Unique continuation for the elasticity system and a counterexample for second order elliptic systems

Unique continuation for the elasticity system and a counterexample for second order elliptic systems

Carlos Kenig

Jenn-Nan Wang

Abstract

In this paper we study the global unique continuation property for the elasticity system and the general second order elliptic system in two dimensions. For the isotropic and the anisotropic systems with measurable coefficients, under certain conditions on coefficients, we show that the global unique continuation property holds. On the other hand, for the anisotropic system, if the coefficients are Lipschitz, we can prove that the global unique continuation is satisfied for a more general class of media. In addition to the positive results, we also present counterexamples to unique continuation and strong unique continuation for general second elliptic systems.

1 Introduction

In this work, we study the unique continuation property for the elasticity system and the general second order elliptic system in two dimensions. We begin with the elasticity system.

Let u = (u₁, u₂)^T be a vector-valued function satisfy

∂_j(a_ijkl(x)∂_ku_l) = 0 in R², (1.1) where a_ijkl(x) is a rank four tensor satisfying the symmetry properties:

a_ijkl = a_klij = a_jikl. (1.2)

Throughout, the Latin indices range from 1 to 2. Also, the summation convention is imposed.

For isotropic media, we have that a_ijkl = λδ_ijδ_kl+ µ(δ_ikδ_jl+ δ_ilδ_jk), where λ and µ are called Lam´e coefficients. In this case, (1.1) is written as

∇ · (µ(∇u + (∇u)^T)) + ∇(λ∇ · u) = 0 in R². (1.3)

∗Department of Mathematics, University of Chicago, Chicago, IL 60637, USA. Email:

cek@math.uchicago.edu. Supported in part by NSF Grant DMS-1265249.

†Institute of Applied Mathematical Sciences, NCTS (Taipei), National Taiwan University, Taipei 106, Taiwan. Email: jnwang@math.ntu.edu.tw. Supported in part by MOST Grant 102-2918-I-002-009 and 102-2115-M-002-009-MY3.

We say that u, a solution of (1.3), satisfies the global unique continuation property if whenever u that vanishes in the lower half plane, it vanishes identically in R². Recall that any solution u of the partial differential equation defined in an open connected set Ω is said to satisfy the unique continuation property if whenever u vanishes in a non-empty open subset of Ω, it is zero in Ω. On the other hand, u satisfies the strong unique continuation property if whenever u vanishes of infinite order at any point of Ω, it vanishes in Ω.

For the isotropic system (1.3) with nice Lam´e coefficients, there are a lot of results on the unique continuation property and the strong unique continuation property (for dimension n ≥ 2). We will not review the detailed development here. To motivate our study, we only mention the recent result in [LNUW11], where the strong unique continuation property was proved for µ ∈ W^1,∞ and λ ∈ L^∞, which is the best known assumption on the coefficients by far. For the scalar second order elliptic equation in non-divergence or divergence form

A∇²u = 0 in R² (1.4)

or

∇ · (A(x)∇u) = 0 in R², (1.5)

the strong unique continuation property is satisfied for A ∈ L^∞ (see, for example, [Al92], [Al12], [AE08], [BN54], [Sc98]). The proof is based on the intimate connection between (1.4) or (1.5) and quasiregular mappings. Therefore, it is a natural question to ask whether the unique continuation or the strong unique continuation hold for (1.3) or even for (1.1) when all coefficients are only measurable.

When µ of (1.3) is Lipschitz, it is known that (1.3) is weakly coupled. Hence, the usual Carleman method will lead us to the unique continuation properties. However, if µ is only measurable, (1.3) is strongly coupled. To the best of our knowledge, the Carleman method has never been successfully applied to strongly coupled systems. The general elasticity system (1.1) is always strongly coupled, regardless of the regularity of coefficients.

In this work we would like to show that solutions u of (1.1) satisfy the global unique continuation property under some restrictions on the measurable coefficients a_ijkl. Our approach to prove this result relies on the the connection between (1.1) and the Beltrami system with matrix-valued coefficients (see (2.16)). When this matrix-valued coefficient is sufficiently small (which is satisfied when the coefficients do not deviate too much from a set of constant coefficients), we can follow the arguments in [IVV02] and use the L^p-norm of the Beurling-Ahlfors transform to conclude the result. When the coefficients are only measurable, the set of constant coefficients is rather restricted, see the conditions in Theorem 2.1 and 2.3.

If some coefficients of the general system (1.1) are Lipschitz, the global unique continuation is true for coefficients near a larger set of constant coefficients, see Theorem 3.1.

In addition to the positive results mentioned above, we also present a counterexample to unique continuation for a second order elliptic system (in the sense of (4.26)) with measurable coefficients based on the example derived in [IVV02]. The main idea in the construction of the counterexample is to convert the second order elliptic system to a first order elliptic system

and match coefficients of the first order system obtained from the example in [IVV02]. Based on the example given in [CP11] (also see related article [Ro09]), using the same argument, we can also construct a counterexample to strong unique continuation for second order elliptic systems with continuous coefficients satisfying the Lagendre-Hadamard condition (2.21) or even the strong convexity condition:

a_ijkl(x)ξ_k^lξ_jⁱ ≥ c|ξ|² (1.6) for any 2 × 2 matrix ξ = (ξ_k^l), where c is a positive constant. Note that (1.6) implies the Legendre-Hadamard condition.

We would like to remark that the nontrivial solution u of the counterexample to unique continuation described above vanishes in the lower half plane. It was shown in [MS03] that there exist nontrivial W^1,2(R²) solution u or Lipschitz solution u whose supports are compact solving second order elliptic system with measurable coefficients satisfying the Legendre- Hadamard condition. This is another counterexample to unique continuation for second order elliptic systems with measurable coefficients. We want to point out that the examples in [MS03] do not exist in second order elliptic systems satisfying the strong convexity condition (1.6). This can be easily seen by the integration by parts. Nonetheless, the strong convexity condition (1.6) does not rule out the existence of the counterexample to unique continuation we constructed in this paper since this nontrivial solution does not necessarily have compact support.

The paper is organized as follows. In Section 2, we prove the global unique continuation property for the Lam´e and general anisotropic systems when the measurable elastic coeffi- cients are close to some constant values. In Section 3, we expand the set of constant values when the elastic coefficients are Lipschitz. Finally, in Section 4, we construct counterexam- ples to unique continuation and strong unique continuation for general second order elliptic systems with measurable coefficients and continuous coefficients, respectively.

2 Elasticity system with measurable coefficients

It is instructive to begin with the isotropic system, i.e., Lam´e system (1.3). Assume that λ, µ ∈ L^∞ satisfy the ellipticity condition

µ ≥ δ > 0, λ + 2µ ≥ δ, ∀ x ∈ R². (2.1) The key to proving the global unique continuation property lies in arranging the coefficients nicely. We write (1.3) componentwisely

∂₁(2µ∂₁u₁) + ∂₂(µ(∂₁u₂+ ∂₂u₁)) + ∂₁(λ∇ · u) = 0 (2.2) and

∂₁(µ(∂₁u₂+ ∂₂u₁)) + ∂₂(2µ∂₂u₂) + ∂₂(λ∇ · u) = 0. (2.3)

Let v = ∇ · u = ∂₁u₁+ ∂₂u₂ and w = ∇ × u = ∂₁u₂− ∂₂u₁, then (2.2) is written as

∂₁(2µ∂₁u₁+ λv) + ∂₂(2µ∂₂u₁+ µw) = 0. (2.4) Similarly, (2.3) is equivalent to

∂₁(2µ∂₁u₂− µw) + ∂₂(2µ∂₂u₂+ λv) = 0. (2.5) By taking advantage of the relation

∆u1 = ∂1v − ∂2w and ∆u2 = ∂2v + ∂1w, we obtain from (2.4) that

∂₁((2µ − 1)∂₁u₁+ (λ + 1)v) + ∂₂((2µ − 1)∂₂u₁+ (µ − 1)w) = 0 (2.6) and from (2.5) that

∂₁((2µ − 1)∂₁u₂ − (µ − 1)w) + ∂₂((2µ − 1)∂₂u₂+ (λ + 1)v) = 0. (2.7) Therefore, there exist u⁰₁ and u⁰₂ such that

(∂₂u⁰₁ = (2µ − 1)∂₁u₁+ (λ + 1)v,

−∂₁u⁰₁ = (2µ − 1)∂₂u₁+ (µ − 1)w, (2.8)

and (

∂₂u⁰₂ = (2µ − 1)∂₁u₂− (µ − 1)w,

−∂₁u⁰₂ = (2µ − 1)∂₂u₂+ (λ + 1)v. (2.9) Setting f1 = u1 + iu⁰₁ and f2 = u2+ iu⁰₂, (2.8) and (2.9) become

( ¯∂f₁ = σ∂f₁+ h,

∂f¯ ₂ = σ∂f₂+ ih, (2.10)

where

σ = 1 − µ

µ and h = −λ + 1

2µ v − iµ − 1 2µ w.

As usual, we define

∂ =¯ 1

2(∂₁+ i∂₂), ∂ = 1

2(∂₁− i∂₂).

Using the obvious relations

∂₁u₁ = 1

2( ¯∂f₁+ ∂f₁+ ∂ ¯f₁+ ∂f₁), ∂₁u₂ = 1

2( ¯∂f₂+ ∂f₂+ ∂ ¯f₂+ ∂f₂),

∂₂u₁ = 1

2i( ¯∂f₁− ∂f₁− ∂ ¯f₁+ ∂f₁), ∂₂u₂ = 1

2i( ¯∂f₂− ∂f₂− ∂ ¯f₂+ ∂f₂),

we can compute h = −λ + 1

2µ v − iµ − 1 2µ w

= −λ + 1

4µ ( ¯∂f1+ ∂f1+ ∂ ¯f1+ ∂f1) − i( ¯∂f2− ∂f2− ∂ ¯f2+ ∂f2)

− iµ − 1

4µ ( ¯∂f₂+ ∂f₂+ ∂ ¯f₂+ ∂f₂) + i( ¯∂f₁− ∂f₁− ∂ ¯f₁+ ∂f₁)

= µ − λ − 2 4µ



∂f¯ ₁+ −λ − µ 4µ



∂f₁+ −λ − µ 4µ



∂ ¯f₁+ µ − λ − 2 4µ



∂f₁ + i λ − µ + 2

4µ



∂f¯ ₂+ i −λ − µ 4µ



∂f₂+ i −λ − µ 4µ



∂ ¯f₂+ i λ − µ + 2 4µ



∂f₂. For simplicity, let us denote

α = µ − λ − 2

4µ , β = −λ − µ 4µ , then

h = α ¯∂f₁+ β∂f₁+ β∂ ¯f₁+ α∂f₁− iα ¯∂f₂+ iβ∂f₂+ iβ∂ ¯f₂− iα∂f₂ and

ih = iα ¯∂f₁+ iβ∂f₁+ iβ∂ ¯f₁+ iα∂f₁ + α ¯∂f₂− β∂f₂− β∂ ¯f₂+ α∂f₂. Therefore, (2.10) is equivalent to



I₂ + −α iα

−iα −α



∂¯f₁ f2



+ −β −iβ

−iβ β



∂

f¯₁ f¯2



= β iβ iβ −β



∂f₁ f₂



+σ + α −iα iα σ + α



∂f₁ f₂

 ,

(2.11)

where I_n denotes the n × n unit matrix. Setting ^{∂ ¯}_∂f¯^f = 0 if ¯∂f = 0 and ^∂f_∂f = 0 if ∂f = 0, (2.11) can be written as



I₂+ −α iα

−iα −α



∂¯f₁ f₂



+ −β −iβ

−iβ β

 ∂ ¯f1

∂f¯ 1 0 0 ^{∂ ¯}_∂f¯^f2

2

!

∂¯f₁ f₂



= β iβ iβ −β



∂f₁ f₂



+σ + α −iα iα σ + α

 ∂f1

∂f1 0 0 ^∂f_∂f²

2

!

∂f₁ f₂

 .

(2.12)

Finally, let U and V be two 2 × 2 matrices U = I₂+ −α iα

−iα −α



+ −β −iβ

−iβ β

 ∂ ¯f1

∂f¯ 1 0 0 ^{∂ ¯}_∂f¯^f2

2

! ,

V = β iβ iβ −β



+σ + α −iα iα σ + α

 ∂f1

∂f1 0 0 ^∂f_∂f²

2

! ,

then (2.12) writes as

U ¯∂f₁ f2



= V ∂f₁ f2



. (2.13)

It is clear that

kU − I2kL^∞ ≤ C(kαkL^∞ + kβkL^∞) and kV kL^∞ ≤ C(kβkL^∞+ kσkL^∞ + kαkL^∞), (2.14) where C is an absolute constant. Hereafter, for any matrix-valued function A(x) : Cⁿ→ Cⁿ with x ∈ R², the norm kAkL^∞ is defined by

kAk_L^∞ = sup

x∈R²

kA(x)k,

where k · k is the usual matrix norm derived by treating Cⁿ as an inner-product space.

Theorem 2.1 There exists an ε > 0 such that if

kµ − 1k_L^∞ ≤ ε and kλ + 1k_L^∞ ≤ ε, (2.15) then for any Lipschitz solution u of (1.3) vanishing in the lower half plane, we must have u ≡ 0.

Remark 2.2 It is clear that if the Lam´e coefficients λ and µ satisfy (2.15), then the ellip- ticity condition (2.1) holds.

Proof. Since u is Lipschitz and vanishes in the lower half plane, so does the vector-valued function

F =f₁ f₂

 .

In view of the definitions of σ, α, β, if ε of (2.15) is sufficiently small, then all of them are sufficiently small as well. By (2.14), U is invertible and V is small, consequently, kU⁻¹V k_L^∞ ≤ ε⁰  1. Therefore, from (2.13) we have

∂F = Ψ∂F¯ in C, (2.16)

where Ψ = U⁻¹V and we have identified R² as the complex plane C. Note that here F : C → C². (2.16) is a Beltrami system studied in [IVV02]. It was proved in [IVV02] that if kΨk_L^∞ is sufficiently small and F vanishes in the lower half plane, then F is trivial, i.e., u is trivial. For the sake of completeness, we sketch the proof here. We refer to [IVV02, Section 7] for more details. Without loss of generality, we assume that F vanishes for =z ≤ 1.

Define G(z) = F (√

z). Then G satisfies

∂G =¯ z

|z|Ψ(√

z)∂G. (2.17)

We can see that the differential of G, DG, lies in L^p(C) for some p > 4 (see (7.13) in [IVV02]).

In other words, we have

k ¯∂Gk_L^p ≤ ε⁰k∂Gk_L^p. (2.18) Let S be the Beurling-Ahlfors transform, i.e., S ¯∂ = ∂. It is known from the Calder´on- Zygmund theory that

kSk_L^p_→L^p ≤ a_p (2.19)

for some constant a_p, depending only on p (see [BJ08] for a more precise bound on a_p).

Hence, (2.19) implies

k∂GkL^p ≤ apk ¯∂GkL^p. (2.20) Combining (2.18) and (2.20), we conclude that k ¯∂Gk_L^p = k∂Gk_L^p = 0 provided a_pε⁰ < 1 for

some p > 4. The proof of theorem then follows.

2

Now we turn to the general elasticity system (1.1). Assume that a_ijkl ∈ L^∞ satisfies the ellipticity condition

a_ijkl(x)ξ_iξ_lρ_jρ_k≥ δ|ξ|²|ρ|² ∀ ξ, ρ ∈ R², (2.21) i.e., the Legendre-Hadamard condition. Due to the symmetry properties (1.2), for simplicity, we denote







a₁₁₁₁ = a, a₂₂₂₂ = b, a₁₁₁₂ = a₁₂₁₁ = a₂₁₁₁ = a₁₁₂₁ = c, a₁₁₂₂ = a₂₂₁₁ = d, a₁₂₁₂ = a₂₁₁₂ = a₁₂₂₁ = a₂₁₂₁ = e, a₁₂₂₂ = a₂₁₂₂ = a₂₂₁₂ = a₂₂₂₁ = g.

Componentwise, (1.1) is written as

(∂₁(a∂₁u₁ + c∂₁u₂+ c∂₂u₁+ d∂₂u₂) + ∂₂(c∂₁u₁+ e∂₁u₂+ e∂₂u₁+ g∂₂u₂) = 0,

∂₁(c∂₁u₁+ e∂₁u₂+ e∂₂u₁+ g∂₂u₂) + ∂₂(d∂₁u₁+ g∂₁u₂+ g∂₂u₁+ b∂₂u₂) = 0. (2.22) For our purpose, we will express (2.22) as











∂₁((a − d − 1)∂₁u₁+ (d + 1)v + c∂₁u₂+ c∂₂u₁) + ∂₂((2e − 1)∂₂u₁+ (e − 1)w + c∂₁u₁+ g∂₂u₂) = 0,

∂₁((2e − 1)∂₁u₂ − (e − 1)w + c∂₁u₁+ g∂₂u₂)

+ ∂₂((b − d − 1)∂₂u₂+ (d + 1)v + g∂₁u₂+ g∂₂u₁) = 0.

(2.23)

Comparing (2.23) with (2.6), (2.7), it is not difficult to see that (2.23) can be transformed to (2.16) with similar smallness condition on Ψ if |a − d − 2|  1, |b − d − 2|  1, |d + 1|  1, |e − 1|  1, |c|  1, |g|  1. Therefore, we can prove that

Theorem 2.3 There exists ε > 0 such that if

(ka − d − 2k_L^∞ ≤ ε, kb − d − 2k_L^∞ ≤ ε, kd + 1k_L^∞ ≤ ε,

ke − 1k_L^∞ ≤ ε, kck ≤ ε, kgk_L^∞ ≤ ε, (2.24)

then if u is a Lipschitz function solving (1.1) and vanishes in the lower half plane, then u vanishes identically.

Remark 2.4 Under the assumptions (2.24), (1.1) is a slightly perturbed system of the Lam´e system with λ, µ satisfying (2.15). Therefore, the ellipticity condition (2.21) holds.

3 Anisotropic system with regular coefficients

One may wonder if |d + 1|  1 and |e − 1|  1 in Theorem 2.3 can be replaced by

|d + k₀|  1 and |e − k₀|  1 for k₀ 6= 1. For measurable coefficients, it is not possible since the requirement of |e − k₀|  1 and 2e − k₀ ∼ 1 will force k₀ = 1. However, if a, b, d, e are Lipschitz, we can extend Theorem 2.3 to a larger class of system. Let k0 be any fixed constant. Similarly to (2.23), we obtain that











∂₁((a − d − k₀)∂₁u₁+ (d + k₀)v + c∂₁u₂+ c∂₂u₁) + ∂₂((2e − k₀)∂₂u₁+ (e − k₀)w + c∂₁u₁+ g∂₂u₂) = 0,

∂1((2e − k0)∂1u2 − (e − k0)w + c∂1u1+ g∂2u2)

+ ∂₂((b − d − k₀)∂₂u₂+ (d + k₀)v + g∂₁u₂+ g∂₂u₁) = 0.

(3.1)

Denote ˜a = a − d − k₀, ˜e = 2e − k₀, ˜b = b − d − k₀. Suppose that ˜a 6= 0 and ˜b 6= 0. Then (3.1) is equivalent to















∂₁(∂₁(˜au₁) − ∂₁˜au₁+ (d + k₀)v + c∂₁u₂+ c∂₂u₁)

+ ∂₂(˜e˜a⁻¹(∂₂(˜au₁) − ∂₂au˜ ₁) + (e − k₀)w + c∂₁u₁+ g∂₂u₂) = 0,

∂₁(˜e˜b⁻¹(∂₁(˜bu₂) − ∂₁˜bu₂) − (e − k₀)w + c∂₁u₁+ g∂₂u₂) + ∂₂(∂₂(˜bu₂) − ∂₂˜bu₂+ (d + k₀)v + g∂₁u₂+ g∂₂u₁) = 0.

(3.2)

We set ˜u1 = ˜au1, ˜u2 = ˜bu2, then (3.2) becomes















∂₁(∂₁(˜u₁) − ∂₁˜a˜a⁻¹u˜₁+ (d + k₀)v + c∂₁(˜b⁻¹u˜₂) + c∂₂(˜a⁻¹u˜₁))

+ ∂₂(˜e˜a⁻¹(∂₂(˜u₁) − ∂₂˜a˜a⁻¹u˜₁) + (e − k₀)w + c∂₁(˜a⁻¹u˜₁) + g∂₂(˜b⁻¹u˜₂)) = 0,

∂₁(˜e˜b⁻¹(∂₁(˜u₂) − ∂₁˜b˜b⁻¹u˜₂) − (e − k₀)w + c∂₁(˜a⁻¹u˜₁) + g∂₂(˜b⁻¹u˜₂)) + ∂₂(∂₂(˜u₂) − ∂₂˜b˜b⁻¹u˜₂+ (d + k₀)v + g∂₁(˜b⁻¹u˜₂) + g∂₂(˜a⁻¹u˜₁)) = 0.

(3.3)

We then express

( v = ∂₁u₁+ ∂₂u₂ = ∂₁˜a⁻¹u˜₁+ ˜a⁻¹∂₁u˜₁ + ∂₂˜b⁻¹u˜₂+ ˜b⁻¹∂₂u˜₂ w = ∂₁u₂− ∂₂u₁ = ∂₁˜b⁻¹u˜₂+ ˜b⁻¹∂₁u˜₂− ∂₂˜a⁻¹u˜₁− ˜a⁻¹∂₂u˜₁.

Theorem 3.1 Let k₀ > 0. Assume that a, b, d, e are Lipschitz and c, g are measurable.

Moreover, suppose that a, b, d, e are constants in R²\ K, where K is compact set. Then there exists an ε = ε(k₀, K) > 0 such that if

(k∇(a − d)k_L^∞ ≤ ε, k∇(b − d)k_L^∞ ≤ ε, ka − d − 2ek_L^∞ ≤ ε, kb − d − 2ek_L^∞ ≤ ε, kd + k₀k_L^∞ ≤ ε, ke − k₀k_L^∞ ≤ ε, kck_L^∞ ≤ ε, kgk_L^∞ ≤ ε,

(3.4) then if u vanishes in the lower half plane, then u is identically zero.

Proof. We first note that when ε, depending on k₀, is sufficiently small, ˜a and ˜b are strictly positive. Let f₁ = ˜u₁+ ˜u⁰₁ and f₂ = ˜u₂+ i˜u⁰₂, where ˜u⁰₁ and ˜u⁰₂ are conjugate functions of ˜u₁ and ˜u₂ defined as above. In view of (3.4), (3.3) is reduced to

∂F = ˜¯ Ψ∂F + HF + ˜H ¯F , (3.5)

where k ˜ΨkL^∞ ≤ ε⁰, kHkL^∞ ≤ ε⁰, k ˜HkL^∞ ≤ ε⁰, and H, ˜H are supported in K. Note that ε⁰ → 0 as ε → 0. As before, let G(z) = F (√

z), then G satisfies

∂G =¯ z

|z|Ψ(˜ √

z)∂G + H(√

z)G + ˜H(√ z) ¯G.

By the Poincar´e inequality, we have that

kHGk_L^p+ k ˜H ¯Gk_L^p ≤ ε⁰C(k ¯∂Gk_L^p+ k∂Gk_L^p) (3.6) for p ≥ 2, where C depends on K (and p). Using (3.6), we have from (3.5) that

k ¯∂Gk_L^p ≤ ε⁰⁰k∂Gk_L^p

with ε⁰⁰→ 0 as ε → 0. Next, using the same arguments as in the proof of Theorem 2.1, the

result follows.

2

Remark 3.2 From the ellipticity condition (2.1) for isotropic media, it is readily seen that if k0 > 0 and ε is sufficiently small, then the ellipticity condition (2.21) is satisfied.

4 Counterexample to unique continuation

In this section we will construct a counterexample to the unique continuation property, which vanishes in the lower half plane, for second order elliptic systems with measurable coefficients. Precisely, we consider

∂_j(a_ijkl(x)∂_ku_l) = 0 in R², (4.1)

where the coefficients a_ijkl do not necessarily satisfy the symmetry conditions (1.2). For simplicity, we use the following short-hand notations:

11 → 1, 12 → 2, 21 → 3, 22 → 4, i.e.,

a₁₁₁₁ = a₁₁, a₁₁₁₂ = a₁₂, a₁₁₂₁ = a₁₃, a₁₁₂₂ = a₁₄, · · · etc.

So, the system (4.1) is written as











∂₁(a₁₁∂₁u₁+ a₁₂∂₁u₂+ a₁₃∂₂u₁+ a₁₄∂₂u₂)

+ ∂₂(a₂₁∂₁u₁+ a₂₂∂₁u₂+ a₂₃∂₂u₁+ a₂₄∂₂u₂) = 0,

∂₁(a₃₁∂₁u₁+ a₃₂∂₁u₂+ a₃₃∂₂u₁+ a₃₄∂₂u₂)

+ ∂₂(a₄₁∂₁u₁+ a₄₂∂₁u₂+ a₄₃∂₂u₁+ a₄₄∂₂u₂) = 0.

(4.2)

As before, we can find v1 and v2 such that

(∂2v1 = a11∂1u1+ a13∂2u1+ a12∂1u2+ a14∂2u2,

−∂1v1 = a21∂1u1 + a23∂2u1+ a22∂1u2+ a24∂2u2, (4.3)

and (

∂₂v₂ = a₃₂∂₁u₂+ a₃₄∂₂u₂+ a₃₁∂₁u₁+ a₃₃∂₂u₁,

−∂₁v₂ = a₄₂∂₁u₂ + a₄₄∂₂u₂+ a₄₁∂₁u₁+ a₄₃∂₂u₁. (4.4) Here we will use a different reduction from the one used in Section 2. The method is inspired by Bojarski’s work [Bo57]. Denote

α₁ = (a₁₁+ a₂₃) + i(a₂₁− a₁₃)

2 , β₁ = (a₁₁− a₂₃) + i(a₂₁+ a₁₃)

2 ,

ζ₁ = a₁₂+ ia₂₂, η₁ = a₁₄+ ia₂₄.

Let f₁ = u₁+ iv₁ and f₂ = u₂+ iv₂, then we can compute that

0 = (1 + α₁) ¯∂f₁+ β₁∂f₁+ β₁∂f₁− (1 − α₁)∂f₁+ ζ₁∂₁u₂+ η₁∂₂u₂

= (1 + α₁) ¯∂f₁+ β₁∂f₁+ β₁∂f₁− (1 − α₁)∂f₁+ ζ₁

2( ¯∂f₂+ ∂f₂+ ∂f₂+ ∂f₂) + η₁

2i( ¯∂f₂− ∂f₂− ∂f₂+ ∂f₂)

= (1 + α₁) ¯∂f₁+ β₁∂f₁+ β₁∂f₁− (1 − α₁)∂f₁+ (ζ₁ 2 +η₁

2i) ¯∂f₂+ (ζ₁ 2 −η₁

2i)∂f₂ + (ζ₁

2 − η₁

2i)∂f₂+ (ζ₁ 2 + η₁

2i)∂f₂.

(4.5)

Likewise, we denote

α2 = (a₃₂+ a₄₄) + i(a₄₂− a₃₄)

2 , β2 = (a₃₂− a₄₄) + i(a₄₂+ a₃₄)

2 ,

ζ₂ = a₃₁+ ia₄₁, η₂ = a₃₃+ ia₄₃, then we obtain

0 = (1 + α₂) ¯∂f₂+ β₂∂f₂+ β₂∂f₂− (1 − α₂)∂f₂+ (ζ₂ 2 +η₂

2i) ¯∂f₁+ (ζ₂ 2 −η₂

2i)∂f₁ + (ζ2

2 − η2

2i)∂f₁+ (ζ2

2 + η2

2i)∂f₁.

(4.6)

Putting (4.5) and (4.6) in matrix form gives

A ¯∂F + B∂F + C∂F + D∂F = 0 in C, (4.7)

where F =f₁ f₂

 and

A = 1 + α₁ ^ζ₂¹ − i^η₂¹

ζ2

2 − i^η₂² 1 + α₂



, B =

 β₁ ^ζ₂¹ + i^η₂¹

ζ2

2 + i^η₂² β₂

 , C =

 β₁ ^ζ₂¹ + i^η₂¹

ζ2

2 + i^η₂² β₂



(= B), D =−1 + α₁ ^ζ₂¹ − i^η₂¹

ζ2

2 − i^η₂² −1 + α₂

 .

(4.8)

Note that D = A − 2I₂. Conversely, it is easy to see that, given any 2 × 2 complex-valued matrices A, B, C, D satisfying B = C and D = A − 2I₂ and (4.7) with F = u₁+ iv₁

u₂+ iv₂

 , then, writing A, B, C, D as in (4.8), we can find real numbers a₁₁, a₁₂, · · · , a₄₄ such that (4.3), (4.4) hold, and hence (4.2) is satisfied.

It was proved in [IVV02] that there exists a 2 × 2 complex-valued matrix Q ∈ L^∞(C) with

kQk_L^∞_(C)≤ κ < 1 (4.9)

and a nontrivial Lipschitz function ˜F : C → C² vanishing in the lower half plane of C such that

∂ ˜¯F + Q∂ ˜F = 0 in C. (4.10)

Adding A × (4.10) and B × (4.10), for any A, B, gives

A ¯∂ ˜F + B∂ ˜F + AQ∂ ˜F + BQ ∂ ˜F = 0 in C. (4.11) Comparing (4.7) with (4.11), we hope to find A, B, C, D satisfying

B = C = AQ, D = BQ, D = A − 2I₂. (4.12)

To fulfill (4.12), we begin with

A − 2I₂ = D = AQQ, which implies

A(I₂− QQ) = 2I₂. (4.13)

In view of (4.9), we have that kQQk_L^∞_(C) ≤ κ² < 1 and hence I₂− QQ is invertible. In other words, (4.13) gives

A = 2(I₂− QQ)⁻¹.

Once A is determined, we can find C and, of course, B. Hence the relations in (4.12) hold. Finally, in view of the definitions of α_j, β_j, ζ_j, η_j, j = 1, 2, there exists a unique fourth rank-tensor (a_ijkl(x)) producing A, B, C, D which were determined above.

With such A, B, C, D obtained above, there exists a nontrivial solution F : C → C², i.e., F = ˜F , vanishing in the lower half plane of C and satisfying (4.7) (and hence, (4.11)), i.e.,

A ¯∂F + AQ∂F + AQ∂F + AQQ ∂F = 0 in C. (4.14) As mentioned above, (4.14) is equivalent to the second order system (4.2) with corresponding coefficients (aijkl(x)). Now we would like to verify that this second order system is elliptic.

The meaning of ellipticity will be specified later. We first show that L₀F := ¯∂F + Q∂F is equivalent to a first order uniformly elliptic system. Let us denote

F = u₁+ iv₁ u₂+ iv₂

 , then

2 ¯∂F =(∂₁u₁− ∂₂v₁) + i(∂₁v₁+ ∂₂u₁) (∂₁u₂− ∂₂v₂) + i(∂₁v₂+ ∂₂u₂)

 and

2∂F = (∂₁u1+ ∂2v1) + i(∂1v1− ∂₂u1) (∂₁u₂+ ∂₂v₂) + i(∂₁v₂− ∂₂u₂)

 .

Let Q = Q_r+ iQ_i, then 2L₀F can be put into the following equivalent system

L₁







 u₁ u₂ v₁ v₂









:=I₂+ Q_r −Q_i Q_i I₂+ Q_r



∂₁







 u₁ u₂ v₁ v₂







 +

 Q_i −I₂ + Q_r I₂− Q_r Q_i



∂₂







 u₁ u₂ v₁ v₂









, (4.15)

i.e., 2L₀F = G =g₁ g2



=w₁ + iz₁ w2 + iz2



is equivalent to

L1







 u₁ u2

v₁ v₂









=







 w₁ w2

z₁ z₂







 .

For simplicity, we denote

R =I₂+ Qr −Qi

Q_i I₂+ Q_r



and S =

 Qi −I2+ Qr

I₂− Q_r Q_i

 . Now we want to show that (4.15) is uniformly elliptic, i.e.,

det(αR + βS) ≥ c(α²+ β²)², ∀ z ∈ C, (α, β) ∈ R² 6= 0, (4.16) where c = c(κ) > 0. To prove (4.16), we first observe that

S =

 Qi −I2+ Qr

I₂− Q_r Q_i



=I₂ − Qr Qi

−Q_i I₂− Q_r

 J where

J = 0 −I₂ I2 0

 . Therefore, we obtain that

αR + βS = α(I₄+ E) + β(I₄− E)J = (αI₄+ βJ ) + E(αI₄− βJ), (4.17) where

E =Q_r −Q_i Q_i Q_r

 . From (4.9), we have that for E : R⁴ → R⁴

kEk_L^∞_(R⁴₎ ≤ κ. (4.18)

It is easy to see that

k(αI₄+ βJ )Zk = k(αI₄− βJ)Zk =p

α²+ β² kZk, ∀ Z ∈ R⁴. (4.19) Combining (4.18) and (4.19) gives

det(αR + βS) = det(αI4+ βJ )det(I4+ (αI4+ βJ )⁻¹E(αI4− βJ)) ≥ c(α²+ β²)² with c = c(κ) and (4.16) is proved.

Now we want to consider

2( ¯∂F + Q∂F + Q(∂F + Q ∂F )) = 2L₀F + 2QL₀F . (4.20) It is easy to see that

2L₀F =w₁− iz₁ w₂− iz₂

 ,

which is equivalent to

ILˆ 1







 u₁ u2

v₁ v₂









= ˆI







 w₁ w2

z₁ z₂







 ,

where

I =ˆ I₂ 0 0 −I₂

 . Consequently, (4.20) can be written as

(R + E ˆIR)∂₁







 u₁ u₂ v₁ v₂









+ (S + E ˆIS)∂₂







 u₁ u₂ v₁ v₂







 .

It is clear that

det(α(R + E ˆIR) + β(S + E ˆIS)) = det([αR + βS] + E ˆI[αR + βS])

= det(αR + βS) · det(I₄+ E ˆI)

≥ c(α²+ β²)². Finally, let us denote A = A_r+ iA_i and

A =ˆ A_r −A_i A_i A_r

 , then

A ¯∂F + AQ∂F + AQ∂F + AQQ ∂F = 0 (4.21)

is equivalent to the first order system

A(R + E ˆˆ IR)∂₁







 u₁ u₂ v₁ v₂









+ ˆA(S + E ˆIS)∂₂







 u₁ u₂ v₁ v₂









= 0. (4.22)

Since det ˆA ≥ c > 0, we immediately obtain that

det(α ˆA(R + E ˆIR) + β ˆA(S + E ˆIS)) ≥ c(α²+ β²)². (4.23) We would like to remind the reader that (4.21) is equivalent to (4.3), (4.4) (and (4.2)).

Now we return to the system (4.3), (4.4), i.e.,

(∂₂v₁ = a₁₁∂₁u₁+ a₁₃∂₂u₁+ a₁₂∂₁u₂+ a₁₄∂₂u₂,

−∂₁v₁ = a₂₁∂₁u₁ + a₂₃∂₂u₁+ a₂₂∂₁u₂+ a₂₄∂₂u₂,

and (

∂₂v₂ = a₃₂∂₁u₂+ a₃₄∂₂u₂+ a₃₁∂₁u₁+ a₃₃∂₂u₁,

−∂₁v₂ = a₄₂∂₁u₂ + a₄₄∂₂u₂+ a₄₁∂₁u₁+ a₄₃∂₂u₁. We put this system as









a11 a12 0 0 a₂₁ a₂₂ 1 0 a₃₁ a₃₂ 0 0 a41 a42 0 1









∂₁







 u1

u₂ v₁ v2







 +









a13 a14 −1 0 a₂₃ a₂₄ 0 0 a₃₃ a₃₄ 0 −1 a43 a44 0 0









∂₂







 u1

u₂ v₁ v2









= 0, (4.24)

which is equivalent to (4.22). From (4.23), we have that c(α²+ β²)²

≤ det(α ˆA(R + E ˆIR) + β ˆA(S + E ˆIS))

= det







 α









a₁₁ a₁₂ 0 0 a₂₁ a₂₂ 1 0 a31 a32 0 0 a₄₁ a₄₂ 0 1







 + β









a₁₃ a₁₄ −1 0 a₂₃ a₂₄ 0 0 a33 a34 0 −1 a₄₃ a₄₄ 0 0

















= −det







 α









a₁₁ a₁₂ 0 0 a31 a32 0 0 a₂₁ a₂₂ 1 0 a₄₁ a₄₂ 0 1







 + β









a₁₃ a₁₄ −1 0 a33 a34 0 −1 a₂₃ a₂₄ 0 0 a₄₃ a₄₄ 0 0

















= det









a12α + a14β a11α + a13β −β 0 a₃₂α + a₃₄β a₃₁α + a₃₃β 0 −β a₂₂α + a₂₄β a₂₁α + a₂₃β α 0 a42α + a44β a41α + a43β 0 α









= deta₁₂α²+ a₁₄αβ + a₂₂αβ + a₂₄β² a₁₁α²+ a₁₃αβ + a₂₁αβ + a₂₃β² a₃₂α²+ a₃₄αβ + a₄₂αβ + a₄₄β² a₃₁α²+ a₃₃αβ + a₄₁αβ + a₄₃β²

 .

(4.25)

It follows from (4.25) that for any ξ = (ξ₁, ξ₂) 6= 0, the 2 × 2 matrix (P

j,ka_ijkl(z)ξ_jξ_k) satisfies

|det(X

j,k

aijkl(z)ξjξk)| ≥ c|ξ|⁴, ∀ z ∈ C. (4.26) In summary, we have shown that

Theorem 4.1 There exists a nontrivial vector-valued function u = (u₁, u₂)^T : R² → R² vanishing in the lower half plane solving a second order uniformly elliptic system (4.1), in the sense of (4.26), with essentially bounded coefficients.

To prove Theorem 4.1, we used a reduction different from the one given in Section 2. It is natural to investigate whether the reduction used here can be applied to prove positive

results stated in Section 2. We only discuss the Lam´e system. Comparing (2.2), (2.3) and (4.2) implies











a11 = λ + 2µ, a12 = a13= 0, a14= λ, a₂₁ = a₂₄= 0, a₂₂ = a₂₃= µ,

a₃₁ = a₃₄= 0, a₃₂ = a₃₃= µ,

a₄₁ = λ, a₄₂ = a₄₃= 0, a₄₄= λ + 2µ.

By the definitions, we see that







α₁ = λ + 3µ

2 , β₁ = λ + µ

2 , ζ₁ = iµ, η₁ = λ, α₂ = λ + 3µ

2 , β₂ = −λ + µ

2 , ζ₂ = iλ, η₂ = µ, and thus,

A =1 + ^λ+3µ_{2 2}ⁱ(µ − λ)

i

2(λ − µ) 1 + ^λ+3µ₂



, B = C =

 _λ+µ

2

i

2(µ + λ)

i

2(λ + µ) −^λ+µ₂

 ,

D =−1 +^λ+3µ_{2 2}ⁱ(µ − λ)

i

2(λ − µ) −1 + ^λ+3µ₂

 .

Now if µ ≈ 1 and λ ≈ −1 as in Theorem 2.1, then B ≈ 0, C ≈ 0, but A ≈ 2 i

−i 2



, D =  0 i

−i 0

 .

In other words, (4.7) corresponding to the Lam´e system can not be put into the form

∂F = Ψ∂F¯ with kΨkL^∞  1.

On the other hand, if the coefficients (a_pq) of (4.2) satisfy (ka_pq− 1k_L^∞ ≤ ε for pq = 11, 23, 32, 44,

ka_pqk_L^∞ ≤ ε for all other pq⁰s (4.27) with a sufficiently small ε, then

kA − 2I₂k_L^∞ ≤ cε, kBk_L^∞ = kCk_L^∞ ≤ cε, kDk_L^∞ ≤ cε

for some constant c. For this case, we can prove that the global unique continuation property holds as in the proof of Theorem 2.3. It is not hard to see that the second order system (4.2) with coefficients satisfying (4.27) is elliptic in the sense of (4.26). In fact, we can even

show that (a_pq) = (a_ijkl) satisfies the strong convexity condition (1.6) (and, of course, the Legendre-Hadamard condition (2.21)) provided ε is small. To see this, it suffices to consider a₁₁ = a₁₁₁₁ = 1, a₂₃ = a₁₂₂₁ = 1, a₃₂ = a₂₁₁₂ = 1, a₄₄ = a₂₂₂₂ = 1, and all other a_pq’s are zero. Then we have that for any 2 × 2 matrix ξ = (ξ_k^l)

a_ijklξ_k^lξ_jⁱ = a₁₁₁₁ξ₁¹ξ₁¹+ a₁₂₂₁ξ₂¹ξ¹₂ + a₂₁₁₂ξ₁²ξ₁²+ a₂₂₂₂ξ₂²ξ₂² = |ξ|²,

which implies (1.6) for small ε. The class of second order elliptic systems (4.2) satisfying (4.27) contains a special class of hyperelastic materials, where only the major symmetry property a_ijkl = a_klij holds.

A counterexample to the strong unique continuation for (2.16) was constructed in [CP11]

(see also related article [Ro09]). The counterexample given in [CP11] shows that there exists a nontrivial function F vanishing at 0 to infinite order satisfying

∂F + Q∂F = 0,¯

where Q(x) ∈ C^2×2 is continuous and vanishes at 0 to infinite order as well. Based on this example, using the same framework as above, we can construct a counterexample to strong unique continuation for second order elliptic systems with continuous coefficients in the plane. Observe that for the extreme case Q = 0, we have A = 2I and B = C = D = 0.

Consequently, we see that

a₁₁= a₂₃= a₃₂ = a₄₄ = 1

and all other a_pq’s are zero. Therefore, when x is near 0, Q is sufficiently small, which is exactly the case we discussed in (4.27). In other words, for the second order elliptic system with coefficients satisfying (4.27) and the strong convexity condition (1.6), the global unique continuation property holds, in spite of the fact that there are examples showing that the strong unique continuation property fails. Furthermore, we want to point out that the counterexample to the strong unique continuation for (4.1) we constructed is a small perturbation of the Laplacian ∆ near the origin. In Section 2 we have shown that the Lam´e system with λ ≈ −1 and µ ≈ 1 can be written as a small perturbation of the Laplacian.

Therefore, this counterexample strongly suggests that the Lam´e system with measurable coefficients, even when λ ≈ −1 and µ ≈ 1, does not possess the strong unique continuation property. Moreover, this example or an earlier example constructed in [Ali80] also suggest that the strong unique continuation property for the anisotropic elasticity system even with continuous coefficients is most likely not true.

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[Al12] G. Alessandrini, Strong unique continuation for general elliptic equations in 2D, J. Math. Anal. Appl. 386 (2012), 669-676.

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[Ali80] S. Alinhac, Non-unicit´e pour des op´erateurs diff´erentiels `a caract´eristiques com- plexes simples, Ann. Sci. Ecole Norm. Sup. 13 (1980), 385-393.

[Bo57] B. V. Bojarski, Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients, Mat. Sb. N.S. 43 (85), 1957, 451-503.

[BJ08] R. Ban˜uelos and P. Janakiraman, L^p-bounds for the Beurling-Ahlfors transform, Trans. Amer. Math. Soc. 360 (2008), 3603-3612.

[BN54] L. Bers and L. Nirenberg, On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications, In Convegno Internazionale sullen Equazioni Lineari alle Derivate Parziali, Trieste, 1954, 111-140. Edizioni Cremonese, Roma, 1955.

[CP11] A. Coffman and Y. Pan, Smooth counterexamples to strong unique continuation for a Beltrami system in C², arXiv:1102.2462 [math.AP].

[IVV02] T. Iwaniec, G. C. Verchota, and A. L. Vogel, The failure of rank-one connections, Arch. Rational Mech. Anal. 163 (2002), 125-169.

[MS03] S. M¨uller and V. ˇSver´ak, Convex integration for Lipschitz mappings and coun- terexamples to regularity, Annals of Math., 157 (2003), 715-742.

[LNUW11] C. L. Lin, G. Nakamura, G. Uhlmann, and J. N. Wang, Quantitative strong unique continuation for the Lame system with less regular coefficients, Methods Appl. Anal. 18 (2011), 85-92.

[Ro09] J. P. Rosay, Uniqueness in rough almost complex structures and differential inequalities. Preprint: arXiv:0911.0668v1.

[Sc98] F. Schulz, On the unique continuation property of elliptic divergence form equa- tions in the plane, Math. Z. 228 (1998), 201-206.