包圍重構法在非等向性介質下的發展與殘留應力的彈性系統之強唯一連續性

國立臺灣大學理學院數學系 博士論文

Department of Mathematics College of Science

National Taiwan University Doctoral Dissertation

包圍重構法在非等向性介質下的發展與殘留應力的彈 性系統之強唯一連續性

The development of the Enclosure Method in an Anisotropic Background and the Strong Unique Continuation for the Elasticity with Residual Stress

林奕亘 Yi-Hsuan Lin

指導教授：王振男 教授 Advisor: Prof. Jenn-Nan Wang

中華民國 105 年 1 月

January, 2016

誌 謝

本文能順利呈現，要感謝我的指導老師王振男教授，在這當中他給我許多 的幫忙，也不辭辛勞的與我討論這相關的內容。王振男老師在我從碩士念到博 士的這六年的歲月中，不但給我學業上的激勵指點，生涯規劃的提醒以及面對 工作的態度，都讓我獲益良多。在 2014 年的 1 月底時，王老師推薦我代表理學 院去參加在新加坡舉辦的 GYSS-全球青年科學家論壇，這個論壇雖然不是針對 數學家舉辦，但是同為科研的人，互相討論互相激勵是一件很重要的事情。在 那個論壇中我認識了很多來自全世界各地的年輕科學家，也認識了幾位費爾茲 獎得主，這個論壇完全打開了我對科學研究的眼界。而在該年二月，王老師給 了我另一個機會去奧地利的 Linz，RICAM 學術機構訪問 Mourad Sini 教授，也 開啟了我對反問題中另一個問題分支的興趣，也在當時奠定了完成 Maxwell 這 個問題的基礎。在 2014 年的 12 月時，王老師給我一個機會在台灣大學舉辦的 逆問題研討會中做第一次報告，報告的內容是我的第一篇文章。這次的經驗非 常重要，因為這是我第一次在國際性的研討會做一個報告，讓我知道如何在真 正的國際會議中給出比較有水準的演講。

在 2015 年的時候，王老師給了我更多的機會，包括一月份在東京明治大學 的台灣日本年輕學者應用數學研討會，給了一個小演講；在同年五月，到芬蘭 赫爾辛基參加逆問題領域中規模最大的會議：應用逆問題大會。超過五百位研 究反問題的專家學者學生，齊聚赫爾辛基大學，這兩周的收穫完全無法用言語 形容。在六月時，去上海的華東師範大學參加第六屆華東偏微分方程年會，也 學了很多有關方程研究的看法。在同年七月也去了京都大學訪問 Iso 教授，在 京都大學也給了兩個演講。九月底去香港科技大學參加 Prof. Uhlmann 舉辦的 逆問題會議，也是認識很多同行的新朋友還有對於數學的見識更上一層樓。以 上總總的出國增廣見聞機會，皆是王振男老師不遺餘力的對我無私的付出。

在研究所求學的過程得到很多老師的幫助，特別感謝林紹雄教授、林長壽 教授、王藹農教授、陳俊全教授、夏俊雄教授的指導，讓我對數學有不同的觀 點。最後就是我的家人，對於學習數學給予許多的的支持，讓我有勇氣去追尋 我的夢想，不管是低潮、壓力、快樂、喜悅，總是給我最大的溫暖與包容，也 以我為榮。謹以此篇論文獻給一路陪伴我的大家，謝謝你們。

ii

中文摘要

這篇論文的目的是在三維中非等向性介質下重構可穿透與不可穿透障礙 物。我們將會示範如何利用包圍法重構對於以下兩種數學模型：非等向性的橢 圓方程以及非等向性的馬克士威方程。到目前為止，對於非等向性的數學模 型，沒有可以利用的複幾何光學解用來重構未知障礙物。因此我們將會使用另 一種特別解：震盪遞減解使用在我們的逆問題之中。

特別的，在這篇文章中，我們會介紹一種新的轉換法，把非等向性的馬克 士威方程轉變成一個二階線性強橢圓系統。這個方法是用來建構非等向性的馬 克士威方程的震盪遞減解。而在此篇文章的最後，我們將會討論強唯一連續性 質對於 Gevrey 係數的殘留應力系統。

iii

Abstract

The goal of this dissertation is to develop reconstruction schemes to determine penetrable and impenetrable obstacles in a region in 3-dimensional in an anisotropic background. We demonstrate the enclosure-type method for two different

mathematical models: The anisotropic elliptic equation and the anisotropic Maxwell system. So far, in the anisotropic case, there are no complex geometrical optics solutions which we can use to reconstruct the unknown obstacles in a given medium.

Therefore, we use another special type solution: the oscillating decaying solutions, which are useful in our inverse problems.

In particular, for the anisotropic Maxwell system model, we also introduce a new reduction method to transform the Maxwell system into a second order strongly elliptic system. This reduction method is the main tool to construct the oscillating decaying solutions for the anisotropic Maxwell system. In addition, we prove the strong unique continuation for a residual stress system with Gevrey coefficients.

iv

目 錄

口試委員審定書………i

誌謝………ii

中文摘要………iii

英文摘要………iv

1 Preliminaries 1

2 The enclosure method for the second order elliptic equations 4

2.2 Ideas of the enclosure method……….…...6

2.3 Complex geometric optics solutions and related topics……….…8

2.4 The enclosure-type method: Second order anisotropic elliptic equations....12

3 The enclosure method for the Maxwell system . 38

3.1 Basic properties for the Maxwell system ...………...38

3.2 Enclosing unknown obstacles in the isotropic media ...………....41

3.3 Constructing CGO solutions ………...42

3.4 Proof of Theorem 3.3 ………45

3.5 Enclosing unknown obstacles in the anisotropic media ………....51

3.6 A new reduction method: From anisotropic Maxwell system to the second order strongly elliptic system ………55

3.7 Constructing of oscillating-decaying solutions for the anisotropic Maxwell system ………67

3.8 Proof of Theorem 3.13 ………..76

4 Strong unique continuation for a residual stress system with Gevrey Coefficients 100

v

4.1 SUCP for the elliptic equation ………..100

4.2 Basic properties for the Gevrey class ………102

4.3 SUCP for the residual stress system with Gevrey coefficients ………….103

4.4 Reduction to a fourth order elliptic system ………...106

4.5 The asymptotic behavior of u near 0 ……….112

4.6 Proof of the main theorem.……….113

5 Future work 117

5.1 Fundamental solutions for the anisotropic Maxwell system ……….117

5.2 More Lp estimates for the anisotropic Maxwell system ………...118

5.3 Strong unique continuation for the general second order elliptic system .118

Bibliography .119

vi

Chapter 1

Preliminaries

Inverse boundary value problem is a field of discussing the inverse problems of partial differential equations. The inverse boundary value problems have become a popular field since A.P. Calderón published his pioneering work “On an inverse boundary value problem” [1] in 1980s. The problem proposed by Calderón is: “Is possible to determine the electrical conductivity of a medium by making voltage and current measurements on its boundary ?” More specifically, for each volt- age density on the boundary, there would be the corresponding current which can be measured theoretically on the same periphery. In addition, the Calderón problem is also called the inverse conductivity problem.

Under the assumptions of no sources or sinks of current in Ω, a voltage potential f at the bound- ary ∂Ω induces a voltage potential u in Ω, which solves the Dirichlet problem for the conductivity

equation, 









∇ · (γ∇u) = 0 in Ω,

u = f on ∂Ω.

Since γ is positive, there exists a unique weak solution u∈ H¹(Ω) for any boundary value f ∈ H^1/2(∂Ω). One can define the Dirichlet-to-Neumann map (termed as DN map hereafter) formally as

Λγ : f → γ∂u

∂ν|∂Ω.

The question is whether this DN map uniquely determines the conductivity γ in Ω. This problem led to the development of the Electrical Impedance Tomography (EIT), an imaging method with potential applications in medical imaging and nondestructive testing. The ideas of solving Calderón problem is based mainly on gaining information from boundary data, which can be extended to tackle many physical issues in the reality. The questions evolve from theoretical determinations to practical reconstructions. For example, boundary measurements determines the information of

unknown obstacles in a given medium.

By extending Calderón’s ideas, we not only can determine the conductivity γ from the bound- ary information but also we can reconstruct unknown obstacles in a given subject. There are several reconstruction methods to know the information of unknown obstacle inside a given do- main: The enclosure method(Ikehata), the linear sampling method(Colton-Kirsch), the factoriza- tion method(Kirsch) and the singular source method(Potthast). The enclosure method can be applied in the following: the subject contains unknown obstacles and the conductivity is unknown in the unknown obstacle which is different from the background. The enclosure method is not only a theoretical detection method but also provides an algorithm to draw the unknown obstacle. In this article, we will illustrate how to reconstruct unknown obstacles in a known background from boundary information. We will employ a nondestructive method: “the enclosure method”, which was first introduced by Ikehata [19].

The simplest inverse obstacle problem has the following formulation. Let u be a solution of the

conductivity equation 









∇ · (γ∇u) = 0 in Ω,

u = f on ∂Ω,

where γ(x) := 1 + γDχD, where D b Ω is an unknown obstacle in Ω. By defining the DN map ΛD: f→ γ∂u

∂ν|∂Ω, we are able to explore the shape of D through the above reconstruction method and the DN map ΛD. This geometrical inverse problem is quite well studied in the literature see [24] and several methods have been proposed to solve it. In this chapter, we focus on the enclosure method, which is initiated by Ikehata, see for examples [17, 19], and developed by many researchers [27, 30, 44, 53, 55, 61], [26, 55] for the acoustic model, [25, 30] for the Lam´e model and [27, 66] for the Maxwell model. The testing functions used in [27, 66] are complex geometric optics (CGO) solutions of the isotropic Maxwell’s equation. The construction of CGO solutions for isotropic inhomogeneous Maxwell’s equations is first proposed in [51]. After that, the authors in [28] also constructed CGO solutions for some special anisotropic Maxwell’s equations. However, there are not yet of CGO solutions for general anisotropic Maxwell system. Besides, CGO solutions, another kind of special solutions for anisotropic elliptic system was proposed for substitution in [48] and [49]. They are called oscillating-decaying (OD) solutions.

This thesis is organized as follows. In Chapter 2, we first review the idea of the enclosure method for the isotropic scalar elliptic equations and generalize such a concept to the anisotropic scalar elliptic equations and fully examine the enclosure method including the complex geometric optics (CGO) solutions and the indicator functional (or indicator function). Consequently, the theoretical linkage between the enclosure method and the Calderón’s problem will be presented.

In the absence of CGO solutions for the anisotropic elliptic equation inR³, we introduce another

special solutions called the oscillating-decaying solutions and use the Runge approximation property to obtain our reconstruction algorithm in the anisotropic case, which primarily steams from the enclosure method (intuitively, viewed as the enclosure-type method). In addition, the traditional indicator function requires some modifications.

In Chapter 3, we confine the framework of the enclosure method to the isotropic Maxwell system, which has been addressed in [66], in the similar approach of the CGO solutions and suitable choice of the indicator function. Instead, we can define the impedance map, the counterpart for the elliptic case (that is, the DN map). The indicator function and the reconstruction algorithm are adjusted due to the slight differences between the impedance map and the DN map. Stretching the result of isotropic case to the anisotropic case poses plenty of challenges. We thereby propose a new reduction method which transforms the anisotropic Maxwell system into a second order strongly elliptic system in R³. Further, given the relationship between the oscillating-decaying solutions and the strongly elliptic system, we utilize the newly-proposed (reduction) method to convey such the relation to the anisotropic Maxwell systems and, in turn, derive the representation of the oscillating-decaying solutions of the anisotropic Maxwell system.

In the following chapter, we prove the strong unique continuation property (SUCP) for a residual stress system with Gevrey coefficients on the basis of the SUCP for the scalar elliptic equations with coefficients in the Gevrey class. Finally, we provide some guidelines for the future works.

Chapter 2

The enclosure method for second order elliptic equations

The enclosure method is to reconstruct an unknown obstacle in a known background, which was first introduced by Ikehata, see [17]. The main tool of this reconstruction algorithm is by the indi- cator functional and the complex geometric optics (CGO) solutions. The idea of these tools can be traced back to the Calderón’s pioneering work. In the following, we will show the relations between Calderón’s work and these tools. Moreover, if the mathematical models are complicated, we need to introduce appropriate elliptic regularity estimates (C^αestimates or Meyers’ L^p estimates), we will discuss these details in the following sections.

2.1 Calderón’s problem

In 1980s’, Calderón published his pioneer work “On an inverse boundary value problem” [1]. His work affected the development of the inverse boundary value problem and the inverse conductivity problem. We will give a brief introduction about the Calderón problem how to relate to the enclosure method.

Let us begin by giving the mathematical model. Let Ω⊂ Rⁿ be a bounded open subset inRⁿ for n = 2, 3 with C^∞boundary. Assume that γ > 0 is a C² function defined on Ω. Let u∈ H¹(Ω)

satisfy 









∇ · (γ∇u) = 0 in Ω,

u = f on ∂Ω,

(2.1.1)

where f∈ H^1/2(∂Ω). It is well-known that (2.1.1) has a unique weak solution u∈ H¹(Ω). We can

define the Dirichlet-to-Neumann (DN) map formally as

Λγf = γ∂u

∂ν|∂Ω.

More precisely, the DN map is defined weakly as

(Λγf, g)∂Ω= ˆ

Ω

γ∇u · ∇vdx, f, g ∈ H^1/2(∂Ω),

where u is the solution of (2.1.1) and v is any function in H¹(Ω) with v|∂Ω= g. The pairing on the boundary is integration with respect to the surface measure

(f, g)∂Ω= ˆ

∂Ω

f gdS.

With the definition, we know that Λγ is a bounded linear map from H^1/2(∂Ω) into H^−1/2(∂Ω).

The Calderón problem (also called the inverse conductivity problem) is to determine the con- ductivity function γ from the knowledge of the map Λγ. That is, if the measured current Λγf is known for all boundary voltages f ∈ H^1/2(∂Ω), one would like to determine the conductivity γ.

There are several aspects of this inverse problem which are interesting to both the mathematical theory and the practical applications. When Ω⊂ Rⁿ for n≥ 3, we have the following results.

1. Uniqueness. If Λγ1 = Λ_γ2, we have γ1= γ₂. The result was proved by Sylvester-Uhlmann [57] in 1987.

2. Reconstruction. Given the boundary measurements Λγ, find a procedure to reconstruct the conductivity γ. There is a convergent algorithm which was found by Nachman [42].

3. Stability. If Λγ₁ is close to Λγ₂ in a suitable sense, then γ1 and γ2 are close. In 1988, Alessandrini [2] proved that if γj ∈ H^s(Ω) for s >n

2 + 2, ∥γj∥H^s(Ω)≤ M and 1

M ≤ γj≤ M (j = 1, 2). Then

∥γ1− γ2∥L^∞(Ω)≤ ω(∥Λγ1− Λγ2∥H^1/2(∂Ω)→H^−1/2(∂Ω)),

where ω(t) = C| log t|^−σ for small t > 0 and C = C(Ω, M, n, s) > 0, σ = σ(n, s)∈ (0, 1).

4. Partial data. If Γ is a subset of ∂Ω and if Λγ₁f|Γ = Λγ₂f|Γfor all boundary voltages f , show that γ1= γ2. When Ω is convex and Γ is any open subset of ∂Ω, Kenig-Sj¨ostrand-Uhlmann then proved this result in [29].

In order to deal with the Calderón problem, Calderón considered the following nonlinear map

Qγ(f ) :=

ˆ

Ω

γ|∇u|²dx = ˆ

∂Ω

Λγ(f )· fds, (2.1.2)

where ds is the standard surface measure and u solves (2.1.1) with u|∂Ω = f . Calderón proved Qγ is analytic and the Frechet derivative of Qγ at γ0is injective whenever γ0is a constant, which means the map from γ to Qγ is injective for constant conductivity γ.

Calderón’s work has made the huge influences in the inverse problems. The nonlinear map Qγ

can be widely applied to other areas of the inverse problems. For example, Qγ is also called the indicator functional, which is useful in the enclosure method for the reconstruction of unknown obstacles. Moreover, Calderón took a special harmonic function u = e^{x·(ρ+iρ⊥)}to show the injec- tivity of the linearized map, where ρ∈ Cⁿ with ρ· ρ^⊥= 0. In addition, we call e^{x·(ρ+iρ⊥)}to be the complex geometric optics (CGO) solutions and the CGO solution plays an important role in the inverse problem, for more details, we refer readers to [60].

2.2 Ideas of the enclosure method

We give two different examples to demonstrate ideas of the enclosure method. Here is the math- ematical setting: Let Ω ⊂ Rⁿ, for n = 2, 3 and D b Ω be an unknown obstacle. We con- sider the simplest case in the following: Let γ0 ≡ 1 be a given conductivity on the background medium and eγ(x) = γ0 + γD(x)χD = 1 + γDχD be a total conductivity defined on Ω, where

χ_D=









1, if x∈ D 0, otherwise

is the characteristic function of domain D and γD(x) > 0 is a bounded func-

tion on Ω. Then we have two different conductivity equations with boundary value f ∈ H^1/2(∂Ω),









∆u0= 0 in Ω, u₀= f on ∂Ω,

and 









∇ · (eγ(x)∇u) = 0 in Ω,

u = f on ∂Ω,

where u0 is the voltage when D = ∅ (no unknown obstacles in Ω) and u is the voltage when D ̸= ∅(Ω contains unknown obstacles). Then we can define the Dirchlet-to-Neumann maps: For f ∈ H^1/2(∂Ω),

Λγ₀(f ) = γ0

∂u0

∂ν |∂Ω,

and

Λ_eγ(f ) =eγ∂u

∂ν|∂Ω,

where ν is a unit outer normal on ∂Ω. The enclosure method consists of two tools: The indicator functional and the special solutions (CGO solutions).

First, we introduce the indicator functional and the ideas came from the nonlinear map Qγ we have defined in the previous section (see (2.1.2)). We consider the indicator functional

E(f ) :=

ˆ

∂Ω

(Λ_eγ(f )− Λγ0(f ))· fdS.

If given voltage f on ∂Ω, we can regard E(f ) as a difference of currents or energies corresponding to the situations with and without D. Furthermore, due to the positivity property of the equations, we can easily derive

E(f )≈ C ˆ

D

|∇u0|²dx,

where C > 0 is independent of f and u0 and recall that u0solves ∆u0= 0 with u0|∂Ω= f . Second, it is not hard to see for any h > 0,

u₀= e^{1h(ρ·x+iρ⊥·x)}

is a solution of the Laplace equation, where ρ, ρ^⊥∈ Sⁿ⁻¹ (for n = 2, 3) and ρ· ρ^⊥= 0. Note that the special function u0 was also appeared in [1], which was proposed by Calderón. Moreover, let d∈ R be arbitrary,

u0,d,h= e^−dhe^{h1(ρ·x+iρ⊥·x)}

satisfies the Laplace equation, i.e. ∆u0,d,h= 0.

We set f0,d,h = u0,d,h|∂Ω and take f0,d,h into the indicator functional E(f ) = E(f0,d,h), then we have

E(f_0,d,h) ≈ C ˆ

D

|∇u0,d,h|²dx

≈ C^′ 1 h²

ˆ

D

e^{2h(ρ·x−d)}dx

for some positive constants C, C^′ independent of h. Now, we define the support function hD(ρ)

hD(ρ) := sup

x∈Dx· ρ and let d := hD(ρ), then we have the following two situations:

1. If x∈ {ρ · x > d}, then we can see that

u0,d,h→ ∞ as h → 0 + .

In addition, we also have

E(f0,d,h)→ ∞ as h → 0 + .

2. If x∈ {ρ · x < d}, then we can see that

u_0,d,h→ 0 as h → 0 + .

In addition, we also have

E(f0,d,h)→ 0 as h → 0 + .

Then from the limiting behaviors of E(f0,d,h) as h tends to 0, we can conclude that if we choose f_0,d,h to be our testing boundary measurements, then the limit behavior of E(f0,d,h) will tell us whether the level set{x · ρ = d} touches ∂D or not. By varying the direction ρ and the real value d, we can reconstruct a convex hull for the unknown obstacle D theoretically.

Remark 2.1. We call E(f ) to be the indicator functional. In fact, in [20], Ikehata called E(f0,d,h) the indicator function.

Let us summarize the ideas of previous reconstruction procedures. First, we define the indicator function E(f ) from the DN map on the boundary. Second, we construct a sequence of special so- lutions u0,d,h(CGO solutions) for the Laplace equation, and let f0,d,h= u_0,d,h|∂Ωbe the boundary testing functions, then the limit behavior of E(f0,d,h) will tell us whether the level set{x · ρ = d}

touches ∂D or not when h tends to 0. It looks like to use the hyperplanes to enclose the unknown obstacle D in Ω, and named the enclosure method.

2.3 Complex geometric optics solutions and related topics

Since Ikehata proposed the idea of the enclosure method, there are many applications of this method to other physical problems. We will show how to extend the ideas to different physical settings and related results.

Recall that the enclosure method contains two different tools: The indicator function and the special solutions. In different mathematical problems, we can define similar indicator functions via the Dirichlet-to-Neumann map (for the Maxwell system, we define the impedance map, it will be seen in Chapter 3). The main problem lies on how to find a suitable sequence of testing functions, which satisfy the specific partial differential equation. For example, we know that

u_0,d,h = e^{h1(x·ρ−d+ix·ρ⊥)} solves the Laplace equation. Notice that u0,d,h are harmonic functions with complex phases. By using the following form of solutions,

e^i1hρ(x)(a(x) + Rh(x)),

one can construct approprate testing data with a complex phase function ρ(x) and Rh(x)≪ a(x) as h→ 0+. The solutions with this form are so-called the complex geometric optics (CGO) solutions, which play an essential role in the enclosure-type method.

The results to the existence of CGO solutions for various mathematical problems and CGO solutions are useful for the inverse boundary value problem, for example, see [56, 57, 51, 52, 18, 16, 61]. In particular, CGO solutions play an important role of the probing method in the enclosure type method, we refer readers to [16, 17, 19, 22, 23, 43, 55, 54, 58, 61, 66].

aa

From linear phase to general phase

From Ikehata’s previous work, he used the Calderón’s harmonic function e^{x·(ρ+ρ⊥)}to construct the boundary testing data. The phase function x· (ρ + ρ^⊥) is linear and we use it to enclose the unknown obstacle. By using the linear phase type harmonic function, we can only reconstruct the convex hull of the unknown obstacle. One can refer to a survey paper [21] for detailed explanation and early development of this theory. In [54, 45, 16], the writers used the complex spherical wave solutions to detect concave parts of the unknown obstacles. Moreover, in [61], the researchers proposed a framework to construct the CGO solutions with general phases for some elliptic systems in 2 dimension. This work provides more choices for the phase function of the CGO solutions in 2D. They also gave a concrete example: the CGO solutions with complex polynomial phases and apply these CGO solutions for the conductivity equations to determine unknown obstacles with more general shapes. This type of CGO solutions were also applied to elastic system [64] and Helmholtz equation [43].

More results for the Helmholtz type equation

Recall that we know that e^{1h(x·ρ−d+ix·ρ⊥)}are CGO solutions for various h, d∈ R and ρ ∈ Sⁿ⁻¹ for n∈ N (we only consider n = 2, 3). For more general mathematical models, we can consider the

following problem 









∇ · (eγ(x)∇u + k²u = 0 in Ω,

u = f on ∂Ω,

(2.3.1)

where eγ(x) = 1 + γDχD, for some γD > 0, γD ∈ L^∞(D) and χD is the characteristic function defined on D. For the unperturbed case, i.e. when D =∅, we have the Helmholtz equation









∆u₀+ k²u₀= 0 in Ω,

u₀= f on ∂Ω.

(2.3.2)

Now, we want to know the information of the unknown obstacle Db Ω.

In the beginning, we need to define the DN map

ΛDf :=∂u

∂ν|∂Ωand Λ_∅f := ∂u0

∂ν |∂Ω,

where u and u0 are solutions of (2.3.1) and (2.3.2), respectively and ν is a unit outer normal on

∂Ω. Similarly, we can define the indicator function

E(f ) :=

ˆ

∂Ω

(Λ_D− Λ_∅)f· fdS,

and use integration by parts many times, we will obtain the upper bound estimates and the lower bound estimates for E(f ):

E(f )≤ C ˆ

D

|∇u0|²dx + k² ˆ

Ω

|w|²dx

and

E(f )≥ c ˆ

D

|∇u0|²dx− k² ˆ

Ω

|w|²dx,

where c, C are independent of u0, w and w = u− u0is called the reflected solution satisfying









∇ · (eγ(x)∇w) + k²w =−∇ · (eγ(x) − 1)∇u0 in Ω,

w = 0 on ∂Ω.

(2.3.3)

For more calculation details, we refer readers to [43]. Note that the upper and lower bounds only involve u0 and w. Our remaining task is to find appropriate estimates for´

Ω|w|². In fact, there are two different approaches for ´

Ω|w|²dx: One is the C^α-estimates method which was first introduced by [43] and the other is Meyers’ L^p estimates method which was first introduced by [55]. We give a brief comparison with C^α-estimates method and Meyers’ L^pestimate method. Note that in the following estimates, the constants C may change line to line, and they are independent of u0and w.

1. C^α-estimates method: This method was introduced in [43]. Recall that we have an upper

bound for the indicator function

E(f )≤ C ˆ

D

|∇u0|²dx + k² ˆ

Ω

|w|²dx.

By (2.3.3) and the standard elliptic regularity estimates, we have ˆ

Ω

|w|²dx≤ C ˆ

D

|∇u0|²dx,

then we obtain

E(f )≤ C ˆ

D

|∇u0|²dx.

The main problem appears on the lower bound for E(f ). In [43], the authors defined a new function

Ix₀,α :=

ˆ

∂D

∂u0

∂ν

|x − x⁰|^αdS,

for any x0∈ Ω, then they derived ˆ

Ω

|w|²dx≤ Cq,α{Ix²₀,α+ Ix0,α∥∇u0∥L^q(D)+∥u0∥²L²(D)}, (2.3.4)

for any α∈ (0, 1) and q ∈ (2, 4]. The estimate (2.3.4) relies on the C^α-estimates for the elliptic equation, which were proved in the paper [35]. In order to apply this type C^α-estimate, we need to add regularity assumptions on the unknown obstacle D, which is ∂D ∈ C². In addition, we know that

u0:= e^{1h(x·ρ−d)+i√τ2+k2x·ρ⊥} (2.3.5)

are CGO solutions for the Helmholtz equation. Combine the lower bound of E(f ), (2.3.4) and put the CGO solutions (2.3.5) into the indicator function E(d, h) := E(f0,d,h) = E(u0,d,h|∂Ω), then we can obtain









E(d, h)→ 0 as h → 0+ if ω· x < hD(ρ), E(d, h)→ ∞ as h → 0+ if ω· x > hD(ρ),

(2.3.6)

where hD(ρ) = inf_x∈Dx· ρ is the support function we have mentioned before.

2. Meyers’ L^p-estimates method: This method was introduced in [55]. Similarly, since the upper bound of E(f ) can be obtained by the standard elliptic regularity, we only need to take care of the lower bound of E(f ). Recall that w is the reflected solution of (2.3.3), and in [39], the author derived the following estimates (Meyers’ L^p estimates): Assume Db Ω and ∂D is Lipschitz. For every p0> 2, there exists a positive constant Cp₀ independent of w and u0

such that

∥w∥L²(Ω)≤ Cp₀∥u0∥W^1,p(D), (2.3.7)

for p∈ (6

5, p0]. In addition, by (2.3.7), we have

E(f )≥ c ˆ

Ω

|∇u0|²dx− c∥u0∥²W^1,p(D).

In [55], the authors used a decomposition technique to obtain the lower bound of E(f ), and we will give details in the next chapter (section 3.4.2). Note that the key point is that we only need ∂D is Lipschitz. In summary, we can use the Meyers’ L^p estimates to obtain the same result (2.3.6). For more enclosure methods for the Helmholtz-type equations, we refer readers to the survey paper [65].

From Laplacian leading term to general elliptic operator

Until now, we only considered the case when the mathematical models with the Laplacian as the leading order term. For the leading term - Laplacian, we call this mathematical model to be isotropic. In order to consider more general situation, we need to consider the equations or systems with non-Laplacian leading terms and we call the case to be anisotropic. However, the anisotropy of the non-Laplacian prevents us from constructing CGO solutions by using the standard methods.

As a result, in [48], the authors constructed another special type of solutions which is called the oscillating-decaying (OD) solutions. The OD solutions are also useful in the inverse problems, especially for the reconstruction problems. In two-dimensional case, we can use the isothermal coordinates to transform a general second order elliptic equation into Laplacian type equations.

However, for three-dimensional case, we do not know how to construct CGO solutions yet, we will use OD solutions to reconstruct the unknown obstacles. We will give all the details in the next section.

2.4 The enclosure-type method: Second order anisotropic elliptic equations

In this section, we develop an enclosure-type reconstruction scheme to identify penetrable obstacles in acoustic waves with anisotropic medium in R³. The main difficulty of treating this problem lies in the fact that there are no complex geometrical optics solutions available for the acoustic equation with anisotropic medium in R³. Instead, we will use another type of special solutions called oscillating-decaying solutions. Even though that oscillating-decaying solutions are defined only on the half space, we are able to give necessary boundary inputs by the Runge approximation

property. Moreover, since we are considering a Helmholtz-type equation, we turn to Meyers’

L^p estimate to compare the integrals coming from oscillating-decaying solutions and those from reflected solutions.

2.4.1 Problem for the anisotropic elliptic equation

In the study of inverse problems, we are interested in the special type of solutions for elliptic equations or systems which play an essential role since the pioneer work of Cald´eron. Sylvester and Uhlmann [57] introduced complex geometric optics (CGO) solutions to solve the inverse boundary value problems of the conductivity equation. Based on CGO solutions, Ikehata proposed the so called enclosure method to reconstruct the impenetrable obstacle, for more details, see [17, 20, 21].

There are many results concerning this reconstruction algorithm, such as [43, 62]. The researchers constructed CGO-solutions with polynomial-type phase function of the Helmholtz equation ∆u + k²u = 0 or the elliptic system with the Laplacian as the principal part.

When the medium is anisotropic, we need to consider more general elliptic equations, such as anisotropic scalar elliptic equation in a bounded domain Ω⊂ R³,

∇ · (A⁰(x)∇u) + k²u = 0, (2.4.1)

where A⁰(x) = (a⁰_ij(x)), a⁰_ij(x) = a⁰_ji(x), and we assume the uniform ellipticity condition, that is, for all ξ = (ξ1, ξ2,· · · ξn)∈ Rⁿ, λ⁰|ξ|² ≤∑

i,ja⁰_ij(x)ξiξj ≤ Λ⁰|ξ|² and x ∈ Ω. In two dimen- sional case, we can transform (2.4.1) to an isotropic equation by using isothermal coordinates, then we can apply the CGO-solutions for this case, which can be found in [58]. When Ω ⊂ R³, we cannot directly transform (2.4.1) to an isotropic equation as we do inR², thus we need to use the oscillating-decaying solutions in our reconstruction algorithm. In [46], the author introduced oscillating-decaying solutions for the conductivity equation∇ · (γ(x)∇u) = 0 with the isotropic conductivity.

We make the following assumptions.

1. Let Ω⊂ R³ be a bounded C^∞-smooth domain and assume that D is an unknown obstacle with Lipschitz boundary such that Db Ω ⊂ R³ with an inhomogeneous index of refraction subset of a larger domain Ω.

2. Let A(x) = (aij(x)) and A⁰(x) = (a⁰_ij(x)) be symmetric matrices with aij(x) = a⁰_ij(x) + f

aij(x)χD, where each a⁰_ij(x) is bounded C^∞-smooth, eA(x) = (afij(x))∈ L^∞(D) is regarded as a perturbation in the unknown obstacle D and eA(x)ξ· ξ ≥ eλ|ξ|² for any ξ ∈ R³ and x∈ D with some eλ > 0. Further A(x) satisfies λ|ξ|²≤ A(x)ξ · ξ ≤ Λ|ξ|²for some constants

0 < λ≤ Λ.

Now, let k > 0 and consider the steady state anisotropic acoustic wave equation with Dirichlet

boundary condition 









∇ · (A(x)∇u) + k²u = 0 in Ω

u = f on ∂Ω.

(2.4.2)

For the unperturbed case, we have









∇ · (A⁰(x)∇u0) + k²u₀= 0 in Ω

u₀= f on ∂Ω.

(2.4.3)

In this paper, we assume that k² is not a Dirichlet eigenvalue of the operator −∇ · (A∇•) and

−∇ · (A⁰∇•) in Ω. It is known that for any f ∈ H^1/2(∂Ω), there exists a unique solution u to (2.4.2). We define the Dirichlet-to-Neumann map ΛD: H^1/2(∂Ω)→ H^−1/2(∂Ω) in the anisotropic case as the following.

Definition 2.2. Λ_Df := A∇u · ν =∑3

i.j=1a_ij∂_ju· νiand Λ_∅f := A⁰∇u0· ν =∑3

i.j=1a_ij∂_ju₀· νi, where ν = (ν1, ν2, ν3) is a unit outer normal on ∂Ω.

Inverse problem: Identify the location and the convex hull of D from the DN-map ΛD. The domain D can also be considered as an inclusion embedded in Ω. The aim of this work is to give a reconstruction algorithm for this problem. Note that the information on the medium parameter eA(x) = (afij(x)) inside D is not known a priori.

The main tool in our reconstruction method is the oscillating-decaying solutions for the second order anisotropic elliptic differential equations. We use the results from the paper [47] to construct the oscillating-decaying solution. In the next section, we will construct the oscillating-decaying solutions for anisotropic elliptic equations. Note that even if k = 0, which means the equation is ∇ · (A(x)∇u) = 0, we do not know of any CGO-type solutions. Roughly speaking, given a hyperplane, an oscillating-decaying solution is oscillating very rapidly along this plane and decaying exponentially in the direction transverse to the same plane. Oscillating-decaying solutions are special solutions with the imaginary part of the phase function non-negative. Note that the domain of the oscillating-decaying solutions is not over the whole Ω, so we need to extend such solutions to the whole domain. Fortunately, the Runge approximation property provides us a good approach to extend this special solution.

In Ikehata’s work, the CGO-solutions are used to define the indicator function (see [21] for the definition). In order to use the oscillating-decaying solutions to the inverse problem of identifying an inclusion, we employ the Runge approximation property to redefine the indicator function. It

was Lax [31] that first recognized the Runge approximation property is a consequence of the weak unique continuation property. In our case, it is clear that the anisotropic elliptic equation has the weak unique continuation property if the leading part is Lipschitz continuous. Finally, the main theorem and reconstruction algorithm will be presented in the end of this chapter.

2.4.2 Construction of oscillating-decaying solutions

In this section, we follow the paper [47] to construct the oscillating-decaying solution in the anisotropic elliptic equations. In our case, since we only consider a scalar elliptic equation, its construction is simpler than that in [47]. Consider the anisotropic Helmholtz type equation

∇ · (A(x)∇u) + k²u = 0 in Ω. (2.4.4)

Note that the oscillating-decaying solutions of

∇ · (A(x)∇u) = 0 in Ω

will have the same form as the equation (2.4.4), which means the lower order term k²u will not affect the representation of the oscillating-decaying solutions, the following are the construction details. Now, we assume that the domain Ω is an open, bounded smooth domain inR³ and the coefficients A(x) = (aij(x)) is a symmetric 3× 3 matrix satisfying uniformly elliptic condition, which means∑3

i.j=1aij(x)ξiξj ≥ c|ξ|²,∀ξ = (ξ1, ξ2, ξ3)∈ R³ for some c > 0.

Assume that

A(x) = (a_ij(x))∈ B^∞(R³) ={f ∈ C^∞(R³) : ∂^αf ∈ L^∞(R³), ∀α ∈ Z³+}

is the anisotropic coefficients. Note that A(x)∈ B^∞already implies that A is Lipschitz continuous and the Lipschitz continuity property of A(x) will apply the weak unique continuation property of (2.4.4) (see [15] for example).

We give several notations as follows. Assume that Ω⊂ R³is an open set with smooth boundary and ω ∈ S² is given. Let η ∈ S² and ζ ∈ S² be chosen so that {η, ζ, ω} forms an orthonormal system of R³. We then denote x^′ = (x· η, x · ζ). Let t ∈ R, Ωt(ω) = Ω∩ {x · ω > t} and Σt(ω) = Ω∩ {x · ω = t} be a non-empty open set. We consider a scalar function uχ_t,t,b,N,ω(x, τ ) :=

u(x, τ )∈ C^∞(Ωt(ω)\Σt(ω))∩ C⁰(Ωt(ω)) with τ ≫ 1 satisfying:









LAu =∇ · (A(x)∇u) + k²u = 0 in Ωt(ω) u = e^{iτ x·ξ}{χt(x^′)Qt(x^′)b + βχ_t,t,b,N,ω} on Σt(ω),

(2.4.5)

where ξ ∈ S² lying in the span of η and ζ and fixed χt(x^′) ∈ C0^∞(R²) with supp(χt) ⊂ Σt(ω), Qt(x^′) is a nonzero smooth function and 0 ̸= b ∈ C³. Moreover, βχ_t,b,t,N,ω(x^′, τ ) is a smooth function supported in supp(χt) satisfying:

∥βχ_t,b,t,N,ω(·, τ)∥L²(R²)≤ cτ⁻¹

for some constant c > 0. From now on, we use c, c^′ and their capitals to denote general positive constants whose values may vary from line to line. As in the paper [47], uχt,b,t,N,ω can be written as

u_χt,b,t,N,ω = w_χt,b,t,N,ω+ r_χt,b,t,N,ω

with

wχ_t,b,t,N,ω= χt(x^′)Qte^{iτ x·ξ}e^{−τ(x·ω−t)At(x′)}b + γχ_t,b,t,N,ω(x, τ ) (2.4.6)

and rχ_tb,t,N,ω satisfying

∥rχ_t,b,t,N,ω∥H¹(Ω_t(ω))≤ cτ^−N−1/2, (2.4.7)

where At(·) ∈ B^∞(R²) is a complex function with its real part ReAt(x^′) > 0, and γχt,b,t,N,ω is a smooth function supported in supp(χt) satisfying

∥∂x^αγ_χt,b,t,N,ω∥L²(Ω_s(ω))≤ cτ^|α|−3/2e^{−τ(s−t)a} (2.4.8)

for|α| ≤ 1 and s ≥ t, where a > 0 is some constant depending on At(x^′).

Without loss of generality, we consider the special case where t = 0, ω = e3 = (0, 0, 1) and choose η = (1, 0, 0), ζ = (0, 1, 0). The general case can be obtained from this special case by change of coordinates. Define L = LA and fM· = e^{−iτx′·ξ′}L(e^{iτ x′·ξ′}·), where x^′ = (x1, x2) and ξ^′ = (ξ1, ξ2) with|ξ^′| = 1, then fM is a differential operator. To be precise, by using ajl= alj, we calculate fM to be given by

Mf = −τ²∑

jl

a_jlξ_jξ_l+ 2τ∑

jl

a_jl(iξ_l)∂_j+∑

jl

a_jl∂_j∂_l

+∑

jl

(∂_ja_jl)(iτ ξ_l) +∑

jl

(∂_ja_jl)∂_l+ k²

= −τ²∑

jl

ajlξjξl+ 2τ∑

l

a3l(iξl)∂3+ a33∂3∂3

+2τ ∑

j̸=3,l

ajl(iξl)∂j+ ∑

j̸=3,l̸=3

ajl∂j∂l

+∑

jl

(∂jajl)(iτ ξl) +∑

jl

(∂jajl)∂l+ k²

with ξ3= 0. Now, we want to solve

M v = 0,f

which is equivalent to M v = 0, where M = a⁻¹₃₃M . Now, we use the same idea in [47], definef

⟨e, f⟩ =∑

ijaijeifj, where e = (e1, e2, e3), f = (f1, f2, f3) and denote⟨e, f⟩0=⟨e, f⟩ |x₃=0. Let P be a differential operator, and we define the order of P , denoted by ord(P ), in the following sense:

∥P (e^{−τx3A(x′)}φ(x^′)∥L²(R³+)≤ cτ^{ord(P )−1/2},

whereR³+ ={x3> 0}, A(x^′) is a smooth complex function with its real part greater than 0 and φ(x^′)∈ C0^∞(R²). In this sense, similar to [47], we can see that τ , ∂₃ are of order 1, ∂₁, ∂₂ are of order 0 and x3is of order -1.

Now according to this order, the principal part M2 (order 2) of M is:

M2=−{D3²+ 2τ⟨e3, e3⟩⁻¹0 ⟨e3, ρ⟩0D3+ τ²⟨e3, e3⟩⁻¹0 ⟨ρ, ρ⟩0}

with D3 =−i∂3 and ρ = (ξ1, ξ2, 0). Note that the principal part M2 does not involve the lower order term k²·, so we can follow all the constructions in the same procedures as in [47] and we omit details.

2.4.3 Runge approximation property

Definition 2.3. [31] Let L be a second order elliptic operator, solutions of an equation Lu = 0 are said to have the Runge approximation property if, whenever K and Ω are two simply connected domains with K⊂ Ω, any solution in K can be approximated uniformly in compact subsets of K by a sequence of solutions which can be extended as solution to Ω.

There are many applications for the Runge approximation property in inverse problems. Similar results for some elliptic operators can be found in [31], [37]. The following theorem is a classical result for Runge approximation property for second order elliptic equations.

Theorem 2.4. (Runge approximation property) Let L0· = ∇(A⁰(x)∇·) + k²· be a second order elliptic differential operator with A⁰(x) to be Lipschitz. Assume that k²is not a Dirichlet eigenvalue of −∇(A⁰(x)∇·) in Ω. Let O and Ω be two open bounded domains with smooth boundary in R³ such that Ob Ω.

Let u0∈ H¹(O) satisfy

L₀u₀= 0 in O.

Then for any compact subset K⊂ O and any ϵ > 0, there exists U ∈ H¹(Ω) satisfying

L0U = 0 in Ω,

such that

∥u0− U∥H¹(K)≤ ϵ.

Proof. The proof is standard and it is based on the weak unique continuation property for the anisotropic second order elliptic operator L0 and the Hahn-Banach theorem. For more details, how to derive the Runge approximation property from the weak unique continuation, we refer readers to [31]

It remains to use the same ideas which comes from the reflected solutions. Here we use the useful elliptic estimates, which is called the Meyers’ L^p estimates.

2.4.4 Meyers’ L

estimates and some identities

We need some estimates for solutions to some Dirichlet problems which will be used in next section. Recall that, for f∈ H^1/2(∂Ω), let u and u₀be solutions to the Dirichlet problems (2.4.2) and (2.4.3), respectively. Note that aij(x) = a⁰_ij(x) +afij(x)χD and we set w = u− u0, then w satisfies the Dirichlet problem









∇ · (A(x)∇w) + k²w =−∇ · (( eAχ_D)∇u0) in Ω

w = 0 on ∂Ω

(2.4.9)

where A(x) = (aij(x)), A⁰(x) = (a⁰_ij(x)) and eA(x) = (afij(x)). Then we have some estimates for w.

Lemma 2.5. There exists a positive constant C independent of w such that we have

∥w∥L²(Ω)≤ C∥∇w∥L^p(Ω)

for 6

5 ≤ p ≤ 2 if n = 3.

Proof. The proof follow from [55] by Freidrich’s inequality, see [38] p.258 and use a standard elliptic regularity.

Lemma 2.6. There exists ϵ∈ (0, 1), depending only on Ω, A⁰(x) = (a⁰_ij(x)) and eA(x) = (afij(x)) such that

∥∇w∥L^p(Ω)≤ C∥u0∥W^1,p(D)

for max{2 − ϵ,6

5} < p ≤ 2 if n = 3.

Proof. The proof is also followed from [55]. Set f :=−( eAχD)∇u0. Let w0be a solution of









∇ · (A(x)∇w0) =∇ · f in Ω,

w0= 0 on ∂Ω.

(2.4.10)

The following L^p-estimate of w0, known as Meyers estimate, followed from [39], then we can get

∥∇w0∥L^p(Ω)≤ C∥f∥L^p(Ω) (2.4.11)

for p∈ (max{2 − ϵ,6

5}, 2], where ϵ ∈ (0, 1) depends on Ω, A⁰(x) = (a⁰_ij(x)) and eA(x) = (afij(x)).

We set W := w− w0, then since w = w0+ W , we have

∥∇w∥L^p(Ω)≤ C(∥∇w0∥L^p(Ω)+∥∇W ∥L^p(Ω)). (2.4.12)

Moreover, W satisfies 









∇ · (A(x)∇W ) + k²W =−k²w0 in Ω,

W = 0 on ∂Ω.

(2.4.13)

By the standard elliptic regularity, we have

∥W ∥H¹(Ω)≤ C∥w0∥L²(Ω).

Thus, we get for p≤ 2,

∥∇W ∥L^p(Ω)≤ C∥∇W ∥L²(Ω)≤ C∥W ∥H¹(Ω)≤ C∥w0∥L²(Ω). (2.4.14)

By Sobolev embedding theorem, we get

∥w0∥L²(Ω)≤ C∥w0∥W^1.p(Ω) (2.4.15)

for p≥ 6

5 if n = 3. Use Poincar�e’s inequality in L^p spaces (w₀|∂Ω= 0), we have

∥w0∥L²(Ω)≤ C∥∇w0∥L^p(Ω) (2.4.16)

for p≥ 6

5 if n = 3. Combining (2.4.11) with (2.4.12), (2.4.14) and (2.4.16), we can obtain

∥∇w∥L^p(Ω)≤ C∥f∥L^p(Ω)≤ C∥u0∥W^1,p(D)