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THE LANDIS CONJECTURE FOR VARIABLE COEFFICIENT SECOND-ORDER ELLIPTIC PDES

BLAIR DAVEY, CARLOS KENIG, AND JENN-NAN WANG

ABSTRACT. In this work, we study the Landis conjecture for second-order elliptic equations in the plane. Pre- cisely, assume that V ≥ 0 is a measurable real-valued function satisfying ||V ||L(R2)≤ 1. Let u be a real solution to div (A∇u) −Vu = 0 in R2. Assume that |u (z)| ≤ exp (c0|z|) and u (0) = 1. Then, for any R sufficiently large,

|zinf0|=R||u||L(B1(z0))≥ exp (−CR log R) .

In addition to equations with electric potentials, we also derive similar estimates for equations with magnetic potentials. The proofs rely on transforming the equations to Beltrami systems and Hadamard’s three-quasi- circle theorem.

1. INTRODUCTION

In this work, we study the asymptotic uniqueness for general second-order elliptic equations in the whole space. One typical example we have in mind is

Lu−Vu := div (A∇u) −Vu = 0 in Rn, (1.1)

where A is symmetric and uniformly elliptic with Lipschitz continuous coefficients and V is essentially bounded. For (1.1), we are interested in the following Landis type conjecture: assume that kV kL(Rn)≤ 1 and kukL(Rn)≤ C0 satisfies |u(x)| ≤ C exp(−C|x|1+), then u ≡ 0. When L = ∆, counterexamples to the Landis conjecture were constructed by Meshkov in [9] where the exponent 4/3 was shown to be optimal for complex-valued potentials and solutions. A quantitative form of Meshkov’s result was derived by Bourgain and Kenig [2] in their resolution of Anderson localization for the Bernoulli model in higher dimensions. The proof of Bourgain and Kenig’s result was based on Carleman type estimates. Using the Carleman method, other related results for the general second elliptic equation involving the first derivative terms were obtained in [3] and [8].

The known results mentioned above indicate that the exponent 1 in the Landis type conjecture is not true for general coefficients and solutions. Therefore, we want to study the same question when A and V of (1.1) are real-valued and the solution u is also real. In the case where L = ∆, n = 2, and V ≥ 0, a quantitative Landis conjecture was proved in [6]. Precisely, let u be a real solution of ∆u − Vu = 0 in R2 satisfying u(0) = 1, |u(x)| ≤ exp(C0|x|), where kV kL ≤ 1 and V ≥ 0. Then for R sufficiently large,

|xinf0|=R sup

|x−x0|<1

|u(x)| ≥ exp(−CR log R), where C depends on C0.

Here we would like to generalize this result to the second-order elliptic operator L. Let A be symmetric and uniformly elliptic with Lipschitz continuous coefficients. That is, for some λ ∈ (0, 1],

A=

 a11 a12 a21 a22



, a12= a21 (1.2)

λ |ξ |2≤ ai j(x) ξiξj≤ λ−1|ξ |2, for all x ∈ R2, ξ ∈ R2. (1.3)

Kenig is supported in part by DMS-1265429.

Wang is supported in part by MOST 102-2115-M-002-009-MY3.

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Since A is Lipschitz continuous, then there exists µ > 0 such that

∇ai j

≤ µ for each i, j = 1, 2. (1.4)

The ellipticity condition (1.3) implies that

aii≥ λ for each i = 1, 2 (1.5)

ai j≤ Cλ−1 for each i, j = 1, 2. (1.6)

L= div (A∇) . (1.7)

Remark 1.1. We will often use that L is a divergence-form operator. However, it will at times be useful to think of L in non-divergence form:

L= ∂i(ai jju) = ai ji ju+ ∂iai jju:= ai ji ju+ bjju.

It follows from (1.4) that b ∈ Lwith

bj

≤ 2µ for each j = 1, 2.

By building on the techniques developed in [6], we will prove quantitative versions of Landis’ conjecture when the leading operator is L. As in [6], to prove each Landis theorem, we first establish an appropriate order-of-vanishing estimate, then we apply the shift and scale argument from [2]. We use the notation Brto denote a ball of radius r centered at the origin. As defined in Section 2, Qsdenotes a quasi-ball of radius s centered at the origin that is associated to an elliptic operator. Constants b and d are chosen so that Bb⊂ Q1 and Q7/5b Bd. It is shown in Section 2 that such ball exists, and they are bounded in terms of the ellipticity constant. The functions σ and ρ, which are introduced at the end of Section 2 (see (2.1) and (2.2)), are used below to define b and d. The first maximal order-of-vanishing theorem that we will discuss is the following.

Theorem 1.1. Set b = σ (1; λ ), d = ρ 75; λ +25. Let u be a real-valued solution to

Lu−Vu = 0 in Bd⊂ R2, (1.8)

where V≥ 0 and A satisfies assumptions (1.2) and (1.3). Assume that

||u||L(Bd)≤ exp C0

√ M



(1.9)

||u||L(Bb)≥ 1 (1.10)

||V ||L(Bd)≤ M (1.11)

∇ai j

L(Bd)≤ µ√

M, (1.12)

where M≥ 1. Then there exists C = C (C0, λ , µ) so that

||u||L(Br)≥ rC

M. (1.13)

As in [2], a scaling argument shows that the following quantitative form of Landis’ conjecture follows from Theorem 1.1.

Theorem 1.2. Assume that V : R2→ R is measurable and satisfies

||V ||L(R2)≤ 1.

Assume also that V ≥ 0 a.e. in R2. Let u be a real solution to

Lu−Vu = 0 in R2, (1.14)

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where A satisfies the assumptions (1.2) – (1.4). Assume that |u (z)| ≤ exp (c0|z|) and u (0) = 1, where z= (x, y). Let z0= (x0, y0). Then, for any R sufficiently large,

|zinf0|=R||u||L(B1(z0))≥ exp (−CR log R) , (1.15) where C depends on c0, λ , µ.

The second maximal order-of-vanishing theorem applies to equations with a magnetic potential in diver- gence form.

Theorem 1.3. Set b = σ (1; λ ), d = ρ 75; λ +25. Let u be a real-valued solution to

Lu+ ∇ · (Wu) −Vu = 0 in Bd⊂ R2, (1.16)

where V ≥ 0 and A satisfies assumptions (1.2) and (1.3). Assume that for some M ≥ 1, (1.9) – (1.12) from above hold, and

||W ||L(Bd)≤√

M. (1.17)

Then there exists C= C (C0, λ , µ) such that (1.13) holds.

As above, the order-of-vanishing estimate implies the following Landis result.

Theorem 1.4. Assume that V : R2→ R, W : R2→ R2are measurable and satisfy

||W ||L(R2)≤ 1, ||V ||L(R2)≤ 1.

Assume also that V ≥ 0 a.e. in R2. Let u be a real solution to

Lu+ ∇ · (Wu) −Vu = 0 in R2, (1.18)

where A satisfies the assumptions (1.2) – (1.4). Assume that |u (z)| ≤ exp (c0|z|) and u (0) = 1, where z= (x, y). Set z0= (x0, y0). Then, for any R sufficiently large, estimate (1.15) holds where C depends on c0, λ , µ .

The third pair of theorems apply to equations with magnetic potentials in a non-divergence form. For this case, in the local setting, it suffices to work with matrices that have determinant equal to 1. This additional assumption changes the ellipticity constant, which in turn changes how we define b and d.

Theorem 1.5. Set b = σ 1; λ2, d = ρ 75; λ2 +25. Let u be a real-valued solution to

Lu−W · ∇u −Vu = 0 in Bd⊂ R2, (1.19)

where V≥ 0 and A satisfies assumptions (1.2) and (1.3) with λ replaced by λ2, anddet A = 1. Assume that for some M≥ 1, (1.9) – (1.10), and (1.12) from above hold, and

||V ||L(Bd)≤ C1M (1.20)

||W ||L(Bd)≤p

C1M. (1.21)

Then there exists C= C (C0,C1, λ , µ) such that (1.13) holds.

Remark 1.2. For the general coefficient matrix A satisfying (1.2) – (1.4), dividing (1.19) gives div

 A

√detA∇u



− ˜W· ∇u − ˜V u= 0, where

W˜ = A∇

 1

√ detA



+ W

detA, V˜ = V

detA. (1.22)

If W and V satisfy (1.20) and (1.21), then ˜W and ˜V satisfy the similar bounds with a new constant C1

depending on λ , µ. Also, the ellipticity constant of A/√

detA is λ2.

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Again, the local theorem implies the Landis theorem.

Theorem 1.6. Assume that V : R2→ R, W : R2→ R2are measurable and satisfy

||W ||L(R2)≤ 1, ||V ||L(R2)≤ 1.

Assume also that V ≥ 0 a.e. in R2. Let u be a real solution to

Lu−W · ∇u −Vu = 0 in R2, (1.23)

where A satisfies the assumptions (1.2) – (1.4). Assume that |u (z)| ≤ exp (c0|z|) and u (0) = 1, where z= (x, y). Set z0= (x0, y0). Then, for any R ≥ R0, estimate(1.15) holds, where R0depends on λ , µ and C depends on c0, λ , µ.

This article is organized as follows. In Section 2, we discuss fundamental solutions of second-order el- liptic operators that satisfy (1.3). These results apply to second-order elliptic operators with Lcoefficients.

These fundamental solutions lead to the definitions of quasi-balls and quasi-circles, as well as related results.

In Section 3, the shift and scale argument from [2] is applied to show how each quantitative Landis theorem follows from the corresponding order-of-vanishing estimate. A number of useful tools are developed in Sec- tion 4. To start, we introduce some first-order Beltrami operators that generalize ∂ . Then, a few properties that relate first-order Beltrami operators to second-order elliptic operators are established. With these facts, a Hadamard three-quasi-circle theorem is proved. Finally, we present some of the work of Bojarksi from [1] including a similarity principle for solutions to non-homogenous Beltrami equations. In Section 5, the tools developed in the previous section are combined with the framework from [6] to prove Theorem 1.1.

Section 6 shows how to account for a magnetic potential, proving Theorem 1.3. The proof of Theorem 1.5 is contained in Section 7. A technical proof of one of the facts from Section 4 may be found in the appendix.

2. QUASI-BALLS AND QUASI-CIRCLES

LetL (λ) denote the set of all second-order elliptic operators acting on R2that satisfy ellipticity condi- tion (1.3). Throughout this section, assume that L ∈L (λ). We start by discussing the fundamental solutions of L. These results are based on the Appendix of [7].

Definition 2.1. A function G is called a fundamental solution for L with pole at the origin if

• G ∈ Hloc1,2 R2\ 0, G ∈ Hloc1,p R2 for all p < 2 and for every ϕ ∈ C0 R2 Z

ai j(z) DiG(z) Djϕ (z) dz = −ϕ (0) .

• |G (z)| ≤ C log |z|, for some C > 0, |z| ≥ C.

Lemma 2.2 (Theorem A-2, [7]). There exists a unique fundamental solution G for L, with pole at the origin and with the property that lim

|z|→∞G(z) − g (z) = 0, where g is a solution to Lg = 0 in |z| > 1 with g = 0 on

|z| = 1. Moreover, there are constants C1,C2,C3,C4, R1< 1, R2> 1, that depend on λ , such that C1log 1

|z|



≤ −G (z) ≤ C2log 1

|z|



for |z|< R1 C3log |z| ≤ G (z) ≤ C4log |z| for |z| > R2.

As a corollary to this theorem, we have the following.

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Corollary 2.3. There exist additional constants C5,C6, depending on λ , such that

|z|C2≤ exp (G (z)) ≤ |z|C1 for |z|< R1

C5|z|C2 ≤ exp (G (z)) ≤ C6|z|C4 for R1< |z| < R2

|z|C3≤ exp (G (z)) ≤ |z|C4 for |z|> R2.

Proof. Exponentiating the bounds given in Theorem 2.2 gives the first and third line of inequalities. Since Gis a solution to Lu = 0 in the annulus A = {z : R1< |z| < R2}, then by the maximum principle and the bounds given in Lemma 2.2

maxz∈A G(z) ≤ max

z∈∂ AG(z) ≤ max {C4log R2,C1log R1} = C4log R2

minz∈AG(z) ≥ min

z∈∂ AG(z) ≥ min {C3log R2,C2log R1} = C2log R1. It follows that for any z ∈ A,

C2log R1≤ G (z) ≤ C4log R2. Therefore, whenever R1< |z| < R2,

exp (G (z)) ≤ RC24 = R2

|z|

C4

|z|C4≤ R2 R1

C4

|z|C4, and

exp (G (z)) ≥ RC12 = R1

|z|

C2

|z|C2≥ R1 R2

C2

|z|C2,

giving the second line of bounds. 

The level sets of G will be important to us.

Definition 2.4. Define a function ` : R2→ (0, ∞) as follows: ` (z) = s iff G (z) = ln s. Then set Zs=z ∈ R2: G (z) = ln s = z ∈ R2: ` (z) = s .

We refer to these level set of G asquasi-circles. That is, Zs is the quasi-circle of radius s. We also define (closed)quasi-balls as

Qs=z ∈ R2: ` (z) ≤ s .

Openquasi-balls are defined analogously. We may also use the notation QLs and ZLs to remind ourselves of the underlying operator.

The following lemma follows from the bounds given in Corollary 2.3.

Lemma 2.5. There are constants c1, c2, c3, c4, c5, c6, S1< 1, S2> 1, that depend on λ , such that if z ∈ Zs, then

sc1 ≤ |z| ≤ sc2 for s≤ S1

c5sc1 ≤ |z| ≤ c6sc4 for S1< s < S2 sc3 ≤ |z| ≤ sc4 for s≥ S2.

Thus, the quasi-circle Zsis contained in an annulus whose inner and outer radii depend on s and λ . For future reference, it will be helpful to have a notation for the bounds on these inner and outer radii.

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Definition 2.6. Define

σ (s; λ ) = sup

r> 0 : Br\

L∈L (λ)

QLs

(2.1)

ρ (s; λ ) = inf

r> 0 : [

L∈L (λ)

QLs ⊂ Br

. (2.2)

Remark 2.1. These functions are defined so that for any operator L inL (λ), Bσ (s;λ )⊂ QLs ⊂ Bρ (s;λ ). The quasi-balls and quasi-circles just defined above are centered at the origin since G is a fundamental solution with a pole at the origin. We may sometimes use the notation Zs(0) and Qs(0) as a reminder that these sets are centered around the origin. If we follow the same process for any point z0∈ R2, we may discuss the fundamental solutions with pole at z0, and we may similarly define the quasi-circles and quasi- balls associated to these functions. We will denote the quasi-circle and quasi-ball of radius s centred at z0by Zs(z0) and Qs(z0), respectively. Although Qs(z0) is not necessarily a translation of Qs(0) for z06= 0, both sets are contained in annuli that are translations.

Throughout, we will often work with quasi-balls in addition to standard balls.

3. THE SHIFT AND SCALE ARGUMENTS

The bulk of the paper is devoted to proving the order-of-vanishing estimates stated in Theorems 1.1, 1.3, and 1.5. Before we get to those details, we show how Theorems 1.2, 1.4, and 1.6 follow from the local estimates and the shift and scale arguments in [2].

Proof of Theorem 1.2. Let u be a real-valued solution to (1.14). Let z0∈ R2be such that |z0| = R for some R≥ 1. For a constant a yet to be determined, define

uR(z) = u (z0+ aRz) , AR(z) = A (z0+ aRz) , VR(z) = (aR)2V(z0+ aRz) , and set

LR= div (AR∇) .

Since A satisfies (1.2) and (1.3), then so too does AR. By construction, uRis a solution to LRuR−VRuR= 0.

Since |u (z)| ≤ exp (c0|z|), it follows that

||uR||L(Bd)≤ exp (c0(1 + ad) R) ,

where d = ρ 75; λ +25 depends on λ . We choose a > 0 so that 1a ≤ b, where b = σ (1; λ ) depends on λ . Then z1:= −aRz0 ∈ Bb, uR(z1) = u (0) = 1 and it follows that

||uR||L(Bb)≥ 1.

Since ||V ||L ≤ 1, then ||VR||L(Bd)≤ (aR)2. The condition

∇ai j

L ≤ µ implies that

∇aR,i j

L(Bd)≤ aRµ. Hence, the assumptions of Theorem 1.1 are satisfied for uRwith M = (aR)2. Therefore,

||uR||L(Br)≥ rCaR. Setting r =aR1 and rewriting in terms of u, we see that

||u||L(B1(z0))≥ exp − ˜CRlog R ,

as required. 

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Proof of Theorem 1.4. Let u be a real-valued solution to (1.18). Define z0, a, uR, AR, VR, and LR as in the previous proof. If we set

WR(z) = RW (z0+ aRz) , then uR is a solution to

LRuR+ ∇ (WRuR) −VRuR= 0.

Since ||W ||L ≤ 1, then ||WR||L(Bd)≤ aR. The assumptions of Theorem 1.3 are satisfied for uR with M =

(aR)2, and the conclusion follows as above. 

To prove the third version of the theorem, we must account for the additional determinant condition in the statement of Theorem 1.5.

Proof of Theorem 1.6. Let u be a real-valued solution to (1.23). Set ˜A=A

det A so that det ˜A= 1. Now the ellipticity constant of ˜Ais λ2. Then u is a solution to ˜Lu − ˜W· ∇u − ˜V u= 0 in R2, where ˜L = div ˜A∇ and W˜, ˜V are given in (1.22). Note that

˜W

L ≤ C1, and ˜V

L ≤ λ−1, with C1= C1(λ , µ). The rest of the

proof proceeds as above. 

4. USEFULTOOLS

This section contains a number of tools that will be used in the proofs of the order-of-vanishing estimates to be given in the following sections. We first define the Beltrami operator that will play the role of ¯∂ from [6]. Then we present some results that show that such Beltrami operators are related to elliptic operators of the form L in the same way that ∂ related to ∆. These results are proved with elementary (but somewhat lengthly) computations. Once we have the computational results, we will prove an optimal three-balls in- equality, which we call the Hadamard three-quasi-ball inequality. Finally, we present some work of Bojarski from [1], including the similarity principle for equations of the form Du = au + b ¯u.

4.1. The Beltrami operators. We define a Beltrami operator that will play the role of the ¯∂ operator from the original paper [6]. For a complex-valued function f = u + iv, define

D f = ¯∂ f + η (z) ∂ f + ν (z) ∂ f , (4.1)

where

∂ =¯ 12(∂x+ i∂y)

∂ = 12(∂x− i∂y)

η (z) = a11− a22+ 2ia12

det (A + I) (4.2)

ν (z) = det A − 1

det (A + I). (4.3)

Lemma 4.1. For η, ν defined above, we have

|η (z)| + |ν (z)| ≤1 − λ 1 + λ.

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Proof. The proof of this lemma is purely computation.

|η (z)|2=(a11− a22)2+ 4a212

[det (A + I)]2 =(a11+ a22)2− 4a11a22+ 4a212

[det (A + I)]2 = (tr A)2− 4 det A (det A + tr A + 1)2

|η (z)| = λ1− λ21+ 1) (λ2+ 1)

|ν (z)| = |det A − 1|

[det (A + I)] = |λ1λ2− 1|

1+ 1) (λ2+ 1),

where we are using λ1≥ λ2to denote the eigenvalues of A. It follows that

|η (z)| + |ν (z)| = λ1− λ2

1+ 1) (λ2+ 1)+ |λ1λ2− 1|

1+ 1) (λ2+ 1)≤1 − λ 1 + λ.

 A computation shows that for f = u + iv

D f =(a11+ det A) + ia12

det (A + I) ux+a12+ i (a22+ det A) det (A + I) uy +(a11+ 1) + ia12

det (A + I) ivx+a12+ i (a22+ 1)

det (A + I) ivy. (4.4)

When A has determinant equal to 1, ν (z) = 0 and we may write D=(a11+ 1) + ia12

det (A + I) ∂x+a12+ i (a22+ 1)

det (A + I) ∂y. (4.5)

In addition to the operator D, we will also make use of an operator that is related to D through some function w. For a given function w, set

ηw(z) =



η (z) + ν (z)∂ w∂ w for ∂ w 6= 0 η (z) + ν (z) otherwise ,

where η and ν are as defined in (4.2) and (4.3), respectively. By Lemma 4.1, it follows that |ηw| ≤ 1 − λ 1 + λ. Define

Dwf = ∂ f + ηw(z) ∂ f . (4.6)

If ηw(z) = αw(z) + iβw(z), then

Dw=1

2[∂x+ i∂y+ (αw+ iβw) (∂x− i∂y)]

=1 + αw+ iβw

2 ∂xw+ i (1 − αw)

2 ∂y (4.7)

Bertrami operators of this form will be used in the proofs of the main theorems.

At times, the dependence on w will not be important to our arguments, so we define Dˆ =1 + α + iβ

2 ∂x+β + i (1 − α )

2 ∂y, (4.8)

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where α, β are assumed to be functions of z such that α2+ β2≤ 1 − λ 1 + λ

2

< 1. Associated to ˆDis the second-order elliptic operator ˆL = div ˆA∇ with

Aˆ=

(1+α)22 1−α2−β2

1−α2−β2

1−α2−β2

(1−α)22 1−α2−β2

=

 aˆ1112 ˆ a1222



. (4.9)

A computation shows that the smallest eigenvalue of ˆAsatisfies

λ= 1 − 2

1 + (α2+ β2)−1/2 =1 −p

α2+ β2 1 +p

α2+ β2 ≥ λ , while the largest eigenvalue of ˆAsatisfies

λ+=1 +p

α2+ β2 1 −p

α2+ β2 ≤ λ−1.

Therefore we can see that ˆAhas the same ellipticity constant, λ . Finally, note that if det A = 1, then D takes the form of ˆD. This means that the rest of the results of this section may be applied to D in this case.

Remark 4.1. Note that if D is given as in (4.1) and D f = 0, then Dwf= 0 with w = f , where Dwis defined in (4.6).

4.2. Computational results for elliptic operators. The following results show that ˆDrelates to ˆL in some of the same ways that ∂ relates to ∆. These properties will allow us to prove the Hadamard three-quasi-circle theorem.

Lemma 4.2. If ˆD f = 0, where f (x, y) = u (x, y) + iv (x, y) for real-valued u and v, then ˆLu = 0 = ˆLv.

Proof. If ˆD f = 0, then it follows from (4.8) that the following Cauchy-Riemann type equations hold

 (1 + α) ux− β vx+ β uy− (1 − α) vy= 0

β ux+ (1 + α) vx+ (1 − α) uy+ β vy= 0. (4.10) Some algebraic manipulations give rise to two more equivalent sets of equations

 aˆ11ux+ ˆa12uy− vy= 0 ˆ

a12ux+ ˆa22uy+ vx= 0, (4.11)

and 

ˆ

a11vx+ ˆa12vy+ uy= 0 ˆ

a12vx+ ˆa22vy− ux= 0, (4.12)

where we have used the definition of ˆAin (4.9). From (4.11), we have

0 = ∂x[ ˆa11ux+ ˆa12uy− vy] + ∂y[ ˆa12ux+ ˆa22uy+ vx] , so that ˆLu = 0. Similarly, by (4.12),

0 = ∂x[ ˆa11vx+ ˆa12vy+ uy] + ∂y[ ˆa12vx+ ˆa22vy− ux] ,

so that ˆLv = 0 as well. 

We find another parallel with the Laplace equation. As in the case of ˆL = ∆, the logarithm of the norm of f is a subsolution to the second-order equation whenever ˆD f = 0. To see this, it suffices to prove that Lemma 4.3. If ˆD f = 0 and f 6= 0, then ˆL [log | f (z)|] = 0.

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Proof. If f = u + iv, where u and v are real-valued, then log | f (z)| = 12log u2+ v2. We have

xlog | f (z)| = uux+ vvx

u2+ v2

ylog | f (z)| = uuy+ vvy

u2+ v2 . Then,

ˆL[log | f (x)|] = ∂x( ˆa11xlog | f (z)| + ˆa12ylog | f (z)|) + ∂y( ˆa12xlog | f (z)| + ˆa22ylog | f (z)|)

= ∂x

 ˆ

a11uux+ vvx

u2+ v2 + ˆa12uuy+ vvy

u2+ v2

 + ∂y

 ˆ

a12uux+ vvx

u2+ v2 + ˆa22uuy+ vvy

u2+ v2



= [∂x( ˆa11ux+ ˆa12uy) + ∂y( ˆa12ux+ ˆa22uy)] u u2+ v2 + [∂x( ˆa11vx+ ˆa12vy) + ∂y( ˆa12vx+ ˆa22vy)] v

u2+ v2 + ∂x

 u

u2+ v2

 ˆ

a11ux+ ∂x

 u

u2+ v2

 ˆ

a12uy+ ∂y

 u

u2+ v2

 ˆ

a12ux+ ∂y

 u

u2+ v2

 ˆ a22uy + ∂x

 v

u2+ v2

 ˆ

a11vx+ ∂x

 v

u2+ v2

 ˆ

a12vy+ ∂y

 v

u2+ v2

 ˆ

a12vx+ ∂y

 v

u2+ v2

 ˆ a22vy. Since ˆLu = 0 = ˆLv by the previous lemma, the top two lines vanish and we have,

ˆL[log | f (x)|] = ∂x

 u

u2+ v2

 ˆ

a11ux+ ∂x

 u

u2+ v2

 ˆ

a12uy+ ∂y

 u

u2+ v2

 ˆ

a12ux+ ∂y

 u

u2+ v2

 ˆ a22uy

+ ∂x

 v

u2+ v2

 ˆ

a11vx+ ∂x

 v

u2+ v2

 ˆ

a12vy+ ∂y

 v

u2+ v2

 ˆ

a12vx+ ∂y

 v

u2+ v2

 ˆ a22vy

= n

ˆ a11

h

(ux)2+ (vx)2 i

+ 2 ˆa12(uxuy+ vxvy) + ˆa22 h

(uy)2+ (vy)2

io 1

u2+ v2

− 2h ˆ

a11(uux+ vvx)2+ 2 ˆa12(uux+ vvx) (uuy+ vvy) + ˆa22(uuy+ vvy)2

i 1

(u2+ v2)2. By the relations (4.11) and (4.12),

ˆ a11

h

(ux)2+ (vx)2i

+ 2 ˆa12(uxuy+ vxvy) + ˆa22 h

(uy)2+ (vy)2i

= ˆa11 h

(ux)2+ (vx)2 i

+ [ux(vy− ˆa11ux) + vx(−uy− ˆa11vx)]

+ [(−vx− ˆa22uy) uy+ (ux− ˆa22vy) vy] + ˆa22 h

(uy)2+ (vy)2 i

= 2 (uxvy− vxuy) , and

ˆ

a11(uux+ vvx)2+ 2 ˆa12(uux+ vvx) (uuy+ vvy) + ˆa22(uuy+ vvy)2

= (uux+ vvx) { ˆa11(uux+ vvx) + [u (vy− ˆa11ux) + v (−uy− ˆa11vx)]}

+ {[u (−vx− ˆa22uy) + v (ux− ˆa22vy)] + ˆa22(uuy+ vvy)} (uuy+ vvy)

= (uux+ vvx) (uvy− vuy) + (−uvx+ vux) (uuy+ vvy)

= (uxvy− uyvx) u2+ v2 . Therefore,

ˆL[log | f (z)|] = 0,

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proving the lemma.  Since ∆ = 4 ¯∂ ∂ = 4∂ ¯∂ is used in [6] to prove the third version of the theorem, we would like a decom- position for our operator L = div (A∇) into first-order operators. Under some additional assumptions on the structure of A, the following lemma shows that this is possible.

Lemma 4.4. Assume that A has determinant equal to 1. Then the operator L may be decomposed as L=

D+ eW

 D,e where

De= [1 + a11− ia12] ∂x+ [a12− i (1 + a22)] ∂y= det (A + I) D

We =(α∂xa11− β ∂xa12+ γ∂ya11+ δ ∂ya12) + i (γ∂xa11+ δ ∂xa12− α∂ya11+ β ∂ya12) a11det (A + I)2

α = a11+ a22+ 2a11a22 β = 2a12(1 + a11) γ = a12(a22− a11) δ = (1 + a11)2− a212, and D is given by(4.5).

The proof of this lemma is straightforward, but tedious. We will prove it in the Appendix.

4.3. A Hadamard three-quasi-circle theorem. Using the fundamental solution ˆGfor the operator ˆL, we can now prove the following.

Theorem 4.5. Let f be a function for which ˆD f= 0. Set

M(s) = max {| f (z)| : z ∈ Zs} . Then for any0 < s1< s2< s3,

log s3 s1



log M (s2) ≤ log s3 s2



log M (s1) + log s2 s1



log M (s3) . (4.13)

Proof. LetAs1,s3= {z : s1≤ ` (z) ≤ s3} = Qs3\ Qs1, where ` is associated to ˆG, the fundamental solution of ˆL. By Lemma 2.5, this set is contained in an annulus with inner and outer radius depending on s1, s3, and λ . In particular, it is bounded and does not contain the origin. Therefore, ˆG(z) is bounded onAs1,s3. Let z0be in the interior ofAs1,s3. If f (z0) = 0, then a ˆG(z0) + log | f (z0)| = −∞ for any a ∈ R. On the other hand, if f(z0) 6= 0, then Lemma 4.3 implies that ˆLa ˆG(z) + log | f (z)| = 0 for z near z0. By the maximum principle, z0 cannot be an extremal point. Therefore, a ˆG(z) + log | f (z)| takes it maximum value on the boundary of As1,s3. We will choose the constant a ∈ R so that

maxa ˆG(z) + log | f (z)| : z ∈ Zs1 = max a ˆG(z) + log | f (z)| : z ∈ Zs3 , or rather

log (sa1M(s1)) = log (sa3M(s3)) . It follows that for any z ∈As1,s3,

a ˆG(z) + log | f (z)| ≤ log (sa1M(s1)) (or log (sa3M(s3))).

Furthermore, for any s2∈ (s1, s3),

maxa ˆG(z) + log | f (z)| : z ∈ Zs2 ≤ log (sa1M(s1)) (or log (sa3M(s3))), or

log (sa2M(s2)) ≤ log (sa1M(s1)) (or log (sa3M(s3))).

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Consequently,

sa2M(s2) ≤ sa1M(s1) (or sa3M(s3)), so that for any τ ∈ (0, 1), since sa1M(s1) = sa3M(s3), then

sa2M(s2) ≤ [sa1M(s1)]τ[sa3M(s3)]1−τ [M (s2)]log

s3 s1



≤ s1 s2

a

M(s1)

τ log

s3 s1



 s3 s2

a

M(s3)

(1−τ) log

s3 s1



. We choose τ so that τ log

s3 s1



= log

s3 s2



. Then (1 − τ) log

s3 s1



= log

s2 s1

 and

 s1 s2

aτ log

s3 s1



 s3 s2

a(1−τ) log

s3 s1



= exp



alog s3 s2



log s1 s2



+ a log s2 s1



log s3 s2



= 1.

Therefore,

M(s2)log

s3 s1



≤ M (s1)log

s3 s2



M(s3)log

s2 s1



.

Taking logarithms completes the proof. 

Corollary 4.6. Let f satisfy ˆD f = 0. Then for 0 < s1< s2< s3

|| f ||L(Qs2) ≤

|| f ||L(Qs1)

θ

|| f ||L(Qs3)

1−θ

, where

θ =log(s3/s2) log(s3/s1).

Remark 4.2. From Remark 4.1, we know that if D f = 0, then Dff = 0. Hence Corollary 4.6 applies to such f .

4.4. The similarity principle. The approach here is based on the work of Bojarksi, as presented in [1]. We will start with a few definitions and facts that will be used below. For simplicity, we work on a bounded domain Ω. Define the operators

T ω (z) = −1 π

ZZ

ω (ζ ) ζ − zdζ Sω (z) = −1

π ZZ

ω (ζ ) (ζ − z)2dζ . We will make use the of the following results, collected from [1].

Lemma 4.7. Suppose that g ∈ Lp for some p> 2. Then T g exists everywhere as an absolutely convergent integral and Sg exists almost everywhere as a Cauchy principal limit. The following relations hold:

∂ (T g) = g¯

∂ (T g) = Sg

|T g (z)| ≤ cp||g||Lp

||Sg||Lp≤ Cp||g||Lp

p→2lim+Cp= 1.

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Lemma 4.8 (see Lemmas 4.1, 4.3 [1]). Let w be a generalized solution (possibly admitting isolated singu- larities) to

∂ w + q¯ 1(z) ∂ w + q2(z) ∂ w = A (z) w + B (z) ¯w

in a bounded domain Ω ⊂ R2. Assume that |q1(z)| + |q2(z)| ≤ α0< 1 in Ω, and A, B are bounded functions.

Then w(z) is given by

w(z) = f (z) eT ω(z)= f (z) eφ (z), where f is a solution to

∂ f + q¯ 0(z) ∂ f = 0 and

φ (z) = T ω (z) . Here, ω solves (4.15) and q0is defined by(4.14).

The proof ideas are available in [1]. For completeness, we include the proof.

Proof. Let w (z) be the generalized solution. Set h(z) =

 A(z) + B (z)ww¯ for w (z) 6= 0 and w (z) 6= ∞ A(z) + B (z) otherwise

q0(z) =

 q1(z) + q2(z)∂ w∂ w for ∂ w 6= 0

q1(z) + q2(z) otherwise . (4.14)

We have |q0(z)| ≤ |q1(z)| + |q2(z)| ≤ α0. Consider the integral equation

ω + q0Sω = h. (4.15)

Let p > 2 be such that Cpq0< 1. Since h (z) ∈ Lp(Ω), then by a fixed point argument, this integral equation has a unique solution ω (z) ∈ Lp. Set φ (z) = T ω (z), then define f (z) = w (z) e−φ (z). We see that

∂ f = ¯¯ ∂ we−φ− ¯∂ φ we−φ= ¯∂ we−φ− ωwe−φ

∂ f = ∂ we−φ− ∂ φ we−φ= ∂ we−φ− Sωwe−φ. It follows that

∂ f + q¯ 0∂ f =∂ w + q¯ 0∂ w − (ω + q0Sω) w e−φ

=∂ w + q¯ 0∂ w − hw e−φ

=h ¯∂ w + q1∂ w + q2∂ w − Aw − B ¯w i

e−φ

= 0.

 Corollary 4.9. Let w be a generalized solution (possibly admitting isolated singularities) to

∂ w + q¯ 1(z) ∂ w + q2(z) ∂ w = A (z) w + B (z) ¯w

in a bounded domain Ω ⊂ R2. Assume that |q1(z)| + |q2(z)| ≤ α0< 1 in Ω, and A, B are bounded functions.

Then w(z) is given by

w(z) = f (z) g (z) , where f is a solution to

∂ f + q¯ 0(z) ∂ f = 0 and

exph

−C

||A||L(Ω)+ ||B||L(Ω)

i≤ |g (z)| ≤ exph C

||A||L(Ω)+ ||B||L(Ω)

i .

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Proof. From the previous lemma, we have that g (z) = exp (T ω (z)), where ω is the unique solution to (4.15).

Since Cpα0< 1, then

||ω||Lp ≤ C ||h||Lp. Therefore,

|T ω (z)| ≤ C ||h||Lp ≤ Ch

||A||L(Ω)+ ||B||L(Ω)

i ,

where C depends on Ω. The conclusion follows. 

5. THE PROOF OF THEOREM1.1

Before we can prove Theorem 1.1, we need to develop a set of results that are analogous to those in [6].

The first step is to show that a positive multiplier exists. We then use this positive multiplier to transform the PDE (1.8) into a divergence-form equation. The divergence-form equation is used to introduce a stream function, which gives rise to a Beltrami equation. Then, using the similarity principle of Bojarski and the Hadamard three-quasi-circle theorem, we are able to prove Theorem 1.1. From now on, unless specified otherwise, all constants C, c depend on λ and µ. Moreover, these constants are allowed to change from line to line. We also use the more compact notation σ (·) and ρ (·) in place of σ (·; λ ) and ρ (·; λ ) where it is understood that these functions depend on the ellipticity constant λ .

The first step is to show that there exists a positive solution φ to (1.8) in the ball Bd, where d = ρ 75 +25. Let η be some constant to be determined and set

φ1(x, y) = exp (ηx) . Then by (1.11), (1.12), and (1.5)

div (A∇φ1) −V φ1=

η (∂xa11+ ∂ya12) + η2a11−V φ1

≥h

λ η2− M − 2η µ√ M

i φ1. If η = c1

Mfor some constant c1depending on λ and µ that is sufficiently large, then φ1is a subsolution.

Now define φ2 = exp c2

√M, where c2 is a constant chosen so that φ2 ≥ φ1 on Bd. Since V ≥ 0, then Lφ2−V φ2≤ 0, so φ2is a supersolution. It follows that there exists a positive solution φ to (1.8) such that

exp



−C1√ M



≤ φ (z) ≤ exp C1

√ M



for all z ∈ Bd, (5.1)

where C1depends on c1, c2, and λ .

Furthermore, (by Theorem 8.32 from [5], for example) for 0 < α < 1, s < 2, φ satisfies the interior estimate

||∇φ ||L(Qα s)≤ C ||φ ||L(Qs), (5.2) where C = C (λ , K, M, s, α), with

K= max

i, j=1,2

ai j

0,γ;Q2= max

i, j=1,2

"

ai j

L(Q2)+ sup

x6=y∈Q2

ai j(x) − ai j(y)

|x − y|γ

# , where 0 < γ < 1 is arbitrary. Note that since

sup

x6=y∈Q2

ai j(x) − ai j(y)

|x − y|γ ≤ sup

x6=y∈Q2

µ |x − y|

|x − y|γ = sup

x6=y∈Q2

µ |x − y|1−γ ≤ µ diam (Q2) , then C = C (λ , µ, M, s, α). Moreover, by scaling considerations and Lemma 2.5,

C(λ , µ, M, s, α) ≤C(λ , µ, M, α)

sc .

Set v = u/φ . Since u and φ are both solutions to (1.8) and A is symmetric by (1.2), we see that

div φ2A∇v = 0 in Bd⊂ R2. (5.3)

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We use (5.3) to define a stream function in Bd. Let ˜v, with ˜v(0) = 0, satisfy the following system of equations

 v˜y = φ2(a11vx+ a12vy)

− ˜vx = φ2(a12vx+ a22vy) . (5.4) Specifically, for any (x, y) ∈ Bd,

˜

v(x, y) = Z y

0

φ2(a11vx+ a12vy) (0,t) dt − Z x

0

φ2(a12vx+ a22vy) (s, y) ds. (5.5) The stream function is used to transform the divergence-free equation into a Beltrami equation. Set w = φ2v+ i ˜v. Then, using (5.4), we see that

Dw= 2φ Dφ v + φ2Dv+ D(i ˜v)

= 2D (log φ ) φ2v

+ φ2 (a11+ det A) + ia21

det (A + I) vx+a12+ i (a22+ det A) det (A + I) vy



+ i (a11+ 1) + ia12

det (A + I) v˜x+a12+ i (a22+ 1) det (A + I) v˜y



= 2D (log φ ) φ2v

+ φ2 (a11+ det A) + ia21

det (A + I) vx+a12+ i (a22+ det A) det (A + I) vy



+ iφ2



−(a11+ 1) + ia12

det (A + I) (a12vx+ a22vy) +a12+ i (a22+ 1)

det (A + I) (a11vx+ a12vy)



= D (log φ ) (w + w) . Therefore,

Dw= α (w + ¯w) , (5.6)

where α = D (log φ ).

The next step is to estimate α. Here we mimic the arguments from [6], making appropriate modifications to account for the variable coefficients of the operator. To understand the behavior of α, we will study ψ = log φ . From (5.1), we see that

|ψ (z)| ≤ C√

M in Bd. (5.7)

Furthermore, a computation shows that ψ solves the following equation

div (A∇ψ) + A∇ψ · ∇ψ = V in Bd. (5.8)

Lemma 5.1. If φ is a solution to (1.8) and ψ = log φ , then

||∇ψ||L(Bρ (7/5)) ≤ C

M, (5.9)

where C depends on λ , µ.

Proof. Recall that d = ρ (7/5) + 2/5. Let θ ∈ C0(Bd) be a cutoff function for which θ ≡ 1 in Bρ (7/5)+1/5. Multiply (5.8) by θ and integrate by parts:

λ Z

θ |∇ψ |2Z

θ A∇ψ · ∇ψ = Z

θV − Z

div (A∇θ ) ψ ≤ C M+√

M

 . It follows that

Z

Bρ (7/5)+1/5

|∇ψ|2≤ CM.

We rescale equation (5.8). Set ϕ = ψ

C

M for some C > 0. Then (5.8) is equivalent to

ε div (A∇ϕ ) + A∇ϕ · ∇ϕ = ˜V in Bd, (5.10)

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where ε = 1

C

M and ˜V =CV2M. Now choose C sufficiently large so that

˜V

L(Bd)≤ 1, ||ϕ||L(Bd)≤ 1, Z

Bρ (7/5)+1/5

|∇ϕ|2≤ 1. (5.11)

Claim 5.2. Let c > 0 be such that for any z ∈ Bρ (7/5), B2c/5(z) ⊂ Bρ (7/5)+1/5. For any z∈ Bρ (7/5), and ε < r < c/5, if (5.10) and (5.11) hold, then

Z

Br(z)

|∇ϕ|2≤ Cr2.

Proof of Claim 5.2. It suffices to take z = 0. Let η ∈ C0(B2r) be a cutoff function such that η ≡ 1 in Br. Set m= |B2r|−1

Z

B2r

ϕ . By the divergence theorem 0 = ε

Z

div A∇(ϕ − m) η2

= ε Z

div (A∇ϕ) η2+ 4ε Z

η A∇ϕ · ∇η + ε Z

(ϕ − m) div A∇ η2 . (5.12) We now estimate each of the three terms. By (5.10) and (5.11),

Z

ε div (A∇ϕ ) η2= − Z

A∇ϕ · ∇ϕη2+ Z

V η˜ 2≤ −λ Z

|∇ϕ|2η2+ ˜V

L(Bd)

Z

B2r

1

≤ −λ Z

|∇ϕ|2η2+Cr2. (5.13)

By Cauchy-Schwarz and Young’s inequality,

Z

η A∇ϕ · ∇η

≤ 4λ ε

Z

|∇ϕ|2η2

1/2Z

|∇η|2

1/2

≤λ 2

Z

|∇ϕ|2η2+Cε2. (5.14) For the third term, we use the Poincar´e inequality to show that

ε

Z

(ϕ − m) div A∇ η2

≤ Cεr−2 Z

B2r

|ϕ − m| ≤ C

Z

B2r

|ϕ − m|2

1/2Z

B2r

ε2r−4

1/2

≤ Cr

Z

B2r

|∇ϕ|2

1/2

ε2r−21/2

≤ Cε2+ 1 400

Z

B2r

|∇ϕ|2. (5.15) Combining (5.12)-(5.15), we see that

Z

Br

|∇ϕ|2≤ Cε2+Cr2+ 1 200λ

Z

B2r

|∇ϕ|2≤ Cr2+ 1 200λ

Z

B2r

|∇ϕ|2. (5.16)

If r22001 , then by the last estimate of (5.11), the inequality above implies that Z

Br

|∇ϕ|2≤ Cr2. Otherwise, if r2<2001 , choose k ∈ N so that

c

5≤ 2kr≤2c 5 .

Clearly, r2≥ C (1/200λ )k. It follows from repeatedly applying (5.16) that Z

Br

|∇ϕ|2≤ Cr2+

 1

200λ

kZ

B2k r

|∇ϕ|2≤ Cr2,

proving the claim. 

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