Stability estimate for an inverse problem for the electro-magnetic wave equation and spectral boundary value problem

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Stability estimate for an inverse problem for the electro-magnetic wave equation and spectral boundary

value problem

Hajer Ben Joud

To cite this version:

Hajer Ben Joud. Stability estimate for an inverse problem for the electro-magnetic wave equation and spectral boundary value problem. 2009. �hal-00426786�

Stability estimate for an inverse problem for the electro-magnetic wave equation and spectral boundary value problem

Hajer Ben Joud¹

1Department of Mathematics, Faculty of Sciences of Bizerte, 7021 Jarzouna Bizerte, Tunisia.

mourad.bellassoued@fsb.rnu.tn

Abstract

In this paper, we prove the stability estimate of the inverse problem for the determination the mag- netic field and the electric potential using the Neumann spectral data. We show that the knowledge of the eigenvalues{λk, k≥1}and the boundary value of the normal derivatives of the corresponding eigenfunctions{∂νϕk, k ≥1}, and of the electro-magnetic Schr¨odinger operator, are sufficient to uniquely determine the magnetic field and the electric potential.

To obtain this result, we establish the stability estimate of the inverse problem of determining the electric potential entering the electro-magnetic wave equation in a bounded smooth domain in R^dfrom boundary observations. This information is enclosed in the hyperbolic (dynamic) Dirichlet- to-Neumann map associated to the solutions to the electro-magnetic wave equation. We prove in dimensiond≥2that the knowledge of the Dirichlet to Neumann map for the electro-magnetic wave equation measured on the boundary determines uniquely the electric potential.

Keywords:Stability estimate, hyperbolic inverse problem, magnetic field, Dirichlet to Neumann map, spectral data.

1 Introduction

Throughout this paper, we assume that the dimensiond≥2. LetΩ⊂R^dbe an open bounded set, with C^∞boundaryΓ =∂Ω. GivenT >0, we consider the following initial boundary value problem for the wave equation with a magnetic and electric potential















∂_t²−∆_A+q(x)

u(t, x) = 0 in Q= (0, T)×Ω, u(0, x) = 0, ∂tu(0, x) = 0 in Ω,

u(t, x) =f(t, x) on Σ = (0, T)×Γ,

(1.1)

where

∆A=

d

X

j=1

(∂j +i aj)²= ∆ + 2i A· ∇+idiv(A)−A·A , (1.2)

andA = (a_j)1≤j≤d ∈ W^3,∞(Ω;R^d) is a magnetic potential and the bounded electric potential q ∈ L^∞(Ω). It is well known (see [24]) that iff(0, x) = 0then (1.1) is a well-posed initial-boundary value problem. Therefore, we may define the operator

Λ_A,q:H¹(Σ) −→ L²(Σ)

f 7−→ (∂ν+i A·ν)u (1.3)

whereν(x)denotes the unit outward normal toΓatx. The operatorΛ_A,q, which is the main subject of this paper, is called the Dirichlet to Neumann map of (1.1) onΣ.

Using energy estimate one can prove that Λ_A,q is continuous from H¹(Σ)toL²(Σ). The inverse problem is whether knowledge of the Dirichlet-to-Neumann map ΛA,q on the boundary determines uniquely the magnetic potentialAand the electric potentialq.

As given in [3] and [12], it is clear that one can not hope to uniquely determine the vector fieldA and that is due to the invariance of the Dirichlet-to-Neumann map by gauge transformation. For that, in geometric term, the vector fieldAdefines the connection given by the one formαA=Pd

j=1ajdxj, and the non-uniqueness says that the best we could hope to reconstruct from the Dirichlet-to-Neumann map Λ_A,qis the connectiondα_Agiven by

dα_A=

d

X

i,j=1

∂a_i

∂xj

−∂a_j

∂xi

dx_j∧dx_i. (1.4)

We denote by

HA,q(x, D) =−∆_A+q(x)

where ∆A as given by (1.2), with domain D(HA,q) = H₀¹(Ω)∩H²(Ω).It is well known that the spectrum ofHA,qconsists of a sequence of the eigenvalues, counted according to their multiplicities

λ_1,A,q ≤λ_2,A,q ≤. . .≤λ_k,A,q →+∞.

The corresponding eigenfunctions is denoted by(ϕ_k,A,q).We may assume that this sequence form an orthonormal basis ofL²(Ω).

We consider the eigenvalue problem







HA_l,q_l(x, D)ϕ(x) =λ ϕ(x) inΩ,

ϕ(x) = 0 onΓ.

(1.5) To simplify the notation we have

(λk,A_l,q_l)k= (λ^l_k)k and(ϕk,A_l,q_l)k= (ϕ^l_k)k forl= 1,2.

We assume that0is not an eigenvalue ofHA,q. Then using the elliptic regularity, we obtain kϕ_kk_H2(Ω)≤Cλ_kkϕ_kk_L2(Ω).

Where,Cis depending only onΩ, M.Therefore

k∂ k ≤

To obtain an asymptotic behavior of eigenvalues ofHA,q, we cannot apply the Weyl’s formula. Then this is owing to the difficulty to find an exact elementary solution of the parabolic equation corresponds to this operator. Since the counting function

N(˜λ) =#{k≥1, λ_k≤λ}˜

depends only on principal part of the symbol ofHA,q(x, D)which equal to|ξ|²(See Reiko and Sigeru [30]), we have

C⁻¹k^2/d≤λ_k ≤Ck^2/d. Then, we can conclude that

k∂_νϕkk_H1/2(Γ)≤Ck^2/d. (1.6)

We recall that`¹ is the usual Banach space of real-valued sequences such that the corresponding series is absolutely convergent. This space is equipped with its natural norm.

We fixd+ 2> m > d

2 + 1such thatw= (w_k)_kis the sequence given byw_k=k^−2md for eachk≥1.

We introduce the following Banach spaces

`¹_w(H^1/2(Γ)) ={g= (g_k)_k; g_k∈H^1/2(Γ); k≥1and

w_kkg_kk_H1/2(Γ)

∈`¹}, and

`<sup>1</sup><sub>w</sub>(C) ={y= (yk)k; yk∈C; k≥1and (wk|y<sub>k</sub>|)∈`¹}.

The natural norms on this spaces are respectively kgk_`1

w(H^1/2(Γ))=X

k≥1

w_kkg_kk_H1/2(Γ), (1.7)

and

kyk_`1

w(C)=X

k≥1

wk|y_k|. (1.8)

In what follows, we shall use the following notations:

D_Ω = infn

R∈R⁺ : Ω⊂B(x₀, R) for some x₀ ∈R^d o

. (1.9)

HereB(x0, R) =

x∈R^d : |x−x0|< R .

The one-dimensional inverse problem of the reconstruction of a differential operator from its spectral data goes back to 40-50 th (B¨org [10], Levinson [23], Gelfand-Levitan [14], Krein [20, 21]).

The issue of stability estimates for multidimensional inverse spectral problems for hyperbolic opera- tor with electric potential was first addressed by Alessandrini and Sylvester [1] and recently Bellassoued, Choulli and Yamamoto [4] found a stability estimate related to the multidimensional Borg-Levinson the- orem using a result of stability in determiningq from a partial Dirichlet to Neumann map provided that qis a priori known in a neighborhood of the boundary of spatial domain and satisfies on additional con- dition. And this result is an extension of result in [11] which itself is a variant of a theorem in [1].

The inverse spectral problem for the Schr¨odinger operator with analytic potential was considered by Berezanskii [8, 9]. In 1987 Sylvester and Uhlmann [33] and Novikov and Henkin [15] proved the

uniqueness in the nonanalytic case. A.Nachman, J.Sylvester, G.Uhlmann [25] solved the inverse bound- ary spectral problem for the Schr¨odinger operator by reducing it to the inverse boundary value problem in fixed frequency and then using the method of complex geometric optics [33].

For more general literature see Katchalov, Kurylev and Lassas [22] studies of the inverse boundary spectral problem. The main aim of this book is to develop a rigorous theory to solve several types of inverse problem exactly, rather than to discuss applied numerical aspects of these problems.

In this paper, we show how to obtain estimates fordαAandqby some spectral data. More precisely, we will show that the knowledge of the eigenvalues{λ_k, k≥1}and the boundary value of the normal derivatives of the corresponding eigenfunctions{∂_νϕ_k, k ≥ 1}is sufficient to uniquely determine the magnetic fielddαAand the electric potentialq.Our main result is given by the following theorem Theorem 1 LetA1, A2 be two realW^3,∞(Ω;R^d)vector fields onΩandq1, q2 ∈L^∞(Ω), α > ^d₂ + 3 such thatkA_lk_Hα(Ω) ≤M andkq_lk_L∞(Ω) ≤M forl = 1,2. We assume thatA1 =A2andq1 =q2 on the boundaryΓ. Then there exist a constantC >0and0< θ <1such that

kq₁−q2k_H−1(Ω)+kdα_A1 −dαA2k_L∞(Ω) ≤C

λ¹−λ²

`¹_w(C)+

∂νϕ¹−∂νϕ²

`¹_w(H^1/2(Γ))

θ

. (1.10) The left-hand side of the inequality is assumed to be small andCdepends onΩ,Mandd.

To prove Theorem 1, we shall make use of the Dirichlet to Neumann given by (1.3). So we are going to establish a stability result for the inverse problem consisting in the determination of magnetic field dα_Aand the potentialqfrom the norm of the D-to-N mapΛ_A,qinL(H¹(Σ), L²(Σ)).

Theorem 2 LetA₁, A₂ be two realW^3,∞(Ω;R^d)vector fields onΩandq₁, q₂ ∈ L^∞(Ω),α > ^d₂ + 3 such thatkA_lk_Hα(Ω) ≤M andkq_lk_L∞(Ω) ≤M forl = 1,2. We assume thatA1 =A2andq1 =q2 on the boundaryΓandT > DΩ. Then there exist a constantC >0andµ⁰ ∈(0,1)such that

kq₁−q₂k_H−1(Ω)+kdα_A1−dα_A2k_L∞(Ω)≤CkΛ_A1,q1 −Λ_A2,q2k^µ0. (1.11) WhereCdepends onΩ,M,T andd.

In order to study the spectral stability, we denote by Λ˜_A,q the restriction of Λ_A,q in the space H^2d+4(0, T;H^3/2(Γ))which given by

Λ^]_A,q:H^2d+4(0, T;H^3/2(Γ))−→L²((0, T);H^s(Γ)), (1.12) for anys∈[0,1/2]. We denotek.k_sthe norm inL(H^2d+4(0, T;H^3/2(Γ)), L²((0, T);H^s(Γ))).

Theorem 3 LetA₁, A₂ be two realW^3,∞(Ω;R^d)vector fields onΩandq₁, q₂ ∈L^∞(Ω), α > ^d₂ + 3 such thatkA_lk_Hα(Ω) ≤M andkq_lk_L∞(Ω) ≤M forl = 1,2. We assume thatA₁ =A₂andq₁ =q₂ on the boundaryΓandT > DΩ. Then there exists a constantC >0andµ∈(0,1)such that

kq₁−q2k_H−1(Ω)+kdα_A1−dαA2k_L∞(Ω)≤C Λ^]_A

1,q1 −Λ^]_A

2,q2

µ

s. (1.13)

The inverse problem given by Theorem 2 and 3 is to recover information about the magnetic and electric potential from the D-to-N map measured on the whole boundary.

The hyperbolic inverse problem often occurs in applications, they have been extensively studied and there are several methods to solve them. Most methods are based on the geometrical ideas and finite speed of propagation e.g Isakov [18]. The results devoted to the uniqueness in these problems can be found in Eskin [13], Rakesh-Symes [28] , Ramm-Sjostrand [29].

Although stability in the hyperbolic inverse problem is a less studied subject than uniqueness, there are already many interesting results in this direction. These results Bellassoued [2], Bellassoued-Ben Joud [3], Bellassoued-Jellali-Yamamoto [5, 6], Imanuvilov-Yamamoto [16], [19], [32].

The paper is organized as follows. Section 2 is devoted to prove the Theorem 1. Section 3 deals with the construction of geometrical optics solutions, Hodge decomposition and contains the proof of Theorem 2 and 3. Finally, we give a proof of some important Lemmas considered in the Appendix.

2 Proof of Theorem 1

LetA ∈W^3,∞(Ω,R^d)andq ∈L^∞(Ω).We denoteσ(HA,q) = {λ_k}_k≥1 be the spectrum ofHA,q and ρ(HA,q) = C\σ(HA,q) be resolvent set ofσ(HA,q). For anyλ ∈ ρ(HA,q) andf ∈ H^3/2(Γ),we introduce the elliptic problem







(−∆_A+q−λ)u= 0 inΩ,

u=f onΓ.

(2.1) We define the D-to-N map associated to the above problem given by

ΠA,q:H^3/2(Γ)→H^s(Γ) (2.2)

for0< s < ¹₂,such thatΠ_A,q(f) = ∂u

∂ν+iA.νu. We denote byk.k3

2,sthe norm inL(H^3/2(Γ), H^s(Γ)).

We are going to introduce the following lemmas, which are given in [1], [4] and [11], in case we have a laplace instead of the magnetic laplace. For more detail, we shall sketch the proofs to Lemma 2.2 and 2.3 in an appendix.

Lemma 2.1 Let A ∈ W^3,∞(Ω;R^d) andq ∈ L^∞(Ω). Then for any m > ^d₂, f ∈ H^3/2(Γ)andλ ∈ ρ(HA,q),we have

d^m dλ^m

ΠA,q(λ)

f =−m!X

k≥1

1

(λ_k−λ)^m+1hf, ∂_νϕki∂νϕk|_Γ, whereh ,idenote the inner product inL²(Γ).

Lemma 2.2 Let0< s < ¹₂. We fixj≥0then we have

d dλ

j

Π_A1,q1(λ)−Π_A2,q2(λ) 3

2,s

≤C|λ|^{−j+14(1+2s)}.

Remark 1 The difference between the result of the previous lemma and what was expressed in [1] is the power of theλand this is due to the presence of the magnetic potential.

Lemma 2.3 For anyf ∈H^2(d+2)(0, T, H^1/2(Γ))satisfies ∂

∂t j

f(0, x) = 0 forx∈Γandj= 0,1, . . . ,2d+ 3, we have

Λ^]_A,q(f) =

d+1

X

j=0

"

d dλ

j

ΠA,q(λ)

#

λ=0

−∂²

∂t² j

f+RA,qf, (2.3)

where,

RA,q(f) =

∞

X

k=1

(λ_k)^−d−5/2 ∂ϕ_k

∂ν Z t

0

sinp

λ_k(t−s)

*

−∂²

∂s² d+2

f(s, .), ∂_νϕ_k(.) +

ds+iA.νf,

whereh ,idenote the inner product inL²(Γ).

2.1 Preliminaries estimations

In this subsection, We will introduce the estimates, which are the keys of our result. For that we note P^(j)(λ) =

d dλ

j

Π_A1,q1(λ)−Π_A2,q2(λ) , R^](f) =RA1,q1(f)−RA2,q2(f) and

δ=

λ¹−λ²

`¹_w(C)+

∂_νϕ¹−∂_νϕ²

`¹_w(H^1/2(Γ)).

Lemma 2.4 There exists a constantC >0such that the following estimates holds true

P^(d+1)(λ) 3

2,s ≤Cδ. (2.4)

and

P^(j)(0) 3

2,s ≤Cδ^θ, (2.5)

for anyλ≤0. Whereθis equal to1−_4d+5−2s^4d−4 .

Proof . We first address the case wherej =d+ 1. We suppose thatA₁ =A₂ onΓ. Forf ∈H^3/2(Ω), we have the following equation

P^(d+1)(λ)(f) = (d+ 1)!



−X

k≥1

1 (λ¹_k−λ)^d+2

f, ∂νϕ¹_k

∂νϕ¹_k+X

k≥1

1 (λ²_k−λ)^d+2

f, ∂νϕ²_k

∂νϕ²_k





= −(d+ 1)!X

k≥1

1

(λ¹_k−λ)^d+2 − 1 (λ²_k−λ)^d+2

f, ∂νϕ¹_k

∂νϕ¹_k

−(d+ 1)!X

k≥1

1

(λ²_k−λ)^d+2 < f, ∂νϕ¹_k−∂νϕ²_k> ∂νϕ¹_k

−(d+ 1)!X

k≥1

1 (λ²_k−λ)^d+2

f, ∂_νϕ²_k

∂_νϕ¹_k−∂_νϕ²_k

= I1+I2+I3. Where

I1 =−(d+ 1)!X

k≥1

1

(λ¹_k−λ)^d+2 − 1 (λ²_k−λ)^d+2

f, ∂νϕ¹_k

∂νϕ¹_k I2 =−(d+ 1)!X

k≥1

1 (λ²_k−λ)^d+2

f, ∂_νϕ¹_k−∂_νϕ²_k

∂_νϕ¹_k I3 =−(d+ 1)!X

k≥1

1 (λ²_k−λ)^d+2

f, ∂νϕ²_k

∂νϕ¹_k−∂νϕ²_k . We have the following estimates

kI1k_H1/2(Γ)≤(d+ 1)!X

k≥1

1

(λ¹_k−λ)^d+2 − 1 (λ²_k−λ)^d+2

∂ϕ¹_k

2

H^1/2(Γ)kfk_L2(Γ). (2.6) Using the asymptotic behavior of the eigenvalue and elliptic regularity we have

1

(λ¹_k−λ)^d+2 − 1 (λ²_k−λ)^d+2

≤ Cmax

k≥1

1

(λ¹_k)^d+3, 1 (λ²_k)^d+3

λ¹_k−λ²_k

≤ C

k^2d(d+3)

λ¹_k−λ²_k

(2.7)

and

∂ϕ¹_k

H^1/2(Γ)≤k^2d. (2.8)

Combining (2.6), (2.7) and (2.8) we obtain

kI1k_H1/2(Γ)≤(d+ 1)!X

k≥1

1 k^2d(d+2)

λ¹_k−λ²_k

kfk_L2(Γ). (2.9)

Using the fact that ²_d(d+ 2)> ²_dm, then we have kI1k_H1/2(Γ)≤C

λ¹−λ²

`¹_w(C)kfk_L2(Γ). (2.10)

Using expression ofI2, we have kI2k_H1/2(Γ) ≤(d+ 1)!X

k≥1

1 λ²_k−λ

d+2

∂_νϕ¹_k H^1/2(Γ)

∂_νϕ¹_k−∂_νϕ²_k

H^1/2(Γ)kfk_L2(Γ). By (2.8) we have

kI2k_H1/2(Γ) ≤ (d+ 1)!X

k≥1

1 k^d2(d+2)

∂νϕ¹_k−∂νϕ²_k

H^1/2(Γ)kfk_L2(Γ)

≤ C

∂νϕ¹−∂νϕ²

`¹_w(H^1/2(Γ))kfk_L2(Γ). (2.11) Using the same argument, we obtain

kI3k_H1/2(Γ)≤C

∂_νϕ¹−∂_νϕ²

`¹_w(H^1/2(Γ))kfk_L2(Γ). (2.12) Combining (2.10), (2.11) and (2.12), we have the desired inequality given by (2.4).

Now, we will prove the second inequality given by (2.5). So to obtain the remaining cases(j < d+1), we write Taylor’s formula with remainder, we may write, for1≤j≤d.

P^(j)(0) =

d

X

h=j

P^(h)(λ)

(h−j)!(−λ)^h−j+ Z 0

λ

(−τ)^d−j

(d−j)!P^(d+1)(τ)dτ. (2.13) Using Lemma 2.2, we have

P^(h)(λ) ₃

2,s ≤C|λ|^{−h+14(1+2s)}. Then

P^(h)(0) 3

2,s

≤ C





d

X

h=j

|λ|^h−j|λ|^−h+14(1+s)+|λ|^d−j+1

P^(d+1) 3

2,s





≤ C





d

X

h=j

|λ|^{−j+14(1+2s)}+|λ|^d−j+1

P^(d+1) 3

2,s



. (2.14)

Using (2.4), we obtain

P^(h)(0) 3

2,s ≤C





d

X

h=j

|λ|^{−j+14(1+2s)}+|λ|^d−j+1δ



. (2.15)

We suppose that|λ| ≥1, so it easy to see that|λ|^−j+12 ≤1and|λ|^−j ≤1forj ≥1.Then we have

P^(h)(0) 3

2,s

≤C

|λ|^{−14(1−2s)}+|λ|^d+1δ

for |λ| ≥1. (2.16) For|λ|=δ^{−4d+5−2s4} and forθ= 1− _4d+5−2s^4d−4 .

This completes the proof.

Lemma 2.5 There existsC >0such that the following estimate is holds

R^]

s

≤Cδ. (2.17)

We remind thatk.k_sis the norm inL(H^2d+4(0, T;H^3/2(Γ)), L²((0, T);H^s(Γ))).

Proof . We denotefj = −∂²

∂s²

j

f, sinceA1=A2onΓ, then R^](f) = X

k≥0

λ¹_k−d−5/2

∂_νϕ¹_k Z t

0

sin q

λ¹_k(t−s)

f_d+2(s, .), ∂_νϕ¹_k(.) ds

−X

k≥0

λ²_k−d−5/2

∂νϕ²_k Z t

0

sin q

λ²_k(t−s)

f_d+2(s, .), ∂νϕ²_k(.) ds.

We splitR^](f)into three termsR^](f) =F1+F2+F3,where F1 =X

k≥0

∂_νϕ¹_k Z t

0







 sin

q

λ¹_k(t−s) λ¹_kd+5/2 −sin

q

λ²_k(t−s) λ²_kd+5/2





f_d+2(s, .), ∂_νϕ¹_k(.)



ds

F2 =X

k≥0

1

(λ²_k)^d+5/2 ∂_νϕ¹_k−∂_νϕ²_k Z t

0

sin q

λ²_k(t−s)

f_d+2(s, .), ∂_νϕ¹_k(.) ds

F3 = X

k≥0

1 (λ²_k)^d+5/2

∂ϕ²_k

∂ν . Z t

0

sin q

λ²_k(t−s)

f_d+2(s, .), ∂νϕ¹_k(.)−∂νϕ²_k(.) ds.

So we have the following estimates

kF1k_H1/2(Γ)≤X

k≥0

∂ϕ¹_k

∂ν

2

H^1/2(Γ)

Z t 0

sin q

λ¹_k(t−s) λ¹_kd+5/2 −

sin q

λ²_k(t−s) λ²_kd+5/2

kf_d+2(s, .)k_L2(Γ)ds.

On the other hand we have

sin q

λ¹_k(t−s) (λ¹_k)^d+5/2 −

sin q

λ²_k(t−s) (λ²_k)^d+5/2

≤ max

k≥1

1

(λ¹_k)^d+3, 1 (λ²_k)^d+3

λ¹_k−λ²_k

≤ 1

k^2d(d+3)

λ¹_k−λ²_k . So we have

kF1k_H1/2(Γ)≤X

k≥0

∂ϕ¹_k

∂ν

2 H^1/2(Γ)

1 k^2d(d+3)

λ¹_k−λ²_k

Z t 0

kf_d+2(s, .)k_L2(Γ)ds. (2.18)

When we use (2.8), we obtain

kF1k_H1/2(Γ) ≤ X

k≥0

1 k^d2(d+3)

λ¹_k−λ²_k

Z _t

0

kf_d+2(s, .)k_L2(Γ)ds

≤ C

λ¹−λ²

`¹_w(C)kfk_H2d+4(0,T;H^1/2(Γ)). Furthermore

kF2k_H1/2(Γ) ≤ X

k≥0

1 k^2d(d+5/2)

∂ϕ¹_k

∂ν −∂ϕ²_k

∂ν

∂ϕ¹_k

∂ν H^1/2(Γ)

Z t 0

kf_d+2(s, .)k_L2(Γ)ds

≤ X

k≥0

1 k^2d(d+5/2)

∂ϕ¹_k

∂ν −∂ϕ²_k

∂ν

Z t 0

kf_d+2(s, .)k_L2(Γ)ds

≤

∂νϕ¹−∂νϕ²

`¹_w(H^1/2(Γ))kfk_H2d+4(0,T;H^1/2(Γ)). Using the same calculus we obtain

kF3k_H1/2(Γ)≤

∂_νϕ¹−∂_νϕ²

`¹_w(H^1/2(Γ))kfk_H2d+4(0,T;H^1/2(Γ)). As a conclusion we found the following estimate

R^]

s

≤Cδ.

This completes the proof of the Lemma.

2.2 End of the proof of Theorem 1

Using the result of Lemma 2.4, we obtain the following estimate for anyϕ∈H^3/2(Γ)

"

d dλ

j

Π_A1,q1(λ)−Π_A2,q2(λ)

#

λ=0

ϕ H^s(Γ)

≤δ^θkϕk_H3/2(Γ). (2.19)

We rely on the decomposition given (2.3), it follows from (2.19) that for anyψ∈H^2d+4(0, T;H^3/2(Γ))

"

d dλ

j

ΠA1,q1(λ)−ΠA2,q2(λ)

#

λ=0

ψ

L²(0,T;H^s(Γ))

≤δ^θkψk_H2d+4(0,T;H^3/2(Γ)). Using the estimation given above and the result of Lemma 2.5, we have

Λ^]_A

1,q1−Λ^]_A

2,q2

s

≤Cδ^θ, (2.20)

provided thatδis sufficiently small.

Combining this estimate with (1.13), we obtain the result of Theorem 1.

3 Stability estimate for the inverse problem from the D-to-N map

In this section, we are interested in the proof of the Theorems 2 and 3. For this, we will prove the two following estimates which are given in the Theorem 2

kdα_A1 −dα_A2k_L∞(Ω)≤CkΛ_A1,q1−Λ_A2,q2k^µ0. (3.21) kq₁−q₂k_H−1(Ω)≤CkΛ_A1,q1−Λ_A2,q2k^µ0. (3.22) We have the same estimates when we replace||Λ_A1,q1 −ΛA2,q2||by||Λ^]_A

1,q1 −Λ^]_A

2,q2||and this is the result of Thorem 3.

To prove the first inequality, we repeat the same idea in Bellassoued and Ben Joud [3] but the difference between the two works is that in [3], we consider only the wave operator in the magnetic field and in this work we added an electric potential. That does not modify neither the Construction of the geometrical optic solutions nor the proofs, we just add in the termV, given in the lemma 3.1 in [3], the term(q₁ − q2)u2and the following estimate

Z

Q

V(x)u2(t, x).v(t, x)dx dt

≤CN_ω(φ2)N_ω(φ1) is still true. The normN_ω(φ)is given after that.

The estimate for the electric potentials is slightly more involved and the complications would arise when one tries to establish the estimate for the electric potential (lower order) in the presence of the magnetic field (higher order). To remedy to this difficulty, we first show by using the Hodge decomposition that the doperator on differential forms in some sense ”bounded invertible” when restricted to right subspaces.

Then we will combine this fact with the estimate we have fordα_A1 −dα_A2 to obtain the estimate for electric potentials. The rest of this paper is devoted to prove an estimate of the electric potential.

3.1 Construction of geometrical optics solutions

The following result concerning the existence of the geometrical optics solutions for the magnetic wave equation will be important to prove our main result. From the hypothesis there exists% > 0such that T > T −4% > D_ΩandΩlies in the ballB(x₀,^T₂ −2%). We may assume without loss of generality that x₀ is the origin ofR^d, and let

D_%=B

0,T 2

\B

0,T 2 −2%

=

x∈R^d : T

2 −2% <|x|< T 2

. (3.1)

Letφ∈ C₀^∞(R^d)such that

supp(φ)⊂D_%. Thus we have

suppφ∩Ω =∅, (suppφ±T ω)∩Ω =∅, ∀ω ∈S^d−1. (3.2) Moreover the functionΦ(t, x) =φ(x+tω)solves inR×R^dthe transport equation

(∂t−ω· ∇) Φ(t, x) = 0. (3.3) Finally, forω∈S^d−1we set

H²_ω(D%) = n

φ∈H²(R^d), ω· ∇φ∈H²(R^d), and supp(φ)⊂D_% o

(3.4)

and

N_ω(φ) =kφk_H2(R^d)+kω· ∇φk_H2(R^d). (3.5) Now we recall the following Lemma which is proved in [3].

Lemma 3.1 Letω ∈S^d−1,φ∈ H_ω²(D_%),A∈W^3,∞(Ω;R^d),q∈L^∞(Ω)andτ >0, then

∂_t²−∆_A+q

u(t, x) = 0, (t, x)∈Q= (0, T)×Ω has a solution of the form

u(t, x) =φ(x+tω)b(t, x)e^iτ(x.ω+t)+ψτ(t, x), (3.6) where

b(t, x) = exp

i Z t

0

ω·A(x+sω)ds

(3.7) andψ_τ(t, x)satisfies

ψ_τ(t, x) = 0, for all (t, x)∈Σ and

ψτ(θ, x) = 0, x∈Ω, θ= 0 orT.

Moreover

τkψ_τk_L2(Q)+k∇ψ_τk_L2(Q)≤CN_ω(φ) (3.8) whereCis a constant depending only onΩ,T,dandkAk_W3,∞(Ω;R^d).

3.2 Hodge decomposition

We consider(Ω,Γ)to be a Riemannian manifold with boundary and denote byF^k(Ω)to be set ofk- forms onΩandW^m,pF^k(Ω)to be itsW^m,pclosure. Set

E^k(Ω) :={dα;α∈H_D¹F^k−1(Ω)}, C^k(Ω) :={dα;α∈H_N¹F^k−1(Ω)},

whereH_D¹F^k(Ω)andH_N¹F^k(Ω)are the set ofH¹ k-forms with homogenous Dirichlet and Neumann boundary trace, respectively. Furthermore, we denote by H^k(Ω)to be the L² closed of the space of harmonick-forms. The corresponding subspaces inW^m,pF^k(Ω)are denoted by

W^m,pE^k(Ω) :=E^k(Ω)∩W^m,pF^k(Ω), W^m,pC^k(Ω) :=C^k(Ω)∩W^m,pF^k(Ω) and

W^m,pH^k(Ω) :=H^k(Ω)∩W^m,pF^k(Ω).

Finally, we denote

W_D^m,p(Ω;Rⁿ) ={A∈W^m,p(Ω;Rⁿ);A.bτ = 0, ∀bτ ∈TxΓ, x∈Γ}

and

X0 =

W^m,pC¹(Ω)⊕W^m,pH¹(Ω)

∩W_D^m,p(Ω;Rⁿ).

In the classical form the Hodge decomposition theorem states that if Ωis a smooth subdomain of smooth compact Riemannian manifold, then any smooth differential formAinΩcan be decomposed as

A=dα+δβ+κ

whereα, β, κare smooth differential forms inΩ,two of which have a vanishing normal or tangential component on Ωand so that dκ = δκ = 0. Heredis the exterior differential operator and δ is the codifferential.

IfA∈W^m,pF¹andΩhas a smooth boundary, then on has Hodge decomposition given by

A=dα+δβ+κ and α∈W_D^m+1,p(Ω,R), δβ∈W^m,pC¹(Ω) and κ∈W^m,pH¹(Ω). (3.9) Further, the three summands in (3.9) are uniquely determined and mutually orthogonal with respect to the naturalL²inner product.

Based on the Hodge decomposition, Tzou showed in [34], that forp≥2andm≥1 d:X₀ −→W^m−1,pE²(Ω) has a bounded inverse.

This statement means that for allA∈W^m−1,pE²(Ω)we havekAk_Wm,p(Ω,Rⁿ)≤CkdAk_Wm−1,p(Ω,Rⁿ). We would like to apply this result to the vector field(A1−A2)which may not be inX0. For that, pick p > nand apply the Hodge decomposition to(A₁−A₂)in the spaceW^3,∞(Ω;Rⁿ)to getA₁−A₂ = dα+δβ +κ, where α ∈ W^4,∞(Ω)∩H₀¹(Ω). Define A⁰₁ = A₁ − dα

2 , A⁰₂ = A₂ + dα

2 so that A⁰₁ −A⁰₂ ∈ W^3,∞C¹(Ω)⊕W^3,∞H¹(Ω).Since we have already assumed that A1 −A2 = 0inΓ, it is easy to show that it has no tangential component at the boundary andα ∈H₀¹(Ω),we conclude that A⁰₁−A⁰₂has no tangential component. This means that

A⁰₁−A⁰₂ ∈W^3,∞C¹(Ω)⊕W^3,∞H¹(Ω)∩H_D¹(Ω;Rⁿ).

So by the Lemma 6.2 from [34]

A⁰₁−A⁰₂

W^3,∞(Ω) ≤C dα_A⁰

1 −dα_A⁰

2

L^∞(Ω)=kdα_A1 −dαA2k_L∞(Ω). (3.10) Gauge equivalence then implies thatΛ_A⁰

j,qj = Λ_Aj,qjforj= 1,2.

Remark 2 With this choice ofA⁰₁ andA₂⁰, we remake that div(A⁰₁−A⁰₂) = 0. This come from the fact that on the 1-form div=∗d∗, so using this definition, the Lemma 4.1 in Ben Joud [7] and the fact thatκ is an harmonic form we obtain

div(A⁰₁−A⁰₂) =div(δβ+κ) =∗d∗δβ+∗d∗κ=−δδβ−δκ= 0.

We will from now, make the same work again with the magnetic field defined above. In this case, with this special forms ofA⁰₁ andA⁰₂, there is a condition change given byA⁰₁−A⁰₂is not vanish inΓbut it has not tangential component.

As before, we set

A⁰(x) = (A⁰₁−A⁰₂)(x) (3.11) and

V⁰(x) =−idiv(A⁰) + (A⁰₂²−A⁰₁²)(x) = (A⁰₂²−A⁰₁²)(x) = (A⁰₂−A₁⁰)(A⁰₂+A⁰₁). (3.12) Recall that since(A⁰₁−A⁰₂) = 0and(q₁−q₂) = 0onΓ, we can extendA⁰to aH¹(R^d)vector field and qtoL^∞(R^d)by defining it to be zero outside ofΩand we will refer to the extension asA⁰andq. We can extendV toL^∞(R^d)function by defining it to be zero outside ofΩand we will refer to the extension as V.

3.3 Preliminaries estimates

In this subsection, we complete the proof of Theorem 2. We are going to use the geometrical optics solutions and thex-ray transform of the difference of two magnetic potentials.

As before, we letω ∈ S^d−1 andA_l ∈ W^3,∞(Ω;R^d),q_l ∈ L^∞(Ω;R^d)such that kA_lk_W3,∞ ≤ M and kq_lk_L∞ ≤M forl= 1,2. We setA⁰(x) = (A⁰₂−A⁰₁)(x), q(x) = (q2−q1)(x)and

b(t, x) = (b2b1)(t, x) = exp

i Z t

0

ω·A⁰(x+sω)ds

. (3.1)

Recall that since(A⁰₁−A⁰₂) = 0andq1−q2= 0onΓ, we can extendA⁰to aH¹(R^d)vector field and qtoL^∞(R^d)by defining it to be zero outside ofΩand we will refer to the extention asA⁰ andq. With this extention we have thatdα⁰_Ais anL²(R^d)function supported only inΩ.

Lemma 3.2 There exists C > 0 such that for any ω ∈ S^d−1 and φ1, φ2 ∈ H_ω²(D%) the following estimates holds true:

Z T 0

Z

R^d

q(x+tω)b(t, x)dxdt

≤ Cτ³kφ₁k_H1(R^d)kφ₂k_H2(R^d)kΛ_A1,q1 −Λ_A2,q2k +C

τN_ω(φ₁)N_ω(φ₂) +Cτ A⁰

L^∞N_ω(φ₁)N_ω(φ₂) (3.2) for any sufficiently largeτ >0.

Proof . Forτ sufficiently large, Lemma 3.1 guarantees the existence of the geometrical optics solutions u2 to

∂_t²−∆_A⁰

2 +q₂

u(t, x) = 0 inQ, u(0,·) =∂_tu(0,·) = 0 inΩ in the form

u₂(t, x) =φ₂(x+tω)b₂(t, x)e^iτ(x.ω+t)+ψ_2,τ(t, x), (3.3) corresponding to the magnetic potentialA₂ andφ₂, whereψ_2,τ satisfies

τkψ_2,τk_L2(Q)+k∇ψ_2,τk_L2(Q)≤CN_ω(φ₂) (3.4) and

u2∈ C¹(0, T;L²(Ω))∩ C(0, T;H¹(Ω)).

We denoteu1, the solution of















∂_t²−∆_A⁰

1 +q₁

u₁ = 0 (t, x)∈Q, u1(0, x) =∂tu1(0, x) = 0 x∈Ω, u₁(t, x) =u₂(t, x) :=f_τ(t, x) (t, x)∈Σ.

(3.5)

Definingu=u1−u2, one gets















∂_t²u−∆_A⁰

1+q1

u(t, x) = 2iA⁰· ∇u₂(t, x) +V⁰(x)u2(t, x) +q(x) (t, x)∈Q,

u(0, x) =∂tu(0, x) = 0 x∈Ω,

whereV⁰(x)is given in (3.12).

Therefore, we have constructed the special solutionv∈ C¹(0, T;L²(Ω))∩ C(0, T;H¹(Ω))to the back- ward magnetic wave equation

∂_t²−∆_A⁰

1 +q1

v(t, x) = 0, (t, x)∈Q, v(T, x) =∂tv(T, x) = 0, x∈Ω, having the form

v(t, x) =φ1(x+tω)b1(t, x)e^iτ(x·ω+t)+ψ1,τ(t, x), (3.6) corresponding to the magnetic and electric potentialA⁰₁, q₁andφ₁, whereψ_1,τ satisfies

τkψ_1,τk_L2(Q)+k∇ψ_1,τk_L2(Q)≤CN_ω(φ1). (3.7) Integrating by parts and using the Green’s formula, we obtain

Z

Q

∂_t²−∆_A⁰

1+q1

u(t, x)·v(t, x)dx dt= Z

Q

2iA⁰· ∇u₂(t, x)·v(t, x)dxdt +

Z

Q

V⁰(x)u2(t, x)·v(t, x)dxdt+ Z

Q

q(x)u2(t, x)·v(t, x)dxdt

= −

Z

Σ

∂_ν+i A⁰₁·ν

u(t, x)v(t, x)dσ_xdt. (3.8)

Combining (3.8) with (3.5), we obtain Z

Q

q⁰(x)u₂(t, x)·v(t, x)dxdt=− Z

Q

2iA⁰· ∇u₂(t, x)·v(t, x)dxdt

− Z

Q

V⁰(x)u₂(t, x)·v(t, x)dxdt− Z

Σ

(Λ_A1,q1 −Λ_A2,q2) (f_τ)(t, x)g_τ(t, x)dσ_xdt (3.9) where

gτ(t, x) =φ1(x+tω)b1(t, x)e^iτ(x·ω+t), (t, x)∈Σ.

It follows from (3.6) and (3.3) that Z

Q

q(x)u₂(t, x)·v(t, x)dxdt=− Z

Q

q(x)(φ₂φ₁)(x+tω)(b₂b₁)(t, x)dxdt+I_τ. (3.10) By using (3.7) and (3.4) we obtain

|I_τ| ≤ C

τ N_ω(φ₂)N_ω(φ₁). (3.11)

Consequently, by (3.11), (3.10) and (3.9), we obtain

Z

Q

q(x)(φ₂φ₁)(x+tω)(b₂b₁)(t, x)dxdt

≤ Z

Q

2iA⁰· ∇u₂(t, x)·v(t, x)dxdt +C

Z

Q

V(x)u2·vdxdt

+C Z

Σ

(ΛA1,q1 −ΛA2,q2) (fτ)(t, x)g_τ(t, x)dσxdt

+C

τ N_ω(φ2)N_ω(φ1).

Moreover, by (3.7) and (3.4), one gets

Z

Q

V(x)u₂·vdxdt

= Z

Q

A⁰(x)(A₁+A₂)u₂·vdxdt

≤C A⁰

L^∞(Ω)N_ω(φ₂)N_ω(φ₁). (3.12)