## 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 R^{n} 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 C^{1} 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_{R}^{n}

±

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

Let u±∈ C^{∞}(R^{n}). 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}(x)}^{n}_{`,j=1} = {a^{±}_{`j}(x^{0}, x_{n})}^{n}_{`,j=1}, x^{0} ∈ R^{n−1}, x_{n} ∈ R (2.2)
is a Lipschitz symmetric matrix-valued function. Assume that

a^{±}_{`j}(x) = a^{±}_{j`}(x), ∀ `, j = 1, · · · , n, (2.3)

and furthermore

a^{±}_{`j}(x) = M_{`j}^{±}(x) + iγN_{`j}^{±}(x), (2.4)
where (M_{`j}^{±}) and (N_{`j}^{±}) are real-valued matrices and γ > 0. We further assume that
there exist λ_{0}, Λ_{0} > 0 such that for all ξ ∈ R^{n} and x ∈ R^{n} we have

λ_{0}|ξ|^{2} ≤ M^{±}(x)ξ · ξ ≤ Λ_{0}|ξ|^{2} (2.5)
and

λ_{0}|ξ|^{2} ≤ N^{±}(x)ξ · ξ ≤ Λ_{0}|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)| ≤ M_{0}|x − y|. (2.7)
To treat the transmission conditions, we write

h_{0}(x^{0}) := u_{+}(x^{0}, 0) − u−(x^{0}, 0), ∀ x^{0} ∈ R^{n−1}, (2.8)
h_{1}(x^{0}) := A_{+}(x^{0}, 0)∇u_{+}(x^{0}, 0) · ν − A−(x^{0}, 0)∇u−(x^{0}, 0) · ν, ∀ x^{0} ∈ R^{n−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}+ βx^{2}_{n}/2, x_{n} ≥ 0,

ϕ−(x_{n}) := α−x_{n}+ βx^{2}_{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|x^{0}|^{2}, (2.11)

and let

φ_{δ}(x) := ψ_{δ}(δ^{−1}x), δ > 0. (2.12)
For a function h ∈ L^{2}(R^{n}), we define

ˆh(ξ^{0}, x_{n}) =
Z

R^{n−1}

h(x^{0}, x_{n})e^{−ix}^{0}^{·ξ}dx^{0}, ξ^{0} ∈ R^{n−1}.

As usual we denote by H^{1/2}(R^{n−1}) the space of the functions f ∈ L^{2}(R^{n−1}) satisfying
Z

R^{n−1}

|ξ^{0}|| ˆf (ξ^{0})|^{2}dξ^{0} < ∞,
with the norm

kf k^{2}_{H}1/2(R^{n−1})=
Z

R^{n−1}

(1 + |ξ^{0}|^{2})^{1/2}| ˆf (ξ^{0})|^{2}dξ^{0}. (2.13)

Moreover we define

[f ]_{1/2,R}^{n−1} =

Z

R^{n−1}

Z

R^{n−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

R^{n−1}

|ξ^{0}|| ˆf (ξ^{0})|^{2}dξ^{0} ≤ [f ]^{2}_{1/2,R}n−1 ≤ C
Z

R^{n−1}

|ξ^{0}|| ˆf (ξ^{0})|^{2}dξ^{0},

so that the norm (2.13) is equivalent to the norm kf k_{L}^{2}_{(R}^{n−1}_{)}+ [f ]_{1/2,R}^{n−1}. We use
the letters C, C_{0}, C_{1}, · · · 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_{r}^{0}(x^{0}) the (n − 1)-ball centered at x^{0} ∈ R^{n−1} with radius r > 0.

Whenever x^{0} = 0 we denote B_{r}^{0} = B_{r}^{0}(0). Likewise, we denote B_{r}(x) be the n-ball
centered at x ∈ R^{n} 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, δ_{0}, r_{0}, γ_{0}
and C depending on λ_{0}, Λ_{0}, M_{0} such that if γ < γ_{0}, δ ≤ δ_{0} and τ ≥ C, then

X

± 2

X

k=0

τ^{3−2k}
Z

R^{n}±

|D^{k}u±|^{2}e^{2τ φ}^{δ,±}^{(x}^{0}^{,x}^{n}^{)}dx^{0}dx_{n}+X

± 1

X

k=0

τ^{3−2k}
Z

R^{n−1}

|D^{k}u±(x^{0}, 0)|^{2}e^{2φ}^{δ}^{(x}^{0}^{,0)}dx^{0}

+X

±

τ^{2}[e^{τ φ}^{δ}^{(·,0)}u±(·, 0)]^{2}_{1/2,R}n−1+X

±

[D(e^{τ φ}^{δ,±}u±)(·, 0)]^{2}_{1/2,R}n−1

≤C X

±

Z

R^{n}±

|L(x, D)(u±)|^{2}e^{2τ φ}^{δ,±}^{(x}^{0}^{,x}^{n}^{)}dx^{0}dx_{n}+ [e^{τ φ}^{δ}^{(·,0)}h_{1}]^{2}_{1/2,R}n−1

+[D_{x}^{0}(e^{τ φ}^{δ}h_{0})(·, 0)]^{2}_{1/2,R}n−1 + τ^{3}
Z

R^{n−1}

|h_{0}|^{2}e^{2τ φ}^{δ}^{(x}^{0}^{,0)}dx^{0}+ τ
Z

R^{n−1}

|h_{1}|^{2}e^{2τ φ}^{δ}^{(x}^{0}^{,0)}dx^{0}

.
(2.14)
where u = H+u++ H−u−, u± ∈ C^{∞}(R^{n}) and supp u ⊂ B_{δr}^{0} _{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 L_{0}(D), where L_{0}(D) is

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

L_{0}(D)u = L(0, D)u =X

±

H±div(A±(0)∇u±).

Since L_{0} 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 L_{0} with the weight function given in (2.11).

To streamline the presentation, we define Ω_{1} := {x_{n} < 0}, Ω_{2} := {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} D_{`}D_{j}, k = 1, 2,

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

a^{(k)}_{`j} = a^{(k)}_{j`} , (3.1)

a^{(k)}_{`j} = M_{`j}^{(k)}+ iγN_{`j}^{(k)}, (3.2)
λ_{0}|ξ|^{2} ≤ M^{(k)}ξ · ξ ≤ Λ_{0}|ξ|^{2}, (3.3)
λ_{0}|ξ|^{2} ≤ N^{(k)}ξ · ξ ≤ Λ_{0}|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_{k}^{1} = (−1)^{k}, T_{k}^{2} = (−1)^{k} X

1≤j≤n

a^{(k)}_{nj}D_{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|x^{0}|^{2}, (3.5)

where

ϕ(xn) =

(ϕ_{1}(x_{n}), x_{n} < 0
ϕ2(xn), xn ≥ 0,
and

ϕ_{k}(x_{n}) = α_{k}x_{n}+ 1
2βx^{2}_{n}

with α_{1}, α_{2} > 0 (corresponding to α− and α_{+} in (2.10), respectively) and β > 0.

Notice that ϕ is smooth in Ω_{1}, Ω_{2} and is continuous across the interface. Then we
have

∇ψ_{ε}(0) =

((0, · · · , 0, α_{1}), x_{n}< 0
(0, · · · , 0, α_{2}), x_{n}≥ 0.

Following the notations and the calculations in [BL, Section 1.7.1], we have for
ω := (0, ξ^{0}, ν, τ ) with ξ^{0} = (ξ_{1}, · · · , ξ_{n−1}) 6= 0, ν = e_{n} and λ ∈ C,

˜t^{1}_{k,ψ}_{ε}(ω, λ) = (−1)^{k}
and

˜t^{2}_{k,ψ}_{ε}(ω, λ) = (−1)^{k}a^{(k)}_{nn}((−1)^{k}λ + iτ ∂_{x}_{n}ψ_{ε}(0)) + (−1)^{k} X

1≤j≤n−1

a^{(k)}_{nj}(ξ_{j}+ iτ ∂_{x}_{j}ψ_{ε}(0))

= (−1)^{k}a^{(k)}_{nn}((−1)^{k}λ + iτ α_{k}) + (−1)^{k} X

1≤j≤n−1

a^{(k)}_{nj}ξ_{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}(ξ^{0})), (3.6)

where

b_{k}(ξ^{0}) = (a^{(k)}_{nn})^{−2} X

1≤`,j≤n−1

(a^{(k)}_{`j} a^{(k)}_{nn} − a^{(k)}_{n`}a^{(k)}_{nj})ξ_{`}ξ_{j}. (3.7)
We also need to introduce the principal symbol of the conjugate operators

˜

p_{k,ψ}_{ε}(ω, λ) = a^{(k)}_{nn}
h

(−1)^{k}λ + iτ ∂_{x}_{n}ψ_{ε}(0) + X

1≤j≤n−1

a^{(k)}_{nj}
a^{(k)}nn

(ξ_{j}+ iτ ∂_{x}_{j}ψ_{ε}(0))2

+ b_{k}(ξ^{0}+ iτ ∂_{x}^{0}ψ_{ε}(0))i

= a^{(k)}_{nn}h

(−1)^{k}λ + iτ α_{k}+ X

1≤j≤n−1

a^{(k)}_{nj}
a^{(k)}nn

ξ_{j}2

+ b_{k}(ξ^{0})i
.

(3.8)

Let us introduce A^{(k)}, B^{(k)} ∈ R for k = 1, 2 such that
b_{k}(ξ^{0}) = (a^{(k)}_{nn})^{−2} X

1≤`,j≤n−1

(a^{(k)}_{`j} a^{(k)}_{nn} − a^{(k)}_{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,ψ}_{ε} = a^{(2)}_{nn}[(λ + iτ α_{2}+ E^{(2)}+ iF^{(2)})^{2}+ (A^{(2)}− iB^{(2)})^{2}]

= a^{(2)}_{nn}[(λ + iτ α_{2}+ E^{(2)}+ iF^{(2)}+ i(A^{(2)}− iB^{(2)}))

· (λ + iτ α_{2}+ E^{(2)}+ iF^{(2)}− i(A^{(2)}− iB^{(2)}))]

= a^{(2)}_{nn}(λ − σ_{1}^{(2)})(λ − σ_{2}^{(2)}),
where

σ_{1}^{(2)} = −E^{(2)}− B^{(2)}− i(τ α_{2}+ F^{(2)}+ A^{(2)}),
σ_{2}^{(2)} = −E^{(2)}+ B^{(2)}− i(τ α_{2}+ F^{(2)}− A^{(2)}).

On the other hand, we can write

˜

p_{1,ψ}_{ε} = a^{(1)}_{nn}[(−λ + iτ α_{1}+ E^{(1)}+ iF^{(1)})^{2}+ (A^{(1)}− iB^{(1)})^{2}]

= a^{(1)}_{nn}[(λ − iτ α_{1}− E^{(1)}− iF^{(1)}+ i(A^{(1)}− iB^{(1)}))

· (λ − iτ α_{1}− E^{(1)}− iF^{(1)}− i(A^{(1)}− iB^{(1)}))]

= a^{(1)}_{nn}(λ − σ_{1}^{(1)})(λ − σ_{2}^{(1)}),
where

σ_{1}^{(1)} = E^{(1)}+ B^{(1)}+ i(τ α_{1}+ F^{(1)}+ A^{(1)}),
σ_{2}^{(1)} = E^{(1)}− B^{(1)}+ i(τ α_{1}+ F^{(1)}− A^{(1)}).

Let us introduce the polynomial

K_{k,ψ}_{ε}(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 q_{1}(λ), q_{2}(λ), there exist polynomials U_{1}(λ), U_{2}(λ)
and constant c1, c2 such that

(q1(λ) = c1˜t^{1}_{1,ψ}_{ε}(ω, λ) + c2˜t^{2}_{1,ψ}_{ε}(ω, λ) + U1(λ)K1,ψε(ω, λ),
q2(λ) = c1˜t^{1}_{2,ψ}_{ε}(ω, λ) + c2˜t^{2}_{2,ψ}_{ε}(ω, λ) + U2(λ)K2,ψε(ω, λ).

In order to check the transmission conditions, we need to study the polynomial
K_{k,ψ}_{ε}(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(ξ^{0}) = 1
a^{(k)}nn

X

1≤`,j≤n−1

a^{(k)}_{`j} ξ`ξj − (E^{(k)}+ iF^{(k)})^{2}. (3.11)

Since b_{k}plays 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} ξ_{`}ξ_{j} = X

1≤`,j≤n

M_{`j}^{(k)}ξ_{`}ξ_{j}+ iγ X

1≤`,j≤n

N_{`j}^{(k)}ξ_{`}ξ_{j}.

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

λ_{0} ≤ Re (a^{(k)}_{nn}) ≤ Λ_{0} and λ_{0} ≤ Im (a^{(k)}nn)
γ ≤ Λ_{0}
and so that θ ∈ [0, π/2). Let us evaluate

(a^{(k)}_{nn})^{−1}A^{(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 ∈ R^{n}

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

≥ |a^{(k)}_{nn}|^{−1}λ_{0}(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}+
γ^{2}(Nnn^{(k)})^{2}, we have

|a^{(k)}_{nn}|^{−1}(cos θ + γ sin θ) = Mnn^{(k)}+ γ^{2}Nnn^{(k)}

(Mnn^{(k)})^{2}+ γ^{2}(Nnn^{(k)})^{2} ≥ λ0(1 + γ^{2})
Λ^{2}_{0}(1 + γ^{2}) = λ0

Λ^{2}_{0}. (3.14)
Combining (3.13) and (3.14) implies

Re ((a^{(k)}_{nn})^{−1}A^{(k)}ξ · ξ) ≥ λ^{2}_{0}

Λ^{2}_{0}|ξ|^{2} := ˜λ_{1}|ξ|^{2}. (3.15)
Now let us write

λ˜_{1}|ξ|^{2} ≤ Re ((a^{(k)}_{nn})^{−1}A^{(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_{n}^{2}]

=ξ_{n}^{2}+ 2b^{(k)}_{0} (ξ^{0})ξ_{n}+ b^{(k)}_{1} (ξ^{0}),

(3.16)

where

b^{(k)}_{0} (ξ^{0}) = 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} (ξ^{0}) = Re ( X

1≤`,j≤n−1

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

ξ_{`}ξ_{j}).

Substituting ˜ξ_{n} = ξ_{n}= −b^{(k)}_{0} (ξ^{0}) into (3.16) gives

λ˜_{1}(|ξ^{0}|^{2}+ | ˜ξ_{n}|^{2}) ≤ ˜ξ_{n}^{2}− 2b^{(k)}_{0} (ξ^{0}) ˜ξ_{n}+ b^{(k)}_{1} (ξ^{0}) = −(b^{(k)}_{0} (ξ^{0})^{2}+ b^{(k)}_{1} (ξ^{0}),
which implies

˜λ_{1}|ξ^{0}|^{2} ≤ Re ( X

1≤`,j≤n−1

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

ξ_{`}ξ_{j}) − E_{k}^{2}. (3.17)
Putting (3.11) and (3.17) together gives

Re (b_{k}(x_{0}, ξ^{0})) = Re ( X

1≤`,j≤n−1

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

ξ_{`}ξ_{j}) − (E^{(k)})^{2}+ (F^{(k)})^{2}

≥ ˜λ1|ξ^{0}|^{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}|ξ^{0}|^{2} + |F^{(k)}|^{2} > |F^{(k)}|. (3.19)
ProofFrom (3.9), it is easy to see that

A^{(k)} = Rep
bk =

s a +√

a^{2} + b^{2}

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˜λ_{1}|ξ^{0}|^{2}+ (F^{(k)})^{2} > |F^{(k)}|.

### 2

Lemma 3.1 implies

Im σ^{(2)}_{1} = −(τ α_{2}+ F^{(2)}+ A^{(2)}) = −τ α_{2}− F^{(2)}− A^{(2)}

≤ −τ α_{2}− |F^{(2)}| − F^{(2)} ≤ −τ α_{2} < 0 (3.20)
and

Im σ_{1}^{(1)} = τ α_{1}+ F^{(1)}+ A^{(1)} > τ α_{1}+ F^{(1)}+ |F^{(1)}| ≥ τ α_{1} > 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,ψ}_{ε} and K_{2,ψ}_{ε}, that
is, on the number of roots with negative imaginary parts.

Case 1. ˜p_{2,ψ}_{ε} has two roots in {Im z < 0}, i.e., −τ α_{2} − F^{(2)} + A^{(2)} < 0 in view of
(3.20). In this case, we have that

K_{2,ψ}_{ε} = 1, while K_{1,ψ}_{ε} has degree 1 or 2 (note (3.21)).

Since ˜t^{1}_{2,ψ}

ε(ω, λ) = 1 and

˜t^{2}_{2,ψ}_{ε}(ω, λ) = a^{(2)}_{nn}(λ + iτ α_{2}+ X

1≤j≤n−1

a^{(2)}_{nj}
a^{(2)}nn

ξ_{j}),

for any q_{2}(λ), we simply choose

U_{2}(λ) = q_{2}(λ) − c_{1}˜t^{1}_{2,ψ}_{ε} − c_{2}t˜^{2}_{2,ψ}_{ε}.
On the other hand, we have ˜t^{1}_{1,ψ}_{ε}(ω, λ) = −1 and

˜t^{2}_{1,ψ}

ε(ω, λ) = a^{(1)}_{nn}(λ − iτ α_{1}− 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 q_{1} and K_{1,ψ}_{ε}. The remainder term is equal to c_{1}˜t_{1,ψ}_{ε} + c_{2}˜t_{2,ψ}_{ε} with suitable
c_{1}, c_{2}.

Case 2. Assume that Im σ_{2}^{(2)} ≥ 0 and Im σ_{2}^{(1)} ≥ 0, i.e.,

−τ α_{2}− F^{(2)}+ A^{(2)} ≥ 0, τ α_{1}+ F^{(1)}− A^{(1)} ≥ 0.

Then K_{1,ψ}_{ε} has degree 2 and K_{2,ψ}_{ε} has degree 1. In order to avoid this case, we need
to be sure that if −τ α_{2}− F^{(2)}+ A^{(2)} ≥ 0, then τ α_{1}+ F^{(1)}− A^{(1)} < 0, that is,

τ α_{2}+ F^{(2)}− A^{(2)} ≤ 0 ⇒ τ α_{1}+ F^{(1)}− A^{(1)} < 0.

This can be achieved by assuming that α2

α_{1} > A^{(2)}− F^{(2)}

A^{(1)}− F^{(1)}, ∀ ξ^{0} 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 ξ^{0}. Hence (3.22) holds provided

α_{2}
α1

= max

|ξ^{0}|=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 α_{1}+ F^{(1)}− A^{(1)} < 0, −τ α_{2}− F^{(2)}+ A^{(2)} > 0.

In this case, we have

K_{1,ψ}_{ε} = (λ − σ_{1}^{(1)}), K_{2,ψ}_{ε} = (λ − σ_{2}^{(2)}).

Given polynomials q_{1}(λ), q_{2}(λ), there exist U_{1}(λ), U_{2}(λ) such that
q_{1}(λ) = U_{1}(λ)K_{1,ψ}_{ε} + ˜q_{1},

q_{2}(λ) = U_{2}(λ)K_{2,ψ}_{ε} + ˜q_{2},

where ˜q_{1}, ˜q_{2} are constants in λ. The transmission condition is satisfied if there exists
constants µ_{1}, µ_{2}, c_{1}, c_{2} so that

(q˜_{1} = µ_{1}K_{1,ψ}_{ε} + c_{1}˜t^{1}_{1,ψ}_{ε} + c_{2}t˜^{2}_{1,ψ}_{ε},

˜

q_{2} = µ_{2}K_{2,ψ}_{ε} + c_{1}˜t^{1}_{2,ψ}_{ε} + c_{2}t˜^{2}_{2,ψ}_{ε},
namely,

(q˜_{1} = µ_{1}(λ − σ_{1}^{(1)}) − c_{1}+ c_{2}a^{(1)}_{nn}(λ − iτ α_{1}− E^{(1)}− iF^{(1)})

˜

q_{2} = µ_{2}(λ − σ_{2}^{(2)}) + c_{1}+ c_{2}a^{(2)}_{nn}(λ + iτ α_{2}+ E^{(2)}+ iF^{(2)}). (3.24)
System (3.24) is equivalent to

µ_{1}+ c_{2}a^{(1)}_{nn} = 0
µ_{2}+ c_{2}a^{(2)}_{nn} = 0

µ_{1}σ_{1}^{(1)}+ c_{1}+ c_{2}a^{(1)}_{nn}(iτ α_{1} + E^{(1)}+ iF^{(1)}) = −˜q_{1}

− µ_{2}σ_{2}^{(2)}+ c_{1}+ c_{2}a^{(2)}_{nn}(iτ α_{2}+ E^{(2)}+ iF^{(2)}) = ˜q_{2}.

(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

σ_{1}^{(1)} 0 1 ζ1

0 −σ_{2}^{(2)} 1 ζ_{2}

with ζ_{1} = a^{(1)}nn(iτ α_{1}+ E^{(1)}+ iF^{(1)}), ζ_{2} = a^{(2)}nn(iτ α_{2}+ E^{(2)}+ iF^{(2)}), is nonsingular. We

compute

detT =det

1 0 a^{(1)}nn

0 1 a^{(2)}nn

0 −σ^{(2)}_{2} ζ_{2}

− det

1 0 a^{(1)}nn

0 1 a^{(2)}nn

σ_{1}^{(1)} 0 ζ_{1}

=ζ_{2}+ σ_{2}^{(2)}a^{(2)}_{nn} − ζ_{1}+ σ_{1}^{(1)}a^{(1)}_{nn}

=a^{(2)}_{nn}(iτ α_{2}+ E^{(2)}+ iF^{(2)}− E^{(2)}+ B^{(2)}− iτ α_{2} − iF^{(2)}+ iA^{(2)})
+ a^{(1)}_{nn}(−iτ α_{1}− E^{(1)}− iF^{(1)}+ E^{(1)}+ B^{(1)}+ iτ α_{1}+ 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)}_{nn}A^{(2)}+ a^{(1)}_{nn}A^{(1)} > 0
and thus (3.26) holds.

For the complex case, we want to show that there exists γ_{0} > 0 such that if γ < γ_{0},
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 R^{2}, 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

λ_{0} ≤ η^{(k)} ≤ Λ_{0}, λ_{0} ≤ δ^{(k)} ≤ Λ_{0}.
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)}u_{2}+ η^{(1)}u_{1} = γδ^{(2)}v_{2}+ γδ^{(1)}v_{1}. (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 u_{1}, u_{2}are in the first quadrant of the plane and v_{1}, v_{2} are in the second
quadrant of the plane. The sets

C_{u} = {η^{(2)}u_{2}+ η^{(1)}u_{1} : η^{(k)} ≥ 0}, C_{u} = {γδ^{(2)}v_{2}+ γδ^{(1)}v_{1} : 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 u_{1} and u_{2} is less than π/2. Note that u_{1} is the first quadrant and u_{2} is in
the fourth quadrant. So the angle between u_{1} and u_{2} 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)}u_{2}+η^{(1)}u_{1}k from below
and kδ^{(2)}v_{2} + δ^{(1)}v_{1}k from above. We now discuss the estimate of kδ^{(2)}v_{2} + δ^{(1)}v_{1}k
from above. Compute

kδ^{(2)}v_{2}+ δ^{(1)}v_{1}k^{2} = (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}]^{1/2}cos 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 = (ξ^{0}, 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}+ γ^{2}(N^{(k)}ξ · ξ)^{2}]

≤ Λ^{2}(1 + γ^{2})|ξ|^{4}

λ^{2}_{0}(1 + γ^{2}) = ˜λ^{−1}_{1} |ξ|^{4},
where we have used the estimate

λ_{0}(1 + γ^{2})^{1/2}≤ |a^{(k)}_{nn}| ≤ Λ_{0}(1 + γ^{2})^{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^{−1/2}_{1} |ξ^{0}|^{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})|ξ^{0}|^{2} ≤ (n − 1)Λ^{2}_{0}(1 + γ^{2})
λ^{2}_{0}(1 + γ^{2}) |ξ^{0}|^{2}

= (n − 1)˜λ^{−1}_{1} |ξ^{0}|^{2}.

(3.33)

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

(A^{(k)})^{2}+ (B^{(k)})^{2} ≤ (˜λ^{−1/2}_{1} + (n − 1)˜λ^{−1}_{1} )|ξ^{0}|^{2} ≤ nΛ^{2}_{0}

λ^{2}_{0}|ξ^{0}|^{2}. (3.34)
It follows from (3.29) and (3.34) that

kδ^{(2)}v_{2}+ δ^{(1)}v_{1}k^{2} ≤ 2Λ^{2}_{0}nΛ^{2}_{0}

λ^{2}_{0}|ξ^{0}|^{2}. (3.35)
Next, we want to estimate kη^{(2)}u_{2}+ η^{(1)}u_{1}k from below. As above, we have
kη^{(2)}u_{2}+ η^{(1)}u_{1}k^{2} = (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}]^{1/2}cos φ. (3.36)
Recall that B_{1} > 0, B_{2} < 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}_{λ}^{0}

0|ξ^{0}|
p˜λ_{1}|ξ^{0}| =

√nΛ^{2}_{0}

λ^{2}_{0} := ˜λ2 ≥ 1.