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

Authors: Teresa Alberico
Journal: Journal of Function Spaces and Applications
 (2012)

---

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


---

## Body

## 1. Introduction

Let us consider a planar elliptic operator of nondivergence form:(1.1)M=∑ijaij(x)∂2∂xi∂xj,
where aij=aji for i,j=1,2 are measurable and the symmetric matrix(1.2)A(x)=(a11(x)a12(x)a12(x)a22(x))
is uniformly elliptic, that is,
(1.3)|ξ|2K≤〈A(x)ξ,ξ〉≤K|ξ|2,
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 solutionsv to the adjoint equation (adjoint solutions, for short)(1.4)M*[v]=∑ij∂2∂xi∂xj(aij(x)v(x))=0
often occurs in the literature (see Section 3, and for a very recent paper, see [1]).We say that the functionv∈Lloc1(Ω) is a weak solution to (1.4) if (1.5)∫ΩvM[φ]=0∀φ∈C0∞(Ω).
In this paper we make the assumption that the coefficients aij(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 casen=2, we give a generalized form of Theorem  1.2 in which he proves that, in particular,(1.6)v∈⋂q>1Gq,
where Gq 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 theGq classes, introduced by Gehring [3], in connection with local integrability properties of the gradient of quasiconformal mappings.Let us assume thatv 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 set(2.1)vQ=∫Q⨍Qv(x)dx=1|Q|∫Qv(x)dx
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 weightv satisfies the Gq-condition if there exists a constant G≥1 such that, for all cubes Q⊂ℝ2 as above, one has
(2.2)(∫Q⨍Q⨍Q⨍Qvq(x)dx)1/q∫Q⨍Qv(x)dx≤G,
and one refers to (2.2) as a “reverse” Hölder inequality.In the following, we will consider elliptic differential equations with coefficientsaij(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 functionf:ℝ2→ℝ is in VMO if
(2.3)lim⁡r→0∫⨍Br|f(y)-fBr|dy=0,
where Br=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:
(2.4)f(∞)=lim⁡r→∞1πr2∫Br(0)f(y)dy,
(see [5]).


## 3. Examples

In the present section we collect a certain number of examples where solutionsv to the adjoint equation(3.1)M*[v]=∑ij∂2∂xi∂xj(aij(x)v(x))=0
occur. The first example deals with adjoint solutions which are partial derivatives hx1 and hx2 of a very weak solution h∈Wloc1,2 to a particular diagonal divergence type equation, and an interesting relation comes out between regularity results.Example 3.1.
For1/K≤β≤K, we consider the following elliptic operators in ℝ2:
(3.2)M=∂2∂x12+β∂2∂x22,L=∂2∂x12+∂2∂x2(β∂∂x2).
Fortuitous relations occur between adjoint solutions toℳ and solutions h to ℒ[h]=0, as the following Lemma reveals (see [6]).Lemma 3.2 (see [6]).
Leth∈Wloc1,2(Br), where Br denotes the open ball in ℝ2 centered at 0 with radius r, such that
(3.3)L[h]=0.
Set w=∂h/∂x1, v=∂h/∂x2 Then
(3.4)M*[w]=0,M*[v]=0.Proof.
We proceed similarly as in [6]. If ϕ∈C0∞(Br), we have
(3.5)∫Brw⋅Mϕdx=∫Br∂h∂x1(∂2ϕ∂x12+β∂2ϕ∂x22)dx1dx2=∫Br(∂h∂x1⋅∂2ϕ∂x12+β∂h∂x2⋅∂2ϕ∂x1∂x2)dx1dx2=0.
Thus ℳ*[w]=0. In analogous way one checks that ℳ*[v]=0.Corollary 3.3.
Leth∈Wloc1,2 such that ℒ[h]=0. If ∂h/∂x1≥0 and ∂h/∂x2≥0, then for any ball Br⊂B2r⊂ℝ2, one has
(3.6)(∫Br⨍Br|∇h|pdx)1/p≤c(K,p)∫Br⨍Br|∇h|dx,
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]).
Leth∈Wloc1,2(Ω) be a local solution to the equation
(3.7)div⁡(A(x)∇h(x))=0,x∈Ω,
where A is a real symmetric matrix satisfying the ellipticity bounds,
(3.8)|ξ|2K≤〈A(x)ξ,ξ〉≤K|ξ|2,∀ξ∈R2,fora.e.x∈Ω.
Then, for any ball B2r⊂Ω one has
(3.9)(∫Br⨍Br|∇h|sdx)1/s≤c(K,s)∫Br⨍B2r|∇h|dx,
where 2≤s<2K/(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 Br 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 Wloc1,2 solution to (3.7), is shown to be solution to an adjoint equation for the elliptic operator
(3.10)M[v]=∂2v∂x12+c∂2v∂x22,
where 1/K≤c≤K.Example 3.6.
Very recently [12, 13], the reduced Beltrami differential equation
(3.11)∂f∂z¯=λ(z)Im(∂f∂z),|λ(z)|≤k<1,k=K-1K+1
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
(3.12)ux2=b(z)-1b(z)+1vx1
with b(z)=ℐmλ(z). As a consequence of (3.12) in [13], it is proved that ux2≠0 a.e, and it is an adjoint solution for a suitable elliptic operator ℳ=∑ijbij(z)(∂2/∂xi∂xj).Namely, it has been proved [13] that u is a solution to an elliptic equation of divergence form(3.13)div⁡A(z)∇u=0,
where A(z) is of the type(3.14)A(z)=(1a12(z)0a22(z)).
As a consequence, the function v=ux2 is a solution to the adjoint equation(3.15)M*[v]=0,
where(3.16)M=∂2∂x12+a12∂2∂x1∂x2+a22∂2∂x22.
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)B(x)=(1a12(z)2a12(z)2a22(z)).
Notice also that v=ux2>0 a.e. (see [13]), and moreover, by general properties of nonnegative adjoint solutions, v satisfies a reverse Hölder inequality [7, 14, 15](3.18)(∫Br⨍Brv(z)2dz)1/2≤c(K)∫Br⨍Brv(z)dz,
in every ball Br⊂Ω such that B2r⊂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 theG-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
(3.19)M[u]=Tr⁡(AD2u),foru∈W2,2(Ω),Ω⊂R2,
where A=(aij)∈𝕄, the set of all symmetric 2×2 real matrices and satisfy the ellipticity condition (1.3).
The adjoint to the operatorℳ 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]).
Letℳk, k=1,2,…,ℳ be operators whose coefficient matrices Ak, A∈𝕄 and satisfying (1.3). Assume that vk∈L2(Ω) are solutions to the adjoint equations ℳk*[vk]=0 and verify that
(3.20)vk⇀vinL2(Ω),vkAk⇀vAinL2(Ω;M),
where v(x)>0 a.e. in Ω. Then, one has ℳk→Gℳ.In order to prove Proposition3.8, the following lemma is crucial.Lemma 3.9 (see [16]).
Letℳk, k=1,2,…,ℳ be operators with coefficient matrices Ak, A∈𝕄 and satisfy (1.3), vk∈L2(Ω) satisfying ℳk*[vk]=0, and let vk∈Wloc2,2(Ω) be given. If
(3.21)uk⇀uinWloc⁡2,2(Ω),vkAk⇀vAinLloc⁡2(Ω;M),
then
(3.22)Tr⁡(vkAkD2uk)⟶Tr⁡(vAD2u)
in the sense of distributions.Moreover, if we consider the Hessian matrix of anyw∈W2,2(Ω),(3.23)D2w=(wx1x1wx1x2wx1x2wx2x2).
In [17] it is proved that w is a solution to(3.24)M[w]=Tr⁡(BD2w)=0,
where B is a suitable coefficient matrix, if and only if(3.25)(wx1x22-wx1x1wx2x2)(K+1K)≥wx1x12+2wx1x22+wx2x22,
where K, K≤1, is the elliptic constant. In the case  where the Hessian matrix is diagonal, that is, wx1x2=0, it is easy to see that a solution of ℳ*[v]=0 is the positive function v=-detD2w=|wx1x1wx2x2|.


## 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, theW2,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(4.1)M*[v]=0inΩ⊂R2,
as defined in (1.4), with(4.2)A=(aij)=At,|ξ|2K≤〈A(x)ξ,ξ〉≤K|ξ|2,
for a.e. x=(x1,x2)∈Ω and for ξ∈ℝ2.Theorem 4.1.
Ifv∈L1(Ω) is a nonnegative solution to (4.1) and the coefficient matrix A(x) satisfies (4.2), and moreover
(4.3)A(x1,⋅)∈VMO,
then
(4.4)v∈⋂q>1Gq.Let us begin with the followingLp-global regularity (for all p≥2) result for the complex Beltrami equation
(4.5)Fz¯+μ(z)Fz+μ(z)¯Fz¯=H(z)z∈R2,
under a partially-VMO assumption on the Beltrami coefficients μ, as defined in Section 2.Proposition 4.2.
LetBr=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∈Lp(ℝ2)(H(z)=0 for |z|>r), there exists a unique solution F to the Beltrami equation (4.5) such that Fz∈Lp and
(4.6)‖Fz¯‖Lp(R2)≤c(p,k)‖H‖Lp(R2).Remark 4.3.
We note that in general, elliptic Beltrami operator(4.7)I-μT-νT¯,
where T is the Beurling transform defined via the relation
(4.8)T(Fz¯)=Fz
under the assumption
(4.9)μ,ν∈VMO(Ĉ),
is invertible in all Lp(ℂ) spaces, p>1. The proof is much the same [5, 20], considering the complex Beltrami equation
(4.10)Fz¯-μ(z)Fz-ν(z)Fz¯=h,h∈Lp(C).
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 C0∞(ℂ) in BMO(ℂ) and that μ and ν are defined at infinity in the following average sense:
(4.11)μ(∞)=lim⁡r→∞∫Br⨍Br(0)μ(z)|dz|2,ν(∞)=lim⁡r→∞∫Br⨍Br(0)ν(z)|dz|2.
The following example, due to T. Iwaniec, shows that without such condition the result fails.Example 4.4.
There exists a functionf∈VMO(ℝn), 0≤f(x)≤1 everywhere, such that
(4.12)0=liminf⁡|B|→∞∫Br⨍Bf(x)dx<limsup⁡|B|→∞∫Br⨍Bf(x)dx=1,
where B stands for a ball centered at the origin.Preliminaries
LetL:ℝ→[0,1] be a Lipschitz function given by
(4.13)L(t)=12(1+|t|-|t-1|).
The Lipschitz constant of L equals 1, and, therefore, for each φ∈BMO(ℝn), we have
(4.14)‖L∘φ‖BMO≤2‖φ‖BMO.
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 integerk, we set
(4.15)φk(x)=L(7-2-klog⁡|x|),ψk(x)=L(5-2-klog⁡|x|).
We define the building block as fk=φk-ψk. Note that each fk is continuous and supported in the ball |x|≤e7·2k, whereas fk+1 vanishes on this ball. The BMO-norm of fk can be estimated as
(4.16)‖fk‖BMO≤‖φk‖BMO+‖ψk‖BMO≤2⋅2-kC(n)+2⋅2-kC(n)=2-k+2C(n).
Thus the infinite series
(4.17)f=∑k=0∞fk
represents a VMO function.Computation ofL1-Averages
Given any positive integerN, we consider concentric balls Br⊂BR centered at the origin and with radii r=e4·2N<e6·2N=R. Elementary geometric observation reveals that
(4.18)⨍BRf≥1|BR|∫BRfN≥1|BR|∫e5⋅2N≤|x|≤e6⋅2NfN=e6n2N-e5n2Ne6n2N=1-e-n2N⟶1,asN⟶∞.∫BR
On the other hand
(4.19)⨍BRf=1|BR|∑k=1N-1∫|x|≤e4⋅2Nfk(x)dx≤1|BR|∑k=1N-1∫|x|≤e7⋅2kdx≤∑k=1N-1e7n2ke4n2N≤(N-1)e7n2N-1e4n2N=(N-1)e-n2N-1⟶0,asN⟶∞,∫BR
as desired.Proof of Proposition4.2.
If we setK=(1+k)/(1-k), then there exists a symmetric matrix
(4.20)A(z)=aij(z)
such that
(4.21)IK≤A(z)≤KI,μ(z)=12[a11(z)-a22(z)+2a12(z)ia11(z)+a22(z)].
Moreover aij(z)=0 for |z|≥r.
We may assume the following familiar normalization:(4.22)Tr⁡A(z)=a11(z)+a22(z)=1.
With the previous prescriptions we easily check that for w∈Wloc2,1(Br) if we define the complex gradient of w as
(4.23)F(z)=wz=12(wx1-iwx2),
we have
(4.24)Tr⁡(A(z)D2w)=2(Fz¯+μFz+μ¯Fz¯)
(see [5]). Hence (4.5) is equivalent to
(4.25)Tr⁡(A(z)D2w)=HinR2
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∈W2,p to (4.25) for H∈Lp has been established (p≥2), together with the estimate
(4.26)‖D2w‖Lp(Br)≤c(p,k)‖H‖Lp(Br).
Hence (4.6) follows.Let us now give the following sharp version of the Alexandrov-Bakelman-Pucci maximum principle for non divergence elliptic operators(4.27)M[w]=∑aij∂2w∂xi∂xj
with partially VMO coefficients.Lemma 4.5.
Under the assumptions (4.2), (4.3), (4.22) on A, suppose Br=B(0,r)⊂ℝ2 and that w∈Wloc2,1(Br)∩C0(B¯r) satisfies, for h∈Lp(Br), p>1,
(4.28)M[w]=hinBr,w=0on∂Br.
Then one has
(4.29)‖w‖L∞(Br)≤c(K,p)r2-2/p‖h‖Lp(Br).Proof.
In view of [19, Theorem  2.4], we know that the Dirichlet problem (4.28) always has a unique solution w∈W2,p(Br)∩W01,2(Br) for every h∈Lp(Br), p>1.
Defineh(z)=0 for z=(x1,x2)∈ℝ2∖Br and
(4.30)μ(z)=12[a11(z)-a22(z)+2a12(z)i],
and set μ(z)=0 for z∈ℝ2∖Br. According to Proposition 4.2, the equation
(4.31)Fz¯+μFz+μ¯Fz¯=h2
has unique solution F∈Wloc1,p(Br) such that
(4.32)‖Fz¯‖Lp(R2)≤c(p,K)‖h‖Lp(R2).
Now, let us see that(4.33)F|Br∈Wloc⁡1,2(Br).
Define f=wz=(1/2)(wx1-iwx2). Then
(4.34)(f-F)z¯+μ(f-F)z+μ¯(f-F)z¯=0inBr.
This means that the mapping
(4.35)g=f-F
is weakly K-quasiregular, and since F∈Wloc1,p and f∈Wloc1,2, we deduce g∈Wloc1,p and actually g is K-quasiregular and in particular
(4.36)g∈Wloc⁡1,2(Br).
Then (4.33) follows.
Now, let us introduce the solutionU to the problem
(4.37)ΔU=4Fz¯,U(0)=0.
We have ΔU∈Lp(ℝ2)∩Lloc2(Br) and ℳ[U]=h a.e. z∈ℝ2. Moreover, classically
(4.38)‖ΔU‖Lp(R2)≤c(K,p)‖h‖Lp.
(Notice that c(K,p) is optimal in Talenti [21].)
Finally, let us introduce(4.39)u=U-w.
Then u is continuous in B¯r by the Sobolev imbedding, and it is the solution to the Dirichlet problem
(4.40)M[u]=0a.e.z∈Br,u/∂Br=U,
and u∈Wloc2,2(Br)∩C0(B¯r). By the classical maximum principle [22],
(4.41)‖U-w‖L∞(Br)=‖U‖L∞(∂Br).
Hence, we use Sobolev and the condition U(0)=0 to conclude
(4.42)‖w‖L∞(Br)≤2‖U‖L∞(Br)≤cr2-2/p‖ΔU‖Lp(Br)≤cr2-2/p‖h‖Lp(Br).Proof of Theorem4.1.
Let us fixq>1, set p=q/(q-1), and fix a ball Br such that B2r⊂Ω. As in [7, 15], we make use of the dual formulation of the Lq-norm
(4.43)(∫Brvq)1/q=sup⁡{∫Brvh:h≥0,h∈C01(Br),‖h‖Lp(R2)≤1}.
Fix h∈C01(Br),∥h∥Lp(ℝ2)≤1,h≥0, and solve the Dirichlet problem
(4.44)M[w]=∑aij∂2w∂xi∂xj=hinB2r,w=0on⁡∂B2r.
SinceA(x1,·)∈VMO, the problem has a unique solution w∈W2,p(B2r) vanishing on ∂B2r, satisfying the estimate
(4.45)‖D2w‖Lp(B2r)≤c‖h‖Lp(B2r),c=c(K,p,∥A∥VMO).
Fix a nonnegative functionφr∈C01(B3r/2) such that φr=1 on Br and |∂αφr/∂xα|≤C(α)/r|α|.
Then, we have(4.46)∫Brvh≤∫B2rvM[w]φr=-∫B2rvwM[φr]-2∫B2rv〈A∇w,∇φr〉≤cr2‖w‖L∞(B2r)∫B3r/2v+cKr∫B3r/2v|∇w|.
By (4.29) ∥w∥L∞(B2r)≤c(K,p)r2-2/p∥h∥Lp(B2r)≤cr2/q; hence, (4.46) implies
(4.47)∫Brvh≤cr2r2/q∫B3r/2v+cr(∫B3r/2v)1/2(∫B2rv|∇w|2)1/2.
Now, we estimate the last integral in the right-hand side. By (1.3), one has
(4.48)∫B2rv|∇w|2≤K∫B2rv〈A∇w,∇w〉=K∫B2rv[M[w2]2-wh].
Since w2=0 and ∇(w2)=0 on ∂B2r, then we deduce
(4.49)∫B2rvM[w2]=0asM*[v]=0.
Using again (4.29) yields
(4.50)∫B2rv|∇w|2≤2K∫B2rv|w|h≤2K‖w‖L∞(B2r)∫Brvh≤2Kcr2/q∫Brvh.
By (4.47) and (4.50), it follows that
(4.51)∫Brvh≤cr2(1-1/q)∫B3r/2v+cr(1-1/q)(∫B3r/2v)1/2(∫Brvh)1/2.
By elementary inequality ab≤a/2+b/2, we obtain
(4.52)∫Brvh≤cr2(1-1/q)∫B3r/2v+cr2(1-1/q)∫B3r/2v+12∫Brvh.
Rearranging yields(4.53)∫Brvh≤cr2(1-(1/q))∫B3r/2v.
Sinceh is arbitrary, by (4.43) and (4.53), we obtain
(4.54)(∫Br⨍Brvp)1/p≤c∫Br⨍B3r/2v,
with c=c(K,p). An application of [14, Lemma  2.0] concludes the proof.



---

*Source: 208741-2012-02-13.xml*