STRONG UNIQUE CONTINUATION FOR THE LAM ´E SYSTEM WITH LESS REGULAR COEFFICIENTS

BLAIR DAVEY, CHING-LUNG LIN, AND JENN-NAN WANG

Abstract. We study the strong unique continuation property (SUCP)
for the Lam´e system in the plane. The main contribution of our work is
to prove that the SUCP holds when Lam´e coefficients (µ, λ) ∈ W^{2,s}(Ω)×

L^{∞}(Ω) for some s > 4/3. In other words, we establish the SUCP for the
Lam´e system in the plane when λ is bounded and µ belongs to certain
H¨older classes.

1. Introduction

In this paper, we are interested in the strong unique continuation property
(SUCP) for the Lam´e system in the plane. We begin with a short description
of the system. Let Ω ⊂ R^{2}be an open, connected set that contains the origin.

Assume that µ ∈ W^{2,s}(Ω) and λ ∈ L^{∞}(Ω), where
(1.1)

(µ(x) ≥ δ_{0}, λ(x) + 2µ(x) ≥ δ_{0}, ∀ x ∈ Ω,
kµk_{W}2,s(Ω)+ kλk_{L}^{∞}_{(Ω)}≤ M_{0}

for some positive constants δ_{0}, M_{0}. Recall that
kf k_{W}2,s(Ω)= X

|α|≤2

k∂_{x}^{α}f k_{L}^{s}_{(Ω)}.

The Lam´e system or isotropic elasticity equation, which represents the dis- placement of equilibrium, is given by

(1.2) div µ ∇u + (∇u)^{t} + ∇(λ divu) = 0 in Ω,

where u = (u_{1}, u_{2})^{t}is the real-valued displacement vector and (∇u)_{jk} = ∂_{k}u_{j}
for j, k = 1, 2.

Without loss of generality, assume that 0 ∈ Ω. In the sequel, B_{r} denotes
an open ball of radius r > 0 centered at the origin, while B_{r}(x_{0}) denotes an
open ball of radius r centered at x0.

We now state the main results of the paper. Our first result is an optimal three-ball inequality for solutions to (1.2).

2010 Mathematics Subject Classification. 35J47.

Key words and phrases. Carleman estimates, elliptic systems, quantitative unique continuation.

Davey is supported in part by the Simons Foundation Grant number 430198.

Lin is partially supported by the Ministry of Science and Technology of Taiwan.

Wang is partially supported by MOST 105-2115-M-002-014-MY3.

1

Theorem 1.1. Assume that for some s > ^{4}_{3}, the estimates in (1.1) are
satisfied. Then there exists an R_{0} > 0, depending on s, M_{0}, δ_{0}, such that
BR0 ⊂ Ω and for any u ∈ H_{loc}^{1} (BR0) satisfying (1.2) in BR0, if 0 < R1 <

R2 < R3≤ R_{0} and R2/R3 < 1/4, then
(1.3) kuk_{L}2(^{B}_{R2}) ≤ C kuk

τ

L^{2}(^{B}_{R1})kuk^{1−τ}

L^{2}(^{B}_{R3}),

where C = C (s, M_{0}, δ_{0}, R_{2}, R_{3}) and τ = τ (R_{1}/R_{2}, R_{2}/R_{3}, R_{0}) ∈ (0, 1).

Moreover, for fixed R2 and R3, the exponent τ behaves like 1/(− log R1)
whenever R_{1} is sufficiently small.

We emphasize that C is independent of R1 and τ has the asymptotic
behavior like (− log R_{1})^{−1}. These facts are crucial to the derivation of the
vanishing order for nontrivial solutions u to (1.2). Due to the behavior of τ ,
the three-ball inequality is called optimal [11].

Our next result establishes the rate of vanishing for solutions to (1.2), thereby proving the strong unique continuation property (SUCP) in our setting.

Theorem 1.2. Assume that for some s > ^{4}_{3}, (1.1) holds. Let R > 0 be
such that 3R ≤ R0, where R0 is defined in Theorem 1.1. There exist posi-
tive constants K and m, depending on s, M_{0}, δ_{0} and u, such that for all R
sufficiently small

(1.4) kuk_{L}2(BR)≥ KR^{m}.

In Theorems 1.1 and 1.2, our main interest is in the case where s ∈ (4/3, 2].

If s > 2, then by the Sobolev embedding theorem, µ is at least Lipschitz, so the SUCP for (1.2) has already been proved (see below). Moreover, when s ∈ (4/3, 2], we have

µ ∈( C^{0,γ} with γ = 2 − (2/s), if s < 2
C^{0,γ} for any γ ∈ (0, 1), if s = 2.

On the other hand, if µ(x) = |x|^{α}+ 1 for some α ∈ (0, 1), then µ ∈ W^{2,s}
for s < _{2−α}^{2} , but µ is clearly not a Lipschitz function. In other words, we
establish the quantitative SUCP for (1.2) when λ ∈ L^{∞} and µ is H¨older
continuous.

Note that by the results of [16], [19], the unique continuation property fails for elliptic equations in d ≥ 3 when the coefficients are only assumed to be H¨older continuous. Therefore, our results are special to the planar setting.

Theorems 1.1 and 1.2 will be proved together. Estimates (1.3) and (1.4) are quantitative forms of the SUCP for the Lam´e system (1.2). Taking advantage of the isotropy of the coefficients in (1.2), the unique continuation property for (1.2) has been studied rather thoroughly. Here we list some of the known results on the SUCP for (1.2):

• λ, µ ∈ C^{1,1}, n ≥ 2 (quantitative): Alessandrini and Morassi [2].

• λ, µ ∈ C^{0,1}, n = 2 (qualitative): Lin and Wang [17].

• λ ∈ L^{∞}, µ ∈ C^{0,1}, n = 2 (qualitative): Escauriaza [10].

• λ, µ ∈ C^{0,1}, n ≥ 2 (quantitative): Lin, Nakamura, and Wang [18].

• µ ∈ C^{0,1}, λ ∈ L^{∞}, n ≥ 2 (quantitative): Lin, Nakamura, Uhlmann,
and Wang [15].

• µ ∈ C^{0,1}, λ ∈ L^{∞}, n ≥ 2 (doubling inequality): Koch, Lin, and
Wang [14].

We also mention a recent article by the authors [7] where we proved
that a Liouville-type theorem holds for the Lam´e system in the plane when
µ ∈ W^{1,2} and k∇µk_{L}^{2} is small. From this result, we proved a weak unique
continuation property and the uniqueness of the Cauchy problem for such
Lam´e system (see Corollaries 1.3, 1.4 in [7]).

The main theme of this work is to prove the SUCP for (1.2) with the focus
on the regularity of µ. Recall that for scalar second-order elliptic equations
in divergence or non-divergence form with n = 2, the SUCP holds even
when the leading coefficients are only essentially bounded, [1], [3], [4], and
[20]. However, we would like to point out that the SUCP for (1.2) may not
hold when µ is essentially bounded or even continuous, see [5] and [13]. In
this paper, we establish the SUCP for (1.2) when µ ∈ W^{2,s} with s > 4/3 (a
subclass of H¨older continuity) and λ ∈ L^{∞}. An interesting open question is
to prove or disprove the SUCP or even the unique continuation property for
(1.2) in n = 2 for general (µ, λ) ∈ (W^{1,p}(Ω), L^{∞}(Ω)) with p < ∞.

The restriction of the range s > 4/3 is dictated by the application of Carleman estimates. We need this condition in order to absorb unwanted terms, see for example (4.5) and the explanation following it. We think that the condition s > 4/3 is optimal as long as we consider the radially symmetric weight function h in the Carleman estimate.

This article is organized as follows. In the next section, Section 2, we introduce the reduced system. That is, we show that (1.2) is equivalent to a system of second-order elliptic systems, and since we are working with real-valued solutions in the plane, it can also be realized as a ¯∂-equation.

The main Carleman estimates appear in Section 3. The first such theorem applies to first-order operators, while the second applies to second-order op- erators; thereby pairing with the equations derived in Section 2. The proof the latter Carleman estimate also appears in Section 3. Section 4 is devoted to the proofs of the main theorems, Theorems 1.1 and 1.2. The technical proof of the ¯∂-Carleman estimate is provided in Section 5. Finally, Section 6 states and proves the interior estimate associated to (1.2) that finds ap- plications in the proofs of the main theorems.

Acknowledgement. Part of this research was carried out while the first author was visiting the National Center for Theoretical Sciences (NCTS) at National Taiwan University. The first author wishes to thank the NCTS for their financial support and their kind hospitality during her visit to Taiwan.

2. Reduced systems

Within this section, we derive reduced systems associated to (1.2). There are two new systems that we produce. First, we show that (1.2) may be written as a systems of second-order elliptic equations. Then we show that a complex-valued first-order elliptic equation may be derived from the system.

These two representations will be instrumental to the Carleman estimate arguments that are used to prove our main unique continuation theorems.

With

(2.1) v = λ + 2µ

µ div u = λ + 2µ

µ (∂_{1}u_{1}+ ∂_{2}u_{2}),
w = curl u = ∂_{2}u_{1}− ∂_{1}u_{2},

a straightforward computation shows that

(2.2)

∆u_{1} = ∂_{1}

µ

λ + 2µv

+ ∂_{2}w,

∆u2 = ∂2

µ

λ + 2µv

− ∂_{1}w.

Expanding (1.2) gives div

2µ∂_{1}u_{1} µ (∂_{1}u_{2}+ ∂_{2}u_{1})
µ(∂1u2+ ∂2u1) 2µ∂2u2

+

∂_{1}{λ (∂_{1}u_{1}+ ∂_{2}u_{2})}

∂2{λ (∂_{1}u1+ ∂2u2)}

=

∂1(2µ∂1u1) + ∂2{µ (∂_{1}u2+ ∂2u1)}

∂_{1}{µ (∂_{1}u_{2}+ ∂_{2}u_{1})} + ∂_{2}(2µ∂_{2}u_{2})

+

∂1(λ∂1u1+ λ∂2u2)

∂_{2}(λ∂_{1}u_{1}+ λ∂_{2}u_{2})

= 0.

Simplifying each row shows that

∂1[(2µ + λ) ∂1u1+ λ∂2u2] + ∂2[µ (∂1u2+ ∂2u1)] = 0 (2.3)

− ∂_{2}[λ∂_{1}u_{1}+ (2µ + λ) ∂_{2}u_{2}] = ∂_{1}[µ (∂_{1}u_{2}+ ∂_{2}u_{1})] .
(2.4)

For brevity, we define the operator ¯∂ = ∂1+ i∂2. By the definition of v and w, we have

(2.5)

∂ (µv − iµw)¯

=∂_{1}(µv) + ∂_{2}(µw) − i∂_{1}(µw) + i∂_{2}(µv)

=∂1[(λ + 2µ) (∂1u1+ ∂2u2)] + ∂2[µ (∂2u1− ∂_{1}u2)]

−i∂_{1}[µ (∂2u1− ∂_{1}u2)] + i∂2[(λ + 2µ) (∂1u1+ ∂2u2)] .
From the real part of (2.5), we get

(2.6)

∂1[(λ + 2µ) (∂1u1+ ∂2u2)] + ∂2[µ (∂2u1− ∂_{1}u2)]

=∂_{1}[(λ + 2µ) ∂_{1}u_{1}+ λ∂_{2}u_{2}] + ∂_{1}[2µ∂_{2}u_{2}]
+∂_{2}[µ (∂_{2}u_{1}+ ∂_{1}u_{2})] + ∂_{2}[−2µ∂_{1}u_{2}]

=2∂1[µ∂2u2] − 2∂2[µ∂1u2]

=2∂1µ∂2u2− 2∂_{2}µ∂1u2,

where we have used (2.3) to eliminate terms. For the imaginary part of (2.5), using (2.4) shows that

(2.7)

∂_{1}[µ (∂_{2}u_{1}− ∂_{1}u_{2})] − ∂_{2}[(λ + 2µ) (∂_{1}u_{1}+ ∂_{2}u_{2})]

=∂_{1}[µ (∂_{2}u_{1}− ∂_{1}u_{2})] − ∂_{2}[λ∂_{1}u_{1}+ (λ + 2µ) ∂_{2}u_{2}] − ∂_{2}(2µ∂_{1}u_{1})

=∂1[µ (∂2u1− ∂_{1}u2)] + ∂1[µ (∂1u2+ ∂2u1)] − ∂2(2µ∂1u1)

=∂1(2µ∂2u1) − ∂2(2µ∂1u1)

=2∂_{1}µ∂_{2}u_{1}− 2∂_{2}µ∂_{1}u_{1}.

Substituting (2.6) and (2.7) into (2.5) then gives
(2.8) ∂ (µv − iµw) = g¯ 1− ig_{2},
where

g1= 2∂1µ∂2u2− 2∂_{2}µ∂1u2, g2 = 2∂1µ∂2u1− 2∂_{2}µ∂1u1.
Using the observation that ∂2u1= w + ∂1u2, we rewrite g1− ig_{2} as

(2.9)

g_{1}− ig_{2}

=2∂_{1}µ∂_{2}u_{2}− i2∂_{1}µ∂_{2}u_{1}− 2∂_{2}µ∂_{1}u_{2}+ i2∂_{2}µ∂_{1}u_{1}

=2∂1µ (∂2u2− iw − i∂_{1}u2) − 2∂2µ (∂2u1− w − i∂_{1}u1)

= − 2i∂_{1}µ ¯∂u_{2}+ 2i∂_{2}µ ¯∂u_{1}− 2iw ¯∂µ

= ¯∂ (−2i∂1µu2) + ¯∂ (2i∂2µu1) − 2iw ¯∂µ + ¯∂ (2i∂1µ) u2− ¯∂ (2i∂2µ) u1. Combining (2.8) and (2.9) then gives

(2.10) ∂ (µv − iµw + 2i∂¯ _{1}µu_{2}− 2i∂_{2}µu_{1}) = g_{3}+ g_{4}.
where

(2.11) g_{3} = −2i ¯∂µw, g_{4}= ¯∂ (2i∂_{1}µ) u_{2}− ¯∂ (2i∂_{2}µ) u_{1}.

The equations described by (2.2) and (2.10) will be used in the proofs of the main theorems.

3. Carleman estimates

Here we present the main Carleman estimates that will be used in the proofs of our main theorems. These Carleman estimates apply to first- and second-order elliptic operators, corresponding to the reduced systems that were derived in the previous section. Our first estimate applies to the operator ¯∂. The proof of this theorem appears in a subsequent section. The second main estimate is for second-order operators and its proof relies on a similar estimate that appeared previously in [14]. The arguments that connect this theorem from [14] to our new estimate are included within this section.

With the notation t = log |x|, define the weight function
h(log |x|) = h(t) = −βt + βe^{t/4}.

We often abuse notation and write h as a shorthand for h (log |x|). The
prime notation denotes differentiation with respect to the t-variable. That
is, h^{0} = ^{∂h}_{∂t}, and we again interpret the function h^{0} as h^{0}(log |x|). At times,
we will use the notation h = h_{β} to remind ourselves of the dependence on
the constant β.

Recall that we use the notation ¯∂ = ∂1+ i∂2. From now on, the notation X . Y or X & Y means that X ≤ CY or X ≥ CY with some constant C.

We first state the following Carleman estimate for first order operators.

Theorem 3.1. Let 1 < p < 2 and ˜R_{0} ∈ (0, 1). Then if β is sufficiently
large, then for all z ∈ C_{0}^{∞}

BR˜0\ {0}

, it holds that

(3.1) β^{1−}^{1}^{p}

|x| e^{h}z

2

≤ C

|x|^{τ}^{0}e^{h}∂z¯
p,
where τ_{0} = ^{11}_{4} −^{2}_{p} and C = C

p, ˜R_{0}
.

The proof of this theorem appears in Section 5. The next theorem gives a Carleman estimate for second order elliptic operators.

Theorem 3.2. Let 2 < q < q^{0}< ∞. If β ∈ N + ^{1}_{4} is sufficiently large, then
for all z ∈ C_{0}^{∞}(B1\ {0}) and all f = (f_{1}, f2) with f1, f2∈ C_{0}^{∞}(B1\ {0}),

(3.2)

β

1 + h^{00}1/2

e^{h}z
2+ β^{γ}^{1}

|x|^{τ}^{1}e^{h}z
q

≤C

e^{h}|x|^{2}(∆z + ∇f )
2+ β

e^{h}|x|f
2

,
where C = C (q, q^{0}), γ_{1}= ^{2}_{q}

1 −_{q}^{q−2}0−2

+^{1}_{2} and τ_{1}= ^{1}_{8}+

1 −^{2}_{q}

q^{0}
q^{0}−2

. To prove Theorem 3.2, we first recall the following Carleman estimate derived in [14].

Lemma 3.3 (Corollary to Theorem 3.1 in [14]). If β ∈ N +^{1}_{4} is sufficiently
large, then for all z ∈ C_{0}^{∞}(B_{1}\ {0}) and all f = (f_{1}, f_{2}) with f_{1}, f_{2} ∈
C_{0}^{∞}(B1\ {0}),

(3.3)

β

1 + h^{00}1/2

e^{h}z
2+

|x| 1 + h^{00}1/2

e^{h}∇z
2

≤C

e^{h}|x|^{2}(∆z + ∇f )
2+ β

e^{h}|x| f

2

, where C is a universal constant.

The proof follows from Theorem 3.1 in [14] by replacing h with ˜h(log |x|) =

˜hβ(t) = −βt + 2t + βe^{t/4}.

Remark 3.4. The notation ∇f used in [14] is interpreted as any (constant) linear combination of first order derivatives of the entries of f , as long as the resulting function takes the same algebraic structure as ∆z (so that adding these quantities is meaningful).

The next lemma is a straightforward consequence of Lemma 3.3.

Lemma 3.5. Let 2 < q^{0} < ∞. If β ∈ N +^{1}_{4} is sufficiently large, then for all
z ∈ C_{0}^{∞}(B_{1}\ {0}) and all f = (f_{1}, f_{2}) with f_{1}, f_{2} ∈ C_{0}^{∞}(B_{1}\ {0}),

β^{3/2}

|x|^{1/8}e^{h}z

2+ β^{1/2}

|x|^{9/8}e^{h}∇z

2+ β^{1/2}

|x|^{9/8}e^{h}z
q^{0}

≤C

e^{h}|x|^{2}(∆z + ∇f )
2+ β

e^{h}|x| f
2

,
where C = C (q^{0}).

Proof. For any q^{0}∈ (2, ∞), the Sobolev inequality shows that

|x|^{9/8}e^{h}z

q^{0} ≤ C_{q}^{0}
∇

|x|^{9/8}e^{h}z

2

≤ C_{q}^{0}
∇

|x|^{9/8}e^{h}
z

2+ C_{q}^{0}

|x|^{9/8}e^{h}∇z
2

≤ C_{q}^{0}β

|x|^{1/8}e^{h}z
2+ C_{q}^{0}

|x|^{9/8}e^{h}∇z
2.
Since h^{00}∼ β |x|^{1/4}, an application of Lemma 3.3 shows that

β^{3/2}

|x|^{1/8}e^{h}z

2+ β^{1/2}

|x|^{9/8}e^{h}∇z
2

.β

1 + h^{00}1/2

e^{h}z
2+

|x| 1 + h^{00}1/2

e^{h}∇z
2

.

e^{h}|x|^{2}(∆z + ∇f )
2+ β

e^{h}|x| f
2.

Adding these two inequalities completes the proof of the lemma. Lemma 3.5 in combination with the H¨older inequality leads to the proof of Theorem 3.2.

Proof of Theorem 3.2. Define θ = ^{q}_{q}^{0}0^{−q}−2 so that q = 2θ+(1−θ)q^{0}. By H¨older’s
inequality combined with Lemma 3.5, we have

|x|^{τ}^{1}e^{h}z
q

≤ C_{q,q}^{0}

|x|^{1/8}e^{h}z

2θ q

2

·

|x|^{9/8}e^{h}z

q0(1−θ) q

q^{0}

. β^{−γ}^{1}

e^{h}|x|^{2}(∆z + ∇f )
2+ β

e^{h}|x| f
2

.

Adding this estimate to (3.3) leads to the conclusion. 4. The proofs of Theorem 1.1 and Theorem 1.2

With the reduced systems and associated Carleman estimates, we are now prepared to prove Theorems 1.1 and 1.2. An additional tool that we make use of is the interior estimate presented in Section 6.

We first choose R_{0} ≤ ˜R_{0} such that B_{R}_{0} ⊂ Ω. Let χ ∈ C_{0}^{∞}(B_{R}_{0} \ {0}) be
a cutoff function with the property that supp χ ⊂ As∪ A_{m}∪ A_{l}, where χ ≡ 1
on A_{m} and supp ∇χ = A_{s}∪ A_{l}. Moreover, for all x, y ∈ A_{s}, |x| / |y| ∼ 1,
and similarly for A_{l}. The exact definitions of these sets will be given below.

For µ, λ as in (1.1), u1, u2 from (1.2), and v, w from (2.1), define the compactly supported function

z = χ (µv − iµw + 2i∂1µu2− 2i∂_{2}µu1) .

Applying estimate (3.1) from Theorem 3.1 to z with some p ∈ (1, 2) to be determined below, then using the equation (2.10) yields

(4.1)

β^{1−}^{1}^{p}

|x|e^{h}χ (|v| + |w|)

2− β^{1−}^{1}^{p}

|x|e^{h}χ |∇µ| u
2

≤Cβ^{1−}^{1}^{p}

|x|e^{h}χ (µv − iµw + 2i∂_{1}µu_{2}− 2i∂_{2}µu_{1})
2

≤C

|x|^{τ}^{0}e^{h}χ (g3+ g4)
p

+C

|x|^{τ}^{0}e^{h}∂χ (µv − iµw + 2i∂¯ 1µu2− 2i∂_{2}µu1)
p

≤C_{1}

|x|^{τ}^{0}e^{h}χ |∇µ| w
p+ C_{1}

|x|^{τ}^{0}e^{h}χ
D^{2}µ

u p

+C_{1}

|x|^{τ}^{0}e^{h}|∇χ| ∇u
p+ C_{1}

|x|^{τ}^{0}e^{h}|∇χ| |∇µ| u
p,

where τ0 = ^{11}_{4} −^{2}_{p} and we have used the structures of g3 and g4 as defined
in (2.11) in the last line.

An application of (3.2) from Theorem 3.2 to the functions z = χu and f = χ

µ λ+2µv, w

, where we interpret ∇f = (−∂1f1− ∂_{2}f2, −∂2f1+ ∂1f2)
(see Remark 3.4) and the values for 2 < q < q^{0}< ∞ will be specified below,
then using (2.2) and (2.1), shows that if β ∈ N +^{1}_{4} is sufficiently large, then

(4.2) β

(1 + h^{00})^{1/2}e^{h}χu
2+ β^{γ}^{1}

|x|^{τ}^{1}e^{h}χu
q

≤C_{2}

|x|^{2}e^{h} |∇χ| |∇u| +
D^{2}χ

u

2+ C_{2}β

|x|e^{h}χ (|v| + |w|)
2.
Recall that γ_{1} = ^{2}_{q}

q^{0}−q
q^{0}−2

+^{1}_{2} and τ_{1}= ^{1}_{8} +

1 −^{2}_{q}

q^{0}
q^{0}−2

.
Adding 2C_{2}β^{1}^{p}·(4.1) to (4.2) then gives

(4.3) β

(1 + h^{00})^{1/2}e^{h}χu
2+ β^{γ}^{1}

|x|^{τ}^{1}e^{h}χu
q+ β

|x|e^{h}χ (|v| + |w|)
2

≤2C_{2}β

|x|e^{h}χ |∇µ| u

2+ 2C1C2β^{1}^{p}

|x|^{τ}^{0}e^{h}χ |∇µ| w
p

+2C1C2β^{1}^{p}

|x|^{τ}^{0}e^{h}χ
D^{2}µ

u p

+2C_{1}C_{2}β^{1}^{p}

|x|^{τ}^{0}e^{h}|∇χ| |∇µ| u

p+ 2C_{1}C_{2}β^{1}^{p}

|x|^{τ}^{0}e^{h}|∇χ| ∇u
p

+C_{2}

|x|^{2}e^{h} |∇χ| |∇u| +
D^{2}χ

u 2.

With an appropriate choice of indices p, q and q^{0}, the first three terms on
the right of the inequality may be absorbed into the left.

First, we consider the term β

|x|e^{h}χ |∇µ| u

2. From the H¨older inequal- ity, we have that

|x|e^{h}χ |∇µ| u
2

≤ C

|x|^{τ}^{1}e^{h}χu
q

|x|^{1−τ}^{1}

_{∞}k∇µk 2q
q−2

. To absorb this term, we need that

(4.4) γ1 > 1, τ1 ≤ 1, _{q−2}^{2q} ≤ s^{∗},

where s^{∗} denotes the Sobolev conjugate with s^{∗} = _{2−s}^{2s} for s < 2, s^{∗} ∈ (2, ∞)
arbitrary for s = 2, and s^{∗}= ∞ otherwise. Notice that s^{∗}> 4 when s > 4/3.

Next, we consider the term β^{1}^{p}

|x|^{τ}^{0}e^{h}χ |∇µ| w

p. If we restrict p ∈ (1, 4/3), then from the H¨older inequality, we have that

|x|^{τ}^{0}e^{h}χ |∇µ| w

p ≤ C

|x|e^{h}χw
2

|x|^{7}^{4}^{−}^{p}^{2}
2ps∗

2s∗−ps∗−2p

k∇µk_{s}∗.
Observe that

7
4 −^{2}_{p}

2ps^{∗}

2s^{∗}−ps^{∗}−2p > −2 whenever s^{∗}> ^{8}_{3}, independent of the
choice of p. Thus we may absorb this term into the left if s^{∗} > ^{8}_{3}.

Finally, using similar techniques, we have

|x|^{τ}^{0}e^{h}χ
D^{2}µ

u p

≤ C

|x|^{τ}^{1}e^{h}χu
q

|x|^{11}^{4} ^{−τ}^{1}^{−}^{2}^{p}
_{∞}

D^{2}µ
pq

q−p

. To handle this term,we need that

(4.5) τ1+^{2}_{p} ≤ ^{11}_{4}, _{q−p}^{pq} ≤ s.

From the condition that γ_{1}> 1, we require that q ∈ (2, 4). This reduction
to the range of q in combination with the condition (4.5) explains why we
can only handle s > ^{4}_{3}. Since s^{∗} > 4, the second term may be absorbed into
the left. Moreover, since

q→4lim 2q

q − 2 = 4 < s^{∗} lim

q→4,p→1

pq q − p = 4

3 < s

(q,q^{0})→(4,∞)lim γ1 = 1 lim

(p,q,q^{0})→(1,4,∞)

τ1+ 2

p

= 21 8 ,

then we can choose p ∈ (1, 4/3), q ∈ (2, 4) and q^{0} ∈ (q, ∞) so that (4.4) and
(4.5) hold.

From now on, we fix p, q, q^{0} as described above to satisfy (4.4) and (4.5).

That is, if β is sufficiently large then the first three terms on the righthand side of (4.3) are absorbed by the lefthand side of (4.3) to get

(4.6)

e^{h}χu

_{2}

≤C_{3}

|x|^{τ}^{0}e^{h}|∇χ| ∇u
p+ C3

|x|^{τ}^{0}e^{h}|∇χ| |∇µ| u
p

+C_{3}

|x|^{2}e^{h} |∇χ| |∇u| +
D^{2}χ

u 2

≤C_{4}

|x|^{2}e^{h}

|x|^{−1/4}|∇χ| |∇u| +
D^{2}χ

|u| + |x|^{−1}|∇χ| |u|
_{2},

where we have used the H¨older inequality and the structure of supp ∇χ to show that

|x|^{τ}^{0}e^{h}|∇χ| ∇u

p ≤ C

|x|^{7}^{4}e^{h}|∇χ| ∇u
2

supp ∇χ|x|^{1−}^{2}^{p}
2p

2−p

≤ C

|x|^{7}^{4}e^{h}|∇χ| ∇u
2

and

|x|^{τ}^{0}e^{h}|∇χ| |∇µ| u
p

≤ C

|x|e^{h}|∇χ| u
2

|x|^{7}^{4}^{−}^{2}^{p}
2ps∗

2s∗−ps∗−2p

k∇µk_{s}∗

≤ C

|x|e^{h}|∇χ| u
2

in the last inequality.

Define the sets A_{s}= {R_{1}/3 ≤ |x| ≤ 2R_{1}/3}, A_{m} = {2R_{1}/3 ≤ |x| ≤ R_{3}/3},
and Al = {R3/3 ≤ |x| ≤ 2R3/3}. As stated above, χ is a smooth cut-off
function supported in A_{s}∪ A_{m}∪ A_{l} with χ ≡ 1 in A_{m}. Since supp D^{α}χ ⊂
A_{s}∪ A_{l}, then for any multi-index α

(4.7)

|D^{α}χ| = O
R^{−|α|}_{1}

for all x ∈ A_{s}

|D^{α}χ| = O

R3−|α|

for all x ∈ Al.

We replace h(t) by h(t) + t in (4.6), then apply (4.7) and (6.1) with
A_{s}b ˜A_{s}:= {R_{1}/4 ≤ |x| ≤ R_{1}} and A_{l} b ˜A_{l}:= {R_{3}/4 ≤ |x| ≤ R_{3}} to get

R^{2}_{2}e^{2˜}^{h(R}^{2}^{)}
Z

{2R1/3<|x|<R2}

|u|^{2}dx ≤
Z

{2R1/3<|x|<R2}

|x|^{2}e^{2h}|u|^{2}dx

≤C_{5}R^{3/2}_{1} e^{2˜}^{h(R}^{1}^{/3)}
Z

A˜s

|u|^{2}dx + C5R^{3/2}_{3} e^{2˜}^{h(R}^{3}^{/3)}
Z

A˜l

|u|^{2}dx,

where we set ˜h(a) = h(ln a) and used that R2 ≤ R_{3}/4 < R3/3. In the
use of (6.1), we choose R_{0} to be smaller if necessary. Dividing through by
R_{2}^{2}e^{2˜}^{h(R}^{2}^{)} shows that

Z

{2R1/3<|x|<R2}

|u|^{2}

≤C_{5}
R^{2}_{2}

R^{3/2}_{1} e^{2˜}^{h(R}^{1}^{/3)−2˜}^{h(R}^{2}^{)}
Z

A˜s

|u|^{2}+ R^{3/2}_{3} e^{2˜}^{h(R}^{3}^{/3)−2˜}^{h(R}^{2}^{)}
Z

A˜_{l}

|u|^{2}

≤C_{6}R^{−1/2}_{2}

e2(β−3/4) log

3R2 R1

Z

A˜s

|u|^{2}+ e−(3/2)(β−1) log

R3 3R2

Z

A˜_{l}

|u|^{2}

≤C_{6}R^{−1/2}_{2}

e2(β−3/4) log

3R2 R1

Z

A˜s

|u|^{2}+ e−(1/2)(β−3/4) log

R3 3R2

Z

A˜_{l}

|u|^{2}

,

since

2˜h(R1/3) − 2˜h(R2) = 2β log 3R_{2}
R1

− βp

R2−p R1/3

≤ 2β log 3R_{2}
R1

2˜h(R_{3}/3) − 2˜h(R_{2}) = 2h^{0}(log ˆR) log R_{3}
3R_{2}

≤ −3

2β log R_{3}
3R_{2}

,
for some R_{2} < ˆR < R_{3}/3 < 1.

Adding Z

B_{2R1/3}

|u|^{2} to both sides of the previous inequality and replacing β
by β + 1 shows that for any sufficiently large β ∈ N + ^{1}_{2}, it holds that
(4.8)

Z

B_{R2}

|u|^{2}≤ C_{7}R^{−1/2}_{2} e^{βE}
Z

B_{R1}

|u|^{2}+ e^{−βB}
Z

B_{R3}

|u|^{2}

! ,

where we set E = 2 log

3R2

R1

and B = ^{1}_{2}log

R3

3R2

. Note that both E and B are positive.

Let ˜β denote the smallest β ∈ N + ^{1}_{2} for which (4.8) holds. To further
simplify the terms on the right hand side of (4.8), we consider two cases. If
Z

B_{R1}

|u|^{2}6= 0 and

e^{βE}^{˜}
Z

B_{R1}

|u|^{2}≤ e^{− ˜}^{βB}
Z

B_{R3}

|u|^{2},

then we find β_{1} ∈ N + ^{1}_{2} satisfying β_{1} ≥ ˜β + 1 and β_{2} ∈ (β_{1}− 1, β_{1}] such
that

(4.9) e^{β}^{2}^{E}

Z

B_{R1}

|u|^{2} = e^{−β}^{2}^{B}
Z

B_{R3}

|u|^{2}
and

e^{(β}^{1}^{−1)E}
Z

B_{R1}

|u|^{2}≤ e^{−(β}^{1}^{−1)B}
Z

B_{R3}

|u|^{2}.
With these choices of β1 and β2, we derive from (4.8) that

(4.10) Z

B_{R2}

|u|^{2} ≤C_{7}R^{−1/2}_{2} e^{(β}^{1}^{−1)E}
Z

B_{R1}

|u|^{2}+ e^{−(β}^{1}^{−1)B}
Z

B_{R3}

|u|^{2}

!

≤C_{7}R^{−1/2}_{2} e^{β}^{2}^{E}
Z

B_{R1}

|u|^{2}+ e^{(β}^{2}^{+1−β}^{1}^{)B−β}^{2}^{B}
Z

B_{R3}

|u|^{2}

!

≤C_{7} 1 + e^{B} R^{−1/2}_{2} e^{β}^{2}^{E}
Z

B_{R1}

|u|^{2}

=C7 1 + e^{B} R^{−1/2}_{2}
Z

B_{R1}

|u|^{2}

!_{B+E}^{B}
Z

B_{R3}

|u|^{2}

!_{B+E}^{E}
,

where we have used 0 < β2+ 1 − β1 ≤ 1 and the value of β_{2} obtained from
(4.9). If

Z

B_{R1}

|u|^{2}= 0, then letting β → ∞ in (4.8) shows that
Z

B_{R2}

|u|^{2} = 0
as well, and then the three-ball inequality obviously holds.

On the other hand, if
e^{− ˜}^{βB}

Z

B_{R3}

|u|^{2}≤ e^{βE}^{˜}
Z

B_{R1}

|u|^{2},
then since R_{2}≤ R_{3}, it holds that

(4.11)

Z

B_{R2}

|u|^{2}≤
Z

B_{R3}

|u|^{2}

!_{E+B}^{B}
Z

B_{R3}

|u|^{2}

!_{E+B}^{E}

≤e^{βB}^{˜}
Z

B_{R1}

|u|^{2}

!_{E+B}^{B}
Z

B_{R3}

|u|^{2}

!_{E+B}^{E}
.
Putting together (4.10), (4.11), we arrive at

(4.12)

Z

B_{R2}

|u|^{2} ≤ C_{8}
Z

B_{R1}

|u|^{2}

!τ

Z

B_{R3}

|u|^{2}

!1−τ

,

where we have set C_{8} = maxn

C_{7} 1 + e^{B} R_{2}^{−1/2}, e^{βB}^{˜} o

and τ = _{E+B}^{B} . It is
readily seen that _{E+B}^{B} ≈ (log(1/R_{1}))^{−1} when R_{1} tends to 0. The proof of
Theorem 1.1 is complete.

We now turn to the proof of Theorem 1.2. Fix R_{2}, R_{3} ≤ R_{0} so that
Theorem 1.1 is applicable, and assume that R1 R_{2}, R3. Applying (1.3)
to u, then raising both sides to 1/τ yields

kuk^{1/τ}

L^{2}(^{B}_{R2})≤ C^{1/τ}kuk_{L}_{2}(^{B}_{R1}) kuk

−1+1/τ
L^{2}(^{B}_{R3}).

Since ^{1}_{τ} = c1log (1/R1) + c2 for some c1 > 0 and c2 ∈ R depending on R2,
R3 and R0, then,

kuk_{L}2(^{B}_{R1}) ≥ C^{−c}^{2}kuk^{c}^{2}

L^{2}(^{B}_{R2})kuk^{1−c}^{2}

L^{2}(^{B}_{R3})R

c1log

CkukL2(BR3)^{/kuk}^{L2}(BR2)

1 .

This completes the proof of Theorem 1.2.

5. The proof of Theorem 3.1

Here we prove the crucial L^{p}− L^{2} Carleman estimate for ¯∂ that is stated
in Theorem 3.1. We rewrite the operator in polar coordinates then use an
eigenvalue decomposition to establish our stated bounds. The techniques
used here were first implemented in [12] and are very similar to those that
appeared in recent papers [8], [9], [6].

We use standard polar coordinates in R^{2}\{0} by setting x_{1} = |x| cos θ and
x2 = |x| sin θ, where |x| =px^{2}_{1}+ x^{2}_{2} and θ = arctan (x2/x1). With the new
coordinate t = log |x|, we see that

∂_{1} = e^{−t}

cos θ∂

∂t− sin θ ∂

∂θ

, ∂_{2} = e^{−t}

sin θ ∂

∂t+ cos θ ∂

∂θ

so that

L := e^{t−iθ}∂ = ∂¯ t+ i∂θ.
(5.1)

The eigenvalues of ∂_{θ} are ik, k ∈ Z, with corresponding eigenspace Ek =
span {e_{k}}, where e_{k}= ^{√}^{1}

2πe^{ikθ} so that ke_{k}k_{L}2(S^{1})= 1. For any v ∈ L^{2} S^{1},
let Pkv = vk denote the projection of v onto Ek. We remark that the
projection operator, P_{k}, acts only on the angular variables. In particular,
P_{k}v (t, ω) = P_{k}v (t, ·) (ω). We may then rewrite the operator L as

L = ∂_{t}−X

k∈Z

kP_{k}.
(5.2)

Recall that we introduced the radial weight function h above. With |x| =
e^{t}, the weight function is given in terms of t as

h (t) = h_{β}(t) = −βt + βe^{t/4}.

Since our result applies to functions that are supported in BR˜0 \ {0}, then in terms of the new coordinate t, we study the case when t is sufficiently close to −∞.

By a slight modification to the result described by [9, Lemma 2], we get the following lemma.

Lemma 5.1. Let M, N ∈ N and let {ck} be a sequence of numbers such that

|c_{k}| ≤ 1 for all k. For any v ∈ L^{2} S^{1} and every p ∈ [1, 2], we have that

M

X

k=N

c_{k}P_{k}v
L^{2}(S^{1})

≤ C

M

X

k=N

|c_{k}|^{2}

!^{1}_{p}^{−}^{1}_{2}

kvk_{L}p(S^{1}),
(5.3)

where C = C (p).

Proof. Recall that P_{k}v = v_{k} is the projection of v onto E_{k}. Thus, for every
k ∈ Z, Pkv = hv, eki e_{k}, where we use the notation h·, ·i to denote a pairing
of elements in dual spaces. By Parseval’s identity,

∞

X

k=−∞

|he_{k}, vi|^{2} = kvk^{2}_{L}2(S^{1}).
Since ke_{k}k_{L}∞(S^{1}) = ^{√}^{1}

2π for all k ∈ Z, then
kP_{k}vk^{2}_{L}∞(S^{1}) = khv, e_{k}i e_{k}k^{2}_{L}∞(S^{1})= 1

2π|he_{k}, vi|^{2} ≤ 1

2πkvk^{2}_{L}2(S^{1}).

It follows that for any u ∈ L^{2}(S^{1})

hu, P_{k}vi = hP_{k}u, vi ≤ kP_{k}uk_{L}^{∞}_{(S}^{1}_{)}kvk_{L}1(S^{1}) ≤ ckuk_{L}2(S^{1})kvk_{L}1(S^{1}).
By duality, we conclude that

kP_{k}vk_{L}^{2}_{(S}^{1}_{)}≤ ckvk_{L}1(S^{1}).
(5.4)

It follows from the normalization condition in combination with Parseval’s identity that

(5.5) kP_{k}vk_{L}^{2}_{(S}^{1}_{)}≤ kvk_{L}2(S^{1}).
Interpolating (5.4) and (5.5) gives that

kP_{k}vk_{L}2(S^{1}) ≤ C(p)kvk_{L}p(S^{1})

(5.6)

for all 1 ≤ p ≤ 2.

Now we consider a more general setting. Let {c_{k}} be a sequence of num-
bers with |c_{k}| ≤ 1. For all N ≤ M , it follows from orthogonality and H¨older’s
inequality that

M

X

k=N

c_{k}P_{k}v

2

L^{2}(S^{1})

=

M

X

k=N

|c_{k}|^{2}hP_{k}v, vi ≤

M

X

k=N

|c_{k}|^{2}kP_{k}vk_{L}∞(S^{1})kvk_{L}1(S^{1}).

Since each e_{k}is normalized in L^{2} S^{1}, then√

2π ke_{k}k_{L}∞(S^{1})= 1 = ke_{k}k_{L}2(S^{1})

and then kPkvk_{L}∞(S^{1})= ^{√}^{1}_{2π}kP_{k}vk_{L}2(S^{1})≤ c kvk_{L}1(S^{1}), where we have used
observation (5.4). It follows that

M

X

k=N

ckPkv
L^{2}(S^{1})

≤ C

M

X

k=N

|c_{k}|^{2}

!^{1}_{2}

kvk_{L}1(S^{1}).
Clearly, as long as |c_{k}| ≤ 1, then

M

X

k=N

ckPkv
L^{2}(S^{1})

≤ kvk_{L}2(S^{1}).

As before, we interpolate the last two inequalities to reach (5.3). The following proposition is crucial to the proof of Theorem 3.1.

Proposition 5.2. Let 1 < p < 2. There exists a β0 > 0 such that we have
for β > β_{0} and z ∈ C_{0}^{∞} (−∞, 0) × S^{1} that

(5.7)

e^{h}^{β}^{(t)+t/8}z

L^{2}(dtdθ)≤ Cβ^{−1+}^{1}^{p}

e^{h}^{β}^{(t)−t/8}Lz

L^{p}(dtdθ),
where C = C (p).

Proof. To prove this lemma, we introduce the conjugated operator Lβ of L, defined by

L_{β}z = e^{h}^{β}^{(t)}L

e^{−h}^{β}^{(t)}z

.

Replacing z with e^{−h}^{β}^{(t)}z in inequality (5.7) shows that it suffices to prove
that

(5.8)

e^{t/8}z

L^{2}(dtdθ)≤ Cβ^{−1+}^{1}^{p}

e^{−t/8}L_{β}z

L^{p}(dtdθ).
From (5.1) and (5.2), the operator L_{β} takes the form
(5.9) L_{β} =X

k∈Z

∂t− h^{0}_{β}(t) − k P_{k}=X

k∈Z

∂t+ β −β

4e^{t/4}− k

P_{k}.
Since X

k∈Z

P_{k}z = z, we split the sum into three parts. Let M = d2βe and
define

P_{β}^{h} = X

k>M

P_{k}, P_{β}^{l} =

M

X

k=0

P_{k}, P_{β}^{n}=X

k<0

P_{k}.
In order to prove (5.8), it suffices to show that

(5.10)

e^{t/8}P_{β}^{}z

L^{2}(dtdθ)≤ Cβ^{−1+}^{1}^{p}

e^{−t/8}L_{β}z
L^{p}(dtdθ)

for = h, l, n, z ∈ C_{0}^{∞} (−∞, 0) × S^{1}, and 1 < p < 2. The sum of all three
inequalities will yield (5.8), which implies (5.7).

From (5.9), we have the first order differential equation
(5.11) P_{k}L_{β}z = ∂t− h^{0}_{β}(t) − k P_{k}z.

For z ∈ C_{0}^{∞} (−∞, 0) × S^{1}, solving the first order differential equation gives
that

(5.12)

P_{k}z(t, ω) = −
Z ∞

t

e^{k(t−s)+h}^{β}^{(t)−h}^{β}^{(s)}P_{k}L_{β}z(s, ω) ds

= Z t

−∞

e^{k(t−s)+h}^{β}^{(t)−h}^{β}^{(s)}P_{k}L_{β}z(s, ω) ds.

We first establish (5.10) with = h. For k > M ≥ 2β, if −∞ < t ≤ s ≤ 0, then

k(t − s) + h_{β}(t) − h_{β}(s) = − (k − β) |t − s| + β

e^{t/4}− e^{s/4}

≤ −k

2|t − s| .
Taking the L^{2} S^{1}-norm in (5.12) and using this bound gives that

kP_{k}z(t, ·)k_{L}2(S^{1})≤
Z ∞

−∞

e^{−}^{1}^{2}^{k|t−s|}kP_{k}L_{β}z(s, ·)k_{L}2(S^{1}) ds.

With the aid of (5.6), we get
kP_{k}z(t, ·)k_{L}2(S^{1})≤ C

Z ∞

−∞

e^{−}^{1}^{2}^{k|t−s|}kL_{β}z(s, ·)k_{L}p(S^{1}) ds

for any 1 ≤ p ≤ 2. Applying Young’s inequality for convolution then yields
kP_{k}zk_{L}2(dtdθ)≤ C

Z ∞

−∞

e^{−}^{σ}^{2}^{k|z|}dz

^{1}_{σ}

kL_{β}zk_{L}p(dtdθ)≤ Ck^{1}^{p}^{−}^{3}^{2}kL_{β}zk_{L}p(dtdθ),