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

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 𝑎𝑖𝑗(𝑥1,𝑥2) which are measurable with respect to one variable and VMO with respect to the other.


Introduction
Let us consider a planar elliptic operator of nondivergence form: where a ij a ji for i, j 1, 2 are measurable and the symmetric matrix A x a 11 x a 12 x a 12 x a 22 x 1.2 is uniformly elliptic, that is, for all ξ ∈ R 2 and a.e.x x 1 , x 2 ∈ Ω, a bounded open subset of R 2 .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 often occurs in the literature see Section 3, and for a very recent paper, see 1 .
We say that the function v ∈ L 1 loc Ω is a weak solution to 1.4 if In this paper we make the assumption that the coefficients a ij x 1 , x 2 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 D 2 w 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, where G q is the Gehring class, as defined in Section 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 R 2 and consider cubes Q ⊂ R 2 with sides parallel to the coordinate axes.We will set to denote the mean value of v over Q, where |Q| denotes the 2-dimensional Lebesgue measure of a subset Q of R 2 .
Definition 2.1.A weight v satisfies the G q -condition if there exists a constant G ≥ 1 such that, for all cubes Q ⊂ R 2 as above, one has and one refers to 2.2 as a "reverse" H ölder 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
where B r B x, r denotes a ball centered at x ∈ R 2 , with radius r.One will also assume that f is defined at ∞ in the following average sense: f y dy, 2.4 see 5 .

Examples
In the present section we collect a certain number of examples where solutions v to the adjoint equation

3.2
Fortuitous relations occur between adjoint solutions to M and solutions h to L h 0, as the following Lemma reveals see 6 .
Proof.We proceed similarly as in 6 .If φ ∈ C ∞ 0 B r , we have

3.5
Thus M * w 0. In analogous way one checks that M * v 0.
Compare this with the following well-known result of Astala 8 see also Leonetti-Nesi 9 .
loc Ω be a local solution to the equation where A is a real symmetric matrix satisfying the ellipticity bounds, Then, for any ball B 2r ⊂ Ω one has |∇h|dx, 3.9 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 det DU, where U : Ω ⊂ R 2 → R 2 is a locally univalent A-harmonic mapping; that is, its components are W 1,2 loc solution to 3.7 , is shown to be solution to an adjoint equation for the elliptic operator where 1/K ≤ c ≤ K.
Example 3.6.Very recently 12, 13 , the reduced Beltrami differential equation 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 12 with b z Imλ z .As a consequence of 3.12 in 13 , it is proved that u x 2 / 0 a.e, and it is an adjoint solution for a suitable elliptic operator Namely, it has been proved 13 that u is a solution to an elliptic equation of divergence form div A z ∇u 0, 3.13 where A z is of the type As a consequence, the function v u x 2 is a solution to the adjoint equation where

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 matrix 3.17 Notice also that v u x 2 > 0 a.e.see 13 , and moreover, by general properties of nonnegative adjoint solutions, v satisfies a reverse H ölder inequality 7, 14, 15 Example 3.7.The properties of the adjoint solutions are also very useful for studing the Gconvergence 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 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 a

3.20
where v x > 0 a.e. in Ω.Then, one has In order to prove Proposition 3.8, the following lemma is crucial.
in the sense of distributions.
Moreover, if we consider the Hessian matrix of any w ∈ W 2,2 Ω ,

3.23
In 17 it is proved that w is a solution to where B is a suitable coefficient matrix, if and only if where √ K, K ≤ 1, is the elliptic constant.In the case where the Hessian matrix is diagonal, that is, w x 1 x 2 0, it is easy to see that a solution of

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 to Ω is a nonnegative solution to 4.1 and the coefficient matrix A x satisfies 4.2 , and moreover Let us begin with the following L p -global regularity for all p ≥ 2 result for the complex Beltrami equation under a partially-VMO assumption on the Beltrami coefficients μ, as defined in Section 2. 4.6 Journal of Function Spaces and Applications 9 Remark 4.3.We note that in general, elliptic Beltrami operator where T is the Beurling transform defined via the relation under the assumption is invertible in all L p C spaces, p > 1.The proof is much the same 5, 20 , considering the complex Beltrami equation 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:

4.11
The following example, due to T. Iwaniec, shows that without such condition the result fails.

Example 4.4. There exists a function
where B stands for a ball centered at the origin.

Preliminaries
Let L : R → 0, 1 be a Lipschitz function given by

Journal of Function Spaces and Applications
The Lipschitz constant of L equals 1, and, therefore, for each ϕ ∈ BMO R n , we have 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

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•2 k , whereas f k 1 vanishes on this ball.The BMO-norm of f k can be estimated as Thus the infinite series represents a VMO function.

Computation of L 1 -Averages
Given any positive integer N, we consider concentric balls B r ⊂ B R centered at the origin and with radii r e 4•2 N < e 6•2 N R. Elementary geometric observation reveals that

4.18
On the other hand

4.32
Now, let us see that

4.34
This means that the mapping

Lemma 3 .2 see 6 .
Let h ∈ W 1,2 loc B r , where B r denotes the open ball in R 2 centered at 0 with radius r, such that

Proposition 4 . 2 .
Let B r B 0, r ⊂ R 2 , and let μ z μ x 1 , x 2 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 x 1 , • ∈ VMO R, R for a.e.x 1 ∈ R. Then for any p ≥ 2 and for H ∈ L p R 2 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 11 v x 1 x 2 2 a 12 v x 1 x 2 a 22 v x 2 x 2 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 ∈ L 2 Ω are solutions to the adjoint equations M * , x 2 allowed to be only measurable with respect to x 1 and VMO with respect to x 2 ∈ 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 Under the assumptions 4.2 , 4.3 , 4.22 on A, suppose B r B 0, r ⊂ R 2 and that w ∈ W 2,1 loc B r ∩ C 0 B r satisfies, for h ∈ L p B r , p > 1, ∞ B r ≤ 2 U L ∞ B r ≤ cr 2−2/p ΔU L p B r ≤ cr 2−2/p h L p B r .4.42Proof 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 B r , h L p R 2 ≤ 1, h ≥ 0, and solve the Dirichlet problem , • ∈ VMO, the problem has a unique solution w ∈ W 2,p B 2r vanishing on ∂B 2r , satisfying the estimate VMO .Fix a nonnegative function ϕ r ∈ C 1 0 B 3r/2 such that ϕ r 1 on B r and |∂ α ϕ r /∂x α | ≤ C α /r |α| .Then, we have By 4.29 w L ∞ B 2r ≤ c K, p r 2−2/p h L p B 2r ≤ c r 2/q ; hence, 4.46 implies Now, we estimate the last integral in the right-hand side.By 1.3 , one has p R 2 ∩ L 2 loc B r and M U h a.e.z ∈ R 2 .Moreover, classically ΔU L p R 2 ≤ c K, p h L p .