## Propagation of smallness and size estimate in the second order elliptic equation with discontinuous

## complex Lipschitz conductivity

### Elisa Francini

^{∗}

### Sergio Vessella

^{†}

### Jenn-Nan Wang

^{‡}

Abstract

In this paper, we would like to derive three-ball inequalities and propagation of smallness for the complex second order elliptic equation with discontinuous Lipschitz coefficients. As an application of such estimates, we study the size estimate problem by one pair of Cauchy data on the boundary. The main ingre- dient in the derivation of three-ball inequalities and propagation of smallness is a local Carleman proved in our recent paper [FVW].

Keywords: Complex second order elliptic operators, discontinuous Lipschitz co- efficients, propagation of smallness, size estimate.

MSC2020: 35B60, 35R30.

### 1 Introduction

In our recent paper [FVW], we derived a Carleman estimate for the second order elliptic equation with piecewise complex-valued Lipschitz coefficients. The theme of this paper is to prove some interesting results based on the Carleman estimate obtained in [FVW]. The ultimate goal is to give upper and lower bounds of the size of the inclusion embedded inside of a conductive body with discontinuous complex conductivity by only one pair of boundary measurements. A typical application of this study is to estimate the size of a cancerous tumor inside an organ by the electric impedance tomography (EIT).

For the conductivity equation with piecewise real Lipschitz coefficients, the same size estimate problem was considered in [FLVW]. We want to point out that, 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

∗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@math.ntu.edu.tw

electromagnetic 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. In the human body, different organs have different conductivities. For instance, the conductivities of heart, liver, intestines are 0.70 (S/m), 0.10 (S/m), 0.03 (S/m), respectively. Therefore, to model the current flow in the human body, it is more reasonable to consider an anisotropic complex- valued conductivity with jump-type discontinuities [MPH].

In the size estimate problem studied in [FLVW], the essential tool is a three-region inequality which is obtained by applying the Carleman estimate for the second order elliptic equation with piecewise real Lipschitz coefficients derived in [DFLVW]. Since we have the similar Carleman estimate available for the case of piecewise complex- valued Lipschitz coefficients, we can proceed the method used in [FLVW] to prove the three-region inequality. In treating the size estimate problem, the three-region inequality is enough since one only needs to cross the interface once in propagating the information in the interior to the boundary. However, the three-region inequality is inconvenient in deriving the general propagation of smallness. Therefore, in this paper, we want to derive the usual three-ball inequality for the complex second order elliptic operator even when the coefficients are piecewise Lipschitz. We will follow the ideas outlined in [CW] where the three-ball inequality was proved for the real second order elliptic operator with piecewise Lipschitz coefficients. We then apply the three-ball inequality to derive a general propagation of smallness for the second order elliptic equation with piecewise complex Lipschitz coefficients.

We would like to raise an issue in the investigation of the size estimate problem when the background medium is complex valued. Following the method used in [ARS], an important step is to derive certain energy inequalities controlling the power gap. It was noted in [BFV] that when the current flows inside and outside of the inclusion obey the usual Ohm’s law (the relation between current and voltage is linear) and the imaginary part of the conductivity outside of the inclusion is a nonzero variable function, energy inequalities (5.9), (5.10) are not likely to hold. Precisely, in this case, an example with δW = 0, but |D| 6= 0, is constructed in [BFV]. On the other hand, if both conductivities inside and outside of the inclusion are complex constants or the conductivity outside of the inclusion is real valued, then energy inequalities (5.9), (5.10) were obtained in [BFV].

A key observation found in [CNW] is that if the current flow inside of the inclusion obeys certain nonlinear Ohm’s law, we can restore the energy inequalities (5.9), (5.10).

In particular, our size estimate result applies to the case of a non-chiral medium with a chiral inclusion having real valued chirality.

The paper is organized as follows. In Section 2, we introduce some notations and state the Carleman estimate proved in [FVW]. In Section 3, we plan to prove a three-region inequality across the interface based on the Carleman estimate given in Section 2. We then combine the classical three-ball inequality and the three-region inequality to derive a three-ball inequality in Section 4. There, we also prove the propagation of smallness. Finally, we study the size estimate problem in Section 5.

### 2 Notations and Carleman estimate

In this section, we will state the Carleman estimate proved in [FVW] where the
interface is assumed to be flat. Since our Carleman estimate is local near any point
at the interface, for a 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±),

where

A±(x) = {a^{±}_{`j}(x)}^{n}_{`,j=1} = {a^{±}_{`j}(x^{0}, xn)}^{n}_{`,j=1}, x^{0} ∈ R^{n−1}, xn ∈ R (2.1)
is a Lipschitz symmetric matrix-valued function. Assume that

a^{±}_{`j}(x) = a^{±}_{j`}(x), ∀ `, j = 1, · · · , n, (2.2)
and furthermore

a^{±}_{`j}(x) = M_{`j}^{±}(x) + iγN_{`j}^{±}(x), (2.3)
where (M_{`j}^{±}) and (N_{`j}^{±}) are real-valued matrices and γ > 0. We further assume that
there exists λ_{0} > 0 such that for all ξ ∈ R^{n} and x ∈ R^{n} we have

λ0|ξ|^{2} ≤ M^{±}(x)ξ · ξ ≤ λ^{−1}_{0} |ξ|^{2} (2.4)
and

λ0|ξ|^{2} ≤ N^{±}(x)ξ · ξ ≤ λ^{−1}_{0} |ξ|^{2}. (2.5)
In the paper, we consider Lipschitz coefficients A_{±}, i.e., there exists a constant M_{0} > 0
such that

|A±(x) − A±(y)| ≤ M_{0}|x − y|. (2.6)
To treat the transmission conditions, we write

h_{0}(x^{0}) := u_{+}(x^{0}, 0) − u−(x^{0}, 0), ∀ x^{0} ∈ R^{n−1}, (2.7)
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}, (2.8)

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, (2.9)
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}, (2.10)

and let

φδ(x) := ψδ(δ^{−1}x), δ > 0. (2.11)
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}. (2.12)
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 (2.12) 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 A±(x) satisfy (2.1)-(2.6). There exist α_{+}, α−, β, δ_{0}, r_{0}, γ_{0}, τ_{0}, C
depending on λ_{0}, M_{0} such that if γ ≤ γ_{0}, δ ≤ δ_{0} and τ ≥ τ_{0}, 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}

.
(2.13)
where u = H_{+}u_{+}+ H−u−, u± ∈ C^{∞}(R^{n}) and supp u ⊂ B_{δr}^{0}

0 × [−δr_{0}, δr_{0}], and φ_{δ} is
given by (2.11).

Remark 2.1 In view of the proof of Theorem 2.1 in [FVW], the coefficients α_{+}, α−

are required to satisfy

α_{+}
α−

≥ κ_{0} > 1, (2.14)

where κ_{0} is general constant depending on the values of A±(0) at the interface.

Remark 2.2 It is clear that (2.13) remains valid if can add lower order terms P

±H±(W · ∇u±+ V u±), where W, V are bounded functions, to the operator L.

That is, one can substitute L(x, D)u =X

±

H_{±}div(A_{±}(x)∇u_{±}) +X

±

H_{±}(W · ∇u_{±}+ V u_{±}) (2.15)

in (2.13).

### 3 Three-region inequalities

Based on the Carleman estimate given in Theorem 2.1, we will derive three-region
inequalities across the interface x_{n} = 0. Here we consider u = H_{+}u_{+}+ H−u− satis-
fying

L(x, D)u = 0 in R^{n}, (3.1)

where L is given in (2.15) and

kW k_{L}^{∞}_{(R}^{n}_{)}+ kV k_{L}^{∞}_{(R}^{n}_{)}≤ λ^{−1}_{0} .

Now all coefficients α±, β, δ_{0}, γ_{0}, r_{0}, τ_{0} have been determined in Theorem 2.1.

Theorem 3.1 Let u be a solution of (3.1) and A±(x) satisfy (2.1)-(2.6) with h_{0} =
h_{1} = 0. Moreover, the constant γ in (2.3) satisfies γ ≤ γ_{0} with γ_{0} given in The-
orem 2.1. Then there exist C and R, depending only on λ_{0}, M_{0}, n, such that if
0 < R_{1}, R_{2} ≤ R, then

Z

U2

|u|^{2}dx ≤ (e^{τ}^{0}^{R}^{2} + CR^{−4}_{1} )

Z

U1

|u|^{2}dx

_{2R1+3R2}^{R2} Z

U3

|u|^{2}dx

^{2R1+2R2}_{2R1+3R2}

, (3.2) where

U_{1} =

z ≥ −4R_{2}, R_{1}

8a < x_{n}< R_{1}
a

,
U_{2} =

−R_{2} ≤ z ≤ R_{1}

2a, x_{n} < R_{1}
8a

,
U_{3} =

z ≥ −4R_{2}, x_{n} < R_{1}
a

, a = α+/δ,

z(x) = α−x_{n}

δ +βx^{2}_{n}

2δ^{2} − |x^{0}|^{2}

2δ , (3.3)

and any δ ≤ δ_{0}.

x_{n}

x^{0}

z=−4R2

z=−R2

z=^{R1}_{2a}

xn=^{R1}_{a}

xn=^{R1}_{8a}

U_{1}

U2

Figure 1: U_{1} and U_{2} are shown in pink and yellow, respectively. U_{3} is the region
enclosed by black boundaries. Note that since z is hyperbolic, there are parts similar
to U_{2} and U_{3} lying below x_{n}< −α_{−}δ/β. Here we are only interested in the solution
near x_{n} = 0. Thus we consider the cut-off function relative to U_{2} and U_{3} as in the
figure.

Proof. Here we adopt the proof given in [FLVW]. To apply the estimate (2.13), we need to ensure that u satisfies the support condition. Let r > 0 be chosen satisfying

r ≤ min

r^{2}_{0},13α−

8β , 2δr_{0}
19α−+ 8β

. (3.4)

We then set

R = α−r 16 . It follows from (3.4) that

R ≤ 13α^{2}_{−}

128β. (3.5)

Given 0 < R_{1} < R_{2} ≤ R. Let ϑ_{1}(t) ∈ C_{0}^{∞}(R) satisfy 0 ≤ ϑ1(t) ≤ 1 and

ϑ1(t) =

(1, t > −2R_{2},
0, t ≤ −3R_{2}.
Also, define ϑ_{2}(t) ∈ C_{0}^{∞}(R) satisfying 0 ≤ ϑ2(t) ≤ 1 and

ϑ_{2}(t) =

0, t ≥ R_{1}
2a,
1, t < R1

4a.

Finally, we define ϑ(x) = ϑ(x^{0}, x_{n}) = ϑ_{1}(z(x))ϑ_{2}(x_{n}), where z is defined by (3.3).

We now check the support condition for ϑ. From its definition, we can see that supp ϑ is contained in

z(x) = α_{−}x_{n}

δ + βx^{2}_{n}

2δ^{2} − |x^{0}|^{2}

2δ > −3R2,
x_{n} < R_{1}

2a.

(3.6)

In view of the relation

α_{+} > α− (see (2.14)) and a = α_{+}
δ ,
we have that

R_{1}
2a < δ

2α−

· R_{1} < δ
α−

· α−r
16 < δr,
i.e., x_{n}< δr ≤ δr_{0}^{2} ≤ δr_{0}. Next, we observe that

−3R_{2} > −3R = −3α−r
16 > α−

δ (−δr) + β

2δ^{2}(−δr)^{2},

which gives −δr_{0} < −δr < x_{n} due to (3.4). Consequently, we verify that |x_{n}| < δr <

δr_{0}. One the other hand, from the first condition of (3.6) and (3.4), we see that

|x^{0}|^{2}

2δ < 3R_{2}+α−xn

δ +βx^{2}_{n}

2δ^{2} ≤ 3α−r
16 + α−

δ · δr + β

2δ^{2} · δ^{2}r^{2}

≤ 19α_{−}+ 8β

16 r ≤ 19α_{−}+ 8β

16 r_{0}^{2} ≤ δ
8r_{0}^{2},
which gives |x^{0}| < δr_{0}/2.

Since h_{0} = 0, we have that

ϑ(x^{0}, 0)u_{+}(x^{0}, 0) − ϑ(x^{0}, 0)u−(x^{0}, 0) = 0, ∀ x^{0} ∈ R^{n−1}. (3.7)
Applying (2.13) to ϑu and using (3.7) yields

X

± 2

X

|k|=0

τ^{3−2|k|}

Z

R^{n}±

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

≤CX

±

Z

R^{n}±

|L(x, D)(ϑu_{±})|^{2}e^{2τ φ}^{δ,±}^{(x}^{0}^{,x}^{n}^{)}dx^{0}dx_{n}

+ Cτ Z

R^{n−1}

|A_{+}(x^{0}, 0)∇(ϑu_{+}(x^{0}, 0)) · ν − A_{−}(x^{0}, 0)∇(ϑu_{−})(x^{0}, 0) · ν|^{2}e^{2τ φ}^{δ}^{(x}^{0}^{,0)}dx^{0}
+ C[e^{τ φ}^{δ}^{(x}^{0}^{,0)} A_{+}(x^{0}, 0)∇(ϑu_{+})(x^{0}, 0) · ν − A−(x^{0}, 0)∇(ϑu−)(x^{0}, 0) · ν]^{2}_{1/2,R}n−1.

(3.8)
We now observe that ∇ϑ1(z) = ϑ^{0}_{1}(z)∇z = ϑ^{0}_{1}(z)(−^{x}_{δ}^{0},^{α}_{δ}^{−} + ^{βx}_{δ}2^{n}) and it is nonzero
only when

−3R_{2} < z < −2R_{2}.
Therefore, when x_{n} = 0, we have

2R_{2} < |x^{0}|^{2}

2δ < 3R_{2}.
Thus, we can see that

|∇ϑ(x^{0}, 0)|^{2} ≤ CR^{−2}_{2} 6R_{2}
δ +α^{2}_{−}

δ^{2}

≤ CR_{2}^{−2}. (3.9)

By h_{0}(x^{0}) = h_{1}(x^{0}) = 0, (3.9), and the easy estimate of [DFLVW, Proposition 4.2],

we can estimate τ

Z

R^{n−1}

|A+(x^{0}, 0)∇(ϑu+(x^{0}, 0)) · ν − A−(x^{0}, 0)∇(ϑu−)(x^{0}, 0) · ν|^{2}e^{2τ φ}^{δ}^{(x}^{0}^{,0)}dx^{0}
+ [e^{τ φ}^{δ}^{(x}^{0}^{,0)} A_{+}(x^{0}, 0)∇(ϑu_{+})(x^{0}, 0) · ν − A_{−}(x^{0}, 0)∇(ϑu_{−})(x^{0}, 0) · ν]^{2}_{1/2,R}n−1

≤ CR^{−2}_{2} e^{−4τ R}^{2}

τ

Z

{√

4δR2≤|x^{0}|≤√
6δR2}

|u_{+}(x^{0}, 0)|^{2}dx^{0}+ [u_{+}(x^{0}, 0)]^{2}_{1/2,{}^{√}_{4δR}

2≤|x^{0}|≤√
6δR2}

+ Cτ^{2}R_{2}^{−3}e^{−4τ R}^{2}
Z

{√

4δR2≤|x^{0}|≤√
6δR2}

|u+(x^{0}, 0)|^{2}dx^{0}

≤ Cτ^{2}R^{−3}_{2} e^{−4τ R}^{2}E,

(3.10) where

E = Z

{√

4δR2≤|x^{0}|≤√
6δR2}

|u_{+}(x^{0}, 0)|^{2}dx^{0}+ [u_{+}(x^{0}, 0)]^{2}_{1/2,{}^{√}_{4δR}

2≤|x^{0}|≤√
6δR2}.

Writing out L(x, D)(ϑu±) and considering the set where ∇ϑ 6= 0, it is not hard to estimate

X

± 1

X

|k|=0

τ^{3−2|k|}

Z

{−2R2≤z≤^{R1}_{2a}, xn<^{R1}_{4a}}

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

≤ CX

± 1

X

|k|=0

R^{2(|k|−2)}_{2}
Z

{−3R_{2}≤z≤−2R_{2}, xn<^{R1}_{2a}}

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

+ C

1

X

|k|=0

R^{2(|k|−2)}_{1}
Z

{−3R2≤z,^{R1}_{4a}<xn<^{R1}_{2a}}

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

+ Cτ^{2}R^{−3}_{2} e^{−4τ R}^{2}E

≤ CX

± 1

X

|k|=0

R^{2(|k|−2)}_{2} e^{−4τ R}^{2}e^{2τ}^{(α+−α−)}^{δ} ^{R1}^{4a}
Z

{−3R2≤z≤−2R2, xn<^{R1}_{4a}}

|D^{k}u±|^{2}dx^{0}dx_{n}

+

1

X

|k|=0

R^{2(|k|−2)}_{1} e^{2τ}^{α+}^{δ} ^{R1}^{2a}e^{2τ}^{2δ2}^{β} ^{(}^{R1}^{2a}^{)}^{2}
Z

{z≥−3R2,^{R1}_{4a}<xn<^{R1}_{2a}}

|D^{k}u_{+}|^{2}dx^{0}dx_{n}
+ Cτ^{2}R^{−3}_{2} e^{−4τ R}^{2}E.

(3.11)

Let us recall U_{1} = {z ≥ −4R_{2}, ^{R}_{8a}^{1} < x_{n} < ^{R}_{a}^{1}}, U_{2} = {−R_{2} ≤ z ≤ ^{R}_{2a}^{1}, x_{n} < ^{R}_{8a}^{1}}.

From (3.11) and interior estimates (Caccioppoli’s type inequality), we can derive that

τ^{3}e^{−2τ R}^{2}
Z

U2

|u|^{2}dx^{0}dx_{n}= τ^{3}e^{−2τ R}^{2}
Z

{−R2≤z≤^{R1}_{2a}, xn<^{R1}_{8a}}

|u|^{2}dx^{0}dx_{n}

≤ X

±

τ^{3}
Z

{−2R2≤z≤^{R1}_{2a}, xn<^{R1}_{4a}}

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

≤ CX

± 1

X

|k|=0

R^{2(|k|−2)}_{2} e^{−4τ R}^{2}e^{2τ}^{(α+−α−)}^{δ} ^{R1}^{4a}
Z

{−3R_{2}≤z≤−2R_{2}, xn<^{R1}_{4a}}

|D^{k}u±|^{2}dx^{0}dx_{n}

+

1

X

|k|=0

R^{2(|k|−2)}_{1} e^{2τ}^{α+}^{δ} ^{R1}^{2a}e^{2τ}^{2δ2}^{β} ^{(}^{R1}^{2a}^{)}^{2}
Z

{z≥−3R2,^{R1}_{4a}<xn<^{R1}_{2a}}

|D^{k}u_{+}|^{2}dx^{0}dx_{n}
+ Cτ^{2}R^{−3}_{2} e^{−4τ R}^{2}E

≤ CR^{−4}_{1} e^{−3τ R}^{2}
Z

{−4R2≤z≤−R2, xn<^{R1}_{a} }

|u|^{2}dx^{0}dx_{n}+ Cτ^{2}R^{−3}_{2} e^{−4τ R}^{2}E

+ CR^{−4}_{1} e^{(1+}

βR1 4α2−

)τ R1Z

{z≥−4R2,^{R1}_{8a}<xn<^{R1}_{a} }

|u|^{2}dx^{0}dx_{n}

≤CR^{−4}_{1}

e^{2τ R}^{1}

Z

U1

|u|^{2}dx^{0}dx_{n}+ τ^{2}e^{−3τ R}^{2}F

,

(3.12)

where

F = Z

U3

|u|^{2}dx^{0}dx_{n}
and we used the inequality ^{βR}_{4α}2^{1}

− < 1 in view of (3.5). Remark that we estimate E by F using the trace estimate and the interior estimate.

Dividing τ^{3}e^{−2τ R}^{2} on both sides of (3.12) gives
Z

U2

|u|^{2}dx ≤ CR^{−4}_{1}

e^{2τ (R}^{1}^{+R}^{2}^{)}
Z

U1

|u|^{2}dx^{0}dx_{n}+ e^{−τ R}^{2}F

. (3.13)

We now discuss two cases. If R

U1|u|^{2}dx^{0}dx_{n}6= 0 and
e^{2τ}^{0}^{(R}^{1}^{+R}^{2}^{)}

Z

U1

|u|^{2}dx^{0}dx_{n}< e^{−τ}^{0}^{R}^{2}F,
then we can choose a τ > τ_{0} so that

e^{2τ (R}^{1}^{+R}^{2}^{)}
Z

U1

|u|^{2}dx^{0}dx_{n}= e^{−τ R}^{2}F.

With such τ , it follows from (3.13) that Z

U2

|u|^{2}dx ≤ CR_{1}^{−4}e^{2τ (R}^{1}^{+R}^{2}^{)}
Z

U1

|u|^{2}dx^{0}dx_{n}

= CR^{−4}_{1}

Z

U1

|u|^{2}dx^{0}dx_{n}

_{2R1+3R2}^{R2}

(F )^{2R1+2R2}^{2R1+3R2}.

(3.14)

If R

U1|u|^{2}dx^{0}dx_{n} = 0, then letting τ → ∞ in (3.13) we have R

U2|u|^{2}dx^{0}dx_{n} = 0 as
well. The three-regions inequality (3.2) obviously holds.

On the other hand, if

e^{−τ}^{0}^{R}^{2}F ≤ e^{2τ}^{0}^{(R}^{1}^{+R}^{2}^{)}
Z

U1

|u|^{2}dx^{0}dx_{n},
then we have

Z

U2

|u|^{2}dx^{0}dx_{n} ≤ (F )^{2R1+3R2}^{R2} (F )^{2R1+2R2}^{2R1+3R2}

≤ exp (τ0R2)

Z

U1

|u|^{2}dx^{0}dxn

_{2R1+3R2}^{R2}

(F )^{2R1+2R2}^{2R1+3R2} .

(3.15)

Putting together (3.14), (3.15) implies Z

U2

|u|^{2}dx^{0}dx_{n}≤ (exp (τ_{0}R_{2}) + CR_{1}^{−4})

Z

U1

|u|^{2}dx^{0}dx_{n}

_{2R1+3R2}^{R2}

(F )^{2R1+2R2}^{2R1+3R2} . (3.16)

### 2

### 4 Propagation of smallness

In this section, we will derive a general propagation of smallness for solutions satis-
fying (3.1), L(x, D)u = 0 in R^{n}, using the ideas given in [CW]. For the region away
from the interface, classical three-ball inequalities are shown to hold for the com-
plex second order elliptic operators [CNW]. We will mainly focus on the inequalities
across the interface. Let us first fix some notations. Assume that Ω ⊂ R^{n} is an open
bounded domain with Lipschitz boundary and Σ ⊂ Ω is a C^{1,1} hypersurface. Fur-
thermore, assume that Ω \ Σ only has two connected components, which we denote
Ω±. Let A±(x) = (a_{`j}(x)^{±})^{n}_{`,j=1}, W (x), V (x) be bounded measurable complex valued
coefficients defined in Ω. We say that

ζ^{±} := (A±, W, V ) ∈V (Ω±, λ_{0}, M_{0}, K_{1}, K_{2})
if A± satisfy (2.2)-(2.6) for x ∈ Ω± and W, V satisfy

kW k_{L}^{∞}_{(Ω)} ≤ K_{1}, kV k_{L}^{∞}_{(Ω)} ≤ K_{2}.
We will use the notation Lζ to denote

L_{ζ}(x, D)u =X

±

H±div(A±(x)∇u±) +X

±

H±(W · ∇u±+ V u±) in Ω.

Here, by an abuse of notation, we denote H_{±} = χ_{Ω}_{±}.

For an open set U ⊂ R^{n} and a number s > 0, we define
U^{s} = {x ∈ R^{n} : dist(x, U ) < s},
U_{s} = {x ∈ U : dist(x, ∂U ) > s},
and

sU = {sx : x ∈ U }.

Definition 4.1 We say that Ω ∈ C^{k,1}, k ∈ N with constants ρ^{1}, M1 if for any point
P ∈ ∂Ω, after a rigid transformation, P = 0 and

Ω ∩ Γ_{ρ}_{1}_{,M}_{1}(0) = {(x^{0}, x_{n}) : x^{0} ∈ R^{n−1}, |x^{0}| < ρ_{1}, x_{n}∈ R, xn > Φ(x^{0})},
where Φ is a C^{k,1} function such that Φ(0) = 0, kΦk_{C}^{α,1}_{(B}_{ρ1}_{(0))} ≤ M_{1}, and

Γ_{ρ}_{1}_{,M}_{1}(0) = {(x^{0}, x_{n}) : x^{0} ∈ R^{n−1}, |x^{0}| < ρ_{1}, |x_{n}| ≤ M_{1}}.

Throughout this paper, when saying that a domain is C^{k,1}, we will mean that it is
C^{k,1} with constants ρ_{1} and M_{1}.

Definition 4.2 We say that Σ is C^{1,1} with constants ρ_{0}, K_{0} if for any point P ∈ Σ,
after a rigid transformation, P = 0 and

Ω±∩ C_{ρ}_{0}_{,K}_{0}(0) = {(x^{0}, x_{n}) : x^{0} ∈ R^{n−1}, |x^{0}| < ρ_{0}, x_{n}∈ R, xn≷ ψ(x)},

where ψ is a C^{1,1} function such that ψ(0) = 0, ∇_{x}^{0}ψ(0) = 0, kψk_{C}^{1,1}_{(B}_{ρ0}_{(0))} ≤ K_{0},
and

Cρ0,K0(0) = {(x^{0}, xn) : x^{0} ∈ R^{n−1}, |x^{0}| < ρ0, |xn| ≤ 1

2K0ρ^{2}_{0}}.

If Σ is as above, then we may ”flatten” the boundary around the point P (without
loss of generality P = 0) via the local C^{1,1}-diffeomeorphism

Ψ_{P}(x, y) = (x, y − ψ(x)).

With these assumptions, we will prove a propagation of smallness result as follows.

Theorem 4.1 Suppose u ∈ H^{1}(Ω) solves

L_{ζ}u = 0 in Ω.

Then there exist γ_{0}, depending on λ_{0}, M_{0}, and h_{0}, depending on λ_{0}, M_{0}, K_{1}, K_{2},
ρ_{0}, K_{0}, such that if γ < γ_{0} and 0 < h < h_{0}, r/2 > h, D ⊂ Ω is connected, open,
and D \ Σ has two connected components, denoted by D±, such that B_{r}(x_{0}) ⊂ D,
dist(D, ∂Ω) ≥ h, then

kuk_{L}^{2}_{(D)}≤ Ckuk^{δ}_{L}2(Br(x0))kuk^{1−δ}_{L}2(Ω),
where

C = C_{1} |Ω|

h^{n}

^{1}_{2}

e^{C}^{3}^{h}^{−s}, δ ≥ τ^{C2|Ω|}^{hn} ,

with s = s(λ_{0}, K_{1}, K_{2}), C_{1}, C_{2} > 0, τ ∈ (0, 1) depending on λ_{0}, M_{0}, K_{1}, K_{2}, ρ_{0}, K_{0}.

We would like to remark that the propagation of smallness in Theorem 4.1 is
valid regardless the locations of D and B_{r}_{0}(x_{0}), which may intersect the interface Σ.

The strategy of proving Theorem 4.1 consists two parts. When we are at one side of the interface, we can use the usual propagation of smallness for equations with Lipschitz complex coefficients based on [CNW]. When near the interface, we then use the three-region inequality derived above to propagate the smallness across the interface. The rest of this section is devoted to the proof of Theorem 4.1.

### 4.1 Propagation of smallness away from the interface

In this subsection, we want to derive a propagation of smallness for second order complex elliptic operators with Lipschitz leading coefficients. We consider ζ :=

(A, W, V ) ∈ V (U, λ0, M_{0}, K_{1}, K_{2}), where U is an open bounded domain and the
value of γ in (2.3) is irrelevant. Note that here A is Lipschitz without jumps in U .
The following three-ball inequality was proved in [CNW, Theorem 3].

Proposition 4.1 Assume that ζ := (A, W, V ) ∈ V (U, λ0, M_{0}, K_{1}, K_{2}) and u ∈
H_{loc}^{1} (Ω) solves L_{ζ}u = 0 in U . Then there exist positive constants R = R(n, λ_{0}, M_{0})
and s = s(λ_{0}, K_{1}, K_{2}) such that if 0 < r_{0} < r_{1} < λ_{0}r_{2}/2 < √

λ_{0}R/2 with B_{r}_{2}(x_{0}) ⊂ U ,
then

kuk_{L}^{2}_{(B}_{r1}_{(x}_{0}_{))}≤ Ckuk^{τ}_{L}2(B_{r0}(x0))kuk^{1−τ}_{L}2(B_{r2}(x0)), (4.1)
where C is explicitly given by

C = e^{C}^{1}^{(r}^{−s}^{0} ^{−r}^{2}^{−s}^{)}
with C1 > depending on λ0, M0, K1, K2, and

τ = (2r_{1}/λ_{0})^{−s}− r^{−s}_{2}

r_{0}^{−s}− r_{2}^{−s} = (2r_{1}/r_{2}λ_{0})^{−s}− 1
(r0/r2)^{−s}− 1 .

The proof of Proposition4.1 relies on the Carleman estimate derived in [CGT].

Having established the three-ball inequality (4.1), we can prove the following prop- agation of smallness based on the chain of balls argument in [ARRV, Theorem 5.1].

We will not repeat the argument here.

Proposition 4.2 Assume that ζ := (A, W, V ) ∈ V (U, λ0, M_{0}, K_{1}, K_{2}) and u ∈
H_{loc}^{1} (Ω) solves Lζu = 0 in U . Let 0 < h < r/2 with r ≤ √

λ0R/2, D ⊂ U con-
nected, open, and such that B_{r}(x_{0}) ⊂ D, dist(D, ∂U ) ≥ h. Then

kuk_{L}^{2}_{(D)} ≤ Ckuk^{δ}_{L}2(Br(x0))kuk^{1−δ}_{L}2(U ),
where

C = C_{2} |U |
h^{n}

^{1}_{2}

e^{C}^{3}^{h}^{−s}, δ ≥ τ^{C4|U|}^{hn} ,
with C2, C3, C4 > 0 depending on λ0, M0, K1, K2.

### 4.2 Propagation of smallness – an intermediate result

Here we would like to prove an intermediate propagation of smallness result in which
the small ball lies entirely on one side of the interface. Assume that D ⊂⊂ Ω is open
and connected. Recall that we have assumed that Ω \ Σ and D \ Σ both have two
connected components, denoted by Ω± and D±, respectively. Let ω_{Σ} be the surface
measure induced on Σ by the Lebesgue measure on R^{n}. We will consider coefficients

ζ = (A±, W, V ) ∈V (Ω±, λ0, M0, K1, K2).

We can now prove the following propagation of smallness result.

Theorem 4.2 Suppose u ∈ H^{1}(Ω) solves Lζu = 0 in Ω. Then there exist γ0, de-
pending on λ_{0}, M_{0}, and h_{0}, depending on λ_{0}, M_{0}, K_{1}, K_{2}, ρ_{0}, K_{0}, such that if γ < γ_{0}
and 0 < h ≤ h_{0}, h < r/2, B_{r}(x_{0}) ⊂ D_{+}, and dist(D, ∂Ω) ≥ h, then

kuk_{L}^{2}_{(D)}≤ Ckuk^{δ}_{L}2(Br(x0))kuk^{1−δ}_{L}2(Ω),
where

C = C_{1} |Ω|

h^{n}

"

1 + ω_{Σ}(Σ ∩ Ω)
h^{n−1}

^{1}_{2}#

e^{C}^{3}^{h}^{−s}, δ ≥ τ^{C2|Ω|}^{hn} ,
with C_{1}, C_{2}, C_{3} > 0, τ ∈ (0, 1) depending on λ_{0}, M_{0}, K_{1}, K_{2}, ρ_{0}, K_{0}.

The difficult part of proving Theorem4.2 is to obtain L^{2} estimates of the solution
in a neighborhood of Σ. We will use Theorem3.1to overcome this difficulty. However,
we cannot apply Theorem3.1directly. The family of regions given in Theorem3.1has
one serious drawback. If we choose the parameters R_{1} = θ ¯R_{1}, R_{2} = θ ¯R_{2}, θ ∈ (0, 1),
the vertical sizes of the regions would scale like θ, while their horizontal sizes would
scale like θ^{1}^{2}. Using just these two parameters in the proof would then lead to
constants in the propagation of smallness inequality (i.e. the constants C and δ
in Theorem 4.1) that depend on the geometry of Ω, D, and B_{r}(x_{0}) in a way that
is not invariant under a rescaling of these sets. Therefore, we will study how the
three-region inequality (3.2) behaves under scaling.

Let us first introduce the scaled coefficients ζ =˜ ˜A±, ˜W , ˜V

∈V (R^{n}±, λ_{0}, M_{0}, K_{1}, K_{2}).

For 0 < θ ≤ 1, let
L^{θ}_{ζ}_{˜}(·, D)v =X

±

H±div( ˜A±(θ·)∇v±) +X

±

H±

θ ˜W (θ·) · ∇v±+ θ^{2}V (θ·)v˜ ±

.

Note that if ˜ζ = ˜A±, ˜W , ˜V

∈V (R^{n}±, λ0, M0, K1, K2), then

˜A±(θ·), θ ˜W (θ·), θ^{2}V (θ·)˜

∈V (R^{n}±, λ_{0}, θM_{0}, θK_{1}, θ^{2}K_{2}).

It is also clear that if Lζ˜u = 0 in Ω, then u^{θ}(x) = θ^{−2}u(θx) solves
L^{θ}_{ζ}_{˜}u^{θ} = 0 in θ^{−1}Ω.

Moreover, if U ⊂ θ^{−1}Ω, we have
Z

θU

|u(x)|^{2}dx = θ^{n+4}
Z

U

|u^{θ}(y)|^{2}dy.

We therefore obtain, by scaling, the following corollary to Theorem 3.1.

Proposition 4.3 Assume that the assumptions in Theorem3.1hold. Let 0 < R_{1}, R_{2} ≤
R, θ ∈ (0, 1], and

L˜γu = 0 in θU3, then

Z

θU2

|u|^{2} ≤ (e^{τ}^{0}^{R}^{2} + CR^{−4}_{1} )

Z

θU1

|u|^{2}

_{2R1+3R2}^{R2} Z

θU3

|u|^{2}

^{2R1+2R2}_{2R1+3R2}
.

In order to adapt that result to the possibly curved surface Σ, we need to first
consider how the three regions transform under a local boundary flattening diffeo-
morphism ΨP. Pick a point P ∈ Σ and set P = 0 without loss of generality. Let
(x^{0}, x_{n}) ∈ C_{ρ}_{0}_{,K}_{0}(0). We will try to determine when (x^{0}, x_{n}) ∈ Ψ^{−1}_{P} (θU_{2}). To this end,
we introduce the notation

y^{0} = x^{0}, y_{n}= x_{n}− ψ(x^{0}).

It is clear that (x^{0}, xn) ∈ Ψ^{−1}_{P} (θU2) if and only if θ^{−1}(y^{0}, yn) ∈ U2. We denote
η(x^{0}) = ψ(x^{0})

|x^{0}|^{2} ,

which is a bounded function due to the regularity assumption of Σ. It was proved in [CW, Lemma 3.1] that if

r < θ min δR_{1}

6aα_{−},2δR_{2}
3α_{−} , R_{1}

12a, θ^{−1}ρ_{0}, ρ_{1}, ρ_{2}, ρ_{3}

, (4.2)

then Ψ_{P}(B_{r}(P )) ⊂ θU_{2}, where

ρ_{1} = α−δ
δ + β,

ρ_{2} is chosen such that 2kηkρ_{2}+ kηk^{2}ρ^{2}_{2} < 1

2, kηk = kηk_{L}^{∞}_{(B}^{0}

ρ0(0)),
ρ_{3} = 2α−δ

β .

In [CW, Lemma 3.2], the following relation was established.

Lemma 4.1 Ψ^{−1}_{P} (θU_{3}) is contained in a ball of radius

θ

(1 + 2kηk^{2}) 2α−

a R1+ 8δR2

+ 1

a^{2}

2 + (1 + 2kηk^{2})β
δ

R^{2}_{1}+ 128δ^{2}R^{2}_{2}
h

α−+ q

α^{2}_{−}− 8βR2

i2

1/2

centered at P .

Finally, we need to estimate the distance from Ψ^{−1}_{P} (θU_{1}) to Σ ∩ C_{ρ}_{0}_{,K}_{0}. It was
proved in [CW, Lemma 3.3] that

dist(Ψ^{−1}_{P} (θU_{1}), Σ) > θR_{1}

16a. (4.3)

We are now ready to prove Theorem 4.2. We will follow the arguments used in the proof of Theorem 3.1 in [CW]. Since we have slightly different constants here, we provide the proof for the sake of completeness.

Proof of Theorem 4.2. By the assumption, we may take D to be the set D = {x ∈ Ω : dist(x, ∂Ω) > h}.

We want to point out that even though the choice of α± in Theorem 2.1 depends
on A_{±}(P ) for P ∈ Σ, we can choose a pair of α_{±} such that Carleman estimate
(2.13) holds near all P ∈ Σ in view of the regularity assumptions of A± and Σ.

Consequently, we can pick R_{1}, R_{2} so that we can apply Proposition4.3 at any point
P ∈ Σ ∩ D. By Lemma 4.1, there is a constant d > 0, independent of P , such that
Ψ^{−1}_{P} (θU_{3}) ⊂ B_{θd}(P ). We then choose θ such that θd = ^{h}_{2}, which implies Ψ^{−1}_{P} (θU_{3}) ⊂ Ω
for any P ∈ Σ ∩ D. Of course, this choice is not possible if h is too large. Therefore,
we need to set h_{0} small enough, depending on ρ_{0}, K_{0}, λ_{0}, M_{0}, K_{1}, K_{2}.

With this choice of parameters, by (4.3), there is a constant 0 < µ < 1, also independent on P , so that

dist(Ψ^{−1}_{P} (θU_{1}), Σ) > µh.

Note that, depending on the geometry of Σ, we again need to set h_{0} and R small
enough so that Ψ^{−1}_{P} (θU_{1}) ∩ Σ^{µh}= ∅, for any P ∈ Σ ∩ D.

It follows from (4.2) that there exists a constant ν > 0, and without loss of
generality ν < µ < 1, such that B_{5νh}(P ) ⊂ Ψ^{−1}_{P} (θU_{2}). By Vitali’s covering lemma,
there exist finitely many P_{1}, . . . , P_{N} ∈ Σ ∩ D so that

Σ^{νh}∩ D ⊂

N

[

j=1

Ψ^{−1}_{P}

j(θU_{2}), (4.4)

and the balls B_{νh}(P_{j}) are pairwise disjoint. By this last property, since for small h
we have ωΣ(Σ^{νh}∩ D) ∼ νhωΣ(Σ ∩ D), it follows that there is a constant C such that

N ≤ Cω_{Σ}(Σ ∩ D)

h^{n−1} ≤ Cω_{Σ}(Σ ∩ Ω)

h^{n−1} . (4.5)