Carleman estimate for complex second order elliptic operators with discontinuous Lipschitz coefficients

Carleman estimate for complex second order elliptic operators with discontinuous Lipschitz coefficients

E. Francini

S. Vessella

J-N. Wang

February 16, 2020

Abstract

In this paper, we derive a local Carleman estimate for the complex second order elliptic operator with Lipschitz coefficients having jump discontinuities.

Combing the result in [BL] and the arguments in [DcFLVW], we present an ele- mentary method to derive the Carleman estimate under the optimal regularity assumption on the coefficients.

1 Introduction

Carleman estimates are important tools for proving the unique continuation property for partial differential equations. Additionally, Carleman estimates have been suc- cessfully applied to study inverse problems and controllability of partial differential equations. Most of Carleman estimates are proved under the assumption that the leading coefficients possess certain regularity. For example, for general second or- der elliptic operators, Carleman estimates were proved when the leading coefficients are at least Lipschitz [Ho3]. In general, the Lipschitz regularity assumption is the optimal condition for the unique continuation property to hold in Rⁿ with n ≥ 3 (see counterexamples constructed by Pli´s [P] and Miller [M]). Therefore, Carleman estimates for second order elliptic operators with general discontinuous coefficients are most likely not valid. Nonetheless, recently, in the case of coefficients having jump discontinuities at an interface with homogeneous or non-homogeneous trans- mission conditions, one can still prove useful Carleman estimates, see, for example, Le Rousseau-Robbiano [LR1], [LR2], Le Rousseau-Lerner [LL], and [DcFLVW].

Above mentioned results are proved for real coefficients. In many real world problems, the case of complex-valued coefficients arises naturally. The modeling of the current flows in biological tissues or the propagation of the electromagnetic

∗Universit`a di Firenze, Italy. Email: elisa.francini@unifi.it

†Universit`a di Firenze, Italy. Email: sergio.vessella@unifi.it

‡National Taiwan University, Taiwan. Email: jnwang@ntu.edu.tw

waves in conductive media are typical examples. In these cases, the conductivities are complex-valued functions. On the other hand, in some situations, the conductivities are not continuous functions. For instance, in the human body, different organs have different conductivities. Therefore, to model the current flow in the human body, it is more reasonable to consider an anisotrotopic complex-valued conductivity with jump-type discontinuities [MPH].

With potential applications in mind, our goal in this paper is to derive a Car- leman estimate for the second order elliptic equations with complex-valued leading coefficients having jump-type discontinuities. Although such a Carleman estimate has been derived in [BL], we want to remark that the method used in [BL], also in [LR1], [LR2], and [LL], are based on the technique of pseudodifferential operators and hence requires C^∞ coefficients and interface; while the method in [DcFLVW]

(and its parabolic counterpart, [FV]) relies on the Fourier transform and a version of partition of unity which requires only Lipschitz coefficients and C^1,1 interface. Hence, the main purpose of the paper is to extend the method in [DcFLVW], [FV] to second order elliptic operators with complex-valued coefficients. It is important to point out that even though second order elliptic operators with complex-valued coefficients can be written as a coupled second order elliptic system with real coefficients, neither the method in [LR1], [LR2], [LL] nor that in [DcFLVW] can be applied to coupled elliptic systems. Therefore, we need to work on operators with complex-valued coefficients directly.

Our strategy to derive the Carleman estimate consists of two major steps. In the first step, we treat second order elliptic operators with constant complex coeffi- cients. Based on [BL], by checking the strong pseudoconvexity and the transmission conditions in a neighborhood of a fixed point at the interface, we can derive a Car- leman estimate for second order elliptic operators with constant complex coefficients from [BL, Theorem 1.6]. We would like to mention that the result in [BL] is stated for quite general complex coefficients, but here we can only verify the transmission condition with our choice of weight functions for complex coefficients having small imaginary parts. So in this paper we will consider this case. In the second step, we extend the Carleman estimate to the operator with non-constant complex coefficients with small imaginary parts. This method in this step is taken from the argument in [DcFLVW, Section 4]. The key tool is a version of partition of unity.

Furthermore, in the second step, we need an interior Carleman estimate for sec- ond order elliptic operators having Lipschitz leading coefficients and with the weight function ψ_ε. An interior Carleman estimate was proved in [Ho1, Theorem 8.3.1], but for operators with C¹ leading coefficients. Another interior estimate was established in [Ho3, Proposition 17.2.3] for operators with Lipschitz leading coefficients, but with a different weight function. H¨ormander remarked in [Ho4] (page 703, line 7-8) that

”Inspection of proof of Theorem 8.3.1 in [Ho1] shows that only Lipschitz continu- ity was actually used in the proof.” But, as far as we can check, there is no formal proof of this statement in literature. To make the paper self contained, we would like give a detailed proof of interior Carleman estimate for second order elliptic opera-

tor with Lipschitz leading coefficients and with a rather general weight function, see Proposition4.1. This interior Carleman estimate may be useful on other occasions.

In this paper, we present a detailed and elementary derivation of the Carleman es- timate for the second order elliptic equations with complex-valued coefficients having jump-type discontinuities following our method in [DcFLVW]. Having established the Carleman estimate, we then can apply the ideas in [FLVW] to prove a three-region inequality and those in [CW] to prove a three-ball inequality across the interface.

With the help of the three-ball inequality, we can study the size estimate problem for the complex conductivity equation following the ideas in [CNW]. We will present these quantitative uniqueness results and the application to the size estimate in the forthcoming paper.

The paper is organized as follows. In Section 2, we introduce notations that will be used in the paper and the statement of the theorem. In Section 3, we derive a Carleman estimate for the operator having discontinuous piecewise constant coeffi- cients. This Carleman estimate is a special case of [BL, Theorem 1.6]. Therefore, the main task of Section 3 is to check the transmission condition and the strong pseu- doconvexity condition. Finally, the main Carleman estimate is proved in Section 4.

The key ingredient is a partition of unity introduced in [DcFLVW].

2 Notations and statement of the main theorem

We will state and prove the Carleman estimate for the case where the interface is flat. Since our Carleman estimate is local near any point at the interface, for general C^1,1 interface, it can be flatten by a suitable change of coordinates. Moreover, the transformed coefficients away from the interface remain Lipschitz. Define H_± = χ_Rⁿ

±

where Rⁿ± = {(x⁰, x_n) ∈ Rⁿ⁻¹× R|xn ≷ 0} and χRⁿ± is the characteristic function of Rⁿ±. In places we will use equivalently the symbols ∂, ∇ and D = −i∇ to denote the gradient of a function and we will add the index x⁰ or x_n to denote gradient in Rⁿ⁻¹ and the derivative with respect to x_n respectively. We further denote ∂_{`</sub> = ∂/∂x<sub>`}, D_{`</sub> = −i∂<sub>`}, and ∂_{ξ`</sub> = ∂/∂ξ<sub>`}.

Let u±∈ C^∞(Rⁿ). We define

u = H₊u₊+ H₋u₋=X

±

H_±u_±, hereafter, we denote P

±a±= a₊+ a−, and L(x, D)u := X

±

H±div(A±(x)∇u±), (2.1) where

A±(x) = {a^±_{`j</sub>(x)}<sup>n</sup><sub>`,j=1} = {a^±_{`j</sub>(x<sup>0</sup>, x<sub>n</sub>)}<sup>n</sup><sub>`,j=1}, x⁰ ∈ Rⁿ⁻¹, x_n ∈ R (2.2) is a Lipschitz symmetric matrix-valued function. Assume that

a^±_{`j</sub>(x) = a<sup>±</sup><sub>j`}(x), ∀ `, j = 1, · · · , n, (2.3)

and furthermore

a^±_{`j</sub>(x) = M<sub>`j}^±(x) + iγN_{`j</sub><sup>±</sup>(x), (2.4) where (M<sub>`j}^±) and (N_`j^±) are real-valued matrices and γ > 0. We further assume that there exist λ₀, Λ₀ > 0 such that for all ξ ∈ Rⁿ and x ∈ Rⁿ we have

λ₀|ξ|² ≤ M^±(x)ξ · ξ ≤ Λ₀|ξ|² (2.5) and

λ₀|ξ|² ≤ N^±(x)ξ · ξ ≤ Λ₀|ξ|². (2.6) In the paper, we consider Lipschitz coefficients A±, i.e., there exists a constant M0 > 0 such that

|A±(x) − A±(y)| ≤ M₀|x − y|. (2.7) To treat the transmission conditions, we write

h₀(x⁰) := u₊(x⁰, 0) − u−(x⁰, 0), ∀ x⁰ ∈ Rⁿ⁻¹, (2.8) h₁(x⁰) := A₊(x⁰, 0)∇u₊(x⁰, 0) · ν − A−(x⁰, 0)∇u−(x⁰, 0) · ν, ∀ x⁰ ∈ Rⁿ⁻¹, (2.9) where ν = e_n.

Let us now introduce the weight function. Let ϕ be ϕ(x_n) =

( ϕ₊(x_n) := α₊x_n+ βx²_n/2, x_n ≥ 0,

ϕ−(x_n) := α−x_n+ βx²_n/2, x_n < 0, (2.10) where α₊, α₋ and β are positive numbers which will be determined later. In what follows we denote by ϕ₊ and ϕ− the restriction of the weight function ϕ to [0, +∞) and to (−∞, 0) respectively. We use similar notation for any other weight functions.

For any ε > 0 let

ψ_ε(x) := ϕ(x_n) −ε

2|x⁰|², (2.11)

and let

φ_δ(x) := ψ_δ(δ⁻¹x), δ > 0. (2.12) For a function h ∈ L²(Rⁿ), we define

ˆh(ξ⁰, x_n) = Z

Rⁿ⁻¹

h(x⁰, x_n)e^−ix0·ξdx⁰, ξ⁰ ∈ Rⁿ⁻¹.

As usual we denote by H^1/2(Rⁿ⁻¹) the space of the functions f ∈ L²(Rⁿ⁻¹) satisfying Z

Rⁿ⁻¹

|ξ⁰|| ˆf (ξ⁰)|²dξ⁰ < ∞, with the norm

kf k²_H1/2(Rⁿ⁻¹)= Z

Rⁿ⁻¹

(1 + |ξ⁰|²)^1/2| ˆf (ξ⁰)|²dξ⁰. (2.13)

Moreover we define

[f ]_1/2,Rⁿ⁻¹ =

Z

Rⁿ⁻¹

Z

Rⁿ⁻¹

|f (x) − f (y)|²

|x − y|ⁿ dydx

1/2

,

and recall that there is a positive constant C, depending only on n, such that C⁻¹

Z

Rⁿ⁻¹

|ξ⁰|| ˆf (ξ⁰)|²dξ⁰ ≤ [f ]²_1/2,Rn−1 ≤ C Z

Rⁿ⁻¹

|ξ⁰|| ˆf (ξ⁰)|²dξ⁰,

so that the norm (2.13) is equivalent to the norm kf k_L²_(Rⁿ⁻¹₎+ [f ]_1/2,Rⁿ⁻¹. We use the letters C, C₀, C₁, · · · to denote constants. The value of the constants may change from line to line, but it is always greater than 1.

We will denote by B_r⁰(x⁰) the (n − 1)-ball centered at x⁰ ∈ Rⁿ⁻¹ with radius r > 0.

Whenever x⁰ = 0 we denote B_r⁰ = B_r⁰(0). Likewise, we denote B_r(x) be the n-ball centered at x ∈ Rⁿ with radius r > 0 and B_r = B_r(0).

Theorem 2.1 Let u and A±(x) satisfy (2.1)-(2.9). There exist α₊, α−, β, δ₀, r₀, γ₀ and C depending on λ₀, Λ₀, M₀ such that if γ < γ₀, δ ≤ δ₀ and τ ≥ C, then

X

± 2

X

k=0

τ^3−2k Z

Rⁿ±

|D^ku±|²e^{2τ φδ,±(x0,xn)}dx⁰dx_n+X

± 1

X

k=0

τ^3−2k Z

Rⁿ⁻¹

|D^ku±(x⁰, 0)|²e^2φδ(x0,0)dx⁰

+X

±

τ²[e^{τ φδ(·,0)}u±(·, 0)]²_1/2,Rn−1+X

±

[D(e^{τ φδ,±}u±)(·, 0)]²_1/2,Rn−1

≤C X

±

Z

Rⁿ±

|L(x, D)(u±)|²e^{2τ φδ,±(x0,xn)}dx⁰dx_n+ [e^{τ φδ(·,0)}h₁]²_1/2,Rn−1

+[D_x⁰(e^{τ φδ}h₀)(·, 0)]²_1/2,Rn−1 + τ³ Z

Rⁿ⁻¹

|h₀|²e^{2τ φδ(x0,0)}dx⁰+ τ Z

Rⁿ⁻¹

|h₁|²e^{2τ φδ(x0,0)}dx⁰

 . (2.14) where u = H+u++ H−u−, u± ∈ C^∞(Rⁿ) and supp u ⊂ B_δr⁰ ₀ × [−δr0, δr0], and φδ is given by (2.12).

Remark 2.2 Estimate (2.14) is a local Carleman estimate near x_n = 0. As men- tioned above, by flattening the interface, we can derive a local Carleman estimate near a C^1,1 interface from (2.14). Nonetheless, an estimate like (2.14) is sufficient for some applications such as the inverse problem of estimating the size of an inclu- sion by one pair of boundary measurement (see, for example, [FLVW]).

3 Carleman estimate for operators with constant coefficients

The purpose of this section is to derive (2.14) for L(x, D) with discontinuous piecewise constant coefficients. More precisely, we derive (2.14) for L₀(D), where L₀(D) is

obtained from L(x, D) by freezing the variable x at (x⁰₀, 0). Without loss of generality, we take (x⁰₀, 0) = (0, 0) = 0 and thus

L₀(D)u = L(0, D)u =X

±

H±div(A±(0)∇u±).

Since L₀ has piecewise constant coefficients, to prove (2.14), we will apply [BL, The- orem 1.6]. So the task here is to verify the strong pseudoconvexity and transmission conditions for operator L₀ with the weight function given in (2.11).

To streamline the presentation, we define Ω₁ := {x_n < 0}, Ω₂ := {x_n > 0}. On each side of the interface, we have complex second order elliptic operators. We denote

P_k= X

1≤j,`≤n

a^(k)_{`j</sub> D<sub>`}D_j, k = 1, 2,

where a⁽¹⁾_{`j</sub> = a<sup>−</sup><sub>`j} and a⁽²⁾_{`j</sub> = a<sup>+</sup><sub>`j}. Here we denote a^(k)_{`j</sub> = a<sup>(k)</sup><sub>`j} (0). Corresponding to (2.3)-(2.6), we have

a^(k)_{`j</sub> = a<sup>(k)</sup><sub>j`} , (3.1)

a^(k)_{`j</sub> = M<sub>`j}^(k)+ iγN_`j^(k), (3.2) λ₀|ξ|² ≤ M^(k)ξ · ξ ≤ Λ₀|ξ|², (3.3) λ₀|ξ|² ≤ N^(k)ξ · ξ ≤ Λ₀|ξ|². (3.4) Since some computations in the verification of the transmission conditions are useful in proving the strong pseudoconvexity condition, we will begin with the discussion of the transmission conditions at the interface {x_n= 0}.

3.1 Transmission conditions

We consider the natural transmission conditions that use the interface operators T_k¹ = (−1)^k, T_k² = (−1)^k X

1≤j≤n

a^(k)_njD_j

that correspond to the continuity of the solution and of the normal flux, respectively.

We now write the weight function

ψε(x) = ϕ(xn) − ε

2|x⁰|², (3.5)

where

ϕ(xn) =

(ϕ₁(x_n), x_n < 0 ϕ2(xn), xn ≥ 0, and

ϕ_k(x_n) = α_kx_n+ 1 2βx²_n

with α₁, α₂ > 0 (corresponding to α− and α₊ in (2.10), respectively) and β > 0.

Notice that ϕ is smooth in Ω₁, Ω₂ and is continuous across the interface. Then we have

∇ψ_ε(0) =

((0, · · · , 0, α₁), x_n< 0 (0, · · · , 0, α₂), x_n≥ 0.

Following the notations and the calculations in [BL, Section 1.7.1], we have for ω := (0, ξ⁰, ν, τ ) with ξ⁰ = (ξ₁, · · · , ξ_n−1) 6= 0, ν = e_n and λ ∈ C,

˜t¹_k,ψε(ω, λ) = (−1)^k and

˜t²_k,ψε(ω, λ) = (−1)^ka^(k)_nn((−1)^kλ + iτ ∂_xnψ_ε(0)) + (−1)^k X

1≤j≤n−1

a^(k)_nj(ξ_j+ iτ ∂_xjψ_ε(0))

= (−1)^ka^(k)_nn((−1)^kλ + iτ α_k) + (−1)^k X

1≤j≤n−1

a^(k)_njξ_j.

The principal symbols of P_k, k = 1, 2, can be written as p_k(ξ) = a^(k)_nn((ξ_n+ X

1≤j≤n−1

a^(k)_nj a^(k)nn

ξ_j)²+ b_k(ξ⁰)), (3.6)

where

b_k(ξ⁰) = (a^(k)_nn)⁻² X

1≤`,j≤n−1

(a^(k)_{`j</sub> a<sup>(k)</sup><sub>nn</sub> − a<sup>(k)</sup><sub>n`}a^(k)_nj)ξ_`ξ_j. (3.7) We also need to introduce the principal symbol of the conjugate operators

˜

p_k,ψε(ω, λ) = a^(k)_nn h

(−1)^kλ + iτ ∂_xnψ_ε(0) + X

1≤j≤n−1

a^(k)_nj a^(k)nn

(ξ_j+ iτ ∂_xjψ_ε(0))
2

+ b_k(ξ⁰+ iτ ∂_x⁰ψ_ε(0))i

= a^(k)_nnh

(−1)^kλ + iτ α_k+ X

1≤j≤n−1

a^(k)_nj a^(k)nn

ξ_j
2

+ b_k(ξ⁰)i .

(3.8)

Let us introduce A^(k), B^(k) ∈ R for k = 1, 2 such that b_k(ξ⁰) = (a^(k)_nn)⁻² X

1≤`,j≤n−1

(a^(k)_{`j</sub> a<sup>(k)</sup><sub>nn</sub> − a<sup>(k)</sup><sub>n`}a^(k)_nj)ξ_`ξ_j

= (A^(k)− iB^(k))²,

(3.9)

where A^(k)≥ 0. We also denote X

1≤j≤n−1

a^(k)_nj a^(k)nn

ξ_j = E^(k)+ iF^(k), (3.10)

where E^(k), F^(k)∈ R. Using (3.8), (3.9), and (3.10), we can write

˜

p_2,ψε = a⁽²⁾_nn[(λ + iτ α₂+ E⁽²⁾+ iF⁽²⁾)²+ (A⁽²⁾− iB⁽²⁾)²]

= a⁽²⁾_nn[(λ + iτ α₂+ E⁽²⁾+ iF⁽²⁾+ i(A⁽²⁾− iB⁽²⁾))

· (λ + iτ α₂+ E⁽²⁾+ iF⁽²⁾− i(A⁽²⁾− iB⁽²⁾))]

= a⁽²⁾_nn(λ − σ₁⁽²⁾)(λ − σ₂⁽²⁾), where

σ₁⁽²⁾ = −E⁽²⁾− B⁽²⁾− i(τ α₂+ F⁽²⁾+ A⁽²⁾), σ₂⁽²⁾ = −E⁽²⁾+ B⁽²⁾− i(τ α₂+ F⁽²⁾− A⁽²⁾).

On the other hand, we can write

˜

p_1,ψε = a⁽¹⁾_nn[(−λ + iτ α₁+ E⁽¹⁾+ iF⁽¹⁾)²+ (A⁽¹⁾− iB⁽¹⁾)²]

= a⁽¹⁾_nn[(λ − iτ α₁− E⁽¹⁾− iF⁽¹⁾+ i(A⁽¹⁾− iB⁽¹⁾))

· (λ − iτ α₁− E⁽¹⁾− iF⁽¹⁾− i(A⁽¹⁾− iB⁽¹⁾))]

= a⁽¹⁾_nn(λ − σ₁⁽¹⁾)(λ − σ₂⁽¹⁾), where

σ₁⁽¹⁾ = E⁽¹⁾+ B⁽¹⁾+ i(τ α₁+ F⁽¹⁾+ A⁽¹⁾), σ₂⁽¹⁾ = E⁽¹⁾− B⁽¹⁾+ i(τ α₁+ F⁽¹⁾− A⁽¹⁾).

Let us introduce the polynomial

K_k,ψε(ω, λ) := Y

Im σ^(k)_j ≥0

(λ − σ^(k)_j ).

Now we state the definition of transmission conditions given in [BL, Definition 1.4].

Definition 3.1 The pair {P_k, ψ_ε, T_k^j, k = 1, 2, j = 1, 2} satisfies the transmission condition at ω if for any polynomials q₁(λ), q₂(λ), there exist polynomials U₁(λ), U₂(λ) and constant c1, c2 such that

(q1(λ) = c1˜t¹_1,ψε(ω, λ) + c2˜t²_1,ψε(ω, λ) + U1(λ)K1,ψε(ω, λ), q2(λ) = c1˜t¹_2,ψε(ω, λ) + c2˜t²_2,ψε(ω, λ) + U2(λ)K2,ψε(ω, λ).

In order to check the transmission conditions, we need to study the polynomial K_k,ψε(ω, λ). For this reason, we need to determine the signs of the imaginary parts of the roots σ^(k)_j defined above. Note that we can write

bk(ξ⁰) = 1 a^(k)nn

X

1≤`,j≤n−1

a^(k)<sub>`j</sub> ξ`ξj − (E^(k)+ iF^(k))². (3.11)

Since b_kplays an essential role, we begin by working some calculations on the matrix

1 a^(k)nn

A^(k), where A^(k) is the matrix (a^(k)_`j ). Let a^(k)nn = |a^(k)nn|e^iθ. Choosing ξ = e_n, we have that

a^(k)_nn = X

1≤`,j≤n

a^(k)_{`j</sub> ξ<sub>`}ξ_j = X

1≤`,j≤n

M_{`j</sub><sup>(k)</sup>ξ<sub>`}ξ_j+ iγ X

1≤`,j≤n

N_{`j</sub><sup>(k)</sup>ξ<sub>`}ξ_j.

Hence, from (3.3), (3.4), we have that

λ₀ ≤ Re (a^(k)_nn) ≤ Λ₀ and λ₀ ≤ Im (a^(k)nn) γ ≤ Λ₀ and so that θ ∈ [0, π/2). Let us evaluate

(a^(k)_nn)⁻¹A^(k) = |a^(k)_nn|⁻¹(M^(k)+ iγN^(k))(cos θ − i sin θ)

= |a^(k)_nn|⁻¹[cos θM^(k)+ γ sin θN^(k)+ i(− sin θM^(k)+ γ cos θN^(k))].

(3.12) Using (3.3), (3.4) again, we see that for ξ ∈ Rⁿ

Re ((a^(k)_nn)⁻¹A^(k)ξ · ξ) = |a^(k)_nn|⁻¹[cos θM^(k)ξ · ξ + γ sin θN^(k)ξ · ξ]

≥ |a^(k)_nn|⁻¹λ₀(cos θ + γ sin θ)|ξ|². (3.13) In fact, since cos θ = Mnn^(k)|a^(k)nn|⁻¹ and sin θ = γNnn^(k)|a^(k)nn|⁻¹, while |a^(k)nn|² = (Mnn^(k))²+ γ²(Nnn^(k))², we have

|a^(k)_nn|⁻¹(cos θ + γ sin θ) = Mnn^(k)+ γ²Nnn^(k)

(Mnn^(k))²+ γ²(Nnn^(k))² ≥ λ0(1 + γ²) Λ²₀(1 + γ²) = λ0

Λ²₀. (3.14) Combining (3.13) and (3.14) implies

Re ((a^(k)_nn)⁻¹A^(k)ξ · ξ) ≥ λ²₀

Λ²₀|ξ|² := ˜λ₁|ξ|². (3.15) Now let us write

λ˜₁|ξ|² ≤ Re ((a^(k)_nn)⁻¹A^(k)ξ · ξ)

=Re [ X

1≤`,j≤n−1

a^(k)_`j a^(k)nn

ξ`ξj+ 2 X

1≤j≤n−1

a^(k)_nj a^(k)nn

ξnξj+ ξ_n²]

=ξ_n²+ 2b^(k)₀ (ξ⁰)ξ_n+ b^(k)₁ (ξ⁰),

(3.16)

where

b^(k)₀ (ξ⁰) = Re ( X

1≤j≤n−1

a^(k)_nj a^(k)nn

ξ_j) = Re (E^(k)+ iF^(k)) = E^(k)

and

b^(k)₁ (ξ⁰) = Re ( X

1≤`,j≤n−1

a^(k)_`j a^(k)nn

ξ_`ξ_j).

Substituting ˜ξ_n = ξ_n= −b^(k)₀ (ξ⁰) into (3.16) gives

λ˜₁(|ξ⁰|²+ | ˜ξ_n|²) ≤ ˜ξ_n²− 2b^(k)₀ (ξ⁰) ˜ξ_n+ b^(k)₁ (ξ⁰) = −(b^(k)₀ (ξ⁰)²+ b^(k)₁ (ξ⁰), which implies

˜λ₁|ξ⁰|² ≤ Re ( X

1≤`,j≤n−1

a^(k)_`j a^(k)nn

ξ_`ξ_j) − E_k². (3.17) Putting (3.11) and (3.17) together gives

Re (b_k(x₀, ξ⁰)) = Re ( X

1≤`,j≤n−1

a^(k)_`j a^(k)nn

ξ_`ξ_j) − (E^(k))²+ (F^(k))²

≥ ˜λ1|ξ⁰|²+ (F^(k))² > 0.

(3.18)

The following lemma guarantees the positivity of A^(k). Lemma 3.1 Assume that (3.3) and (3.4) hold. Then

A^(k) ≥

qλ˜₁|ξ⁰|² + |F^(k)|² > |F^(k)|. (3.19) ProofFrom (3.9), it is easy to see that

A^(k) = Rep bk =

s a +√

a² + b²

2 ,

where a = Re b_k and b = Im b_k. We have from (3.18) that a > 0 and thus A^(k) ≥√

a ≥

q˜λ₁|ξ⁰|²+ (F^(k))² > |F^(k)|.

2

Lemma 3.1 implies

Im σ⁽²⁾₁ = −(τ α₂+ F⁽²⁾+ A⁽²⁾) = −τ α₂− F⁽²⁾− A⁽²⁾

≤ −τ α₂− |F⁽²⁾| − F⁽²⁾ ≤ −τ α₂ < 0 (3.20) and

Im σ₁⁽¹⁾ = τ α₁+ F⁽¹⁾+ A⁽¹⁾ > τ α₁+ F⁽¹⁾+ |F⁽¹⁾| ≥ τ α₁ > 0. (3.21)

We are now ready to check the transmission condition defined in Definition 3.1.

Being able to satisfy this condition depends on the degree of K_1,ψε and K_2,ψε, that is, on the number of roots with negative imaginary parts.

Case 1. ˜p_2,ψε has two roots in {Im z < 0}, i.e., −τ α₂ − F⁽²⁾ + A⁽²⁾ < 0 in view of (3.20). In this case, we have that

K_2,ψε = 1, while K_1,ψε has degree 1 or 2 (note (3.21)).

Since ˜t¹_2,ψ

ε(ω, λ) = 1 and

˜t²_2,ψε(ω, λ) = a⁽²⁾_nn(λ + iτ α₂+ X

1≤j≤n−1

a⁽²⁾_nj a⁽²⁾nn

ξ_j),

for any q₂(λ), we simply choose

U₂(λ) = q₂(λ) − c₁˜t¹_2,ψε − c₂t˜²_2,ψε. On the other hand, we have ˜t¹_1,ψε(ω, λ) = −1 and

˜t²_1,ψ

ε(ω, λ) = a⁽¹⁾_nn(λ − iτ α₁− X

1≤j≤n−1

a⁽¹⁾_nj a⁽¹⁾nn

ξ_j).

Then for any polynomial q1(λ), we choose U1(λ) to be the quotient of the division between q₁ and K_1,ψε. The remainder term is equal to c₁˜t_1,ψε + c₂˜t_2,ψε with suitable c₁, c₂.

Case 2. Assume that Im σ₂⁽²⁾ ≥ 0 and Im σ₂⁽¹⁾ ≥ 0, i.e.,

−τ α₂− F⁽²⁾+ A⁽²⁾ ≥ 0, τ α₁+ F⁽¹⁾− A⁽¹⁾ ≥ 0.

Then K_1,ψε has degree 2 and K_2,ψε has degree 1. In order to avoid this case, we need to be sure that if −τ α₂− F⁽²⁾+ A⁽²⁾ ≥ 0, then τ α₁+ F⁽¹⁾− A⁽¹⁾ < 0, that is,

τ α₂+ F⁽²⁾− A⁽²⁾ ≤ 0 ⇒ τ α₁+ F⁽¹⁾− A⁽¹⁾ < 0.

This can be achieved by assuming that α2

α₁ > A⁽²⁾− F⁽²⁾

A⁽¹⁾− F⁽¹⁾, ∀ ξ⁰ 6= 0. (3.22) Recall that A^(k)− F^(k)> 0, k = 1, 2. We remark that all A^(k) and F^(k) are homoge- neous of degree 1 in ξ⁰. Hence (3.22) holds provided

α₂ α1

= max

|ξ⁰|=1

 A⁽²⁾− F⁽²⁾ A⁽¹⁾− F⁽¹⁾



+ 1. (3.23)

Hence, if we assume (3.23), then the transmission condition is satisfied.

Case 3. Each symbol has exactly one root in {Im z < 0}, i.e., τ α₁+ F⁽¹⁾− A⁽¹⁾ < 0, −τ α₂− F⁽²⁾+ A⁽²⁾ > 0.

In this case, we have

K_1,ψε = (λ − σ₁⁽¹⁾), K_2,ψε = (λ − σ₂⁽²⁾).

Given polynomials q₁(λ), q₂(λ), there exist U₁(λ), U₂(λ) such that q₁(λ) = U₁(λ)K_1,ψε + ˜q₁,

q₂(λ) = U₂(λ)K_2,ψε + ˜q₂,

where ˜q₁, ˜q₂ are constants in λ. The transmission condition is satisfied if there exists constants µ₁, µ₂, c₁, c₂ so that

(q˜₁ = µ₁K_1,ψε + c₁˜t¹_1,ψε + c₂t˜²_1,ψε,

˜

q₂ = µ₂K_2,ψε + c₁˜t¹_2,ψε + c₂t˜²_2,ψε, namely,

(q˜₁ = µ₁(λ − σ₁⁽¹⁾) − c₁+ c₂a⁽¹⁾_nn(λ − iτ α₁− E⁽¹⁾− iF⁽¹⁾)

˜

q₂ = µ₂(λ − σ₂⁽²⁾) + c₁+ c₂a⁽²⁾_nn(λ + iτ α₂+ E⁽²⁾+ iF⁽²⁾). (3.24) System (3.24) is equivalent to















µ₁+ c₂a⁽¹⁾_nn = 0 µ₂+ c₂a⁽²⁾_nn = 0

µ₁σ₁⁽¹⁾+ c₁+ c₂a⁽¹⁾_nn(iτ α₁ + E⁽¹⁾+ iF⁽¹⁾) = −˜q₁

− µ₂σ₂⁽²⁾+ c₁+ c₂a⁽²⁾_nn(iτ α₂+ E⁽²⁾+ iF⁽²⁾) = ˜q₂.

(3.25)

System (3.25) has a unique solution if and only if the matrix

T =











1 0 0 a⁽¹⁾nn

0 1 0 a⁽²⁾nn

σ₁⁽¹⁾ 0 1 ζ1

0 −σ₂⁽²⁾ 1 ζ₂











with ζ₁ = a⁽¹⁾nn(iτ α₁+ E⁽¹⁾+ iF⁽¹⁾), ζ₂ = a⁽²⁾nn(iτ α₂+ E⁽²⁾+ iF⁽²⁾), is nonsingular. We

compute

detT =det







1 0 a⁽¹⁾nn

0 1 a⁽²⁾nn

0 −σ⁽²⁾₂ ζ₂





− det







1 0 a⁽¹⁾nn

0 1 a⁽²⁾nn

σ₁⁽¹⁾ 0 ζ₁







=ζ₂+ σ₂⁽²⁾a⁽²⁾_nn − ζ₁+ σ₁⁽¹⁾a⁽¹⁾_nn

=a⁽²⁾_nn(iτ α₂+ E⁽²⁾+ iF⁽²⁾− E⁽²⁾+ B⁽²⁾− iτ α₂ − iF⁽²⁾+ iA⁽²⁾) + a⁽¹⁾_nn(−iτ α₁− E⁽¹⁾− iF⁽¹⁾+ E⁽¹⁾+ B⁽¹⁾+ iτ α₁+ iF⁽¹⁾+ iA⁽¹⁾)

=a⁽²⁾_nn(B⁽²⁾+ iA⁽²⁾) + a⁽¹⁾_nn(B⁽¹⁾+ iA⁽¹⁾).

Therefore, if

a⁽²⁾_nn(B⁽²⁾+ iA⁽²⁾) + a⁽¹⁾_nn(B⁽¹⁾+ iA⁽¹⁾) 6= 0, (3.26) then the transmission condition holds.

We now verify (3.26). In the real case where a⁽²⁾nn, a⁽¹⁾nn are positive real numbers, it is easy to see that

a⁽²⁾_nnA⁽²⁾+ a⁽¹⁾_nnA⁽¹⁾ > 0 and thus (3.26) holds.

For the complex case, we want to show that there exists γ₀ > 0 such that if γ < γ₀, then (3.26) is satisfied. Let uk= A^(k)+ iB^(k) and vk = iuk = −B^(k)+ iA^(k). We will consider u_k and v_k as vectors in R², i.e., u_k = (A^(k), B^(k)), v_k = u^⊥_k = (−B^(k), A^(k)).

Let a^(k)nn = η^(k)+ iγδ^(k) for η^(k), δ^(k) ∈ R. By the ellipticity conditions (3.3), (3.4), we have

λ₀ ≤ η^(k) ≤ Λ₀, λ₀ ≤ δ^(k) ≤ Λ₀. Notice that detT = 0 if and only if

(η⁽²⁾+ iγδ⁽²⁾)(B⁽²⁾+ iA⁽²⁾) + (η⁽¹⁾+ iγδ⁽¹⁾)(B⁽¹⁾+ iA⁽¹⁾) = 0, i.e.,

(η⁽²⁾B⁽²⁾− γδ⁽²⁾A⁽²⁾+ η⁽¹⁾B⁽¹⁾− γδ⁽¹⁾A⁽¹⁾)

+ i(η⁽²⁾A⁽²⁾+ γδ⁽²⁾B⁽²⁾+ η⁽¹⁾A⁽¹⁾+ γδ⁽¹⁾B⁽¹⁾) = 0, which is equivalent to

η⁽²⁾A⁽²⁾ B⁽²⁾



+ η⁽¹⁾A⁽¹⁾ B⁽¹⁾



= γδ⁽²⁾−B⁽²⁾ A⁽²⁾



+ γδ⁽¹⁾−B⁽¹⁾ A⁽¹⁾



(3.27) or simply

η⁽²⁾u₂+ η⁽¹⁾u₁ = γδ⁽²⁾v₂+ γδ⁽¹⁾v₁. (3.28) Recall that A^(k) ≥ |F^(k)| > 0. Therefore, in the real case γδ^(k) = 0, then (3.27) will never be satisfied. If B⁽¹⁾ and B⁽²⁾ have the same sign, that is, either B^(k) ≥ 0 or B^(k) ≤ 0 for k = 1, 2, (3.28) can not hold. To see this, let us consider B^(k) ≥ 0,

k = 1, 2. Then u₁, u₂are in the first quadrant of the plane and v₁, v₂ are in the second quadrant of the plane. The sets

C_u = {η⁽²⁾u₂+ η⁽¹⁾u₁ : η^(k) ≥ 0}, C_u = {γδ⁽²⁾v₂+ γδ⁽¹⁾v₁ : γδ^(k)≥ 0}

can only intersect at the original. Same thing happens if B^(k) ≤ 0 for k = 1, 2.

The only case we need to investigate is when B⁽¹⁾ and B⁽²⁾ have different signs.

For example, let us assume

B⁽¹⁾ > 0, B⁽²⁾ < 0.

Even in this case, the intersection between Cu and Cv is non-trivial if the angle φ between u₁ and u₂ is less than π/2. Note that u₁ is the first quadrant and u₂ is in the fourth quadrant. So the angle between u₁ and u₂ is less than π. We would like to show that (3.28) cannot hold for φ ∈ [π/2, π) if we choose γ0 small enough.

Note that in this case cos φ ≤ 0. To do so, we estimate kη⁽²⁾u₂+η⁽¹⁾u₁k from below and kδ⁽²⁾v₂ + δ⁽¹⁾v₁k from above. We now discuss the estimate of kδ⁽²⁾v₂ + δ⁽¹⁾v₁k from above. Compute

kδ⁽²⁾v₂+ δ⁽¹⁾v₁k² = (δ⁽²⁾)²[(A⁽²⁾)²+ (B⁽²⁾)²] + (δ⁽¹⁾)²[(A⁽¹⁾)²+ (B⁽¹⁾)²] + 2δ⁽¹⁾δ⁽²⁾(−B⁽²⁾, A⁽²⁾) · (−B⁽¹⁾, A⁽¹⁾)

= (δ⁽²⁾)²[(A⁽²⁾)²+ (B⁽²⁾)²] + (δ⁽¹⁾)²[(A⁽¹⁾)²+ (B⁽¹⁾)²] + 2δ⁽¹⁾δ⁽²⁾[(A⁽²⁾)²+ (B⁽²⁾)²]^1/2[(A⁽¹⁾)² + (B⁽¹⁾)²]^1/2cos φ

≤ (δ⁽²⁾)²[(A⁽²⁾)²+ (B⁽²⁾)²] + (δ⁽¹⁾)²[(A⁽¹⁾)²+ (B⁽¹⁾)²].

(3.29)

In view of (3.9) and (3.11), we have (A^(k))²+ (B^(k))² = |bk| = | X

1≤`,j≤n−1

a^(k)_`j a^(k)nn

ξ`ξj− (E^(k)+ iF^(k))²|

≤ | X

1≤`,j≤n−1

a^(k)_`j a^(k)nn

ξ_`ξ_j| + |(E^(k)+ iF^(k))²|.

(3.30)

By (3.3), (3.4), and (3.12), we can obtain

| X

1≤`,j≤n−1

a^(k)_`j a^(k)nn

ξ_`ξ_j|² = | 1 a^(k)nn

A^(k)ξ · ξ|² (with ξ = (ξ⁰, 0))

= |a^(k)_nn|⁻²| cos θM^(k)ξ · ξ + γ sin θN^(k)ξ · ξ + i(− sin θM^(k)ξ · ξ + γ cos θN^(k)ξ · ξ)|²

= |a^(k)_nn|⁻²[(M^(k)ξ · ξ)²+ γ²(N^(k)ξ · ξ)²]

≤ Λ²(1 + γ²)|ξ|⁴

λ²₀(1 + γ²) = ˜λ⁻¹₁ |ξ|⁴, where we have used the estimate

λ₀(1 + γ²)^1/2≤ |a^(k)_nn| ≤ Λ₀(1 + γ²)^1/2 (3.31)

in deriving the inequality above. We thus obtain

| X

1≤`,j≤n−1

a^(k)_`j a^(k)nn

ξ_`ξ_j| ≤ ˜λ^−1/2₁ |ξ⁰|². (3.32)

Furthermore, we can estimate

|(E^(k)+ iF^(k))²| = |( X

1≤j≤n−1

a^(k)_nj a^(k)nn

ξ_j)²| = | X

1≤j≤n−1

a^(k)_nj a^(k)nn

ξ_j|²

≤ ( X

1≤j≤n−1

|a^(k)_nj a^(k)nn

|²)|ξ⁰|² ≤ (n − 1)Λ²₀(1 + γ²) λ²₀(1 + γ²) |ξ⁰|²

= (n − 1)˜λ⁻¹₁ |ξ⁰|².

(3.33)

Substituting (3.32), (3.33) into (3.30) gives

(A^(k))²+ (B^(k))² ≤ (˜λ^−1/2₁ + (n − 1)˜λ⁻¹₁ )|ξ⁰|² ≤ nΛ²₀

λ²₀|ξ⁰|². (3.34) It follows from (3.29) and (3.34) that

kδ⁽²⁾v₂+ δ⁽¹⁾v₁k² ≤ 2Λ²₀nΛ²₀

λ²₀|ξ⁰|². (3.35) Next, we want to estimate kη⁽²⁾u₂+ η⁽¹⁾u₁k from below. As above, we have kη⁽²⁾u₂+ η⁽¹⁾u₁k² = (η⁽²⁾)²[(A⁽²⁾)²+ (B⁽²⁾)²] + (η⁽¹⁾)²[(A⁽¹⁾)²+ (B⁽¹⁾)²]

+ 2η⁽¹⁾η⁽²⁾[(A⁽²⁾)²+ (B⁽²⁾)²]^1/2[(A⁽¹⁾)²+ (B⁽¹⁾)²]^1/2cos φ. (3.36) Recall that B₁ > 0, B₂ < 0. Thus,

cos φ = A⁽¹⁾A⁽²⁾+ B⁽¹⁾B⁽²⁾

[(A⁽²⁾)²+ (B⁽²⁾)²]^1/2[(A⁽¹⁾)²+ (B⁽¹⁾)²]^1/2

= A⁽¹⁾A⁽²⁾− |B⁽¹⁾||B⁽²⁾|

[(A⁽²⁾)²+ (B⁽²⁾)²]^1/2[(A⁽¹⁾)²+ (B⁽¹⁾)²]^1/2

= 1 −^|B_A(1)^(1)|

|B⁽²⁾| A⁽²⁾

(1 + (^B_A⁽²⁾(2))²)^1/2(1 + (^B_A(1)⁽¹⁾)²)^1/2. Notice that by (3.19) and (3.35)

0 ≤ |B^(k)| A^(k) ≤

p(A^(k))²+ (B^(k))²

A^(k) ≤

√n^Λ_λ⁰

0|ξ⁰| p˜λ₁|ξ⁰| =

√nΛ²₀

λ²₀ := ˜λ2 ≥ 1.