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}∞(R^{2})≤ 1. Let u be a real solution
to div (A∇u) −Vu = 0 in R^{2}. Assume that |u (z)| ≤ exp (c0|z|) and u (0) = 1. Then, for any R sufficiently large,

|zinf_{0}|=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 R^{n}, (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^{∞}(R^{n})≤ 1
and kuk_{L}∞(R^{n})≤ C_{0} 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 R^{2} satisfying
u(0) = 1, |u(x)| ≤ exp(C_{0}|x|), where kV kL^{∞} ≤ 1 and V ≥ 0. Then for R sufficiently large,

|xinf_{0}|=R sup

|x−x_{0}|<1

|u(x)| ≥ exp(−CR log R),
where C depends on C_{0}.

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=

a_{11} a_{12}
a_{21} a_{22}

, a12= a21 (1.2)

λ |ξ |^{2}≤ a_{i j}(x) ξiξj≤ λ^{−1}|ξ |^{2}, for all x ∈ R^{2}, ξ ∈ R^{2}. (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⊂ Q_{1}
and Q_{7/5}b 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 ^{7}_{5}; λ +^{2}_{5}. Let u be a real-valued solution to

Lu−Vu = 0 in Bd⊂ R^{2}, (1.8)

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

||u||_{L}∞(B_{d})≤ exp
C_{0}

√ 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 (C_{0}, 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 : R^{2}→ R is measurable and satisfies

||V ||_{L}∞(R^{2})≤ 1.

Assume also that V ≥ 0 a.e. in R^{2}. Let u be a real solution to

Lu−Vu = 0 in R^{2}, (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 z_{0}= (x_{0}, y_{0}). Then, for any R sufficiently large,

|zinf0|=R||u||_{L}∞(B1(z0))≥ exp (−CR log R) , (1.15)
where C depends on c_{0}, 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 ^{7}_{5}; λ +^{2}_{5}. Let u be a real-valued solution to

Lu+ ∇ · (Wu) −Vu = 0 in Bd⊂ R^{2}, (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 : R^{2}→ R, W : R^{2}→ R^{2}are measurable and satisfy

||W ||_{L}∞(R^{2})≤ 1, ||V ||_{L}∞(R^{2})≤ 1.

Assume also that V ≥ 0 a.e. in R^{2}. Let u be a real solution to

Lu+ ∇ · (Wu) −Vu = 0 in R^{2}, (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 z_{0}= (x_{0}, y_{0}). Then, for any R sufficiently large, estimate (1.15) holds where C depends on c_{0},
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; λ^{2}, d = ρ ^{7}_{5}; λ^{2} +^{2}_{5}. Let u be a real-valued solution to

Lu−W · ∇u −Vu = 0 in Bd⊂ R^{2}, (1.19)

where V≥ 0 and A satisfies assumptions (1.2) and (1.3) with λ replaced by λ^{2}, anddet A = 1. Assume that
for some M≥ 1, (1.9) – (1.10), and (1.12) from above hold, and

||V ||_{L}∞(Bd)≤ C_{1}M (1.20)

||W ||_{L}∞(Bd)≤p

C_{1}M. (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 λ^{2}.

Again, the local theorem implies the Landis theorem.

Theorem 1.6. Assume that V : R^{2}→ R, W : R^{2}→ R^{2}are measurable and satisfy

||W ||_{L}∞(R^{2})≤ 1, ||V ||_{L}∞(R^{2})≤ 1.

Assume also that V ≥ 0 a.e. in R^{2}. Let u be a real solution to

Lu−W · ∇u −Vu = 0 in R^{2}, (1.23)

where A satisfies the assumptions (1.2) – (1.4). Assume that |u (z)| ≤ exp (c_{0}|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 c_{0}, 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 R^{2}that 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} R^{2}\ 0, G ∈ H_{loc}^{1,p} R^{2} for all p < 2 and for every ϕ ∈ C_{0}^{∞} R^{2}
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 C_{1},C_{2},C_{3},C_{4}, R_{1}< 1, R_{2}> 1, that depend on λ , such that
C_{1}log 1

|z|

≤ −G (z) ≤ C_{2}log 1

|z|

for |z|< R_{1}
C_{3}log |z| ≤ G (z) ≤ C_{4}log |z| for |z| > R_{2}.

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|^{C}^{2}≤ exp (G (z)) ≤ |z|^{C}^{1} for |z|< R1

C_{5}|z|^{C}^{2} ≤ exp (G (z)) ≤ C_{6}|z|^{C}^{4} for R_{1}< |z| < R_{2}

|z|^{C}^{3}≤ exp (G (z)) ≤ |z|^{C}^{4} 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| < R_{2}}, 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,

C_{2}log R_{1}≤ G (z) ≤ C_{4}log R_{2}.
Therefore, whenever R_{1}< |z| < R_{2},

exp (G (z)) ≤ R^{C}_{2}^{4} = R_{2}

|z|

C4

|z|^{C}^{4}≤ R_{2}
R_{1}

C4

|z|^{C}^{4},
and

exp (G (z)) ≥ R^{C}_{1}^{2} = R_{1}

|z|

C2

|z|^{C}^{2}≥ R_{1}
R_{2}

C2

|z|^{C}^{2},

giving the second line of bounds.

The level sets of G will be important to us.

Definition 2.4. Define a function ` : R^{2}→ (0, ∞) as follows: ` (z) = s iff G (z) = ln s. Then set
Z_{s}=z ∈ R^{2}: G (z) = ln s = z ∈ R^{2}: ` (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 ∈ R^{2}: ` (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, c_{2}, c_{3}, c_{4}, c_{5}, c_{6}, S_{1}< 1, S_{2}> 1, that depend on λ , such that if z ∈ Zs,
then

s^{c}^{1} ≤ |z| ≤ s^{c}^{2} for s≤ S_{1}

c_{5}s^{c}^{1} ≤ |z| ≤ c_{6}s^{c}^{4} for S_{1}< s < S_{2}
s^{c}^{3} ≤ |z| ≤ s^{c}^{4} for s≥ S_{2}.

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∈ R^{2}, we may
discuss the fundamental solutions with pole at z_{0}, 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}(z_{0}) and Qs(z_{0}), respectively. Although Qs(z_{0}) is not necessarily a translation of Qs(0) for z_{0}6= 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 z_{0}∈ R^{2}be such that |z_{0}| = R for some
R≥ 1. For a constant a yet to be determined, define

u_{R}(z) = u (z_{0}+ aRz) , A_{R}(z) = A (z_{0}+ aRz) , V_{R}(z) = (aR)^{2}V(z_{0}+ 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 ^{7}_{5}; λ +^{2}_{5} depends on λ . We choose a > 0 so that ^{1}_{a} ≤ b, where b = σ (1; λ ) depends on λ .
Then z1:= −_{aR}^{z}^{0} ∈ 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 =_{aR}^{1} 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_{R}u_{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 λ^{2}. Then u is a solution to ˜Lu − ˜W· ∇u − ˜V u= 0 in R^{2}, where ˜L = div ˜A∇ and
W˜, ˜V are given in (1.22). Note that

˜W

L^{∞} ≤ C_{1}, 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

∂ =¯ ^{1}_{2}(∂x+ i∂y)

∂ = ^{1}_{2}(∂x− i∂y)

η (z) = a_{11}− a_{22}+ 2ia_{12}

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}=(a_{11}− a_{22})^{2}+ 4a^{2}_{12}

[det (A + I)]^{2} =(a_{11}+ a_{22})^{2}− 4a11a_{22}+ 4a^{2}_{12}

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

|η (z)| = λ_{1}− λ_{2}
(λ_{1}+ 1) (λ_{2}+ 1)

|ν (z)| = |det A − 1|

[det (A + I)] = |λ_{1}λ_{2}− 1|

(λ_{1}+ 1) (λ_{2}+ 1),

where we are using λ1≥ λ_{2}to denote the eigenvalues of A. It follows that

|η (z)| + |ν (z)| = λ1− λ2

(λ_{1}+ 1) (λ_{2}+ 1)+ |λ1λ2− 1|

(λ_{1}+ 1) (λ_{2}+ 1)≤1 − λ
1 + λ.

A computation shows that for f = u + iv

D f =(a11+ det A) + ia12

det (A + I) u_{x}+a_{12}+ i (a22+ det A)
det (A + I) u_{y}
+(a_{11}+ 1) + ia_{12}

det (A + I) iv_{x}+a_{12}+ i (a_{22}+ 1)

det (A + I) iv_{y}. (4.4)

When A has determinant equal to 1, ν (z) = 0 and we may write
D=(a_{11}+ 1) + ia_{12}

det (A + I) ∂x+a_{12}+ i (a_{22}+ 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 α^{2}+ β^{2}≤ 1 − λ
1 + λ

2

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

Aˆ=

(1+α)^{2}+β^{2}
1−α^{2}−β^{2}

2β
1−α^{2}−β^{2}
2β

1−α^{2}−β^{2}

(1−α)^{2}+β^{2}
1−α^{2}−β^{2}

=

aˆ_{11} aˆ_{12}
ˆ
a_{12} aˆ_{22}

. (4.9)

A computation shows that the smallest eigenvalue of ˆAsatisfies

λ−= 1 − 2

1 + (α^{2}+ β^{2})^{−1/2} =1 −p

α^{2}+ β^{2}
1 +p

α^{2}+ β^{2} ≥ λ ,
while the largest eigenvalue of ˆAsatisfies

λ+=1 +p

α^{2}+ β^{2}
1 −p

α^{2}+ β^{2} ≤ 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

aˆ_{11}ux+ ˆa_{12}uy− v_{y}= 0
ˆ

a_{12}ux+ ˆa_{22}uy+ vx= 0, (4.11)

and

ˆ

a_{11}vx+ ˆa_{12}vy+ uy= 0
ˆ

a_{12}v_{x}+ ˆa_{22}v_{y}− u_{x}= 0, (4.12)

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

0 = ∂x[ ˆa_{11}u_{x}+ ˆa_{12}u_{y}− v_{y}] + ∂y[ ˆa_{12}u_{x}+ ˆa_{22}u_{y}+ vx] ,
so that ˆLu = 0. Similarly, by (4.12),

0 = ∂x[ ˆa_{11}v_{x}+ ˆa_{12}v_{y}+ uy] + ∂y[ ˆa_{12}v_{x}+ ˆa_{22}v_{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)| = ^{1}_{2}log u^{2}+ v^{2}. We have

∂xlog | f (z)| = uux+ vvx

u^{2}+ v^{2}

∂ylog | f (z)| = uuy+ vvy

u^{2}+ v^{2} .
Then,

ˆL[log | f (x)|] = ∂x( ˆa_{11}∂xlog | f (z)| + ˆa_{12}∂ylog | f (z)|) + ∂y( ˆa_{12}∂xlog | f (z)| + ˆa_{22}∂ylog | f (z)|)

= ∂x

ˆ

a_{11}uux+ vvx

u^{2}+ v^{2} + ˆa_{12}uu_{y}+ vvy

u^{2}+ v^{2}

+ ∂y

ˆ

a_{12}uux+ vvx

u^{2}+ v^{2} + ˆa_{22}uu_{y}+ vvy

u^{2}+ v^{2}

= [∂x( ˆa_{11}ux+ ˆa_{12}uy) + ∂y( ˆa_{12}ux+ ˆa_{22}uy)] u
u^{2}+ v^{2}
+ [∂x( ˆa_{11}vx+ ˆa_{12}vy) + ∂y( ˆa_{12}vx+ ˆa_{22}vy)] v

u^{2}+ v^{2}
+ ∂x

u

u^{2}+ v^{2}

ˆ

a_{11}u_{x}+ ∂x

u

u^{2}+ v^{2}

ˆ

a_{12}u_{y}+ ∂y

u

u^{2}+ v^{2}

ˆ

a_{12}u_{x}+ ∂y

u

u^{2}+ v^{2}

ˆ
a_{22}u_{y}
+ ∂x

v

u^{2}+ v^{2}

ˆ

a_{11}vx+ ∂x

v

u^{2}+ v^{2}

ˆ

a_{12}vy+ ∂y

v

u^{2}+ v^{2}

ˆ

a_{12}vx+ ∂y

v

u^{2}+ v^{2}

ˆ
a_{22}vy.
Since ˆLu = 0 = ˆLv by the previous lemma, the top two lines vanish and we have,

ˆL[log | f (x)|] = ∂x

u

u^{2}+ v^{2}

ˆ

a_{11}ux+ ∂x

u

u^{2}+ v^{2}

ˆ

a_{12}uy+ ∂y

u

u^{2}+ v^{2}

ˆ

a_{12}ux+ ∂y

u

u^{2}+ v^{2}

ˆ
a_{22}uy

+ ∂x

v

u^{2}+ v^{2}

ˆ

a_{11}v_{x}+ ∂x

v

u^{2}+ v^{2}

ˆ

a_{12}v_{y}+ ∂y

v

u^{2}+ v^{2}

ˆ

a_{12}v_{x}+ ∂y

v

u^{2}+ v^{2}

ˆ
a_{22}v_{y}

= n

ˆ
a_{11}

h

(ux)^{2}+ (vx)^{2}
i

+ 2 ˆa_{12}(uxu_{y}+ vxv_{y}) + ˆa_{22}
h

(uy)^{2}+ (vy)^{2}

io 1

u^{2}+ v^{2}

− 2h ˆ

a_{11}(uux+ vvx)^{2}+ 2 ˆa_{12}(uux+ vvx) (uuy+ vvy) + ˆa_{22}(uuy+ vvy)^{2}

i 1

(u^{2}+ v^{2})^{2}.
By the relations (4.11) and (4.12),

ˆ
a_{11}

h

(ux)^{2}+ (vx)^{2}i

+ 2 ˆa_{12}(uxuy+ vxvy) + ˆa_{22}
h

(uy)^{2}+ (vy)^{2}i

= ˆa_{11}
h

(ux)^{2}+ (vx)^{2}
i

+ [ux(vy− ˆa_{11}u_{x}) + vx(−uy− ˆa_{11}v_{x})]

+ [(−vx− ˆa_{22}u_{y}) uy+ (ux− ˆa_{22}v_{y}) vy] + ˆa_{22}
h

(uy)^{2}+ (vy)^{2}
i

= 2 (uxv_{y}− v_{x}u_{y}) ,
and

ˆ

a_{11}(uux+ vvx)^{2}+ 2 ˆa_{12}(uux+ vvx) (uuy+ vvy) + ˆa_{22}(uuy+ vvy)^{2}

= (uux+ vvx) { ˆa_{11}(uux+ vvx) + [u (vy− ˆa_{11}ux) + v (−uy− ˆa_{11}vx)]}

+ {[u (−vx− ˆa_{22}u_{y}) + v (ux− ˆa_{22}v_{y})] + ˆa_{22}(uuy+ vvy)} (uuy+ vvy)

= (uux+ vvx) (uvy− vu_{y}) + (−uvx+ vux) (uuy+ vvy)

= (uxv_{y}− u_{y}v_{x}) u^{2}+ v^{2} .
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 + a_{11}− ia_{12}] ∂x+ [a_{12}− i (1 + a22)] ∂y= det (A + I) D

We =(α∂xa_{11}− β ∂xa_{12}+ γ∂ya_{11}+ δ ∂ya_{12}) + i (γ∂xa_{11}+ δ ∂xa_{12}− α∂ya_{11}+ β ∂ya_{12})
a_{11}det (A + I)^{2}

α = a_{11}+ a_{22}+ 2a_{11}a_{22} β = 2a_{12}(1 + a_{11})
γ = a12(a22− a_{11}) δ = (1 + a11)^{2}− a^{2}_{12},
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 < s_{1}< s_{2}< s_{3},

log s_{3}
s_{1}

log M (s2) ≤ log s_{3}
s_{2}

log M (s1) + log s_{2}
s_{1}

log M (s3) . (4.13)

Proof. LetAs_{1},s3= {z : s_{1}≤ ` (z) ≤ s_{3}} = Qs_{3}\ Q_{s}_{1}, 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 onAs_{1},s3. Let z_{0}be
in the interior ofAs_{1},s_{3}. If f (z_{0}) = 0, then a ˆG(z_{0}) + log | f (z_{0})| = −∞ for any a ∈ R. On the other hand, if
f(z_{0}) 6= 0, then Lemma 4.3 implies that ˆLa ˆG(z) + log | f (z)| = 0 for z near z_{0}. By the maximum principle,
z_{0} cannot be an extremal point. Therefore, a ˆG(z) + log | f (z)| takes it maximum value on the boundary of
As_{1},s3. We will choose the constant a ∈ R so that

maxa ˆG(z) + log | f (z)| : z ∈ Zs_{1} = max a ˆG(z) + log | f (z)| : z ∈ Zs_{3} ,
or rather

log (s^{a}_{1}M(s_{1})) = log (s^{a}_{3}M(s_{3})) .
It follows that for any z ∈As1,s3,

a ˆG(z) + log | f (z)| ≤ log (s^{a}_{1}M(s_{1})) (or log (s^{a}_{3}M(s_{3}))).

Furthermore, for any s2∈ (s_{1}, s_{3}),

maxa ˆG(z) + log | f (z)| : z ∈ Zs_{2} ≤ log (s^{a}_{1}M(s_{1})) (or log (s^{a}_{3}M(s_{3}))),
or

log (s^{a}_{2}M(s2)) ≤ log (s^{a}_{1}M(s1)) (or log (s^{a}_{3}M(s3))).

Consequently,

s^{a}_{2}M(s_{2}) ≤ s^{a}_{1}M(s_{1}) (or s^{a}_{3}M(s_{3})),
so that for any τ ∈ (0, 1), since s^{a}_{1}M(s_{1}) = s^{a}_{3}M(s_{3}), then

s^{a}_{2}M(s2) ≤ [s^{a}_{1}M(s1)]^{τ}[s^{a}_{3}M(s3)]^{1−τ}
[M (s_{2})]^{log}

s3 s1

≤ s_{1}
s_{2}

a

M(s_{1})

τ log

s3 s1

s_{3}
s_{2}

a

M(s_{3})

(1−τ) log

s3 s1

. We choose τ so that τ log

s_{3}
s_{1}

= log

s_{3}
s_{2}

. Then (1 − τ) log

s_{3}
s_{1}

= log

s_{2}
s_{1}

and

s_{1}
s_{2}

aτ log

s3 s1

s_{3}
s_{2}

a(1−τ) log

s3 s1

= exp

alog s_{3}
s_{2}

log s_{1}
s_{2}

+ a log s_{2}
s_{1}

log s_{3}
s_{2}

= 1.

Therefore,

M(s_{2})^{log}

s3 s1

≤ M (s_{1})^{log}

s3 s2

M(s_{3})^{log}

s2 s1

.

Taking logarithms completes the proof.

Corollary 4.6. Let f satisfy ˆD f = 0. Then for 0 < s_{1}< s_{2}< s_{3}

|| f ||_{L}_{∞}(^{Q}_{s2}) ≤

|| f ||_{L}_{∞}(^{Q}_{s1})

θ

|| f ||_{L}_{∞}(^{Q}_{s3})

1−θ

, where

θ =log(s3/s2)
log(s_{3}/s_{1}).

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)^{2}dζ .
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||_{L}p

||Sg||_{L}p≤ C_{p}||g||_{L}p

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 + q_{2}(z) ∂ w = A (z) w + B (z) ¯w

in a bounded domain Ω ⊂ R^{2}. Assume that |q_{1}(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

q_{0}(z) =

q_{1}(z) + q_{2}(z)^{∂ w}_{∂ w} for ∂ w 6= 0

q_{1}(z) + q2(z) otherwise . (4.14)

We have |q0(z)| ≤ |q_{1}(z)| + |q_{2}(z)| ≤ α_{0}. Consider the integral equation

ω + q0Sω = h. (4.15)

Let p > 2 be such that Cpq_{0}< 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 + q_{0}Sω) 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 ⊂ R^{2}. Assume that |q_{1}(z)| + |q_{2}(z)| ≤ α_{0}< 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 Cpα_{0}< 1, then

||ω||_{L}p ≤ C ||h||_{L}p.
Therefore,

|T ω (z)| ≤ C ||h||_{L}p ≤ 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 ^{7}_{5} +^{2}_{5}.
Let η be some constant to be determined and set

φ_{1}(x, y) = exp (ηx) .
Then by (1.11), (1.12), and (1.5)

div (A∇φ_{1}) −V φ_{1}=

η (∂xa_{11}+ ∂ya_{12}) + η^{2}a_{11}−V
φ_{1}

≥h

λ η^{2}− 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 c_{2} is a constant chosen so that φ2 ≥ φ1 on Bd. Since V ≥ 0, then
Lφ2−V φ_{2}≤ 0, so φ_{2}is a supersolution. It follows that there exists a positive solution φ to (1.8) such that

exp

−C_{1}√
M

≤ φ (z) ≤ exp
C_{1}

√ M

for all z ∈ Bd, (5.1)

where C_{1}depends on c_{1}, c_{2}, 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∈Q_{2}

ai j(x) − ai j(y)

|x − y|^{γ}

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

sup

x6=y∈Q_{2}

a_{i j}(x) − ai j(y)

|x − y|^{γ} ≤ sup

x6=y∈Q_{2}

µ |x − y|

|x − y|^{γ} = sup

x6=y∈Q_{2}

µ |x − y|^{1−γ} ≤ µ diam (Q_{2}) ,
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 φ^{2}A∇v = 0 in B_{d}⊂ R^{2}. (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} = φ^{2}(a_{11}v_{x}+ a_{12}v_{y})

− ˜vx = φ^{2}(a12v_{x}+ a22v_{y}) . (5.4)
Specifically, for any (x, y) ∈ Bd,

˜

v(x, y) = Z y

0

φ^{2}(a11vx+ a12vy) (0,t) dt −
Z x

0

φ^{2}(a12vx+ a22vy) (s, y) ds. (5.5)
The stream function is used to transform the divergence-free equation into a Beltrami equation. Set w =
φ^{2}v+ i ˜v. Then, using (5.4), we see that

Dw= 2φ Dφ v + φ^{2}Dv+ D(i ˜v)

= 2D (log φ ) φ^{2}v

+ φ^{2} (a_{11}+ det A) + ia21

det (A + I) v_{x}+a_{12}+ i (a22+ det A)
det (A + I) v_{y}

+ i (a_{11}+ 1) + ia12

det (A + I) v˜_{x}+a_{12}+ i (a22+ 1)
det (A + I) v˜_{y}

= 2D (log φ ) φ^{2}v

+ φ^{2} (a_{11}+ det A) + ia_{21}

det (A + I) v_{x}+a_{12}+ i (a_{22}+ det A)
det (A + I) v_{y}

+ iφ^{2}

−(a11+ 1) + ia12

det (A + I) (a12vx+ a22vy) +a_{12}+ 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}∞(^{B}ρ (7/5)) ≤ C

√

M, (5.9)

where C depends on λ , µ.

Proof. Recall that d = ρ (7/5) + 2/5. Let θ ∈ C_{0}^{∞}(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}^{V}2M. 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}≤ Cr^{2}.

Proof of Claim 5.2. It suffices to take z = 0. Let η ∈ C_{0}^{∞}(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) η^{2}

= ε Z

div (A∇ϕ) η^{2}+ 4ε
Z

η A∇ϕ · ∇η + ε Z

(ϕ − m) div A∇ η^{2} . (5.12)
We now estimate each of the three terms. By (5.10) and (5.11),

Z

ε div (A∇ϕ ) η^{2}= −
Z

A∇ϕ · ∇ϕη^{2}+
Z

V η˜ ^{2}≤ −λ
Z

|∇ϕ|^{2}η^{2}+
˜V

L^{∞}(Bd)

Z

B_{2r}

1

≤ −λ Z

|∇ϕ|^{2}η^{2}+Cr^{2}. (5.13)

By Cauchy-Schwarz and Young’s inequality,

4ε Z

η A∇ϕ · ∇η

≤ 4λ ε

_{Z}

|∇ϕ|^{2}η^{2}

1/2_{Z}

|∇η|^{2}

1/2

≤λ 2

Z

|∇ϕ|^{2}η^{2}+Cε^{2}. (5.14)
For the third term, we use the Poincar´e inequality to show that

ε

Z

(ϕ − m) div A∇ η^{2}

≤ Cεr^{−2}
Z

B_{2r}

|ϕ − m| ≤ C

Z

B_{2r}

|ϕ − m|^{2}

1/2Z

B_{2r}

ε^{2}r^{−4}

1/2

≤ Cr

_{Z}

B_{2r}

|∇ϕ|^{2}

1/2

ε^{2}r^{−2}1/2

≤ Cε^{2}+ 1
400

Z

B_{2r}

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

Z

B_{r}

|∇ϕ|^{2}≤ Cε^{2}+Cr^{2}+ 1
200λ

Z

B_{2r}

|∇ϕ|^{2}≤ Cr^{2}+ 1
200λ

Z

B_{2r}

|∇ϕ|^{2}. (5.16)

If r^{2}≥_{200}^{1} , then by the last estimate of (5.11), the inequality above implies that
Z

Br

|∇ϕ|^{2}≤ Cr^{2}.
Otherwise, if r^{2}<_{200}^{1} , choose k ∈ N so that

c

5≤ 2^{k}r≤2c
5 .

Clearly, r^{2}≥ C (1/200λ )^{k}. It follows from repeatedly applying (5.16) that
Z

B_{r}

|∇ϕ|^{2}≤ Cr^{2}+

1

200λ

kZ

B2k r

|∇ϕ|^{2}≤ Cr^{2},

proving the claim.