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We study the strong unique continuation property (SUCP) for the Lam´e system in the plane

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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) ∈ W2,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 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 ⊂ R2be an open, connected set that contains the origin.

Assume that µ ∈ W2,s(Ω) and λ ∈ L(Ω), where (1.1)

(µ(x) ≥ δ0, λ(x) + 2µ(x) ≥ δ0, ∀ x ∈ Ω, kµkW2,s(Ω)+ kλkL(Ω)≤ M0

for some positive constants δ0, M0. Recall that kf kW2,s(Ω)= X

|α|≤2

k∂xαf kLs(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 = (u1, u2)tis the real-valued displacement vector and (∇u)jk = ∂kuj for j, k = 1, 2.

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

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Theorem 1.1. Assume that for some s > 43, the estimates in (1.1) are satisfied. Then there exists an R0 > 0, depending on s, M0, δ0, such that BR0 ⊂ Ω and for any u ∈ Hloc1 (BR0) satisfying (1.2) in BR0, if 0 < R1 <

R2 < R3≤ R0 and R2/R3 < 1/4, then (1.3) kukL2(BR2) ≤ C kuk

τ

L2(BR1)kuk1−τ

L2(BR3),

where C = C (s, M0, δ0, R2, R3) and τ = τ (R1/R2, R2/R3, R0) ∈ (0, 1).

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

We emphasize that C is independent of R1 and τ has the asymptotic behavior like (− log R1)−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 > 43, (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, M0, δ0 and u, such that for all R sufficiently small

(1.4) kukL2(BR)≥ KRm.

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 ∈( C0,γ with γ = 2 − (2/s), if s < 2 C0,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 ∈ W2,s for s < 2−α2 , 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 ∈ C1,1, n ≥ 2 (quantitative): Alessandrini and Morassi [2].

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• λ, µ ∈ C0,1, n = 2 (qualitative): Lin and Wang [17].

• λ ∈ L, µ ∈ C0,1, n = 2 (qualitative): Escauriaza [10].

• λ, µ ∈ C0,1, n ≥ 2 (quantitative): Lin, Nakamura, and Wang [18].

• µ ∈ C0,1, λ ∈ L, n ≥ 2 (quantitative): Lin, Nakamura, Uhlmann, and Wang [15].

• µ ∈ C0,1, 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 ∈ W1,2 and k∇µkL2 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 ∈ W2,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) ∈ (W1,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.

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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 (∂1u1+ ∂2u2), w = curl u = ∂2u1− ∂1u2,

a straightforward computation shows that

(2.2)

∆u1 = ∂1

 µ

λ + 2µv



+ ∂2w,

∆u2 = ∂2

 µ

λ + 2µv



− ∂1w.

Expanding (1.2) gives div

 2µ∂1u1 µ (∂1u2+ ∂2u1) µ(∂1u2+ ∂2u1) 2µ∂2u2

 +

 ∂1{λ (∂1u1+ ∂2u2)}

2{λ (∂1u1+ ∂2u2)}



=

 ∂1(2µ∂1u1) + ∂2{µ (∂1u2+ ∂2u1)}

1{µ (∂1u2+ ∂2u1)} + ∂2(2µ∂2u2)

 +

 ∂1(λ∂1u1+ λ∂2u2)

2(λ∂1u1+ λ∂2u2)



= 0.

Simplifying each row shows that

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

− ∂2[λ∂1u1+ (2µ + λ) ∂2u2] = ∂1[µ (∂1u2+ ∂2u1)] . (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− ∂1u2)]

−i∂1[µ (∂2u1− ∂1u2)] + 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− ∂1u2)]

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

=2∂1[µ∂2u2] − 2∂2[µ∂1u2]

=2∂1µ∂2u2− 2∂2µ∂1u2,

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

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

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

=∂1(2µ∂2u1) − ∂2(2µ∂1u1)

=2∂1µ∂2u1− 2∂2µ∂1u1.

Substituting (2.6) and (2.7) into (2.5) then gives (2.8) ∂ (µv − iµw) = g¯ 1− ig2, where

g1= 2∂1µ∂2u2− 2∂2µ∂1u2, g2 = 2∂1µ∂2u1− 2∂2µ∂1u1. Using the observation that ∂2u1= w + ∂1u2, we rewrite g1− ig2 as

(2.9)

g1− ig2

=2∂1µ∂2u2− i2∂1µ∂2u1− 2∂2µ∂1u2+ i2∂2µ∂1u1

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

= − 2i∂1µ ¯∂u2+ 2i∂2µ ¯∂u1− 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∂¯ 1µu2− 2i∂2µu1) = g3+ g4. where

(2.11) g3 = −2i ¯∂µw, g4= ¯∂ (2i∂1µ) u2− ¯∂ (2i∂2µ) u1.

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 + βet/4.

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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, h0 = ∂h∂t, and we again interpret the function h0 as h0(log |x|). At times, we will use the notation h = hβ 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 ˜R0 ∈ (0, 1). Then if β is sufficiently large, then for all z ∈ C0



BR˜0\ {0}

, it holds that

(3.1) β1−1p

|x| ehz

2

≤ C

|x|τ0eh∂z¯ p, where τ0 = 1142p and C = C

p, ˜R0 .

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 < q0< ∞. If β ∈ N + 14 is sufficiently large, then for all z ∈ C0(B1\ {0}) and all f = (f1, f2) with f1, f2∈ C0(B1\ {0}),

(3.2)

β

1 + h001/2

ehz 2+ βγ1

|x|τ1ehz q

≤C

eh|x|2(∆z + ∇f ) 2+ β

eh|x|f 2

 , where C = C (q, q0), γ1= 2q

1 −qq−20−2



+12 and τ1= 18+

1 −2q 

q0 q0−2

 . 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 +14 is sufficiently large, then for all z ∈ C0(B1\ {0}) and all f = (f1, f2) with f1, f2 ∈ C0(B1\ {0}),

(3.3)

β

1 + h001/2

ehz 2+

|x| 1 + h001/2

eh∇z 2

≤C

eh|x|2(∆z + ∇f ) 2+ β

eh|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 + βet/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.

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Lemma 3.5. Let 2 < q0 < ∞. If β ∈ N +14 is sufficiently large, then for all z ∈ C0(B1\ {0}) and all f = (f1, f2) with f1, f2 ∈ C0(B1\ {0}),

β3/2

|x|1/8ehz

2+ β1/2

|x|9/8eh∇z

2+ β1/2

|x|9/8ehz q0

≤C

eh|x|2(∆z + ∇f ) 2+ β

eh|x| f 2

 , where C = C (q0).

Proof. For any q0∈ (2, ∞), the Sobolev inequality shows that

|x|9/8ehz

q0 ≤ Cq0 ∇

|x|9/8ehz

 2

≤ Cq0 ∇

|x|9/8eh z

2+ Cq0

|x|9/8eh∇z 2

≤ Cq0β

|x|1/8ehz 2+ Cq0

|x|9/8eh∇z 2. Since h00∼ β |x|1/4, an application of Lemma 3.3 shows that

β3/2

|x|1/8ehz

2+ β1/2

|x|9/8eh∇z 2

1 + h001/2

ehz 2+

|x| 1 + h001/2

eh∇z 2

.

eh|x|2(∆z + ∇f ) 2+ β

eh|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 = qq00−q−2 so that q = 2θ+(1−θ)q0. By H¨older’s inequality combined with Lemma 3.5, we have

|x|τ1ehz q

≤ Cq,q0

|x|1/8ehz

q

2

·

|x|9/8ehz

q0(1−θ) q

q0

. β−γ1

eh|x|2(∆z + ∇f ) 2+ β

eh|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 R0 ≤ ˜R0 such that BR0 ⊂ Ω. Let χ ∈ C0(BR0 \ {0}) be a cutoff function with the property that supp χ ⊂ As∪ Am∪ Al, where χ ≡ 1 on Am and supp ∇χ = As∪ Al. Moreover, for all x, y ∈ As, |x| / |y| ∼ 1, and similarly for Al. The exact definitions of these sets will be given below.

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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∂2µu1) .

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)

β1−1p

|x|ehχ (|v| + |w|)

2− β1−1p

|x|ehχ |∇µ| u 2

≤Cβ1−1p

|x|ehχ (µv − iµw + 2i∂1µu2− 2i∂2µu1) 2

≤C

|x|τ0ehχ (g3+ g4) p

+C

|x|τ0eh∂χ (µv − iµw + 2i∂¯ 1µu2− 2i∂2µu1) p

≤C1

|x|τ0ehχ |∇µ| w p+ C1

|x|τ0ehχ D2µ

u p

+C1

|x|τ0eh|∇χ| ∇u p+ C1

|x|τ0eh|∇χ| |∇µ| u p,

where τ0 = 1142p 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− ∂2f2, −∂2f1+ ∂1f2) (see Remark 3.4) and the values for 2 < q < q0< ∞ will be specified below, then using (2.2) and (2.1), shows that if β ∈ N +14 is sufficiently large, then

(4.2) β

(1 + h00)1/2ehχu 2+ βγ1

|x|τ1ehχu q

≤C2

|x|2eh |∇χ| |∇u| + D2χ

u

2+ C2β

|x|ehχ (|v| + |w|) 2. Recall that γ1 = 2q

q0−q q0−2



+12 and τ1= 18 +

1 −2q 

q0 q0−2

 . Adding 2C2β1p·(4.1) to (4.2) then gives

(4.3) β

(1 + h00)1/2ehχu 2+ βγ1

|x|τ1ehχu q+ β

|x|ehχ (|v| + |w|) 2

≤2C2β

|x|ehχ |∇µ| u

2+ 2C1C2β1p

|x|τ0ehχ |∇µ| w p

+2C1C2β1p

|x|τ0ehχ D2µ

u p

+2C1C2β1p

|x|τ0eh|∇χ| |∇µ| u

p+ 2C1C2β1p

|x|τ0eh|∇χ| ∇u p

+C2

|x|2eh |∇χ| |∇u| + D2χ

u 2.

With an appropriate choice of indices p, q and q0, the first three terms on the right of the inequality may be absorbed into the left.

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First, we consider the term β

|x|ehχ |∇µ| u

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

|x|ehχ |∇µ| u 2

≤ C

|x|τ1ehχu q

|x|1−τ1

k∇µk 2q q−2

. To absorb this term, we need that

(4.4) γ1 > 1, τ1 ≤ 1, q−22q ≤ s,

where s denotes the Sobolev conjugate with s = 2−s2s 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 β1p

|x|τ0ehχ |∇µ| w

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

|x|τ0ehχ |∇µ| w

p ≤ C

|x|ehχw 2

|x|74p2 2ps∗

2s∗−ps∗−2p

k∇µks. Observe that

7 42p

2ps

2s−ps−2p > −2 whenever s> 83, independent of the choice of p. Thus we may absorb this term into the left if s > 83.

Finally, using similar techniques, we have

|x|τ0ehχ D2µ

u p

≤ C

|x|τ1ehχu q

|x|114 −τ12p

D2µ pq

q−p

. To handle this term,we need that

(4.5) τ1+2p114, q−ppq ≤ s.

From the condition that γ1> 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 > 43. 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,q0)→(4,∞)lim γ1 = 1 lim

(p,q,q0)→(1,4,∞)

 τ1+ 2

p



= 21 8 ,

then we can choose p ∈ (1, 4/3), q ∈ (2, 4) and q0 ∈ (q, ∞) so that (4.4) and (4.5) hold.

From now on, we fix p, q, q0 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)

ehχu

2

≤C3

|x|τ0eh|∇χ| ∇u p+ C3

|x|τ0eh|∇χ| |∇µ| u p

+C3

|x|2eh |∇χ| |∇u| + D2χ

u 2

≤C4

|x|2eh

|x|−1/4|∇χ| |∇u| + D2χ

|u| + |x|−1|∇χ| |u| 2,

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

|x|τ0eh|∇χ| ∇u

p ≤ C

|x|74eh|∇χ| ∇u 2

supp ∇χ|x|1−2p 2p

2−p

≤ C

|x|74eh|∇χ| ∇u 2

and

|x|τ0eh|∇χ| |∇µ| u p

≤ C

|x|eh|∇χ| u 2

|x|742p 2ps∗

2s∗−ps∗−2p

k∇µks

≤ C

|x|eh|∇χ| u 2

in the last inequality.

Define the sets As= {R1/3 ≤ |x| ≤ 2R1/3}, Am = {2R1/3 ≤ |x| ≤ R3/3}, and Al = {R3/3 ≤ |x| ≤ 2R3/3}. As stated above, χ is a smooth cut-off function supported in As∪ Am∪ Al with χ ≡ 1 in Am. Since supp Dαχ ⊂ As∪ Al, then for any multi-index α

(4.7)

|Dαχ| = O R−|α|1 

for all x ∈ As

|Dαχ| = 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 Asb ˜As:= {R1/4 ≤ |x| ≤ R1} and Al b ˜Al:= {R3/4 ≤ |x| ≤ R3} to get

R22eh(R2) Z

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

|u|2dx ≤ Z

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

|x|2e2h|u|2dx

≤C5R3/21 eh(R1/3) Z

A˜s

|u|2dx + C5R3/23 eh(R3/3) Z

A˜l

|u|2dx,

where we set ˜h(a) = h(ln a) and used that R2 ≤ R3/4 < R3/3. In the use of (6.1), we choose R0 to be smaller if necessary. Dividing through by R22eh(R2) shows that

Z

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

|u|2

≤C5 R22



R3/21 eh(R1/3)−2˜h(R2) Z

A˜s

|u|2+ R3/23 eh(R3/3)−2˜h(R2) Z

A˜l

|u|2



≤C6R−1/22



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



≤C6R−1/22



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

 ,

(11)

since

2˜h(R1/3) − 2˜h(R2) = 2β log 3R2 R1



− βp

R2−p R1/3



≤ 2β log 3R2 R1



2˜h(R3/3) − 2˜h(R2) = 2h0(log ˆR) log R3 3R2



≤ −3

2β log R3 3R2

 , for some R2 < ˆR < R3/3 < 1.

Adding Z

B2R1/3

|u|2 to both sides of the previous inequality and replacing β by β + 1 shows that for any sufficiently large β ∈ N + 12, it holds that (4.8)

Z

BR2

|u|2≤ C7R−1/22 eβE Z

BR1

|u|2+ e−βB Z

BR3

|u|2

! ,

where we set E = 2 log

3R2

R1



and B = 12log

R3

3R2



. Note that both E and B are positive.

Let ˜β denote the smallest β ∈ N + 12 for which (4.8) holds. To further simplify the terms on the right hand side of (4.8), we consider two cases. If Z

BR1

|u|26= 0 and

eβE˜ Z

BR1

|u|2≤ e− ˜βB Z

BR3

|u|2,

then we find β1 ∈ N + 12 satisfying β1 ≥ ˜β + 1 and β2 ∈ (β1− 1, β1] such that

(4.9) eβ2E

Z

BR1

|u|2 = e−β2B Z

BR3

|u|2 and

e1−1)E Z

BR1

|u|2≤ e−(β1−1)B Z

BR3

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

(4.10) Z

BR2

|u|2 ≤C7R−1/22 e1−1)E Z

BR1

|u|2+ e−(β1−1)B Z

BR3

|u|2

!

≤C7R−1/22 eβ2E Z

BR1

|u|2+ e2+1−β1)B−β2B Z

BR3

|u|2

!

≤C7 1 + eB R−1/22 eβ2E Z

BR1

|u|2

=C7 1 + eB R−1/22 Z

BR1

|u|2

!B+EB Z

BR3

|u|2

!B+EE ,

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

Z

BR1

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

BR2

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

On the other hand, if e− ˜βB

Z

BR3

|u|2≤ eβE˜ Z

BR1

|u|2, then since R2≤ R3, it holds that

(4.11)

Z

BR2

|u|2≤ Z

BR3

|u|2

!E+BB Z

BR3

|u|2

!E+BE

≤eβB˜ Z

BR1

|u|2

!E+BB Z

BR3

|u|2

!E+BE . Putting together (4.10), (4.11), we arrive at

(4.12)

Z

BR2

|u|2 ≤ C8 Z

BR1

|u|2

!τ

Z

BR3

|u|2

!1−τ

,

where we have set C8 = maxn

C7 1 + eB R2−1/2, eβB˜ o

and τ = E+BB . It is readily seen that E+BB ≈ (log(1/R1))−1 when R1 tends to 0. The proof of Theorem 1.1 is complete.

We now turn to the proof of Theorem 1.2. Fix R2, R3 ≤ R0 so that Theorem 1.1 is applicable, and assume that R1  R2, R3. Applying (1.3) to u, then raising both sides to 1/τ yields

kuk1/τ

L2(BR2)≤ C1/τkukL2(BR1) kuk

−1+1/τ L2(BR3).

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

kukL2(BR1) ≥ C−c2kukc2

L2(BR2)kuk1−c2

L2(BR3)R

c1log



CkukL2(BR3)/kukL2(BR2)



1 .

This completes the proof of Theorem 1.2.

5. The proof of Theorem 3.1

Here we prove the crucial Lp− L2 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].

(13)

We use standard polar coordinates in R2\{0} by setting x1 = |x| cos θ and x2 = |x| sin θ, where |x| =px21+ x22 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 := et−iθ∂ = ∂¯ t+ i∂θ. (5.1)

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

eikθ so that kekkL2(S1)= 1. For any v ∈ L2 S1, let Pkv = vk denote the projection of v onto Ek. We remark that the projection operator, Pk, acts only on the angular variables. In particular, Pkv (t, ω) = Pkv (t, ·) (ω). We may then rewrite the operator L as

L = ∂t−X

k∈Z

kPk. (5.2)

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

h (t) = hβ(t) = −βt + βet/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

|ck| ≤ 1 for all k. For any v ∈ L2 S1 and every p ∈ [1, 2], we have that

M

X

k=N

ckPkv L2(S1)

≤ C

M

X

k=N

|ck|2

!1p12

kvkLp(S1), (5.3)

where C = C (p).

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

X

k=−∞

|hek, vi|2 = kvk2L2(S1). Since kekkL(S1) = 1

for all k ∈ Z, then kPkvk2L(S1) = khv, eki ekk2L(S1)= 1

2π|hek, vi|2 ≤ 1

2πkvk2L2(S1).

(14)

It follows that for any u ∈ L2(S1)

hu, Pkvi = hPku, vi ≤ kPkukL(S1)kvkL1(S1) ≤ ckukL2(S1)kvkL1(S1). By duality, we conclude that

kPkvkL2(S1)≤ ckvkL1(S1). (5.4)

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

(5.5) kPkvkL2(S1)≤ kvkL2(S1). Interpolating (5.4) and (5.5) gives that

kPkvkL2(S1) ≤ C(p)kvkLp(S1)

(5.6)

for all 1 ≤ p ≤ 2.

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

M

X

k=N

ckPkv

2

L2(S1)

=

M

X

k=N

|ck|2hPkv, vi ≤

M

X

k=N

|ck|2kPkvkL(S1)kvkL1(S1).

Since each ekis normalized in L2 S1, then√

2π kekkL(S1)= 1 = kekkL2(S1)

and then kPkvkL(S1)= 1kPkvkL2(S1)≤ c kvkL1(S1), where we have used observation (5.4). It follows that

M

X

k=N

ckPkv L2(S1)

≤ C

M

X

k=N

|ck|2

!12

kvkL1(S1). Clearly, as long as |ck| ≤ 1, then

M

X

k=N

ckPkv L2(S1)

≤ kvkL2(S1).

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 > β0 and z ∈ C0 (−∞, 0) × S1 that

(5.7)

ehβ(t)+t/8z

L2(dtdθ)≤ Cβ−1+1p

ehβ(t)−t/8Lz

Lp(dtdθ), where C = C (p).

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

Lβz = ehβ(t)L

e−hβ(t)z

 .

(15)

Replacing z with e−hβ(t)z in inequality (5.7) shows that it suffices to prove that

(5.8)

et/8z

L2(dtdθ)≤ Cβ−1+1p

e−t/8Lβz

Lp(dtdθ). From (5.1) and (5.2), the operator Lβ takes the form (5.9) Lβ =X

k∈Z

t− h0β(t) − k Pk=X

k∈Z



t+ β −β

4et/4− k

 Pk. Since X

k∈Z

Pkz = z, we split the sum into three parts. Let M = d2βe and define

Pβh = X

k>M

Pk, Pβl =

M

X

k=0

Pk, Pβn=X

k<0

Pk. In order to prove (5.8), it suffices to show that

(5.10)

et/8Pβz

L2(dtdθ)≤ Cβ−1+1p

e−t/8Lβz Lp(dtdθ)

for  = h, l, n, z ∈ C0 (−∞, 0) × S1, 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) PkLβz = ∂t− h0β(t) − k Pkz.

For z ∈ C0 (−∞, 0) × S1, solving the first order differential equation gives that

(5.12)

Pkz(t, ω) = − Z

t

ek(t−s)+hβ(t)−hβ(s)PkLβz(s, ω) ds

= Z t

−∞

ek(t−s)+hβ(t)−hβ(s)PkLβz(s, ω) ds.

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

k(t − s) + hβ(t) − hβ(s) = − (k − β) |t − s| + β

et/4− es/4

≤ −k

2|t − s| . Taking the L2 S1-norm in (5.12) and using this bound gives that

kPkz(t, ·)kL2(S1)≤ Z

−∞

e12k|t−s|kPkLβz(s, ·)kL2(S1) ds.

With the aid of (5.6), we get kPkz(t, ·)kL2(S1)≤ C

Z

−∞

e12k|t−s|kLβz(s, ·)kLp(S1) ds

for any 1 ≤ p ≤ 2. Applying Young’s inequality for convolution then yields kPkzkL2(dtdθ)≤ C

Z

−∞

eσ2k|z|dz

1σ

kLβzkLp(dtdθ)≤ Ck1p32kLβzkLp(dtdθ),

參考文獻

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