For both systems, we provide explicit formulas to reconstruct the parameters on the boundary as well as its rate of convergence formula

AIMS’ Journals

Volume X, Number 0X, XX 200X pp. X–XX

BOUNDARY DETERMINATION OF ELECTROMAGNETIC AND

LAM ´E PARAMETERS WITH CORRUPTED DATA

Pedro Caro

Basque Center for Applied Mathematics, Bilbao, Spain

Ru-Yu Lai

School of Mathematics, University of Minnesota, USA

Yi-Hsuan Lin

Department of Applied Mathematics, National Chiao Tung University, Taiwan

Ting Zhou

Department of Mathematics, Northeastern University, USA

(Communicated by the associate editor name)

Abstract. We study boundary determination for an inverse problem associ- ated to the time-harmonic Maxwell equations and another associated to the isotropic elasticity system. We identify the electromagnetic parameters and the Lam´e moduli for these two systems from the corresponding boundary measure- ments. In a first step we reconstruct Lipschitz magnetic permeability, electric permittivity and conductivity on the surface from the ideal boundary mea- surements. Then, we study inverse problems for Maxwell equations and the isotropic elasticity system assuming that the data contains measurement errors.

For both systems, we provide explicit formulas to reconstruct the parameters on the boundary as well as its rate of convergence formula.

1. Introduction. In this work we consider the inverse problem of determining the material parameters, specifically electromagnetic parameters or elasticity pa- rameters, at the boundary of a body from knowledge of certain boundary maps of electromagnetic fields or elastic waves. Such boundary determination is usually the preliminary step in solving the inverse problem of determining these parameters inside the body. The prototypical study that inspired that of the electromagnetic and elastic inverse problems is for electrostatics, known as the Calder´on problem.

In the Calder´on problem, one aims at determining the conductivity function σ from the Dirichlet-to-Neumann (DN) map (also known as the voltage-to-current map) associated to the diffusion conductivity equation ∇ · (σ∇u) = 0 for the electrical potential. Extensive studies have been devoted to show the unique determination of the conductivity inside the body, see [26, 21, 2, 9, 11] for example. The work on internal unique determination of electromagnetic and elastic parameters can be found in [22, 23,10] and [19], respectively. An important direction of generalizing

2020 Mathematics Subject Classification. Primary: 35R30; Secondary: 78A46, 86A22.

Key words and phrases. Inverse boundary value problems; uniqueness; boundary determina- tion; electromagnetism; Lam´e system; corrupted data; Stroh formalism.

1

the uniqueness result is to obtain uniqueness for parameters with lower regular- ity, such as Lipschitz conductivities discussed in [9, 11], Lipschitz electromagnetic parameters discussed in [10,24] and Lipschitz Lam´e moduli in [19].

The boundary determination for the Calder´on problem was first shown in [18]

for smooth conductivities, and later generalized in a series of papers [1,3,4,5]. In particular, the methods in [3,5] are constructive. A fundamental insight obtained in [27], is that the DN-map Λ_σ is a first order pseudo-differential operator whose full symbol carries all information of the conductivity σ and its derivatives on the boundary. In the case of systems, the available results in this context are due to Joshi-McDowall [15, 20], and Salo-Tzou [25]. In [19], the boundary determination of the Lam´e parameters for an isotropic elasticity system has been investigated.

In this paper, we consider the boundary determination of parameters when the boundary of the body is non-smooth, but Lipschitz. Under this assumption, the principal symbol approach in [27, 15] does not directly apply. Instead, we follow the scheme in [3] to show that the boundary values of the electromagnetic parame- ters in the time-harmonic Maxwell’s equations can be uniquely determined by the ideal (noiseless) boundary measurements of electromagnetic fields, formulated as the admittance map for Maxwell’s equations.

The second goal of the paper is to provide theoretical analysis for reconstructing the values of the parameters on the boundary assuming corrupted boundary mea- surements. We consider both inverse problems for electromagnetics and elasticity.

The corruption of the data is usually a result of discretized approximation by real data with errors. A formulation of such measurements was introduced in [7] for the Dirichlet-to-Neumann map in solving the Calder´on problem, where the random white noise was modeled by a random perturbation in the energy bilinear form, that depends on the intensity of the boundary potential and current. Other approaches in handling noises in boundary measurements can be found in [16,12,13,17]. Based on our boundary reconstruction result with ideal data for Maxwell’s equations and [19] for elasticity equation, we adopt the approach in [7] to show that the given type of Gaussian noises can be filtered using highly concentrated and oscillatory wave- packets, precisely those used in the ideal reconstruction scheme, when the noise variance is small. The observation in [7] and our work here inspired another paper [8] where the authors present a general framework and theory to solve the inverse problem of recovering the symbol of a pseudo-differential operator from its bilinear form, corrupted by Gaussian white noise that is modeled as a perturbation. More detailed exposition of the idea and the insights of the method are summarized in the following subsections for the two inverse problems individually.

1.1. Maxwell system. We first formulate the inverse problem for Maxwell’s equa- tions. Let Ω ⊂ R³ be a bounded domain with a Lipschitz boundary ∂Ω. Consider real-valued functions µ, ε, σ, first in the space L^∞(Ω), representing the magnetic permeability, electric permittivity and electric conductivity, respectively. Further- more, they satisfy

µ(x) ≥ µ₀> 0, ε(x) ≥ ε₀> 0 and σ(x) ≥ 0, (1) almost everywhere (a.e.) x ∈ Ω, for some positive constants µ0 and ε0. Suppose that we have access to the boundary measurements of all electromagnetic waves that are time-harmonic with angular frequency ω > 0. Then, let (E, H) be an

electromagnetic field satisfying time-harmonic Maxwell system, either







curl E − iωµH = 0 in Ω, curl H + iωγE = 0 in Ω,

ν × E = f on ∂Ω,

(2)

or







curl E − iωµH = 0 in Ω, curl H + iωγE = 0 in Ω,

ν × H = g on ∂Ω,

(3)

where γ :=  + iσ/ω. It is known that (2) and (3) are well-posed except at a discrete set of frequencies. Note that for real parameters (i.e. σ = 0), one needs to consider either the vacuum of eigenvalues for the Maxwell operator or replace the following well-defined boundary maps by the Cauchy data set. For the complex parameters (i.e. σ > 0), there are no real eigenvalues. Throughout this paper, we assume that ω > 0 is not an eigenvalue of (2) and (3). Then the boundary admittance map Λ^A_µ,γ can be defined by

Λ^A_µ,γ(f ) = ν × H|∂Ω,

where (E, H) ∈ H(curl; Ω) × H(curl; Ω) satisfies the boundary value problem (2).

Here ν ∈ (L^∞(∂Ω))³denotes the unit outer normal vector to ∂Ω and H(curl; Ω) =u ∈ (L²(Ω))³ | curl u ∈ (L²(Ω))³ . Similarly, one can define the boundary impedance map Λ^I_µ,γ by

Λ^I_µ,γ(g) = ν × E|_∂Ω,

where (E, H) ∈ H(curl; Ω) × H(curl; Ω) satisfies the boundary value problem (3).

In order to reconstruct γ and µ, we need to use the whole boundary information Λ^A_µ,γ and Λ^I_µ,γ.

The main result for the ideal data case is the unique boundary identifiability of Lip(Ω)-parameters µ, γ at frequency ω from boundary measurements

Λ^A_µ,γ, Λ^I_µ,γ : H^−1/2(Div ; ∂Ω) → H^−1/2(Div ; ∂Ω).

See (14) in Section2for the definition of H^−1/2(Div ; ∂Ω).

The following result contains the boundary determination of the electromagnetic parameters without noise.

Theorem 1.1 (Boundary identifiability of electromagnetic parameters). Let Ω be a bounded domain in R³, where the boundary ∂Ω is locally described by the graphs of Lipschitz functions, and ω > 0. Assume that two sets of parameters µ_j and γ_j for j ∈ {1, 2} belong to Lip(Ω), then we have

(1) Unique determination.

Λ^A_µ

1,γ₁= Λ^A_µ

2,γ₂ implies that γ₁= γ₂ a.e. on ∂Ω and

Λ^I_µ1,γ1 = Λ^I_µ2,γ2 implies that µ1= µ2 a.e. on ∂Ω.

(2) Pointwise reconstruction. For almost every P ∈ ∂Ω, there exists an ex- plicit sequence of localized boundary data {fN}^∞_{N =1} supported around P such that

lim

N →∞

i ω

ˆ

∂Ω

Λ^A_µ,γ(fN|∂Ω) × ν · fN dS = γ(P ) (4) and

lim

N →∞

i ω

ˆ

∂Ω

h

Λ^I_µ,γ(fN|∂Ω)i

· (fN × ν) dS = µ(P ). (5) Remark 1. In Theorem1.1, the conclusion (2) will imply (1) immediately. There- fore, we only prove (2). Note that the boundary data {f_N}^∞_{N =1} stands for electric and magnetic fields on ∂Ω in (4) and (5), respectively.

As mentioned above, since the boundary ∂Ω is Lipschitz, the principal symbol approach in [15] does not directly apply. Therefore, we adopt the idea from [3].

However, one of the novelties and key ingredients in [3] is the use of Hardy’s in- equality which seems not to have a clear counterpart in the problem for Maxwell’s equations. Instead, we handle the issue by a new technique that involves a duality argument. See the proof of Theorem2.1.

Our next result provides the analysis for reconstructing the values of the param- eters on the boundary assuming corrupted boundary measurements. To be more specific about the modeling of the noise, we consider a complete probability space (Π, H, P), and a countable family {X^α: α ∈ N²} of independent complex Gaussian random variables Xα: $ ∈ Π 7→ Xα($) ∈ C such that

EXα= 0, E(XαXα) = 1, E(XαXα) = 0 ∀α ∈ N², (6) with standard expectation of a random variable defined by

EX = ˆ

Π

XdP.

In [7], the noisy data for the Calder´on problem is defined as Nσ(f, g) =

ˆ

∂Ω

Λσf g dS + X

α∈N²

(f |eα₁)(g|eα₂)Xα f, g ∈ H^1/2(∂Ω), where α = (α₁, α₂) and {e_n: n ∈ N} is an orthonormal basis of L²(∂Ω) and (φ|ψ) denotes the inner product in L²(∂Ω, C). Here Λσdenotes the Dirichlet-to-Neumann map from H^1/2(∂Ω) to H^−1/2(∂Ω)

Λ_σ : f 7→ ν · σ∇u|_∂Ω,

where u is the solution to ∇ · (σ∇u) = 0 and u|∂Ω = f , and ν is the unit outer normal vector on ∂Ω. It is shown in [7] that at almost every point P ∈ ∂Ω, with a single realization of N_σ at explicit oscillatory boundary inputs f_N (such as the traces of (23)) (N ∈ N), the boundary value of σ at the point P can be recovered almost surely by

lim

N →∞N_σ(f_N, f_N) = σ(P ).

Note that the noise introduced in the energy form for the Dirichlet-to-Neumann map above is modeled on L²(∂Ω). In the case of Maxwell’s equations, we will see that similar type of noise could be introduced at two different levels: the H⁻¹(∂Ω)- level which guaranties decay of kf_Nk_(H−1(∂Ω))³ in Lipschitz domains, and L²(∂Ω)- level where there is not decay of kf_Nk(L²(∂Ω))³ and we need extra regularity for ∂Ω.

Starting by defining the corrupted data at the H⁻¹(∂Ω)-level:

N_µ,γ^A (f, g) :=

ˆ

∂Ω

(Λ^A_µ,γ(f ) × ν) · g dS + X

α∈N²

(f |eα₁)(g|eα₂)Xα

N_µ,γ^I (f, g) :=

ˆ

∂Ω

Λ^I_µ,γ(f ) · (g × ν) dS + X

α∈N²

(f |eα₁)(g|eα₂)Xα

(7)

for f, g ∈ H^−1/2(Div ; ∂Ω) ⊂ (H⁻¹(∂Ω))³, where {e_n : n ∈ N} is an orthonormal basis of the Hilbert space (H⁻¹(∂Ω))³ and (φ|ψ) here denotes the inner product in (H⁻¹(∂Ω))³. Then we have the following reconstruction formula for the Maxwell system with corrupted data.

Theorem 1.2. Let Ω ⊂ R³ be a bounded Lipschitz domain and µ, , σ be Lipschitz continuous functions satisfying (1). Let N_µ,γ^A and N_µ,γ^I be the quadratic form given by (7), then for almost every P ∈ ∂Ω, one has

(1) Unique determination. There exists explicit boundary data {fN}^∞_{N =1} in the space H^−1/2(Div ; ∂Ω) such that

lim

N →∞N_µ,γ^A (fN, fN) = γ(P ), lim

N →∞N_µ,γ^I (fN, fN) = µ(P ) almost surely.

(2) Rates of convergence. There exist positive constants C_γ (depending on ∂Ω and bounds for γ) and C_µ (depending on ∂Ω and bounds for µ), such that, for every 0 < θ < 1 and  > 0, we have

P

n|N_µ,γ^A (fN, fN) − γ(P )| ≤ CγN^−θ/2o

≥ 1 −  for any N ≥ c^−1−θ1 , where the constant c only depends on C∂Ωand θ. A similar estimate holds for µ, that is,

P

n|N_µ,γ^I (fN, fN) − µ(P )| ≤ CµN^−θ/2o

≥ 1 −  for any N ≥ c^−1−θ1 , where the constant c > 0 only depends on C∂Ω and θ.

Next we consider the problem with error modeled at the L²(∂Ω)-level. That is, in the definition (7), we choose {en: n ∈ N} to be an orthonormal basis of (L²(∂Ω))³ with the inner product (φ|ψ) =´

∂Ωφ · ψdS and f, g ∈ (L²(∂Ω))³. To make rigorous sense of this definition, we will assume in this discussion that the boundary of the domain is locally defined by the graph of C^1,1 functions. In this case, the boundary impedance and admittance maps are well-defined for f, g ∈ H^1/2(Div , ∂Ω). Unlike the previous case of (H⁻¹(∂Ω))³ perturbations, the norm kf_Nk_(L2(∂Ω))³ does not decay as N increases. We actually have kf_Nk_(L2(∂Ω))³ ≤ C_∂Ω where C_∂Ω is a constant depending on the boundary. This is similar to the reconstruction of the normal derivative of the conductivity with corrupted data in [7]; and similarly, our family of solutions can filter out the noise when averaged with respect to the parameter N^1/2. We then obtain the following result.

Theorem 1.3. Let Ω ⊂ R³ be a bounded domain whose boundary can be defined by the graphs of C^1,1-functions, and µ, ε, σ be Lipschitz continuous functions satisfying (1). Let N_µ,γ^A and N_µ,γ^I be the quadratic form given by (7) at the L²(∂Ω)-level. Then for every P ∈ ∂Ω, there exists an explicit family {ft : t ≥ 1} in H^1/2(Div , ∂Ω) such that for N ∈ N\{0} and TN := N^3+3θ/2 with θ ∈ (0, 1),

(1) Unique determination.

lim

N →∞

1 TN

ˆ 2T_N TN

N_µ,γ^A (f_t2, f_t2) dt = γ(P ), lim

N →∞

1 TN

ˆ 2T_N TN

N_µ,γ^I (f_t2, f_t2) dt = µ(P ) almost surely.

(2) Rates of convergence. Set Y_N^A= 1

TN

ˆ 2T_N T_N

N_µ,γ^A (ft², ft²) dt, Y_N^I = 1 TN

ˆ 2T_N T_N

N_µ,γ^I (ft², ft²) dt.

There exist positive constants C_γ > 0 (depending on ∂Ω and bounds for γ) and C_µ > 0 (depending on ∂Ω and bounds for µ), such that, for every 0 < θ < 1 and  > 0, we have

P

n|Y_N^A− γ(P )| ≤ CγN^−θ/2o

≥ 1 −  for any N ≥ cγ^−1−θ1 , and

P

n|Y_N^I − µ(P )| ≤ CµN^−θ/2o

≥ 1 −  for any N ≥ cµ^−1−θ1 ,

where the constants cγ and cµ depend on θ, ∂Ω, lower bounds for ε0 and µ0, and upper bounds for kγk_Lip(Ω) and kµk_Lip(Ω), respectively.

Remark 2. The reconstruction in Theorem 1.2 can only be ensured for almost every point at the boundary because of the regularity of the domain. However, the reconstruction formula of Theorem 1.3 holds for every point since the domain is assumed to have a C^1,1 boundary.

If we compare Theorem 1.2 and Theorem 1.3 with the results in [7] for the reconstruction of the conductivity and its normal derivative at the boundary, we can see a couple of similarities. When modeled the noise at the H⁻¹-level, no averaging is required for the reconstruction, as it happened in [7] for the reconstruction of the conductivity. In [7], this was a consequence of the rate of concentration of the supports of the family {fN} around the point to be reconstructed. However, in our Theorem 1.2 this is due to the regularizing effect of the covariance operator associated to the noise in the H⁻¹(∂Ω)-level. On the other hand, when modeling the noise at the L²(∂Ω)-level, we require to perform an average in the parameter

√

N (since the radius of the support of fN shrinks as 1/√

N ) to overcome the lack of decay of kfNk_(L2(∂Ω))³. This was exactly the same situation as in [7] for the reconstruction of the normal derivative of the conductivity at the boundary. In these situations, we have to analyze an oscillatory integral, and isolate appropriately the stationary points. These are the contents of Lemma3.5. Note that the decaying rate in this lemma suggests that we might still obtain decays in average even if the norms of f_N are increasing as N grows. Consequently, errors modeled in spaces of higher regularities might be potentially filtered.

1.2. Elasticity system. For the second system, we consider the boundary deter- mination of the Lam´e parameters for the isotropic elasticity equations. Let Ω ⊂ R³ be a bounded domain, λ(x) and µ(x) be the Lam´e parameters satisfying the ellip- ticity condition

µ(x) > 0 and 3λ(x) + 2µ(x) > 0 for all x ∈ Ω. (8)

The boundary value problem for the isotropic elasticity system is given by ((∇ · (C∇u))i=P3

j,k,l=1

∂

∂x_j

 Cijkl ∂

∂x_luk



= 0 (i = 1, 2, 3) in Ω,

u = f on ∂Ω,

(9)

where u = (u1, u2, u3) is the displacement vector, C = (Cijkl)1≤i,j,k,l≤3 and C_ijkl= λδ_ijδ_kl+ µ(δ_ikδ_jl+ δ_ilδ_jk) for 1 ≤ i, j, k, l ≤ 3 (10) is the isotropic elastic four tensor with Kronecker delta δij. One can easily see that Cijkl given by (10) satisfies the major and minor symmetries, i.e.,

Cijkl= Cklij= Cjikl, for 1 ≤ i, j, k, l ≤ 3.

The Dirichlet-to-Neumann (DN) map for the isotropic elasticity system is defined by

ΛC : (H^1/2(∂Ω))³→ (H^−1/2(∂Ω))³with (ΛCf )_i =

3

X

j,k,l=1

νjCijkl

∂uk

∂xl

     _∂Ω

(11)

for i = 1, 2, 3, where u ∈ (H¹(Ω))³ is the solution to (9) and ν = (ν1, ν2, ν3) is the unit outer normal on ∂Ω. The inverse problem is whether the elastic tensor C is uniquely determined by ΛC, and to calculate C of ΛC if C is determined by ΛC. Note that the global uniqueness for the isotropic elasticity system stays open for the three-dimensional case and it was solved in [14] for the two-dimensional case.

The boundary determination of the zeroth order and higher order Lam´e moduli was studied by [28] and [19], respectively. In other words, given any P ∈ ∂Ω (when

∂Ω and the Lam´e moduli are sufficiently smooth), one can derive reconstruction formulas for the Lam´e moduli λ and µ and their derivatives at P ∈ ∂Ω, from the localized DN map. Now, our goal is to give a similar reconstruction algorithm for the Lam´e parameters with corrupted data.

Due to the existence of elliptic regularity theory for this system, the corrupted data for the elastic system is similar to that of the scalar conductivity equation discussed in [7], namely, the random noise is introduced at (L²(∂Ω))³ vector level by introducing the bilinear form with corrupted data

NC(f, g) :=

ˆ

∂Ω

ΛCf · g dS + X

α∈N²

(f |eα₁)(g|eα₂)Xα,

for f, g ∈ (H^1/2(∂Ω))³, where {en : n ∈ N} is an orthonormal basis of the Hilbert space (L²(∂Ω))³and (φ|ψ) here denotes the inner product in (L²(∂Ω))³. Then our results for the elasticity system is as follows:

Theorem 1.4. Let Ω ⊂ R³ be a bounded Lipschitz domain. Let C be a Lipschitz continuous elastic four tensor in Ω. Then for almost every P ∈ ∂Ω, one has

(1) Unique determination. There exists an explicit boundary data {fN}^∞_{N =1} in (H^1/2(∂Ω))³ such that

lim

N →∞N_C(f_N, f_N) = Z(P )

almost surely, where Z(P ) = (Zij)1≤i,j≤3(P ) with Zij = Zji for 1 ≤ i, j ≤ 3, and

Zii= µ

λ + 3µ 2(λ + 2µ) − (λ + µ)ι²_i
, Z_ij= µ

λ + 3µ − (λ + µ)ι_iι_j+√

−1(−1)^k2µ ι_k
, 1 ≤ i < j ≤ 3 (12) with (ι1, ι2, ι3) = (ω2, −ω1, 0) and the index k ∈ N satisfies the condition 1 ≤ k ≤ 3, k 6= i, j.

(2) Rates of convergence. There exists a constant C > 0, independent of N , such that, for every 0 < θ < 1 and  > 0, we have

P

n|NC(fN, fN) − Z(P )| ≤ CN^−θ/2o

≥ 1 −  for any N ≥ c^−1−θ1 , (13) where the constant c > 0 depends only on C∂Ω and θ.

Theorem 1.4 shows that when the domain Ω is Lipschitz and C is Lipschitz continuous, then one can reconstruct the Lam´e moduli at almost every boundary point P ∈ ∂Ω in a constructive way.

1.3. Outline. The rest of this paper is organized as follows. The reconstruction formulas for Lipschitz parameters µ and γ in Maxwell’s equations on a Lipschitz boundary ∂Ω are given in Section2. In Section3, we analyze the reconstruction with corrupted data by random white noise for the Maxwell equations. The analysis for the reconstruction of the Lipschitz Lam´e moduli for the isotropic elasticity system with corrupted data is given in Section4.

2. Boundary determination of electromagnetic parameters. First, let us define several function spaces and notations.

2.1. Preliminaries. Let us begin with some definitions of function spaces, where the impedance map is well-defined. For a bounded Lipschitz domain Ω, we adopt Tartar’s definition (see [29] or [6]) of the space

H^−1/2(Div ; ∂Ω) :=n

u ∈ (H^−1/2(∂Ω))³| ∃ η ∈ H^−1/2(∂Ω), s.t., ˆ

∂Ω

u · ∇φ dS = ˆ

∂Ω

ηφ dS for φ ∈ H²(Ω)o ,

(14)

where (H^−1/2(∂Ω))³is the dual space of (H^1/2(∂Ω))³. This implies in a weak sense that η = −Div u, where Div denotes the surface divergence, and that ν · u|_∂Ω= 0, based on the identity for u smooth

− ˆ

∂Ω

(Div u)φ dS = ˆ

∂Ω

u · ∇φ dS − ˆ

∂Ω

(u · ν)(∇φ · ν) dS.

We will also define in the same spirit the space for the surface scalar curl H^−1/2(Curl ; ∂Ω) :=n

u ∈ (H^−1/2(∂Ω))³ | ∃ ξ ∈ H^−1/2(∂Ω), s.t., ˆ

∂Ω

(ν × u) · ∇φ dS = ˆ

∂Ω

ξφ dS for φ ∈ H²(Ω) and

ˆ

∂Ω

u · ∇ψ dS = 0 for ψ ∈ H²(Ω) ∩ H₀¹(Ω)o .

(15)

Note that the first condition implies in the weak sense that ξ = −Curl u, where Curl denotes the surface scalar curl, and the second condition in the definition implies weakly the tangentiality ν · u|_∂Ω= 0.

Moreover, H^−1/2(Curl ; ∂Ω) is the dual of H^−1/2(Div ; ∂Ω). It is then shown in [6,29] that the tangential trace map

τ_t: H(curl ; Ω) → H^−1/2(Div ; ∂Ω) u 7→ ν × u|∂Ω

and the projection map

πt: H(curl ; Ω) → H^−1/2(Curl ; ∂Ω) u 7→ (ν × u|_∂Ω) × ν are both surjective.

In order to reconstruct the values of the parameters, we begin with the following energy identity, which is obtained by integration by parts

i ω

ˆ

∂Ω

(ν × (ν × H)) · (ν × E) dS = ˆ

Ω

γ|E|²− µ|H|²dx (16) for the solution (E, H) ∈ H(curl ; Ω) × H(curl ; Ω) to the Maxwell’s equations. Here the boundary integral is the parity of H^−1/2(Div ; ∂Ω) and H^−1/2(Curl ; ∂Ω).

In the following we use d to denote the dimension number so one can trace the dependence of the convergence rate on d. In all cases considered in this paper including Maxwell system and elasticity system, d = 3. We denote by B(x, r) the ball centered at x of radius r > 0 and adopt the coordinate notation x = (x⁰, xd) ∈ R^d−1×R in d dimensions. Since we will use some results of Brown [3], we will follow his notation.

Given a Lipschitz domain Ω ⊂ R^d, for each P := (p⁰, pd) ∈ ∂Ω, we consider a change of variable that flattens the boundary near P

(z⁰, zd) = F (x⁰, xd) = x⁰+ p⁰, xd+ φ(x⁰+ p⁰)
, (17) where φ : R^d−1→ R is Lipschitz such that

B(P, ρ) ∩ ∂Ω = B(P, ρ) ∩ {zd = φ(z⁰)}

B(P, ρ) ∩ Ω = B(P, ρ) ∩ {zd> φ(z⁰)}

for some ρ > 0. Let eΩ = F⁻¹(Ω) ⊂ R^d and ∂ eΩ be its boundary. There exists a r > 0 such that

B(0, 2r) ∩ {xd= 0} ⊂ F⁻¹ B(P, ρ) ∩ ∂Ω
⊂ ∂Ω.e

Since we are interested in the coefficients at the point P , we focus on reconstructing µ(F (0, 0)) = µ(p⁰, φ(p⁰)) and γ(F (0, 0)) = γ(p⁰, φ(p⁰)).

Denote

M (x) := DF⁻¹(F (x)) = dx_i dzj



i,j

(F (x)). (18)

By the change of coordinates (17), we have the right hand side of (16) to be I :=

ˆ

Ω

γ|E|²− µ|H|²dz = ˆ

Ωe

(eγ eE) · eE − (µ eeH) · eH dx, (19) where

µ(x) := µ(F (x))M (x)M (x)e ^t, eγ(x) := γ(F (x))M (x)M (x)^t, and

E(x) := (M (x)e ^t)⁻¹E(F (x)), H(x) := (M (x)e ^t)⁻¹H(F (x)).

Furthermore, the electromagnetic field ( eE, eH) (defined as the pull-back of (E, H) by F : eΩ → Ω) satisfies the Maxwell’s equations (in the weak sense)

curl eE − iωeµ eH = 0, curl eH + iωeγ eE = 0 in eΩ. (20) This last point can be justified by checking that curl eE(x) = M (x)(curl E)(F (x)).

We now list a couple of properties of the parameters that are required to apply some results of Brown [3]. First, let us note that µ, γ ∈ Lip(Ω) satisfy the hypothesis (H1) in [3], that is,

|µ(F (x⁰, x_d)) − µ(F (x⁰, 0))| + |γ(F (x⁰, x_d)) − γ(F (x⁰, 0))| . |xd| (21) for all |x⁰| < 2r. Regarding the hypothesis H2 in [3], note that

s^1−d ˆ

|y⁰|<s

|γ(0, 0) −e γ(ye ⁰, 0)|² dy⁰+ s^1−d ˆ

|y⁰|<s

|µ(0, 0) −e µ(ye ⁰, 0)|² dy⁰

. s²+ s^1−d ˆ

|y⁰|<s

|∇⁰φ(y⁰+ p⁰) − ∇⁰φ(p⁰)|² dy⁰,

where the limit of the last term on the right-hand side vanishes, when s goes to zero, for almost every p⁰ by the Lebesgue differentiation theorem. Here we denote

∇⁰φ := (∂1φ, ∂2φ)^t.

Our reconstruction method only will work for points P ∈ ∂Ω so that

s→0lims^1−d ˆ

|y⁰|<s

|∇⁰φ(y⁰+ p⁰) − ∇⁰φ(p⁰)|² dy⁰= 0 (22) for the corresponding φ and p⁰. As pointed out before, for almost every point in P ∈ ∂Ω its corresponding limit in (22) vanishes.

2.2. Reconstruction of γ. We first give an explicit reconstruction formula of γ in an admissible point P ∈ ∂Ω from the knowledge of the admittance map Λ^A_µ,γ. Recall in [3], a family of functions with special decaying property is constructed as the input of the Dirichlet-to-Neumann map for ∇ · σ∇ to reconstruct σ. More specifically, this family was given by

v_N(y) = η(N^1/2|y⁰|)η(N^1/2y_d)e^{N (iα−~ed)·y}, (23) where ~e_d = (0, · · · , 0, 1) ∈ R^d and η : R → [0, 1] is a smooth cutoff function which takes value 1 in B(0, 1/2) and 0 outside B(0, 1), the vector α ∈ R^d can be chosen such that

|M (0)^tα| = |M (0)^t~e_d|,

α · M (0)M (0)^t~e_d= 0. (24)

An explicit choice of α is given in (37).

We will make an essential use of the gradient fields {∇vN}N. More particularly we will choose (E, H) so that their pull-back ( eE, eH) = (∇vN+ w1, w2) with w1and w2 solving







curl w1− iωµwe 2= 0 in eΩ, curl w2+ iωeγw1= −iωeγ∇vN in eΩ,

ν × w₁= 0 on ∂ eΩ.

(25)

Note that eΩ is not necessarily locally described by the graph of Lipschitz functions, so in principle, the theory of well-posedness for (25) should be revisited. In our particular case, the situation is simpler since eΩ is the pull-back of a domain whose

boundary is locally described by the graph of a Lipschitz function. Therefore, it is enough to use the map F to obtain (w1, w2) in eΩ from the corresponding fields in Ω. We will be solving in eΩ in the rest of the paper, and it will always be justified through the map F .

The corresponding energy (19) for ( eE, eH) is then given by I =

ˆ

Ωe

γ(F (y))∇vN · M M^t∇vN dy +

ˆ

Ωe

γ(F (y))2Re(∇vN · M M^tw1) + w1· M M^tw1dy

+ ˆ

Ωe

µ(F (y))w2· M M^tw2dy.

(26)

On the other hand, the tangential boundary condition of the electric field is transformed according to

ν × E(F (x)) = DF (x)eν × eE(x),

whereeν(x) = DF (x)^tν(F (x)). For N^−1/2< 2r the support of ∇v_N is contained on {xd= 0} ∩ B(0, 2r), and the tangential boundary condition there becomes

ν × E(F (x⁰, 0)) = DF (x⁰, 0)~ed× ∇vN(x⁰, 0). (27)

Since H1 and H2 in [3, Lemma 1] are satisfied, the first term of I satisfies

´

Ωeγ(F (y))∇v_N· M M^t∇vN dy N^3−d2

→ γ(p⁰, φ(p⁰))(1 + |∇⁰φ(p⁰)|²) ˆ

R^d−1

η(|x⁰|)²dx⁰, (28) as N → ∞.

It turns out that this first term dominates, hence provides the reconstruction of γ(F (0)) knowing φ and η.

Theorem 2.1. Suppose Ω ⊂ R^d (d = 3) is a bounded Lipschitz domain. Let µ, ε, σ ∈ Lip(Ω) satisfy (1). Let P ∈ ∂Ω be an admissible point with F as in (17).

We define

f_N(z) := c^−1/2₀ (M (y))^t(ν(y) × ∇v_N(y))|_y=F−1(z), (29) where

c₀= (1 + |∇⁰φ(p⁰)|²) ˆ

R^d−1

η(|x⁰|)² dx⁰, and M and v_N are given by (18) and (23), respectively. Then

I(fN|∂Ω) := i ω

ˆ

∂Ω

Λ^A_µ,γ(fN|∂Ω) × ν · fN dS → γ(P ) as N → ∞.

Proof. To show that the last two terms in (26) are lower order terms, it suffices to show that the (L²(eΩ))³-norms of w1and w2 are o(1).

First, we need to consider the dual of the standard regularity estimate for the Maxwell’s equations, targeting the L²-norm of the solution.

Notice that the elliptic condition for the parameters is preserved in the following dual problem: Given (G1, G2) ∈ (L²(eΩ))⁶, except for a discrete set of frequencies, there exists a unique solution (u1, u2) ∈ H(curl ; eΩ) × H(curl ; eΩ) to







curl u₁+ iωeµu₂= G₁ in eΩ, curl u2− iωeγu1= G2 in eΩ, ν × u₁= 0 on ∂ eΩ.

(30)

Furthermore, we have

ku₁k_{H(curl ;eΩ)}+ ku₂k_{H(curl ;eΩ)}. kG1k_(L2(eΩ))³+ kG₂k_(L2(eΩ))³. (31) Then by integration by parts (duality), we have

    ˆ

Ωe

w1· G2+ w2· G1dy    

=     ˆ

Ωe

(−iωeγ∇vN) · u1 dy    

. (32)

It then suffices to show that the right hand side is bounded by o(1)ku₁k_{H(curl ;eΩ)} since this would imply, using (31),

kw1k_(L2(eΩ))³+ kw2k_(L2(eΩ))³≤ o(1).

It is worth noticing that in [3], Brown used Hardy’s inequality to show a similar estimate

k∇ ·eγ∇vNk_H−1(eΩ)= o(1).

The main novelty in our approach is to replace the use of Hardy’s inequality by a duality argument involving the possibility of writingeγ(0)∇e_N as the curl of certain vector field L_N.

Start by writing v_N := ψ_Ne_N with

ψN(y) := η(N^1/2|y⁰|)η(N^1/2yd), eN(y) = e^{N (iα−~ed)·y}. We will estimate the three terms of

eγ∇v_N(y) =eγ(y)∇ψ_Ne_N+ eγ(y) −eγ(0)
ψN∇eN +eγ(0)ψ_N∇eN. (33) For the first two terms, we only need to control their L²-norms by duality. Then we have

keγ∇ψ_Ne_Nk²_(L2(eΩ))3 . k∇ψNe_Nk²

(L²(eΩ))³

= N^2−d2 ˆ

R^d

e^−2N1/2yd(η⁰(|y⁰|)²η(yd)²+ η(|y⁰|)²η⁰(yd)²) dy

. N^2−d2 ˆ 1

0

e^−2N1/2yd+ e^−2N1/2ydη⁰(yd)² dyd

. N^2−d2 

N^−1/2+ O(e^−N1/2)

= O(N^1−d2 ) = O(N⁻¹).

(34)

Similarly, we consider the square of L²-norm of the second term N²

ˆ

eΩ

|(eγ(y) −eγ(0)) (iα − ~e_d)|²ψ_N²e^{−2N yd} dy . N²

ˆ

B(0,N^−1/2)×R⁺

|eγ(y) −eγ(0)|²e^{−2N yd} dy,

(35)

where B(0, N^−1/2) denotes the ball in R^d−1centered at 0 and radius N^−1/2. It is convenient to write,

eγ(y) −eγ(0)

= γ(F (y)) − γ(F (0))
M (y)M (y)^t+ γ(F (0)) M (y)M (y)^t− M (0)M (0)^t
.

Thus, the right-had side of (35) can be bounded by N²

ˆ

B(0,N^−1/2)×R⁺

|y⁰|²e^{−2N yd} dy

+ N² ˆ

B(0,N^−1/2)×R⁺

|∇⁰φ(y⁰+ p⁰) − ∇⁰φ(p⁰)|²e^{−2N yd} dy.

(36)

By the (22), we have that the previous sum is o(1). It remains to prove 

   ˆ

Ωe

−iωγ(0)ψe N∇eN· u1 dx    

≤ o(1)ku1k_{H(curl ;eΩ)}.

The idea will be to writeeγ(0)∇eN as the curl of certain vector field LN. First, we state the explicit expression of the matrices M and M M^t at 0:

M (0) =

 I_d−1 0

−∇⁰φ(p⁰)^t 1



, M (0)M^t(0) =

 I_d−1 −∇⁰φ(p⁰)

−(∇⁰φ(p⁰))^t 1 + |∇⁰φ(p⁰)|²

 .

Since α is chosen such that β = M (0)^t(iα − ~e_d) satisfies β · β = 0, we have that eγ(0)∇e_N is divergence free, namely,

∇ · (eγ(0)∇eN(y)) = 0.

Therefore, there must exist a vector field LN = LN(y) such that

∇ × LN =eγ(0)∇e_N = Neγ(0)(iα − ~e_d)e_N. Next, look for such an LN. We write an ansatz

L_N = γ(F (0))(a + ib)e_N

and find a, b ∈ R^d satisfying the following algebraic equations

~ed× a + α × b = M (0)M (0)^t~ed, α × a − ~ed× b = M (0)M (0)^tα.

It can be verified that in R³, the choice

a = α =









1 + |∇⁰φ|²

|∇⁰φ| ∇⁰φ

|∇⁰φ|









(p⁰), b =











− 1

|∇⁰φ|∂2φ 1

|∇⁰φ|∂₁φ 1











(p⁰), (37)

where ∇⁰φ := (∂₁φ, ∂₂φ)^t, qualifies and satisfies η ·η = 0 and η ·η = 2(1+|∇⁰φ(p⁰)|²).

Finally, ˆ

Ωe

(ψNeγ(0)∇eN) · u1 dy

= ˆ

Ωe

ψN(∇ × LN) · u1 dy

= ˆ

Ωe

LN · (ψN∇ × u1+ ∇ψN× u1) dy .

kψNLNk_(L2(eΩ))³+ k∇ψN · LNk_L2(eΩ))

kuk_{H(curl ;eΩ)},

(38)

where we have used that ν × u1= 0 on ∂ eΩ. It is then easy to verify, similar to that for (34), k∇ψN · LNk_L2(eΩ)= o(1). For the other term,

kψNLNk²

(L²(eΩ))³ . ˆ

Ωe

η(N^1/2|y⁰|)²η(N^1/2yd)²e^{−2N yd} dy

= N^−d2 ˆ

R^d

η(|y⁰|)²η(yd)²e^−2N1/2yd dy

= O(N^−1−d2 ) = O(N⁻²).

This completes the proof.

2.3. Reconstruction of µ. In order to reconstruct µ, the idea is to let the magnetic energy, namely´

Ωµ|H|²dz, dominate. By symmetry of the equations, H should be chosen roughly ∇v_N, for example, by equating them at the boundary. From now on, we utilize the impedance map, that is, the map

Λ^I_µ,γ : ν × H|∂Ω7→ ν × E|∂Ω,

then similarly to the previous section, we define our indicator functional being J (f_N|∂Ω) := i

ω ˆ

∂Ω

hΛ^I_µ,γ(f_N|∂Ω)i

· (fN × ν) dS, (39) where f_N = ν × ∇v_N as before. This implies

J (fN|∂Ω)

= ˆ

Ω

µ|H|²− γ|E|² dx

= ˆ

Ωe

µ(F (y))∇vN · M M^t∇vN dy +

ˆ

Ωe

µ(F (y))2Re(∇vN · M M^tw2) + w2· M M^tw2dy

− ˆ

Ωe

γ(F (y))w1· M M^tw1 dy,

where (w1, w2) := ( eE, eH − ∇vN) in this section and satisfies







curl w1− iωµwe 2= iωeµ∇vN in eΩ, curl w₂+ iωγwe ₁= 0 in eΩ,

ν × w2= 0 on ∂ eΩ.

(40)

Following the proof of Theorem2.1, the equation (32) is replaced by 

   ˆ

Ωe

w₁· G₂+ w₂· G₁ dy    

=     ˆ

eΩ

(−iωµ∇ve _N) · u₂ dy    

(41) for any (G1, G2) ∈ (L²(eΩ))⁶, where (u1, u2) is the unique solution to







curl u1+ iωµue 2= G1 in eΩ, curl u₂− iωeγu₁= G₂ in eΩ, ν × u2= 0 on ∂ eΩ.

Then it is left to show similarly 

   ˆ

eΩ

(−iωµ∇ve _N) · u₂ dy    

= o(1)ku₂k_{H(curl ;eΩ)}.

The proof is the same as in Theorem2.1. In particular, the integration by parts in (38) is still valid in this case using the boundary condition ν × u2|_{∂ eΩ}= 0.

As a result, we obtain the reconstruction formula for µ.

Theorem 2.2. Suppose that Ω, µ, ε, σ, P ∈ ∂Ω and f_N all satisfy the assumptions in Theorem 2.1. Then we have

lim

N →∞J (fN|∂Ω) = µ(P ), where J (fN|∂Ω) is defined by (39).

Proof of Theorem 1.1. By using all results in Section2, we can prove Theorem1.1 immediately.

3. Boundary determination of electromagnetic parameters with corrupted data. The main objective of this part is to stably identify boundary values of the unknown electromagnetic coefficients from the boundary measurement corrupted by errors, modeled and handled similarly to that in [7] for the Calder´on problem.

First, we give a description of the modeling for the random white noise, first introduced in [7] for the Calder´on problem, with modifications adopted to the system of Maxwell’s equations with our electromagnetic boundary maps. In particular, the random white noise is introduced to the boundary data on the H⁻¹(∂Ω)-level as well as on the L²(∂Ω) one.

3.1. Noise modeled on H⁻¹(∂Ω). We start with the fact that (H⁻¹(∂Ω))³ is a Hilbert space and let {e_n: n ∈ N} be an orthonormal basis of (H⁻¹(∂Ω))³. Recall that our bilinear form with corrupted data are defined as

N_µ,γ^A (f, g) = ˆ

∂Ω

(Λ^A_µ,γ(f ) × ν) · g dS + X

α∈N²

(f |e_α1)(g|e_α2)X_α (42)

N_µ,γ^I (f, g) = ˆ

∂Ω

Λ^I_µ,γ(f ) · (g × ν) dS + X

α∈N²

(f |eα₁)(g|eα₂)Xα (43)

for f, g ∈ H^−1/2(Div ; ∂Ω) ⊂ (H⁻¹(∂Ω))³, where α = (α1, α2) ∈ N² and (φ|ψ) denotes the inner product in (H⁻¹(∂Ω))³.

Then we have the following lemma after replacing L²(∂Ω) by (H⁻¹(∂Ω))³ in [7, Lemma 2.3].

Lemma 3.1. There exists a complete probability space (Π, H, P), and a countable family {Xα : α ∈ N²} of independent complex random variables satisfying (6).

Moreover, for every f, g ∈ (H⁻¹(∂Ω))³ we have that

E     

X

α∈N²

(f |eα₁)(g|eα₂)Xα

2

= kf k²_(H−1(∂Ω))³kgk²_(H−1(∂Ω))³.

Since the (H⁻¹(∂Ω))³-norm is bounded by the H^−1/2(Div , ∂Ω)-norm, immedi- ately, we obtain the boundedness of the operators N_µ,γ^A and N_µ,γ^I from the space H^−1/2(Div ; ∂Ω)×H^−1/2(Div ; ∂Ω) to L²(Π, H, P). It gives that

N_µ,γ^A (f, g) ,

N_µ,γ^I (f, g)  are finite almost surely. Moreover, we have the following decay for the covariance.

Lemma 3.2. The following estimate holds

E     

X

α∈N²

(fN|eα₁)(fN|eα₂)Xα

2

= kfNk⁴_(H−1(∂Ω))³ ≤ C∂ΩN⁻². (44) Proof. The first equality directly comes from Lemma3.1and the second inequality is obtained as follows. From (29), one has the equivalent formula

fN(z) = c^−1/2₀ ν(z) × WN(z), z ∈ ∂Ω, where

W_N(z) := (F⁻¹)^∗(∇v_N) = M (y)^t∇yv_N(y)|_y=F−1(z).

Here, ν(z) is the unit outer normal to ∂Ω while ν(y) in (29) is the unit outer normal to ∂ ˜Ω.

It is easy to verify

∇z× WN(z) = 0.

For ϕ ∈ (H¹(∂Ω))³, ˆ

Ω

∇ × WN · ϕe− WN · ∇ × ϕedz = ˆ

∂Ω

fN · ϕ dS,

where ϕe∈ (H^3/2(Ω))³is the extension such that ϕ = ν × ϕe|∂Ω× ν. The first term of the left hand side vanishes by above. For the second term of the left hand side, after a change of variable and passing the derivative, we have

ˆ

∂Ω

fN · ϕ dS = − ˆ

Ω

WN · ∇ × ϕ dz

= − ˆ

Ω

 ∂y

∂z

t

∇vN◦ F⁻¹
(z) · (∇z× ϕ(z)) dz

= − ˆ

Ωe

∇vN(y) · (∇y× ˜ϕ(y)) det ∂z

∂y

 dy

= − ˆ

∂ eΩ

vNν · (∇y×ϕ(y)) dS,e whereϕ is the push-forward of ϕ by F given bye

ϕ = (Me ^t)⁻¹(y)ϕ(F (y)).

Finally, it is not hard to see that ˆ

∂Ω

fN · ϕ dS ≤ CkvNk_(L2(∂ eΩ))³kϕk_(H1(∂Ω))³.

Therefore,

kfNk(H⁻¹(∂Ω))³≤ CkvNk_(L2(∂ eΩ))³≤ CkvNk_(H1/2(eΩ))³≤ C∂ΩN^−1/2 which gives (44).

We state one crucial result from [7] which also works for the vector-valued func- tions in this paper. This result will lead to the unique determination and the rate of convergence of parameters for both Maxwell and elasticity systems with corrupted data.

Proposition 1 (Lemma 2.5 in [7]). Let (X, Σ, m) be a measure space and {fn}^∞_n=1 be a vector-valued sequence in (L^s(X, Σ, m))³for s ∈ [1, ∞). Assume that fn→ f in (L^s(X, Σ, m))³ for some f ∈ (L^s(X, Σ, m))³ and there exists a sequence of positive numbers {λn}^∞_n=1⊂ R+ with λn→ 0 as n → ∞ such that

∞

X

n=1

1 λ^s_n

ˆ

X

|fn− f |^sdm < ∞.

Then one has fn → f for almost every x ∈ X.

Suppose furthermore that m(X) < ∞. Then, for every  > 0, there exists a n₀∈ N such that

m{x ∈ X : |fn(x) − f (x)| ≤ λn} ≥ m(X) − , for n ≥ n0. Remark 3. The n0 in the second part of the statement should satisfy

∞

X

n=n₀

1 λ^s_n

ˆ

X

|fn− f |^sdm ≤ .

Proof of Theorem 1.2. The part (1) is a consequence of the first part of Proposition 1 to the sequenceP(fN|eα₁)(fN|eα₂)Xα : N ∈ N\{0} with λN = N^−θ.

To prove part (2) of Theorem 1.2, again we take λN = N^−θ/2. Applying the second part of Proposition1to the sequence {P(fN|eα1)(f_N|eα2)X_α: N ∈ N\{0}}, and using (44), we obtain

P n

X(fN|eα₁)(fN|eα₂)Xα

≤ N^−θ/2o

≥ 1 −  for N ≥ N0, where N0is as in Remark3, that is, we need

∞

X

N =N₀

C_∂Ω² N^2−θ ≤ .

This holds whenever

(N0− 1)^1−θ> C_∂Ω²

(1 − θ),

which gives N0≥ c^−1−θ1 . Lastly, we see that there exist Cγ > 0 and Cµ > 0 such that

n  

X(fN|eα₁)(fN|eα₂)Xα

≤ N^−θ/2o

⊂n

|N_µ,γ^A (fN, fN) − γ(P )| ≤ CγN^−θ/2o and

n  

X(fN|eα₁)(fN|eα₂)Xα

≤ N^−θ/2o

⊂n

|N_µ,γ^I (fN, fN) − µ(P )| ≤ CµN^−θ/2o ,