions.

Definition 2.2. A locally integrable function 𝑓∶ℝ²→ℝ is in VMO if lim_𝑟→0_𝐵𝑟||𝑓(𝑦)−𝑓_𝐵𝑟||𝑑𝑦=0,(2.3) where 𝐵_𝑟=𝐵(𝑥,𝑟) denotes a ball centered at 𝑥∈ℝ², with radius 𝑟. One will also assume that 𝑓 is defined at ∞ in the following average sense: 𝑓(∞)=lim_𝑟→∞1𝜋𝑟²_𝐵𝑟(0)𝑓(𝑦)𝑑𝑦,(2.4) (see [5]).

3. Examples

In the present section we collect a certain number of examples where solutions 𝑣 to the adjoint equationℳ^∗[𝑣]=_𝑖𝑗𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗𝑎_𝑖𝑗(𝑥)𝑣(𝑥)=0(3.1) occur. The first example deals with adjoint solutions which are partial derivatives ℎ_𝑥1 and ℎ_𝑥2 of a very weak solution ℎ∈𝑊^1,2loc to a particular diagonal divergence type equation, and an interesting relation comes out between regularity results.

Example 3.1. For 1/√𝐾≤𝛽≤√𝐾, we consider the following elliptic operators in ℝ²: ℳ=𝜕²𝜕𝑥²¹+𝛽𝜕²𝜕𝑥²²ℒ=𝜕₂,𝜕𝑥²¹+𝜕²𝜕𝑥₂𝛽𝜕𝜕𝑥₂.(3.2)
Fortuitous relations occur between adjoint solutions to ℳ and solutions ℎ to ℒ[ℎ]=0, as the following Lemma reveals (see [6]).

Lemma 3.2 (see [6]). Let ℎ∈𝑊^1,2loc(𝐵_𝑟), where 𝐵_𝑟 denotes the open ball in ℝ² centered at 0 with radius 𝑟, such that ℒ[ℎ]=0.(3.3) Set 𝑤=𝜕ℎ/𝜕𝑥₁, 𝑣=𝜕ℎ/𝜕𝑥₂ Then ℳ^∗[𝑤]=0,ℳ^∗[𝑣]=0.(3.4)

Proof. We proceed similarly as in [6]. If 𝜙∈𝐶^∞0(𝐵_𝑟), we have _𝐵𝑟𝑤⋅ℳ𝜙𝑑𝑥=_𝐵𝑟𝜕ℎ𝜕𝑥₁𝜕²𝜙𝜕𝑥²¹+𝛽𝜕²𝜙𝜕𝑥²²𝑑𝑥¹𝑑𝑥₂=_𝐵𝑟𝜕𝜕𝑥ℎ₁⋅𝜕²𝜙𝜕𝑥²¹+𝛽𝜕ℎ𝜕𝑥₂⋅𝜕²𝜙𝜕𝑥₁𝜕𝑥₂𝑑𝑥¹𝑑𝑥₂=0.(3.5) Thus ℳ^∗[𝑤]=0. In analogous way one checks that ℳ^∗[𝑣]=0.

Corollary 3.3. Let ℎ∈𝑊^1,2loc such that ℒ[ℎ]=0. If 𝜕ℎ/𝜕𝑥₁≥0 and 𝜕ℎ/𝜕𝑥₂≥0, then for any ball 𝐵_𝑟⊂𝐵_2𝑟⊂ℝ², one has _𝐵𝑟|∇ℎ|^𝑝𝑑𝑥^1/𝑝≤𝑐(𝐾,𝑝)_𝐵𝑟||∇ℎ||𝑑𝑥,(3.6) where 2≤𝑝<2𝐾/(𝐾−1).

Proof. See [7, Theorem  3.1].

Compare this with the following well-known result of Astala [8] (see also Leonetti-Nesi [9]).

Theorem 3.4 (see [8, 9]). Let ℎ∈𝑊^1,2loc(Ω) be a local solution to the equation div(𝐴(𝑥)∇ℎ(𝑥))=0,𝑥∈Ω,(3.7) where 𝐴 is a real symmetric matrix satisfying the ellipticity bounds, ||𝜉||²√𝐾≤⟨𝐴(𝑥)𝜉,𝜉⟩≤√𝐾||𝜉||²,∀𝜉∈ℝ²,fora.e.𝑥∈Ω.(3.8) Then, for any ball 𝐵_2𝑟⊂Ω one has _𝐵𝑟|∇ℎ|^𝑠𝑑𝑥^1/𝑠≤𝑐(𝐾,𝑠)_𝐵2𝑟||∇ℎ||𝑑𝑥,(3.9) where 2≤𝑠<2√𝐾/(√𝐾−1).

Notice that, while the exponent in the left-hand side of the reverse inequality (3.9) may be greater than the exponent in the reverse inequality (3.6), this one is stronger in another sense, because it involves the same support 𝐵_𝑟 at both sides.

Example 3.5. In [10] (see also [11]) the Jacobian 𝑣=det𝐷𝑈, where 𝑈∶Ω⊂ℝ²→ℝ² is a locally univalent 𝐴-harmonic mapping; that is, its components are 𝑊^1,2loc solution to (3.7), is shown to be solution to an adjoint equation for the elliptic operator ℳ[𝑣]=𝜕²𝑣𝜕𝑥²¹+𝑐𝜕²𝑣𝜕𝑥²²,(3.10) where 1/𝐾≤𝑐≤𝐾.

Example 3.6. Very recently [12, 13], the reduced Beltrami differential equation 𝜕𝑓𝜕𝑧=𝜆(𝑧)ℐ𝑚𝜕𝜕𝑓𝑧,||𝜆(𝑧)||≤𝑘<1,𝑘=𝐾𝐾−+11(3.11) has been introduced and studied because it naturally arises in different contexts in the theory of quasiconformal mappings. It turns out that the partial derivatives of the components 𝑢,𝑣 of 𝑓(𝑧)=𝑢(𝑧)+𝑖𝑣(𝑧) solution to (3.11) satisfy the equation 𝑢_𝑥2=𝑏(𝑧)−1𝑏(𝑧)+1𝑣^𝑥1(3.12) with 𝑏(𝑧)=ℐ𝑚𝜆(𝑧). As a consequence of (3.12) in [13], it is proved that 𝑢_𝑥2≠0 a.e, and it is an adjoint solution for a suitable elliptic operator ℳ=∑_𝑖𝑗𝑏_𝑖𝑗(𝑧)(𝜕²/𝜕𝑥_𝑖𝜕𝑥_𝑗).

Namely, it has been proved [13] that 𝑢 is a solution to an elliptic equation of divergence formdiv𝐴(𝑧)∇𝑢=0,(3.13) where 𝐴(𝑧) is of the type𝐴(𝑧)=1𝑎¹²0𝑎₂₂(𝑧)(𝑧).(3.14) As a consequence, the function 𝑣=𝑢_𝑥2 is a solution to the adjoint equationℳ^∗[𝑣]=0,(3.15) whereℳ=𝜕²𝜕𝑥²¹+𝑎₁₂𝜕²𝜕𝑥₁𝜕𝑥₂+𝑎₂₂𝜕²𝜕𝑥²².(3.16) Note that the matrix 𝐴(𝑧) is not symmetric; however, the operator ℳ can also be represented by the symmetric and uniformly elliptic matrix𝐵(𝑥)=⎛⎜⎜⎝1𝑎¹²(𝑧)𝑎₁₂2(𝑧)2𝑎²²(𝑧)⎞⎟⎟⎠.(3.17) Notice also that 𝑣=𝑢_𝑥2>0 a.e. (see [13]), and moreover, by general properties of nonnegative adjoint solutions, 𝑣 satisfies a reverse Hölder inequality [7, 14, 15]_𝐵𝑟𝑣(𝑧)²𝑑𝑧^1/2≤𝑐(𝐾)_𝐵𝑟𝑣(𝑧)𝑑𝑧,(3.18) in every ball 𝐵_𝑟⊂Ω such that 𝐵_2𝑟⊂Ω. Hence 𝑣 is identically zero or 𝑣>0 a.e. [13].

Example 3.7. The properties of the adjoint solutions are also very useful for studing the 𝐺-convergence of non divergence operators, as shown, for example, in a paper of D’Onofrio and Greco [16]. In that paper the authors consider elliptic operators ℳ of non divergence type, defined by ℳ[𝑢]=Tr𝐴𝐷²𝑢,for𝑢∈𝑊^2,2(Ω),Ω⊂ℝ²,(3.19) where 𝐴=(𝑎_𝑖𝑗)∈𝕄, the set of all symmetric 2×2 real matrices and satisfy the ellipticity condition (1.3).
The adjoint to the operator ℳ is given by ℳ^∗[𝑣]=(𝑎₁₁𝑣)_𝑥1𝑥2+2(𝑎₁₂𝑣)_𝑥1𝑥2+(𝑎₂₂𝑣)_𝑥2𝑥2 and reveals useful behaviour with respect to 𝐺-convergence of sequence of operators of the form (3.19).

Proposition 3.8 (see[16]). Let ℳ_𝑘, 𝑘=1,2,…,ℳ be operators whose coefficient matrices 𝐴_𝑘, 𝐴∈𝕄 and satisfying (1.3). Assume that 𝑣_𝑘∈𝐿²(Ω) are solutions to the adjoint equations ℳ^∗𝑘[𝑣_𝑘]=0 and verify that 𝑣𝑣_𝑘⇀𝑣in𝐿²(Ω),_𝑘𝐴_𝑘⇀𝑣𝐴in𝐿²(Ω;𝕄),(3.20) where 𝑣(𝑥)>0 a.e. in Ω. Then, one has ℳ−→_𝑘^Gℳ.

In order to prove Proposition 3.8, the following lemma is crucial.

Lemma 3.9 (see [16]). Let ℳ_𝑘, 𝑘=1,2,…,ℳ be operators with coefficient matrices 𝐴_𝑘, 𝐴∈𝕄 and satisfy (1.3), 𝑣_𝑘∈𝐿²(Ω) satisfying ℳ^∗𝑘[𝑣_𝑘]=0, and let 𝑣_𝑘∈𝑊^2,2loc(Ω) be given. If 𝑣𝑢_𝑘⇀𝑢in𝑊^2,2loc(Ω),_𝑘𝐴_𝑘⇀𝑣𝐴in𝐿^2loc(Ω;𝕄),(3.21) then Tr𝑣_𝑘𝐴_𝑘𝐷²𝑢_𝑘⟶Tr𝑣𝐴𝐷²𝑢(3.22) in the sense of distributions.

Moreover, if we consider the Hessian matrix of any 𝑤∈𝑊^2,2(Ω),𝐷²𝑤=𝑤^𝑥1𝑥1𝑤_𝑥1𝑥2𝑤_𝑥1𝑥2𝑤_𝑥2𝑥2.(3.23) In [17] it is proved that 𝑤 is a solution toℳ[𝑤]=Tr𝐵𝐷²𝑤=0,(3.24) where 𝐵 is a suitable coefficient matrix, if and only if𝑤^2𝑥_1𝑥2−𝑤_𝑥1𝑥1𝑤_𝑥2𝑥2𝐾+𝐾1≥𝑤^2𝑥1𝑥1+2𝑤^2𝑥_1𝑥2+𝑤^2𝑥_2𝑥2,(3.25) where √𝐾, 𝐾≤1, is the elliptic constant. In the case where the Hessian matrix is diagonal, that is, 𝑤_𝑥1𝑥2=0, it is easy to see that a solution of ℳ^∗[𝑣]=0 is the positive function 𝑣=√−det𝐷²𝑤=|𝑤_𝑥1𝑥1𝑤_𝑥2𝑥2|.

4. The Coefficients Measurable with Respect to One Variable and VMO with Respect to the Other

It is well known that, for linear elliptic operators in nondivergence form with continuous coefficients, the 𝑊^2,𝑝 estimates hold for all 𝑝>1. It was shown that these estimates still hold in the same range when the coefficients are in VMO [18] or partially in VMO [19]. Our aim here is to generalize a regularity result of Escauriaza (Theorem  1.2, [2]) for the nonnegative adjoint solutions 𝑣 toℳ^∗[𝑣]=0inΩ⊂ℝ²,(4.1) as defined in (1.4), with||𝜉||₂𝐴=𝑎_𝑖𝑗=^𝑡𝐴,√𝐾≤⟨𝐴(𝑥)𝜉,𝜉⟩≤√𝐾||𝜉||²,(4.2) for a.e. 𝑥=(𝑥₁,𝑥₂)∈Ω and for 𝜉∈ℝ².

Theorem 4.1. If 𝑣∈𝐿¹(Ω) is a nonnegative solution to (4.1) and the coefficient matrix 𝐴(𝑥) satisfies (4.2), and moreover 𝐴𝑥₁,⋅∈VMO,(4.3) then 𝑣∈_𝑞>1𝐺_𝑞.(4.4)

Let us begin with the following 𝐿^𝑝-global regularity (for all 𝑝≥2) result for the complex Beltrami equation 𝐹_𝑧+𝜇(𝑧)𝐹_𝑧+𝜇(𝑧)𝐹_𝑧=𝐻(𝑧)𝑧∈ℝ²,(4.5) under a partially-VMO assumption on the Beltrami coefficients 𝜇, as defined in Section 2.

Proposition 4.2. Let 𝐵_𝑟=𝐵(0,𝑟)⊂ℝ², and let 𝜇(𝑧)=𝜇(𝑥₁,𝑥₂) be measurable, such that |𝜇(𝑧)|+|𝜇(𝑧)|=2|𝜇(𝑧)|≤𝑘<1 with 𝑘=(𝐾−1)/(𝐾+1) and 𝜇(𝑧)=0 for |𝑧|≥𝑟>0. Moreover, assume that 𝜇(𝑥₁,⋅)∈VMO(ℝ,ℝ) for a.e. 𝑥₁∈ℝ. Then for any 𝑝≥2 and for 𝐻∈𝐿^𝑝(ℝ²)(𝐻(𝑧)=0 for |𝑧|>𝑟), there exists a unique solution 𝐹 to the Beltrami equation (4.5) such that 𝐹_𝑧∈𝐿^𝑝 and ‖‖𝐹_𝑧‖‖_{𝐿𝑝(ℝ2)}≤𝑐(𝑝,𝑘)‖𝐻‖_𝐿^𝑝_(ℝ²₎.(4.6)

Remark 4.3. We note that in general, elliptic Beltrami operator 𝐼−𝜇𝑇−𝜈𝑇,(4.7) where 𝑇 is the Beurling transform defined via the relation 𝑇𝐹_𝑧=𝐹_𝑧(4.8) under the assumption 𝜇,𝜈∈VMOℂ,(4.9) is invertible in all 𝐿^𝑝(ℂ) spaces, 𝑝>1. The proof is much the same [5, 20], considering the complex Beltrami equation 𝐹_𝑧−𝜇(𝑧)𝐹_𝑧−𝜈(𝑧)𝐹_𝑧=ℎ,ℎ∈𝐿^𝑝(ℂ).(4.10)
The meaning of the condition (4.9) is that 𝜇 and 𝜈 have vanishing mean oscillation in the usual sense, that is, belong to the closure of 𝐶^∞0(ℂ) in BMO(ℂ) and that 𝜇 and 𝜈 are defined at infinity in the following average sense: 𝜇(∞)=lim_𝑟→∞_𝐵𝑟(0)𝜈(∞)=lim_𝑟→∞𝜇(𝑧)||𝑑𝑧||²,_𝐵𝑟(0)𝜈(𝑧)||𝑑𝑧||².(4.11)
The following example, due to T. Iwaniec, shows that without such condition the result fails.

Example 4.4. There exists a function 𝑓∈VMO(ℝ^𝑛), 0≤𝑓(𝑥)≤1 everywhere, such that 0=liminf_|𝐵|→∞_𝐵𝑓(𝑥)𝑑𝑥<limsup_{||𝐵||→∞}_𝐵𝑓(𝑥)𝑑𝑥=1,(4.12) where 𝐵 stands for a ball centered at the origin.

Preliminaries
Let 𝐿∶ℝ→[0,1] be a Lipschitz function given by 𝐿(𝑡)=121+|𝑡|−||𝑡−1||.(4.13) The Lipschitz constant of 𝐿 equals 1, and, therefore, for each 𝜑∈BMO(ℝ^𝑛), we have ‖𝐿∘𝜑‖_BMO≤2‖𝜑‖_BMO.(4.14) Next, denote by 𝐶(𝑛) the BMO-norm of the function 𝑥→log|𝑥|. We will truncate this function to make building blocks to our construction.

The Building Blocks
For a nonnegative integer 𝑘, we set 𝜑𝜓_𝑘(𝑥)=𝐿7−2^−𝑘log|𝑥|_𝑘,(𝑥)=𝐿5−2^−𝑘log|𝑥|.(4.15) We define the building block as 𝑓_𝑘=𝜑_𝑘−𝜓_𝑘. Note that each 𝑓_𝑘 is continuous and supported in the ball |𝑥|≤𝑒^7⋅2𝑘, whereas 𝑓_𝑘+1 vanishes on this ball. The BMO-norm of 𝑓_𝑘 can be estimated as ‖‖𝑓_𝑘‖‖_BMO≤‖‖𝜑_𝑘‖‖_BMO+‖‖𝜓_𝑘‖‖_BMO≤2⋅2^−𝑘𝐶(𝑛)+2⋅2^−𝑘𝐶(𝑛)=2^−𝑘+2𝐶(𝑛).(4.16) Thus the infinite series 𝑓=^∞_𝑘=0𝑓_𝑘(4.17) represents a VMO function.

Computation of 𝐿¹-Averages
Given any positive integer 𝑁, we consider concentric balls 𝐵_𝑟⊂𝐵_𝑅 centered at the origin and with radii 𝑟=𝑒^4⋅2𝑁<𝑒^6⋅2𝑁=𝑅. Elementary geometric observation reveals that _𝐵𝑅𝑓≥1||𝐵_𝑅||_𝐵𝑅𝑓_𝑁≥1||𝐵_𝑅||_{𝑒𝑁5⋅2≤|𝑥|≤𝑒𝑁6⋅2}𝑓_𝑁=𝑒^6𝑛2𝑁−𝑒^5𝑛2𝑁𝑒^6𝑛2𝑁=1−𝑒^−𝑛2𝑁⟶1,as𝑁⟶∞.(4.18) On the other hand _𝐵𝑅𝑓=1||𝐵_𝑅||^𝑁−1_𝑘=1_{|𝑥|≤𝑒𝑁4⋅2}𝑓_𝑘(𝑥)𝑑𝑥≤1||𝐵_𝑅||^𝑁−1_𝑘=1_{|𝑥|≤𝑒𝑘7⋅2}≤𝑑𝑥_𝑁−1_𝑘=1𝑒^7𝑛2𝑘𝑒^4𝑛2𝑁≤(𝑁−1)𝑒^{7𝑛2𝑁−1}𝑒^4𝑛2𝑁=(𝑁−1)𝑒^{−𝑛2𝑁−1}⟶0,as𝑁⟶∞,(4.19) as desired.

Proof of Proposition 4.2. If we set 𝐾=(1+𝑘)/(1−𝑘), then there exists a symmetric matrix 𝐴(𝑧)=𝑎_𝑖𝑗(𝑧)(4.20) such that 𝐼√𝐾≤𝐴(𝑧)≤√𝜇(𝑧)=1𝐾𝐼,2𝑎¹¹(𝑧)−𝑎₂₂(𝑧)+2𝑎₁₂(𝑧)𝑖𝑎₁₁(𝑧)+𝑎₂₂(𝑧).(4.21) Moreover 𝑎_𝑖𝑗(𝑧)=0 for |𝑧|≥𝑟.
We may assume the following familiar normalization: Tr𝐴(𝑧)=𝑎₁₁(𝑧)+𝑎₂₂(𝑧)=1.(4.22) With the previous prescriptions we easily check that for 𝑤∈𝑊^2,1loc(𝐵_𝑟) if we define the complex gradient of 𝑤 as 𝐹(𝑧)=𝑤_𝑧=12𝑤^𝑥1−𝑖𝑤_𝑥2,(4.23) we have Tr𝐴(𝑧)𝐷²𝑤=2𝐹_𝑧+𝜇𝐹_𝑧+𝜇𝐹_𝑧(4.24) (see [5]). Hence (4.5) is equivalent to Tr𝐴(𝑧)𝐷²𝑤=𝐻inℝ²(4.25) with coefficient matrix 𝐴(𝑧)=𝐴(𝑥₁,𝑥₂) allowed to be only measurable with respect to 𝑥₁ and VMO with respect to 𝑥₂∈ℝ, (thanks to (4.22)).
Under these assumptions, in [19, Theorem 2.4], the existence of a unique solution 𝑤∈𝑊^2,𝑝 to (4.25) for 𝐻∈𝐿^𝑝 has been established (𝑝≥2), together with the estimate ‖‖𝐷²𝑤‖‖_{𝐿𝑝(𝐵𝑟)}≤𝑐(𝑝,𝑘)‖𝐻‖_𝐿^𝑝_(𝐵𝑟).(4.26) Hence (4.6) follows.

Let us now give the following sharp version of the Alexandrov-Bakelman-Pucci maximum principle for non divergence elliptic operatorsℳ[𝑤]=𝑎_𝑖𝑗𝜕²𝑤𝜕𝑥_𝑖𝜕𝑥_𝑗(4.27) with partially VMO coefficients.

Lemma 4.5. Under the assumptions (4.2), (4.3), (4.22) on 𝐴, suppose 𝐵_𝑟=𝐵(0,𝑟)⊂ℝ² and that 𝑤∈𝑊^2,1loc(𝐵_𝑟)∩𝐶⁰(𝐵_𝑟) satisfies, for ℎ∈𝐿^𝑝(𝐵_𝑟), 𝑝>1, ℳ𝑤[𝑤=]0=ℎ𝑜𝑛𝜕𝑖𝑛𝐵𝐵_𝑟𝑟,.(4.28) Then one has ‖𝑤‖_𝐿^∞_(𝐵𝑟)≤𝑐(𝐾,𝑝)𝑟^2−2/𝑝‖ℎ‖_𝐿^𝑝_(𝐵𝑟).(4.29)

Proof. In view of [19, Theorem  2.4], we know that the Dirichlet problem (4.28) always has a unique solution 𝑤∈𝑊^2,𝑝(𝐵_𝑟)∩𝑊^01,2(𝐵_𝑟) for every ℎ∈𝐿^𝑝(𝐵_𝑟), 𝑝>1.
Define ℎ(𝑧)=0 for 𝑧=(𝑥₁,𝑥₂)∈ℝ²⧵𝐵_𝑟 and 𝜇(𝑧)=12𝑎¹¹(𝑧)−𝑎₂₂(𝑧)+2𝑎₁₂(𝑧)𝑖,(4.30) and set 𝜇(𝑧)=0 for 𝑧∈ℝ²⧵𝐵_𝑟. According to Proposition 4.2, the equation 𝐹_𝑧+𝜇𝐹_𝑧+𝜇𝐹_𝑧=ℎ2(4.31) has unique solution 𝐹∈𝑊^1,𝑝loc(𝐵_𝑟) such that ‖‖𝐹_𝑧‖‖_{𝐿𝑝(ℝ2)}≤𝑐(𝑝,𝐾)‖ℎ‖_𝐿^𝑝_(ℝ²₎.(4.32)
Now, let us see that 𝐹_|𝐵𝑟∈𝑊^1,2loc𝐵_𝑟.(4.33) Define 𝑓=𝑤_𝑧=(1/2)(𝑤_𝑥1−𝑖𝑤_𝑥2). Then (𝑓−𝐹)_𝑧+𝜇(𝑓−𝐹)_𝑧+𝜇(𝑓−𝐹)_𝑧=0in𝐵_𝑟.(4.34) This means that the mapping 𝑔=𝑓−𝐹(4.35) is weakly 𝐾-quasiregular, and since 𝐹∈𝑊^1,𝑝loc and 𝑓∈𝑊^1,2loc, we deduce 𝑔∈𝑊^1,𝑝loc and actually 𝑔 is 𝐾-quasiregular and in particular 𝑔∈𝑊^1,2loc𝐵_𝑟.(4.36) Then (4.33) follows.
Now, let us introduce the solution 𝑈 to the problem Δ𝑈𝑈(0=)=4𝐹0_𝑧.,(4.37) We have Δ𝑈∈𝐿^𝑝(ℝ²)∩𝐿^2loc(𝐵_𝑟) and ℳ[𝑈]=ℎ a.e. 𝑧∈ℝ². Moreover, classically ‖Δ𝑈‖_𝐿^𝑝_(ℝ²₎≤𝑐(𝐾,𝑝)‖ℎ‖_𝐿^𝑝.(4.38) (Notice that 𝑐(𝐾,𝑝) is optimal in Talenti [21].)
Finally, let us introduce 𝑢=𝑈−𝑤.(4.39) Then 𝑢 is continuous in 𝐵_𝑟 by the Sobolev imbedding, and it is the solution to the Dirichlet problem ℳ[𝑢]=𝑢0_{/𝜕𝐵𝑟}a.e.𝑧∈𝐵_𝑟,=𝑈,(4.40) and 𝑢∈𝑊^2,2loc(𝐵_𝑟)∩𝐶⁰(𝐵_𝑟). By the classical maximum principle [22], ‖𝑈−𝑤‖_𝐿^∞_(𝐵𝑟)=‖𝑈‖_𝐿^∞_{(𝜕𝐵𝑟)}.(4.41) Hence, we use Sobolev and the condition 𝑈(0)=0 to conclude ‖𝑤‖_𝐿^∞_(𝐵𝑟)≤2‖𝑈‖_𝐿^∞_(𝐵𝑟)≤𝑐𝑟^2−2/𝑝‖Δ𝑈‖_𝐿^𝑝_(𝐵𝑟)≤𝑐𝑟^2−2/𝑝‖ℎ‖_𝐿^𝑝_(𝐵𝑟).(4.42)

Proof of Theorem 4.1. Let us fix 𝑞>1, set 𝑝=𝑞/(𝑞−1), and fix a ball 𝐵_𝑟 such that 𝐵_2𝑟⊂Ω. As in [7, 15], we make use of the dual formulation of the 𝐿^𝑞-norm _𝐵𝑟𝑣^𝑞^1/𝑞=sup_𝐵𝑟𝑣ℎ:ℎ≥0,ℎ∈𝐶¹⁰𝐵_𝑟,‖ℎ‖_𝐿^𝑝_(ℝ²₎≤1.(4.43) Fix ℎ∈𝐶¹⁰(𝐵_𝑟),‖ℎ‖_𝐿^𝑝_(ℝ²₎≤1,ℎ≥0, and solve the Dirichlet problem ℳ[𝑤]=𝑎_𝑖𝑗𝜕²𝑤𝜕𝑥_𝑖𝜕𝑥_𝑗𝑤=0on𝜕𝐵=_2𝑟ℎin𝐵_2𝑟,.(4.44)
Since 𝐴(𝑥₁,⋅)∈VMO, the problem has a unique solution 𝑤∈𝑊^2,𝑝(𝐵_2𝑟) vanishing on 𝜕𝐵_2𝑟, satisfying the estimate ‖‖𝐷²𝑤‖‖_{𝐿𝑝(𝐵2𝑟)}≤𝑐‖ℎ‖_𝐿^𝑝_(𝐵2𝑟),(4.45)𝑐=𝑐(𝐾,𝑝,‖𝐴‖_VMO).
Fix a nonnegative function 𝜑_𝑟∈𝐶¹⁰(𝐵_3𝑟/2) such that 𝜑_𝑟=1 on 𝐵_𝑟 and |𝜕^𝛼𝜑_𝑟/𝜕𝑥^𝛼|≤𝐶(𝛼)/𝑟^|𝛼|.
Then, we have _𝐵𝑟𝑣ℎ≤≤𝑐_𝐵2𝑟𝑣ℳ[𝑤]𝜑_𝑟=−_𝐵2𝑟𝑣𝑤ℳ𝜑_𝑟−2_𝐵2𝑟𝑣⟨𝐴∇𝑤,∇𝜑_𝑟⟩𝑟²‖𝑤‖_𝐿^∞𝐵_2𝑟_𝐵3𝑟/2𝑣+𝑐√𝐾𝑟_𝐵3𝑟/2𝑣||∇𝑤||.(4.46) By (4.29) ‖𝑤‖_𝐿^∞_(𝐵2𝑟)≤𝑐(𝐾,𝑝)𝑟^2−2/𝑝‖ℎ‖_𝐿^𝑝_(𝐵2𝑟)≤𝑐𝑟^2/𝑞; hence, (4.46) implies _𝐵𝑟𝑣ℎ≤c𝑟²𝑟^2/𝑞_𝐵3𝑟/2𝑣+𝑐𝑟_𝐵3𝑟/2𝑣^1/2_𝐵2𝑟𝑣||∇𝑤||²^1/2.(4.47)
Now, we estimate the last integral in the right-hand side. By (1.3), one has _𝐵2𝑟𝑣||∇𝑤||²≤√𝐾_𝐵2𝑟𝑣⟨𝐴∇𝑤,∇𝑤⟩=√𝐾_𝐵2𝑟𝑣𝑀𝑤²2−𝑤ℎ.(4.48) Since 𝑤²=0 and ∇(𝑤²)=0 on 𝜕𝐵_2𝑟, then we deduce _𝐵2𝑟𝑣ℳ𝑤²=0asℳ^∗[𝑣]=0.(4.49) Using again (4.29) yields _𝐵2𝑟𝑣||∇𝑤||²≤2√𝐾_𝐵2𝑟𝑣|𝑤|ℎ≤2√𝐾‖𝑤‖_𝐿^∞_(𝐵2𝑟)_𝐵𝑟≤2√𝑣ℎ𝐾𝑐𝑟^2/𝑞_𝐵𝑟𝑣ℎ.(4.50)
By (4.47) and (4.50), it follows that _𝐵𝑟𝑣ℎ≤𝑐𝑟^{2(1−1/𝑞)}_𝐵3𝑟/2𝑣+𝑐𝑟^(1−1/𝑞)_𝐵3𝑟/2𝑣^1/2_𝐵𝑟𝑣ℎ^1/2.(4.51) By elementary inequality √𝑎√𝑏≤𝑎/2+𝑏/2, we obtain _𝐵𝑟𝑣ℎ≤𝑐𝑟^{2(1−1/𝑞)}_𝐵3𝑟/2𝑣+𝑐𝑟^{2(1−1/𝑞)}_𝐵3𝑟/2𝑣+12_𝐵𝑟𝑣ℎ.(4.52)
Rearranging yields _𝐵𝑟𝑣ℎ≤𝑐𝑟^{2(1−(1/𝑞))}_𝐵3𝑟/2𝑣.(4.53)
Since ℎ is arbitrary, by (4.43) and (4.53), we obtain _𝐵𝑟𝑣^𝑝^1/𝑝≤𝑐_𝐵3𝑟/2𝑣,(4.54) with 𝑐=𝑐(𝐾,𝑝). An application of [14, Lemma  2.0] concludes the proof.

Acknowledgment

The author wish to thank Professor T. Iwaniec for letting her include in this paper Example 4.4, which is a simplified variant of a construction by P. Jones.