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].

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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