Propagation of smallness and size estimate in the second order elliptic equation with discontinuous complex Lipschitz conductivity

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_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 ∂` = ∂/∂x`, D` = −i∂`, and ∂ξ<sub>`</sub> = ∂/∂ξ`. 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±),

where

A±(x) = {a^±_{`j</sub>(x)}<sup>n</sup><sub>`,j=1} = {a^±_{`j</sub>(x<sup>0</sup>, xn)}<sup>n</sup><sub>`,j=1}, x⁰ ∈ Rⁿ⁻¹, xn ∈ R (2.1) is a Lipschitz symmetric matrix-valued function. Assume that

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

a^±_{`j</sub>(x) = M<sub>`j}^±(x) + iγN_{`j</sub><sup>±</sup>(x), (2.3) where (M<sub>`j}^±) and (N_`j^±) are real-valued matrices and γ > 0. We further assume that there exists λ₀ > 0 such that for all ξ ∈ Rⁿ and x ∈ Rⁿ we have

λ0|ξ|² ≤ M^±(x)ξ · ξ ≤ λ⁻¹₀ |ξ|² (2.4) and

λ0|ξ|² ≤ N^±(x)ξ · ξ ≤ λ⁻¹₀ |ξ|². (2.5) 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|. (2.6) To treat the transmission conditions, we write

h₀(x⁰) := u₊(x⁰, 0) − u−(x⁰, 0), ∀ x⁰ ∈ Rⁿ⁻¹, (2.7) h₁(x⁰) := A₊(x⁰, 0)∇u₊(x⁰, 0) · ν − A₋(x⁰, 0)∇u₋(x⁰, 0) · ν, ∀ x⁰ ∈ Rⁿ⁻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+ βx²_n/2, x_n ≥ 0,

ϕ₋(x_n) := α₋x_n+ βx²_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|x⁰|², (2.10)

and let

φδ(x) := ψδ(δ⁻¹x), δ > 0. (2.11) 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. (2.12) 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 (2.12) 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 A±(x) satisfy (2.1)-(2.6). There exist α₊, α−, β, δ₀, r₀, γ₀, τ₀, C depending on λ₀, M₀ such that if γ ≤ γ₀, δ ≤ δ₀ and τ ≥ τ₀, 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

 . (2.13) where u = H₊u₊+ H−u−, u± ∈ C^∞(Rⁿ) and supp u ⊂ B_δr⁰

0 × [−δr₀, δr₀], 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−

≥ κ₀ > 1, (2.14)

where κ₀ 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 Rⁿ, (3.1)

where L is given in (2.15) and

kW k_L^∞_(Rⁿ₎+ kV k_L^∞_(Rⁿ₎≤ λ⁻¹₀ .

Now all coefficients α±, β, δ₀, γ₀, r₀, τ₀ 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 h₀ = h₁ = 0. Moreover, the constant γ in (2.3) satisfies γ ≤ γ₀ with γ₀ given in The- orem 2.1. Then there exist C and R, depending only on λ₀, M₀, n, such that if 0 < R₁, R₂ ≤ R, then

Z

U2

|u|²dx ≤ (e^τ0R2 + CR⁻⁴₁ )

Z

U1

|u|²dx

_2R1+3R2^R2 Z

U3

|u|²dx

^2R1+2R2_2R1+3R2

, (3.2) where

U₁ =



z ≥ −4R₂, R₁

8a < x_n< R₁ a

 , U₂ =



−R₂ ≤ z ≤ R₁

2a, x_n < R₁ 8a

 , U₃ =



z ≥ −4R₂, x_n < R₁ a

 , a = α+/δ,

z(x) = α−x_n

δ +βx²_n

2δ² − |x⁰|²

2δ , (3.3)

and any δ ≤ δ₀.

x_n

x⁰

z=−4R2

z=−R2

z=^R1_2a

xn=^R1_a

xn=^R1_8a

U₁

U2

Figure 1: U₁ and U₂ are shown in pink and yellow, respectively. U₃ is the region enclosed by black boundaries. Note that since z is hyperbolic, there are parts similar to U₂ and U₃ 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 U₂ and U₃ 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



r²₀,13α−

8β , 2δr₀ 19α−+ 8β



. (3.4)

We then set

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

R ≤ 13α²₋

128β. (3.5)

Given 0 < R₁ < R₂ ≤ R. Let ϑ₁(t) ∈ C₀^∞(R) satisfy 0 ≤ ϑ1(t) ≤ 1 and

ϑ1(t) =

(1, t > −2R₂, 0, t ≤ −3R₂. Also, define ϑ₂(t) ∈ C₀^∞(R) satisfying 0 ≤ ϑ2(t) ≤ 1 and

ϑ₂(t) =







0, t ≥ R₁ 2a, 1, t < R1

4a.

Finally, we define ϑ(x) = ϑ(x⁰, x_n) = ϑ₁(z(x))ϑ₂(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 + βx²_n

2δ² − |x⁰|²

2δ > −3R2, x_n < R₁

2a.

(3.6)

In view of the relation

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

R₁ 2a < δ

2α−

· R₁ < δ α−

· α−r 16 < δr, i.e., x_n< δr ≤ δr₀² ≤ δr₀. Next, we observe that

−3R₂ > −3R = −3α−r 16 > α−

δ (−δr) + β

2δ²(−δr)²,

which gives −δr₀ < −δr < x_n due to (3.4). Consequently, we verify that |x_n| < δr <

δr₀. One the other hand, from the first condition of (3.6) and (3.4), we see that

|x⁰|²

2δ < 3R₂+α−xn

δ +βx²_n

2δ² ≤ 3α−r 16 + α−

δ · δr + β

2δ² · δ²r²

≤ 19α₋+ 8β

16 r ≤ 19α₋+ 8β

16 r₀² ≤ δ 8r₀², which gives |x⁰| < δr₀/2.

Since h₀ = 0, we have that

ϑ(x⁰, 0)u₊(x⁰, 0) − ϑ(x⁰, 0)u−(x⁰, 0) = 0, ∀ x⁰ ∈ Rⁿ⁻¹. (3.7) Applying (2.13) to ϑu and using (3.7) yields

X

± 2

X

|k|=0

τ^3−2|k|

Z

Rⁿ±

|D^k(ϑu_±)|²e^{2τ φδ,±(x0,xn)}dx⁰dx_n

≤CX

±

Z

Rⁿ±

|L(x, D)(ϑu_±)|²e^{2τ φδ,±(x0,xn)}dx⁰dx_n

+ Cτ Z

Rⁿ⁻¹

|A₊(x⁰, 0)∇(ϑu₊(x⁰, 0)) · ν − A₋(x⁰, 0)∇(ϑu₋)(x⁰, 0) · ν|²e^{2τ φδ(x0,0)}dx⁰ + C[e^{τ φδ(x0,0)} A₊(x⁰, 0)∇(ϑu₊)(x⁰, 0) · ν − A−(x⁰, 0)∇(ϑu−)(x⁰, 0) · ν
]²_1/2,Rn−1.

(3.8) We now observe that ∇ϑ1(z) = ϑ⁰₁(z)∇z = ϑ⁰₁(z)(−^x_δ⁰,^α_δ⁻ + ^βx_δ2ⁿ) and it is nonzero only when

−3R₂ < z < −2R₂. Therefore, when x_n = 0, we have

2R₂ < |x⁰|²

2δ < 3R₂. Thus, we can see that

|∇ϑ(x⁰, 0)|² ≤ CR⁻²₂  6R₂ δ +α²₋

δ²



≤ CR₂⁻². (3.9)

By h₀(x⁰) = h₁(x⁰) = 0, (3.9), and the easy estimate of [DFLVW, Proposition 4.2],

we can estimate τ

Z

Rⁿ⁻¹

|A+(x⁰, 0)∇(ϑu+(x⁰, 0)) · ν − A−(x⁰, 0)∇(ϑu−)(x⁰, 0) · ν|²e^{2τ φδ(x0,0)}dx⁰ + [e^{τ φδ(x0,0)} A₊(x⁰, 0)∇(ϑu₊)(x⁰, 0) · ν − A₋(x⁰, 0)∇(ϑu₋)(x⁰, 0) · ν
]²_1/2,Rn−1

≤ CR⁻²₂ e^{−4τ R2}

 τ

Z

{√

4δR2≤|x⁰|≤√ 6δR2}

|u₊(x⁰, 0)|²dx⁰+ [u₊(x⁰, 0)]²_1/2,{^√_4δR

2≤|x⁰|≤√ 6δR2}



+ Cτ²R₂⁻³e^{−4τ R2} Z

{√

4δR2≤|x⁰|≤√ 6δR2}

|u+(x⁰, 0)|²dx⁰

≤ Cτ²R⁻³₂ e^{−4τ R2}E,

(3.10) where

E = Z

{√

4δR2≤|x⁰|≤√ 6δR2}

|u₊(x⁰, 0)|²dx⁰+ [u₊(x⁰, 0)]²_1/2,{^√_4δR

2≤|x⁰|≤√ 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^ku_±|²e^{2τ φδ,±(x0,xn)}dx⁰dx_n

≤ CX

± 1

X

|k|=0

R^2(|k|−2)₂ Z

{−3R₂≤z≤−2R₂, xn<^R1_2a}

|D^ku±|²e^{2τ φδ,±(x0,xn)}dx⁰dx_n

+ C

1

X

|k|=0

R^2(|k|−2)₁ Z

{−3R2≤z,^R1_4a<xn<^R1_2a}

|D^ku₊|²e^{2τ φδ,+(x0,xn)}dx⁰dx_n

+ Cτ²R⁻³₂ e^{−4τ R2}E

≤ CX

± 1

X

|k|=0

R^2(|k|−2)₂ e^{−4τ R2}e^{2τ(α+−α−)δ R14a} Z

{−3R2≤z≤−2R2, xn<^R1_4a}

|D^ku±|²dx⁰dx_n

+

1

X

|k|=0

R^2(|k|−2)₁ e^{2τα+δ R12a}e^{2τ2δ2β (R12a)2} Z

{z≥−3R2,^R1_4a<xn<^R1_2a}

|D^ku₊|²dx⁰dx_n + Cτ²R⁻³₂ e^{−4τ R2}E.

(3.11)

Let us recall U₁ = {z ≥ −4R₂, ^R_8a¹ < x_n < ^R_a¹}, U₂ = {−R₂ ≤ z ≤ ^R_2a¹, x_n < ^R_8a¹}.

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

τ³e^{−2τ R2} Z

U2

|u|²dx⁰dx_n= τ³e^{−2τ R2} Z

{−R2≤z≤^R1_2a, xn<^R1_8a}

|u|²dx⁰dx_n

≤ X

±

τ³ Z

{−2R2≤z≤^R1_2a, xn<^R1_4a}

|u±|²e^{2τ φδ,±(x0,xn)}dx⁰dx_n

≤ CX

± 1

X

|k|=0

R^2(|k|−2)₂ e^{−4τ R2}e^{2τ(α+−α−)δ R14a} Z

{−3R₂≤z≤−2R₂, xn<^R1_4a}

|D^ku±|²dx⁰dx_n

+

1

X

|k|=0

R^2(|k|−2)₁ e^{2τα+δ R12a}e^{2τ2δ2β (R12a)2} Z

{z≥−3R2,^R1_4a<xn<^R1_2a}

|D^ku₊|²dx⁰dx_n + Cτ²R⁻³₂ e^{−4τ R2}E

≤ CR⁻⁴₁ e^{−3τ R2} Z

{−4R2≤z≤−R2, xn<^R1_a }

|u|²dx⁰dx_n+ Cτ²R⁻³₂ e^{−4τ R2}E

+ CR⁻⁴₁ e⁽¹⁺

βR1 4α2−

)τ R1Z

{z≥−4R2,^R1_8a<xn<^R1_a }

|u|²dx⁰dx_n

≤CR⁻⁴₁

 e^{2τ R1}

Z

U1

|u|²dx⁰dx_n+ τ²e^{−3τ R2}F

 ,

(3.12)

where

F = Z

U3

|u|²dx⁰dx_n and we used the inequality ^βR_4α2¹

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

Dividing τ³e^{−2τ R2} on both sides of (3.12) gives Z

U2

|u|²dx ≤ CR⁻⁴₁



e^{2τ (R1+R2)} Z

U1

|u|²dx⁰dx_n+ e^{−τ R2}F



. (3.13)

We now discuss two cases. If R

U1|u|²dx⁰dx_n6= 0 and e^2τ0(R1+R2)

Z

U1

|u|²dx⁰dx_n< e^−τ0R2F, then we can choose a τ > τ₀ so that

e^{2τ (R1+R2)} Z

U1

|u|²dx⁰dx_n= e^{−τ R2}F.

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

U2

|u|²dx ≤ CR₁⁻⁴e^{2τ (R1+R2)} Z

U1

|u|²dx⁰dx_n

= CR⁻⁴₁

Z

U1

|u|²dx⁰dx_n

_2R1+3R2^R2

(F )^{2R1+2R22R1+3R2}.

(3.14)

If R

U1|u|²dx⁰dx_n = 0, then letting τ → ∞ in (3.13) we have R

U2|u|²dx⁰dx_n = 0 as well. The three-regions inequality (3.2) obviously holds.

On the other hand, if

e^−τ0R2F ≤ e^2τ0(R1+R2) Z

U1

|u|²dx⁰dx_n, then we have

Z

U2

|u|²dx⁰dx_n ≤ (F )^2R1+3R2R2 (F )^{2R1+2R22R1+3R2}

≤ exp (τ0R2)

Z

U1

|u|²dx⁰dxn

_2R1+3R2^R2

(F )^{2R1+2R22R1+3R2} .

(3.15)

Putting together (3.14), (3.15) implies Z

U2

|u|²dx⁰dx_n≤ (exp (τ₀R₂) + CR₁⁻⁴)

Z

U1

|u|²dx⁰dx_n

_2R1+3R2^R2

(F )^{2R1+2R22R1+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 Rⁿ, 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 ⊂ Rⁿ 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</sub>(x)<sup>±</sup>)<sup>n</sup><sub>`,j=1}, W (x), V (x) be bounded measurable complex valued coefficients defined in Ω. We say that

ζ^± := (A±, W, V ) ∈V (Ω±, λ₀, M₀, K₁, K₂) if A± satisfy (2.2)-(2.6) for x ∈ Ω± and W, V satisfy

kW k_L^∞_(Ω) ≤ K₁, kV k_L^∞_(Ω) ≤ K₂. 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 ⊂ Rⁿ and a number s > 0, we define U^s = {x ∈ Rⁿ : 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 ρ¹, M1 if for any point P ∈ ∂Ω, after a rigid transformation, P = 0 and

Ω ∩ Γ_ρ1,M1(0) = {(x⁰, x_n) : x⁰ ∈ Rⁿ⁻¹, |x⁰| < ρ₁, x_n∈ R, xn > Φ(x⁰)}, where Φ is a C^k,1 function such that Φ(0) = 0, kΦk_C^α,1_(Bρ1(0)) ≤ M₁, and

Γ_ρ1,M1(0) = {(x⁰, x_n) : x⁰ ∈ Rⁿ⁻¹, |x⁰| < ρ₁, |x_n| ≤ M₁}.

Throughout this paper, when saying that a domain is C^k,1, we will mean that it is C^k,1 with constants ρ₁ and M₁.

Definition 4.2 We say that Σ is C^1,1 with constants ρ₀, K₀ if for any point P ∈ Σ, after a rigid transformation, P = 0 and

Ω±∩ C_ρ0,K0(0) = {(x⁰, x_n) : x⁰ ∈ Rⁿ⁻¹, |x⁰| < ρ₀, x_n∈ R, xn≷ ψ(x)},

where ψ is a C^1,1 function such that ψ(0) = 0, ∇_x⁰ψ(0) = 0, kψk_C^1,1_(Bρ0(0)) ≤ K₀, and

Cρ0,K0(0) = {(x⁰, xn) : x⁰ ∈ Rⁿ⁻¹, |x⁰| < ρ0, |xn| ≤ 1

2K0ρ²₀}.

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 ∈ H¹(Ω) solves

L_ζu = 0 in Ω.

Then there exist γ₀, depending on λ₀, M₀, and h₀, depending on λ₀, M₀, K₁, K₂, ρ₀, K₀, such that if γ < γ₀ and 0 < h < h₀, r/2 > h, D ⊂ Ω is connected, open, and D \ Σ has two connected components, denoted by D±, such that B_r(x₀) ⊂ D, dist(D, ∂Ω) ≥ h, then

kuk_L²_(D)≤ Ckuk^δ_L2(Br(x0))kuk^1−δ_L2(Ω), where

C = C₁ |Ω|

hⁿ

¹₂

e^C3h−s, δ ≥ τ^C2|Ω|hn ,

with s = s(λ₀, K₁, K₂), C₁, C₂ > 0, τ ∈ (0, 1) depending on λ₀, M₀, K₁, K₂, ρ₀, K₀.

We would like to remark that the propagation of smallness in Theorem 4.1 is valid regardless the locations of D and B_r0(x₀), 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, M₀, K₁, K₂), 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, M₀, K₁, K₂) and u ∈ H_loc¹ (Ω) solves L_ζu = 0 in U . Then there exist positive constants R = R(n, λ₀, M₀) and s = s(λ₀, K₁, K₂) such that if 0 < r₀ < r₁ < λ₀r₂/2 < √

λ₀R/2 with B_r2(x₀) ⊂ U , then

kuk_L²_(Br1(x0))≤ Ckuk^τ_L2(B_r0(x0))kuk^1−τ_L2(B_r2(x0)), (4.1) where C is explicitly given by

C = e^{C1(r−s0 −r2−s)} with C1 > depending on λ0, M0, K1, K2, and

τ = (2r₁/λ₀)^−s− r^−s₂

r₀^−s− r₂^−s = (2r₁/r₂λ₀)^−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, M₀, K₁, K₂) and u ∈ H_loc¹ (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(x₀) ⊂ D, dist(D, ∂U ) ≥ h. Then

kuk_L²_(D) ≤ Ckuk^δ_L2(Br(x0))kuk^1−δ_L2(U ), where

C = C₂ |U | hⁿ

¹₂

e^C3h−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 Rⁿ. 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 ∈ H¹(Ω) solves Lζu = 0 in Ω. Then there exist γ0, de- pending on λ₀, M₀, and h₀, depending on λ₀, M₀, K₁, K₂, ρ₀, K₀, such that if γ < γ₀ and 0 < h ≤ h₀, h < r/2, B_r(x₀) ⊂ D₊, and dist(D, ∂Ω) ≥ h, then

kuk_L²_(D)≤ Ckuk^δ_L2(Br(x0))kuk^1−δ_L2(Ω), where

C = C₁ |Ω|

hⁿ

"

1 + ω_Σ(Σ ∩ Ω) hⁿ⁻¹

¹₂#

e^C3h−s, δ ≥ τ^C2|Ω|hn , with C₁, C₂, C₃ > 0, τ ∈ (0, 1) depending on λ₀, M₀, K₁, K₂, ρ₀, K₀.

The difficult part of proving Theorem4.2 is to obtain L² 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 R₁ = θ ¯R₁, R₂ = θ ¯R₂, θ ∈ (0, 1), the vertical sizes of the regions would scale like θ, while their horizontal sizes would scale like θ¹². 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(x₀) 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 (Rⁿ±, λ₀, M₀, K₁, K₂).

For 0 < θ ≤ 1, let L^θ_ζ˜(·, D)v =X

±

H±div( ˜A±(θ·)∇v±) +X

±

H±



θ ˜W (θ·) · ∇v±+ θ²V (θ·)v˜ ±

 .

Note that if ˜ζ = ˜A±, ˜W , ˜V



∈V (Rⁿ±, λ0, M0, K1, K2), then

 ˜A±(θ·), θ ˜W (θ·), θ²V (θ·)˜ 

∈V (Rⁿ±, λ₀, θM₀, θK₁, θ²K₂).

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

Moreover, if U ⊂ θ⁻¹Ω, we have Z

θU

|u(x)|²dx = θⁿ⁺⁴ Z

U

|u^θ(y)|²dy.

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

Proposition 4.3 Assume that the assumptions in Theorem3.1hold. Let 0 < R₁, R₂ ≤ R, θ ∈ (0, 1], and

L˜γu = 0 in θU3, then

Z

θU2

|u|² ≤ (e^τ0R2 + CR⁻⁴₁ )

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 (x⁰, x_n) ∈ C_ρ0,K0(0). We will try to determine when (x⁰, x_n) ∈ Ψ⁻¹_P (θU₂). To this end, we introduce the notation

y⁰ = x⁰, y_n= x_n− ψ(x⁰).

It is clear that (x⁰, xn) ∈ Ψ⁻¹_P (θU2) if and only if θ⁻¹(y⁰, yn) ∈ U2. We denote η(x⁰) = ψ(x⁰)

|x⁰|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 δR₁

6aα₋,2δR₂ 3α₋ , R₁

12a, θ⁻¹ρ₀, ρ₁, ρ₂, ρ₃



, (4.2)

then Ψ_P(B_r(P )) ⊂ θU₂, where



















ρ₁ = α−δ δ + β,

ρ₂ is chosen such that 2kηkρ₂+ kηk²ρ²₂ < 1

2, kηk = kηk_L^∞_(B⁰

ρ0(0)), ρ₃ = 2α−δ

β .

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

Lemma 4.1 Ψ⁻¹_P (θU₃) is contained in a ball of radius

θ





(1 + 2kηk²) 2α−

a R1+ 8δR2

 + 1

a²



2 + (1 + 2kηk²)β δ



R²₁+ 128δ²R²₂ h

α−+ q

α²₋− 8βR2

i2







1/2

centered at P .

Finally, we need to estimate the distance from Ψ⁻¹_P (θU₁) to Σ ∩ C_ρ0,K0. It was proved in [CW, Lemma 3.3] that

dist(Ψ⁻¹_P (θU₁), Σ) > θR₁

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 R₁, R₂ 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 (θU₃) ⊂ B_θd(P ). We then choose θ such that θd = ^h₂, which implies Ψ⁻¹_P (θU₃) ⊂ Ω for any P ∈ Σ ∩ D. Of course, this choice is not possible if h is too large. Therefore, we need to set h₀ small enough, depending on ρ₀, K₀, λ₀, M₀, K₁, K₂.

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 (θU₁), Σ) > µh.

Note that, depending on the geometry of Σ, we again need to set h₀ and R small enough so that Ψ⁻¹_P (θU₁) ∩ 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 (θU₂). By Vitali’s covering lemma, there exist finitely many P₁, . . . , P_N ∈ Σ ∩ D so that

Σ^νh∩ D ⊂

N

[

j=1

Ψ⁻¹_P

j(θU₂), (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)

hⁿ⁻¹ ≤ Cω_Σ(Σ ∩ Ω)

hⁿ⁻¹ . (4.5)