Let u be a real solution to div (A∇u) −Vu = 0 in R2

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 kuk_L∞(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≤ a_{i 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.

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

a_ii≥ λ for each i = 1, 2 (1.5)

a_{i j}≤ Cλ⁻¹ for each i, j = 1, 2. (1.6)

We define the leading operator

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 j∂ju) = ai j∂i ju+ ∂iai j∂ju:= ai j∂i ju+ bj∂ju.

It follows from (1.4) that b ∈ L^∞with 

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 Q_7/5b 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∞(B_d)≤ exp C₀

√ M



(1.9)

||u||_L∞(B_b)≥ 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)≥ r^C

√

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)

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.

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 L^∞coefficients.

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 ∈ H_loc^1,2 R²\ 0
, G ∈ H_loc^1,p R²
for all p < 2 and for every ϕ ∈ C₀^∞ R²
Z

a_{i 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.

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)) ≤ R^C₂⁴ = R₂

|z|

C4

|z|^C4≤ R₂ R₁

C4

|z|^C4, and

exp (G (z)) ≥ R^C₁² = 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 ` : R<sup>2</sup>→ (0, ∞) as follows: ` (z) = s iff G (z) = ln s. Then set Z_s=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 Q^L_s and Z^L_s 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

s^c1 ≤ |z| ≤ s^c2 for s≤ S₁

c₅s^c1 ≤ |z| ≤ c₆s^c4 for S₁< s < S₂ s^c3 ≤ |z| ≤ s^c4 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.

Definition 2.6. Define

σ (s; λ ) = sup







r> 0 : Br⊂ ^\

L∈L (λ)

Q^L_s







(2.1)

ρ (s; λ ) = inf







r> 0 : ^[

L∈L (λ)

Q^L_s ⊂ B_r







. (2.2)

Remark 2.1. These functions are defined so that for any operator L inL (λ), Bσ (s;λ )⊂ Q^L_s ⊂ B_{ρ (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 Z_s(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

u_R(z) = u (z₀+ aRz) , A_R(z) = A (z₀+ aRz) , V_R(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

||u_R||_L∞(B_d)≤ exp (c0(1 + ad) R) ,

where d = ρ ⁷₅; λ
+²₅ depends on λ . We choose a > 0 so that ¹_a ≤ b, where b = σ (1; λ ) depends on λ . Then z1:= −_aR^z0 ∈ B_b, uR(z1) = u (0) = 1 and it follows that

||u_R||_L∞(Bb)≥ 1.

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

∇ai j

L^∞ ≤ µ implies that 

∇a_{R,i j}  

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

||u_R||_L∞(Br)≥ r^CaR. Setting r =_aR¹ and rewriting in terms of u, we see that

||u||_L∞(B1(z0))≥ exp − ˜CRlog R
,

as required. 

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

L_Ru_R+ ∇ (WRu_R) −VRu_R= 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.

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− λ₂ (λ₁+ 1) (λ₂+ 1)

|ν (z)| = |det A − 1|

[det (A + I)] = |λ₁λ₂− 1|

(λ₁+ 1) (λ₂+ 1),

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

|η (z)| + |ν (z)| = λ1− λ2

(λ₁+ 1) (λ₂+ 1)+ |λ1λ2− 1|

(λ₁+ 1) (λ₂+ 1)≤1 − λ 1 + λ.

 A computation shows that for f = u + iv

D f =(a11+ det A) + ia12

det (A + I) u_x+a₁₂+ i (a22+ det A) det (A + I) u_y +(a₁₁+ 1) + ia₁₂

det (A + I) iv_x+a₁₂+ i (a₂₂+ 1)

det (A + I) iv_y. (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 ∂x+βw+ 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)

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)²+β² 1−α²−β²

2β 1−α²−β² 2β

1−α²−β²

(1−α)²+β² 1−α²−β²



=

 aˆ₁₁ aˆ₁₂ ˆ a₁₂ aˆ₂₂



. (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− v_y= 0 ˆ

a₁₂ux+ ˆa₂₂uy+ vx= 0, (4.11)

and 

ˆ

a₁₁vx+ ˆa₁₂vy+ uy= 0 ˆ

a₁₂v_x+ ˆa₂₂v_y− u_x= 0, (4.12)

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

0 = ∂x[ ˆa₁₁u_x+ ˆa₁₂u_y− v_y] + ∂y[ ˆa₁₂u_x+ ˆa₂₂u_y+ vx] , so that ˆLu = 0. Similarly, by (4.12),

0 = ∂x[ ˆa₁₁v_x+ ˆa₁₂v_y+ uy] + ∂y[ ˆa₁₂v_x+ ˆa₂₂v_y− u_x] ,

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.

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( ˆa₁₁∂xlog | f (z)| + ˆa₁₂∂ylog | f (z)|) + ∂y( ˆa₁₂∂xlog | f (z)| + ˆa₂₂∂ylog | f (z)|)

= ∂x

 ˆ

a₁₁uux+ vvx

u²+ v² + ˆa₁₂uu_y+ vvy

u²+ v²

 + ∂y

 ˆ

a₁₂uux+ vvx

u²+ v² + ˆa₂₂uu_y+ 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

a₁₁u_x+ ∂x

 u

u²+ v²

 ˆ

a₁₂u_y+ ∂y

 u

u²+ v²

 ˆ

a₁₂u_x+ ∂y

 u

u²+ v²

 ˆ a₂₂u_y + ∂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

a₁₁v_x+ ∂x

 v

u²+ v²

 ˆ

a₁₂v_y+ ∂y

 v

u²+ v²

 ˆ

a₁₂v_x+ ∂y

 v

u²+ v²

 ˆ a₂₂v_y

= n

ˆ a₁₁

h

(ux)²+ (vx)² i

+ 2 ˆa₁₂(uxu_y+ vxv_y) + 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− ˆa₁₁u_x) + vx(−uy− ˆa₁₁v_x)]

+ [(−vx− ˆa₂₂u_y) uy+ (ux− ˆa₂₂v_y) vy] + ˆa₂₂ h

(uy)²+ (vy)² i

= 2 (uxv_y− v_xu_y) , 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− ˆa₂₂u_y) + v (ux− ˆa₂₂v_y)] + ˆa₂₂(uuy+ vvy)} (uuy+ vvy)

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

= (uxv_y− u_yv_x) u²+ v²
. Therefore,

ˆL[log | f (z)|] = 0,

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) ≤ s<sub>3</sub>} = Qs<sub>3</sub>\ Q<sub>s1</sub>, 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 ˆLa 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

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

log (s^a₁M(s₁)) = log (s^a₃M(s₃)) . It follows that for any z ∈As1,s3,

a ˆG(z) + log | f (z)| ≤ log (s^a₁M(s₁)) (or log (s^a₃M(s₃))).

Furthermore, for any s2∈ (s₁, s₃),

maxa ˆG(z) + log | f (z)| : z ∈ Zs₂ ≤ log (s^a₁M(s₁)) (or log (s^a₃M(s₃))), or

log (s^a₂M(s2)) ≤ log (s^a₁M(s1)) (or log (s^a₃M(s3))).

Consequently,

s^a₂M(s₂) ≤ s^a₁M(s₁) (or s^a₃M(s₃)), so that for any τ ∈ (0, 1), since s^a₁M(s₁) = s^a₃M(s₃), then

s^a₂M(s2) ≤ [s^a₁M(s1)]^τ[s^a₃M(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

aτ 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∞(^Q_s2) ≤

|| f ||_L∞(^Q_s1)

θ

|| f ||_L∞(^Q_s3)

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

ω (ζ ) ζ − zdζ Sω (z) = −1

π ZZ

Ω

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

Lemma 4.7. Suppose that g ∈ L^p 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≤ C_p||g||_Lp

p→2lim⁺Cp= 1.

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) e^{T ω(z)}= f (z) e^{φ (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)^w_w^¯ 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) ∈ L^p(Ω), then by a fixed point argument, this integral equation has a unique solution ω (z) ∈ L^p. 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 .

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∞(Q_{α 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

a_{i 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)

s^c .

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 B_d⊂ R². (5.3)

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(a₁₁v_x+ a₁₂v_y)

− ˜vx = φ²(a12v_x+ a22v_y) . (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) v_x+a₁₂+ i (a22+ det A) det (A + I) v_y



+ 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) v_x+a₁₂+ i (a₂₂+ det A) det (A + I) v_y



+ 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 |²≤ Z

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)

where ε = ¹

C√

M and ˜V =_C^V2M. 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 ∈ B_{ρ (7/5)}, B_2c/5(z) ⊂ Bρ (7/5)+1/5. For any z∈ B_{ρ (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^∞(B_2r) be a cutoff function such that η ≡ 1 in Br. Set m= |B2r|⁻¹

Z

B_2r

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

B_2r

1

≤ −λ Z

|∇ϕ|²η²+Cr². (5.13)

By Cauchy-Schwarz and Young’s inequality, 

4ε 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

B_2r

|ϕ − m| ≤ C

Z

B_2r

|ϕ − m|²

1/2Z

B_2r

ε²r⁻⁴

1/2

≤ Cr

_Z

B_2r

|∇ϕ|²

1/2

ε²r⁻²
1/2

≤ Cε²+ 1 400

Z

B_2r

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

Z

B_r

|∇ϕ|²≤ Cε²+Cr²+ 1 200λ

Z

B_2r

|∇ϕ|²≤ Cr²+ 1 200λ

Z

B_2r

|∇ϕ|². (5.16)

If r²≥₂₀₀¹ , 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≤ 2^kr≤2c 5 .

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

B_r

|∇ϕ|²≤ Cr²+

 1

200λ

kZ

B2k r

|∇ϕ|²≤ Cr²,

proving the claim.