Reconstruction of interfaces from the elastic far-field measurements using CGO

www.oeaw.ac.at

www.ricam.oeaw.ac.at

Reconstruction of interfaces from the elastic far-field measurements using CGO

solutions

M. Kar, M. Sini

RICAM-Report 2014-35

MEASUREMENTS USING CGO SOLUTIONS

MANAS KAR^∗ AND MOURAD SINI ^†

Abstract. In this work, we are concerned with the inverse scattering by interfaces for the linearized and isotropic elastic model at a fixed frequency. First, we derive complex geometrical optic solutions with linear or spherical phases having a computable dominant part and anH^α-decaying remainder term withα <3, whereH^α is the classical Sobolev space. Second, based on these properties, we estimate the convex hull as well as non convex parts of the interface using the farfields of only one of the two reflected body waves (pressure waves or shear waves) as measurements. The results are given for both the impenetrable obstacles, with traction boundary conditions, and the penetrable obstacles. In the analysis, we require the surfaces of the obstacles to be Lipschitz regular and, for the penetrable obstacles, the Lam´e coefficients to be measurable and bounded with the usual jump conditions across the interface.

Key words. Scattering, elasticity, farfields, complex geometrical optic solutions, integral equations.

AMS subject classifications. 35P25, 35R30, 78A45.

1. Introduction and statement of the result. LetD be a bounded and open set ofR³ such that R³\D is connected. The boundary ∂D of D is Lipschitz. We denote by λ and µ the Lam´e coefficients andκthe frequency. We assume that those coefficients are measurable, bounded and satisfy the conditions µ >0, 2µ+ 3λ >0 andµ=µ0, λ=λ0forx∈R³\D withµ0 andλ0 being constants. In addition, we set λD:=λ−λ0 andµD:=µ−µ0 and assume that 2µD+ 3λD≥0 andµD>0.¹

The direct scattering problem can be formulated as follows. Let uⁱ be an incident field, i,e. a vector field satisfying µ0∆uⁱ+ (λ0+µ0)∇divuⁱ+κ²uⁱ = 0 inR³ and u^s(uⁱ) be the scattered field associated to the incident fielduⁱ. In the impenetrable case, the scattering problem reads as follows







µ0∆u^s+ (λ0+µ0)∇divu^s+κ²u^s= 0, inR³\D σ(u^s)·ν =−σ(uⁱ)·ν, on∂D

lim_|x|→∞|x|(^∂u

s p

∂|x|−iκ_pu^s_p) = 0 and lim_|x|→∞|x|(^∂u_∂|x|^ss −iκ_su^s_s) = 0,

(1.1)

where the last two limits are uniform in all the directions ˆx:=_|x|^x ∈S², a unit sphere inR³, withσ(u^s)·ν :=

(2µ∂ν+λνdiv +µν×curl)u^sand the unit normal vectorνis directed into the exterior ofD. In the penetrable obstacle case, the total fieldu^t:=u^s+uⁱ satisfies

(∇ ·(σ(u^t)) +κ²u^t= 0, in R³ lim_|x|→∞|x|(^∂u

s p

∂|x|−iκpu^s_p) = 0 and lim_|x|→∞|x|(_∂|x|^∂uss −iκsu^s_s) = 0. (1.2) The two limits in (1.1) and (1.2) are called the Kupradze radiation conditions. For any displacement field v, taken as a column vector, the corresponding stress tensor σ(v) can be represented as a 3×3 matrix:

σ(v) =λ(∇ ·v)I₃+ 2µ(v), where I₃ is the 3×3 identity matrix and(v) = ¹₂(∇v+ (∇v)^>) denotes the infinitesimal strain tensor. Note that for v = (v₁, v₂, v₃)^>,∇v denotes the 3×3 matrix whose j-th row is

∇v_jforj= 1,2,3.Also for a 3×3 matrix functionA,∇ ·Adenotes the column vector whosej-th component is the divergence of thej-th row ofA forj = 1,2,3.

In both (1.1) and (1.2), we denotedu^s_p:=−κ⁻²_p ∇div u^sto be the longitudinal (or the pressure) part of the field u^sand u^s_s:=κ⁻²_s curl curl u^s to be the transversal (or the shear) part of the field u^s. The constants κp:=^√_2µ^κ

0+λ₀ andκs:= ^√κ_µ

0 are known as the longitudinal and the transversal wave numbers respectively.

∗RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz, Austria. (Email:[email protected]) Supported by the Austrian Science Fund (FWF): P22341-N18.

†RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz, Austria. (Email:[email protected]) Partially supported by the Austrian Science Fund (FWF): P22341-N18.

1The assumptions on the jumps can be relaxed. It is needed only at the vicinity of the points on the interface∂D. In addition, we can also consider the case 2µ_D+ 3λ_D≤0 andµ_D<0, see Remark 5.5.

1

From the first equality of (1.1) (and equally the one of (1.2) for x ∈ R³\D), we obtain the well known decomposition of the scattered fieldu^sas the sum of its longitudinal and transversal parts, i.e. u^s=u^s_p+u^s_s. The scattering problems (1.1) and (1.2) are well posed, see for instance [8, 24, 25, 27] and [26]. The scattered fieldu^shas the following asymptotic expansion at infinity:

u^s(x) := e^iκp|x|

|x| u^∞_p (ˆx) +e^iκs|x|

|x| u^∞_s (ˆx) +O( 1

|x|²), |x| → ∞ (1.3) uniformly in all the directions ˆx∈ S², see [2] for instance. The fields u^∞_p (ˆx) andu^∞_s (ˆx) defined on S² are called correspondingly the longitudinal and transversal parts of the far field pattern. The longitudinal part u^∞_p (ˆx) is normal toS² while the transversal partu^∞_s (ˆx) is tangential toS². It is well known that scattering problems in linear elasticity occur when we excite one of the two types of incident plane waves, pressure (or longitudinal) waves and shear (or transversal) waves. They have the analytic formsu^p_i(x, d) :=de^iκpd·x and u^s_i(x, d) :=d^⊥e^iκsd·x respectively, where d^⊥ is any vector in S² orthogonal to d. Remark that u^p_i(·, d) is normal toS² andu^s_i(·, d) is tangential toS². Then we use superpositions of incident pressure and shear waves given by

ui(x, d) :=αde^iκpd·x+βd^⊥e^iκsd·x (1.4) whereα, β∈Candd∈S².

We denote by u^∞(·, d, α, β) the far field pattern associated with the incident waves of the form (1.4).

We also denote by u^∞_p (·, d, α, β) andu^∞_s (·, d, α, β) the corresponding longitudinal and transversal parts of the far field u^∞(·, d, α, β), i.e. u^∞(·, d, α, β) = (u^∞_p (·, d, α, β), u^∞_s (·, d, α, β)). Hence, we obtain the far field measurements

(u^p_i, u^s_i)7→F(u^p_i, u^s_i) :=

u^∞,p_p (·, d) u^∞,s_p (·, d) u^∞,p_s (·, d) u^∞,s_s (·, d)

(1.5) where:

1. (u^∞,p_p (·, d), u^∞,p_s (·, d)) is the far field pattern associated with the pressure incident fieldu^p_i(·, d).

2. (u^∞,s_p (·, d), u^∞,s_s (·, d)) is the far field pattern associated with the shear incident fieldu^s_i(·, d).

Our concern now is to investigate the followinggeometrical inverse problem:

From the knowledge of u^∞(·, d, α, β) for all directionsxˆ andd inS² and a couple (α, β)6= (0,0) in C, determineD.

We can also restate this problem as follows: From the knowledge of the matrix (1.5) for all directionsxˆ and dinS² determineD.

The first uniqueness result was proved by Hahner and Hsiao, for the model (1.1), see [13]. It says that every column of the matrix (1.5) for all directions ˆxanddin S², determinesD. Later Alves and Kress [2], Arens [3], A. Charalambopoulos, D. Gintides and K. Kiriaki [5, 6, 10] proposed sampling types methods to solve the inverse problem using the full matrix (1.5) for all directions ˆxand d in S². We also mention the works by Guzina and his collaborators using the full near fields [4, 12, 32]. We remark that one not only needs the information over all directions of incidence and measurements, but also both pressure and shear far fields are necessary. In recent works we proved that it is possible to reduce the amount of data for detectingD as follows:

The knowledge ofu^∞_p (ˆx, d, α, β)(or respectivelyu^∞_s (ˆx, d, α, β)) for all directionsxˆanddinS²and the couple (α, β) = (1,0) or(α, β) = (0,1)uniquely determines the obstacle D.

In other words, this result says that every component of the matrix (1.5) for all directions ˆx and din S², determinesD. In [11], we assumed aC⁴-regularity of∂D to prove this result for the impenetrable obstacle with free boundary conditions, the model (1.1). ThisC⁴-regularity is used to derive explicitly the first order term in the asymptotic of the indicator functions of the probe or singular sources method in terms of the source points. This regularity is reduced to Lipschitz in [19] for both the models (1.1) and (1.2), see also [15] for the rigid obstacles.

It is known that these probe/singular sources methods are based on the use of approximating domains isolating the source point of the used point sources, see [33]. Since this source point has to move around

and near the interface, it creates extra instabilities of the method. To overcome this difficulty, Ikehata [16]

proposed the enclosure method which has the same principle as the probe/singular sources methods but instead of using point sources, he uses complex geometrical optics type solutions with linear phases, CGOs in short. The price to pay is that, in contrast to the point sources with which we obtain the whole interface

∂D, we can only obtain the intersection of the level-curves of the CGOs with ∂D. In the case of linear phases, we can estimate the convex hull, see [18] for an overview of this method. Later, in the work [22]

by Kenig, Sj¨ostrand and Uhlmann, other CGOs have been proposed to solve the EIT problem using the localized Dirichlet-Neumann map. These CGOs have a phase of quadratic form, i.e. behaving as spherical waves. Inspired from these CGOs, Nakamura and Yoshida proposed in [31] an enclosure method based on CGOs with spherical waves with which they could estimate, in addition to the convex hull of D, some non convex parts of∂D. Another family of CGOs is proposed by Uhlmann and Wang [36] where the phases are the harmonic polynomials in the 2D case. With these CGOs, one can recover all the visible part, by straight rays, of the interface∂D. We refer the reader to [9] for a classification of these CGOs. Regarding the Lam´e model, the CGOs with linear or spherical phases are constructed in [35], for the stationary case, i.e. k= 0, while in [23] the ones with harmonic polynomial phases in the 2D case are investigated. The general form of these CGO’s is

u:=e^{−τ(φ+iψ)}(a+r) (1.6)

whereφis the known and explicit phase we were talking about,ψis also explicitly known,ais either known explicitly or computable and the remainder ris small in the classical SobolevH^α−norm, α <1, in term of the parameterτ, i.e. precisely one has the estimatekrkH^α=O(τ^α−1),α≤1.

In our work, we use p-parts or s-parts of the farfield measurements. The CGOs of the form (1.6) are not enough, in particular when we use mixed measurements, i.e. p-incident (respectively,s-incident) waves ands-parts (respectively,p-parts) of the corresponding farfield patterns. Instead, we construct CGOs of the form

u:=e^{−τ(φ+iψ)}(a₀+τ⁻¹a₁+a₂τ⁻²+r), (1.7) with linear or logarithmic phases, where now a0, a1 and a2 are either known explicitly or computable and the remainder r is small with the H^α−norm, α <3, in term of the parameter τ, i.e. krkH^α =O(τ^α−3), α≤3. Having these CGOs at hand, we state the indicator function of the enclosure method directly from the farfield measurements and use only one of the two body waves (pressure or shear waves). Then we justify the enclosure method with no geometrical assumptions on the interface ∂D. The analysis is based on the use of integral equation methods on the Sobolev spaces H^s(∂D), s∈ R, for the impenetrable case andL^p estimates of the gradients of the solutions of the Lam´e system with discontinuous Lam´e coefficients, i.e. Meyers’s typeL^pestimates for the Lam´e system, for the penetrable case. This is a generalization to the Lam´e system of the previous works [21] and [34] concerning the Maxwell and acoustic cases respectively.

The paper is organized as follows. In section 2, we define the indicator functions via the farfield pattern.

Then using the denseness property of the Herglotz wave functions and the well posedness of the forward problem, we link the far field measurements to the CGOs. In section 3, the construction of the CGOs is discussed while in section 4 we state the main theorem and describe the reconstruction scheme. In section 5, we prove the main theorem and postpone to section 6 and the appendix the justification of the needed estimates for the CGOs.

2. The indicator functions linking the used farfield parts to the CGOs. In this section, we follow the procedure of [20] where we showed the link between farfields and CGOs in the scalar Helmholtz case. We start with the following identity, see for instance Lemma 3.1 in [2]:

Z

∂D

U ·σ(wh)·ν−wh·σ(U)·ν

ds(x) = 4π Z

S²

U_p^∞hp+U_s^∞hs

ds(θ) (2.1)

for all radiating fields U and w_h where w_h is the scattered field associated with the Herglotz field v_h for h = (h_p, h_s) ∈ L²_p(S²)×L²_s(S²), i.e. v_h(x) := R

S²[e^iκpx·dh_p(d) +e^iκsx·dh_s(d)]ds(d) with L²_p(S²) := {U ∈

(L²(S²))³;U(d)×d= 0} whileL²_s(S²) :={U ∈(L²(S²))³;U(d)·d= 0}.

Let v be a CGO solution for the Lam´e system and state Ω := Ωcgo to be its domain of definition with D ⊂⊂Ω. Examples of these CGOs will be described in Section 3. We take itsp-part vp and its s-part vs. We can find sequences of densities (hⁿ_p)n and (hⁿ_s)n such that the Herglotz waves vhⁿ_p and vhⁿ_s converge to vp andvsrespectively on any domain ˜Ω containingDand contained in Ω. These sequences can be obtained as follows.

We defineH : (L²(S²))³ →(L²(∂Ω))³ as (Hg)(x) :=v_g(x). We know thatH is injective and has a dense range if κ² is not an eigenvalue of the Dirichlet-Lam´e operator on Ω. Due to the monotonicity of these eigenvalues in terms of the domains, we change, if needed, Ω slightly so thatκ²is not an eigenvalue anymore.

Hence, we can find a sequence g_n ∈ (L²(S²))³ such that Hg_n → v in (L²(∂Ω))³. Recall that both Hg_n and v satisfy the interior Lam´e problem. By the well-posedness of the interior problem and the interior estimates, we deduce that Hgn → v in C^∞( ˜Ω), since D ⊂⊂ Ω˜ ⊂⊂ Ω. Hence, −κ⁻²_p ∇∇ ·Hgn → vp

and κ⁻²_s curl curlHgn → vs in C^∞( ˜Ω). But −κ⁻²_p ∇∇ ·Hgn = Hhⁿ_p and κ⁻²_s curl curlHgn =Hhⁿ_s, where hⁿ_p :=d(d·gn) andhⁿ_s :=−d∧(d∧gn).

We setu^s(vs) to be the scattered field corresponding to the s incident wavevs. Thep ands parts of the scattered field u^s(vs) areu^s_p(vs) andu^s_s(vs) respectively. Similarly, we set u^s(vp) to be the scattered field corresponding to the p incident wave vp. The p and s parts of the scattered field u^s(vp) are u^s_p(vp) and u^s_s(vp) respectively.

2.1. Using longitudinal waves. Let (u^∞,p_p , u^∞,p_s ) be the farfield associated to the incident field de^iκpd·x.By the principle of superposition, the farfield associated to the incident fieldv_g(x) :=R

S²de^iκpd·x(d·

g(d))ds(d) is given by

u^∞_g (θ) := (u^∞,p_g (θ), u^∞,s_g (θ)) = ( Z

S²

u^∞,p_p (d, θ)(d·g(d))ds(d), Z

S²

u^∞,p_s (d, θ)(d·g(d))ds(d))

where each component is a vector. Replacing in (2.1), using the fact thatHhⁿ_p converges tov_p in C^∞( ˜Ω), withD⊂⊂Ω, the trace theorem and the well-posedness of the scattering problem, we obtain:˜

m,n→∞lim Z

S²

Z

S²

[u^∞,p_p (d, θ)d·g_n(d)]·[θ(θ·g_m(θ))]ds(θ)ds(d)

= Z

∂D

[u^s(v_p)·(σ(v_p)·ν)−v_p·(σ(u^s(v_p))·ν)]ds(x) (2.2)

and lim

m,n→∞

Z

S²

Z

S²

[u^∞,p_s (d, θ)d·g_n(d)]·[θ∧(θ∧g_m(θ))]ds(θ)ds(d)

= Z

∂D

[u^s(v_p)·(σ(v_s)·ν)−v_s·(σ(u^s(v_p))·ν)]ds(x). (2.3) 2.2. Using shear incident waves. Let i1 and i2 be two vectors linearly independent and tangent to S². Then d∧(d∧i1) and d∧(d∧i2) are obviously also tangent to S² and in addition they are linear independent. Indeed,α1d∧(d∧i1) +α2d∧(d∧i2) = 0⇔d∧(d∧(α1i1+α2i2)) = 0⇔α1=α2= 0.Let now (u^∞,s_p (d, θ), u^∞,s_s (d, θ)) be the farfield associated with the incident plane waved∧(d∧ij)e^iκsx·d, j= 1,2.

Hence the farfield associated with the Herglotz wavevg(x) :=R

S²d∧(d∧g)e^iκsx·dds(d) is v^∞_g (θ) :=

2

X

j=1

Z

S²

u^∞,s_p (d, θ)(ij·g)ds(d), Z

S²

u^∞,s_s (d, θ)(ij·g)ds(d)

,

sinced∧(d∧g) =d∧(d∧i₁)(i₁·g) +d∧(d∧i₂)(i₂·g).Hence,

m,n→∞lim

2

X

j=1

Z

S²

Z

S²

[u^∞,s_s (d, θ)(ij·gn)(d)]·[θ∧θ∧gm(θ)]ds(θ)ds(d)

= Z

∂D

[u^s(v_s)·(σ(v_s)·ν)−v_s·(σ(u^s(v_s))·ν)]ds(x), (2.4)

and lim

m,n→∞

2

X

j=1

Z

S²

Z

S²

[u^∞,s_p (d, θ)(ij·gn)(d)]·[θ(θ·gm(θ))]ds(θ)ds(d)

= Z

∂D

[u^s(v_s)·(σ(v_p)·ν)−v_p·(σ(u^s(v_s))·ν)]ds(x). (2.5) 2.3. The indicator functions. We set

Ipp:= lim

m,n→∞

Z

S²

Z

S²

[u^∞,p_p (d, θ)d·gn(d)]·[θ(θ·gm(θ))]ds(θ)ds(d), (2.6) Ips:= lim

m,n→∞

Z

S²

Z

S²

[u^∞,p_s (d, θ)d·gn(d)]·[θ∧(θ∧gm(θ))]ds(θ)ds(d), (2.7) I_ss:= lim

m,n→∞

2

X

j=1

Z

S²

Z

S²

[u^∞,s_s (d, θ)(i_j·g_n)(d)]·[θ∧θ∧g_m(θ)]ds(θ)ds(d), (2.8)

and Isp:= lim

m,n→∞

2

X

j=1

Z

S²

Z

S²

[u^∞,s_p (d, θ)(ij·gn)(d)]·[θ(θ·gm(θ))]ds(θ)ds(d). (2.9) Therefore, the indicator function I_pp is defined based onp-parts of the far field associated to p-incident wave. Correspondingly I_ps depends ons-part of the far field associated top-incident wave,I_ss depends on s-part of the far field associated to the s-incident wave and finally Isp depends on p-part of the far field associated to thes-incident wave.

3. Construction of CGO solutions.

3.1. CGO solutions for Ipp and Iss. Let us assumeuto be a solution for the following Lam´e system µ₀∆u+ (λ₀+µ₀)∇divu+κ²u= 0. (3.1) Applying the identity curl curl =∇∇ · −∆ in (3.1), we have

u=−λ₀+ 2µ₀

κ² ∇∇ ·u+µ₀

κ²curl curlu=:u_p+u_s, whereu_p andu_s are thep-part ands-part of the solutionurespectively.

3.1.1. p-type CGOs for Ipp. Taking∇·in both sides in (3.1), we obtain µ₀∆(∇ ·u) + (λ₀+µ₀)∆(∇ ·u) =−κ²(∇ ·u).

Define U :=∇ ·u. ThereforeU satisfies the Helmholtz equation ∆U +κ²_pU = 0. Hence thep-part ofu is

−^λ0+2µ_κ2 ⁰∇U. This suggest to take the CGO solution for the Lam´e system of the form

∇U (3.2)

whereU is the CGO solution for scalar Helmholtz equation, i.e,U satisfies

∆U+κ²_pU = 0. (3.3)

The resulting vector field∇U, in (3.2), is also a solution of (3.1) and it is ofp-type since itss-part is zero.

3.1.2. s-type CGOs for Iss. Define V := curlu, where u satisfies (3.1). It satisfies the vector Helmholtz equation

∆V +κ²_sV = 0. (3.4)

Thes-part of the solutionuis ^µ_κ⁰₂curlV.This suggests to take the CGO solution for the Lam´e system of the form

curlV. (3.5)

The resulting vector field curlV, in (3.5), is also a solution of (3.1) and it is ofs-type since itsp-part is zero.

3.1.3. The forms of the corresponding CGOs. In the following proposition, we provide the forms of the CGOs for the Helmholtz type equations in (3.3) and (3.4). The CGOs we use forIppandIss are then deduce using (3.2) and (3.5) respectively.

Proposition 3.1.

1. (Linear phase.)

Letρ, ρ^⊥∈S²,withρ·ρ^⊥= 0andt, τ >0.The functionU(x;τ, t) :=e^{τ(x·ρ−t)+i}

√

τ²+κ²_lx·ρ^⊥ satisfies (∆ +κ²_l)U = 0in R³, forl=sor l=p.

2. (Logarithmic phase.)

Let Ωbe a C²-smooth domain and set ch(Ω) to be its convex hull. Choosex0∈R³\ch(Ω) and let ω0∈S² be a vector such that{x∈R³;x−x0=λω0, λ∈R} ∩∂Ω =∅.Then there exists a solution of (∆ +κ²_l)U = 0 of the following form

U(x;τ, t) :=e^{τ(t−log|x−x0|)−iτ ψ(x)}(a0+τ⁻¹a1+r) (3.6) with a0 anda1 are smooth and computable functions which depend on κl, l=sor l=p,andr has the following behavior

krk_H2

scl(Ω)≤Cτ⁻², C >0 is a universal constant, (3.7) where H_scl² (Ω) is the semi-classical Sobolev space defined by H_scl² (Ω) := {V ∈ L²(Ω)/(τ⁻¹∂)^αV ∈ L²(Ω),|α| ≤ 2}, equipped with the norm kVk²_H2

scl(Ω) := P

|α|≤2k(τ⁻¹∂)^αVk²_L2(Ω). From (3.7), we have in particular

krkH^s(Ω)≤Cτ^−(2−s), 0≤s≤2. (3.8) Proof. The CGOs with linear phases are given in [16]. The CGOs with log-phases are given in [22] for κ= 0 of the forme^{τ(t−log|x−x0|)−iτ ψ(x)}(a0+r) andkrk_H1

scl(Ω)≤cτ⁻¹.The CGOs stated in this proposition, with the corresponding estimate (3.7) of the remainder term, are given in [34].

3.2. CGO solutions for I_sp and I_ps. The natural CGOs introduced in section 3.1.1 are useful for Ipp. However, with such functions as incident waves the other indicator functionsIps, IssandIspvanish, see (2.3), (2.4) and (2.5) respectively, since thes-part,vs, of the CGO solution is nulle, and hence are not useful.

Similarly, the CGOs constructed in section 3.1.2 are useful forIss but not forIpp,Isp andIps. To construct CGOs useful forIsp andIps, we need to consider the full Lam´e system, i.e. solutions with non vanishingp andsparts. To construct such CGOs, we follow the approach by Uhlmann-Wang [35]. The non-divergence form of the isotropic elasticity system can be written as

µ₀∆u+ (λ₀+µ₀)∇(∇ ·u) +κ²u= 0 on Ω. (3.9) LetW =

w g

satisfy

P W := ∆ w

g

+A ∇g

∇ ·w

+Q w

g

= 0, (3.10)

where A = 0 0

0 _λ^λ0+µ0

0+2µ₀µ

1 2

0

!

and Q=κ² 1

µ0 0

0 _λ ¹

0+2µ₀

. Then u:= µ⁻₀¹²w+µ₀⁻¹∇g satisfies (3.9).

Consider now the matrix operatorP_τ⁻¹ =−τ⁻²P.Then the operatorPin (3.10) turns out to be the following operator

P_τ−1= (τ⁻¹D)²+iτ⁻¹A₁

τ⁻¹D τ⁻¹D·

+τ⁻²A₀,

whereD:=−i∇, A1:=−AandA0:=−Q.Later on we shall also denote the matrix operator iA1

τ⁻¹D τ⁻¹D·

=A1(τ⁻¹D).

Using semi classical Weyl calculus, the derivation of the Carleman estimate with semiclassicalH⁻²norm forP_τ⁻¹ can be found in [35]. We state it in the following proposition.

Proposition 3.2. Let ϕ(x) be a linear or logarithmic phase. If τ is large enough, then for any F ∈L²(Ω), there existsV ∈H_scl² (Ω)such that

e^{τ ϕ}P_τ−1(e^{−τ ϕ}V) =F, kVk_H2

scl(Ω)≤CτkFk_L2(Ω)

with some constantC >0.

Remark 3.3. In [35], the functions wandg are represented in the form w=e^{−τ(ϕ+iψ)}(l+r)and g=e^{−τ(ϕ+iψ)}(d+s), wherel, r∈C³,l, d are smooth and(r, s)^> satisfy the estimate

k∂^αRk_L2(Ω)≤Cτ^|α|−1 for|α| ≤2.

However, this estimate is not enough for our analysis ofIspandIpscases. Indeed, we need the boundedness of the remainder term in the norm of the Sobolev spaceH³(Ω). In the following proposition the representation of the CGO solutions with stronger estimate is given.

Proposition 3.4.

1. (Linear phase). Choose ρ, ρ^⊥∈S² with ρ·ρ^⊥= 0. Then there existsw andg of the forms (w=eτ((x·ρ−t)+ix·ρ^⊥)(a₀+τ⁻¹a₁+τ⁻²a₂+r),

g=eτ((x·ρ−t)+ix·ρ^⊥)(b0+τ⁻¹b1+τ⁻²b2+s), (3.11) respectively, satisfying (3.10), where τ(> 0) and t ∈ R are parameters, a₀(x) is a smooth non- vanishing 3×1 vector valued complex function on Ω and a₁(x), a₂(x), b₀(x), b₁(x), b₂(x) are all smooth functions inΩwith a1, a2∈C³ andb0, b1, b2∈C.The coefficients aj andbj, j= 0,1,2, are all computable.

2. (Logarithmic phase.) Choosex0∈R³\ch(Ω) and letω0∈S² be a vector such that

{x∈R³;x−x₀=λω₀, λ∈R} ∩∂Ω =∅. (3.12) Then there existsw andg of the form²

(w=e^{τ(t−log|x−x0|)−iτ ψ(x)}(a₀+τ⁻¹a₁+τ⁻²a₂+r),

g=e^{τ(t−log|x−x0|)−iτ ψ(x)}(b0+τ⁻¹b1+τ⁻²b2+s), (3.13) satisfying (3.10) whereτ(>0), t∈Rare parameters, a0(x) is a smooth non-vanishing 3×1 vector valued complex function onΩanda1(x), a2(x), b0(x), b1(x), b2(x)are all smooth functions onΩwith a1, a2∈C³ andb0, b1, b2∈C. ψ(x)is defined by

ψ(x) :=d_S2

x−x0

|x−x₀|, ω₀

with the distance function d_S2(·,·)onS². The coefficientsa_j, b_j, j= 0,1,2, are all computable.

In addition, for both linear and logarithmic phases, the remainder termR:= (r, s)^> enjoys the estimates:

k∂^αRkL²(Ω)≤Cτ^|α|−3 for|α| ≤2 (3.14) and

k∇Rk_H2(Ω0)≤C for any Ω0⊂⊂Ωasτ → ∞. (3.15)

2The parameterajandbj, j= 0,1,2,are not necessary the same as in the case of CGOs with linear phases. However, we keep the same symbols to avoid heavy notations in the sections 6 and 7, in particular.

Finally, for both linear and logarithmic phasesu:=µ0−¹₂w+µ0−1∇gis a complex geometrical optics solution for (3.9). The p-part and s-part of these CGO solutions, denoted by up andus respectively, are represented by

up=−κ⁻²_p [µ⁻

1 2

0 ∇(∇ ·w) +µ⁻¹₀ ∇(∆g)], (3.16) and us=κ⁻²_s µ⁻

1 2

0 [∇(∇ ·w)−∆w]. (3.17)

Proof. First, remark that we can get rid of the constant termse^{−τ t} and e^{τ t} in (3.11) and (3.13). We construct the solution of (3.10) of the form

U =e^{−τ(ϕ+iψ)}(L₀+τ⁻¹L₁+τ⁻²L₂+R),

where (ϕ+iψ) is a phase function,L₀, L₁, L₂ are smooth functions in Ω andR∈H²(Ω) is the remainder.

Applying the WKB method for the conjugate operatore^{τ ϕ}τ⁻²P e^{−τ ψ}, we obtain e^{τ ϕ}τ⁻²P e^{−τ(ϕ+iψ)}(L0+τ⁻¹L1+τ⁻²L2)

=e^{−iτ ψ}h

(−(∇(ϕ+iψ))²+τ⁻¹Q˜+τ⁻²P)(L0+τ⁻¹L1+τ⁻²L2)i

=e^{−iτ ψ}[−(∇(ϕ+iψ))²L0+τ⁻¹{−(∇(ϕ+iψ))²L1+ ˜QL0}

+τ⁻²{−(∇(ϕ+iψ))²L2+ ˜QL1+P L0}+τ⁻³{QL˜ 2+P L1}+τ⁻⁴{P L2}],

(3.18)

where ˜Q:=−∇ψ·D−D· ∇ψ+i∇ϕ·D+iD· ∇ϕ+A₁(i∇ϕ− ∇ψ).We choose ϕ, ψ, L₀, L₁ andL₂ such that

(∇(ϕ+iψ))²= 0, (3.19)

−(∇(ϕ+iψ))²L₁+ ˜QL₀= 0, (3.20)

−(∇(ϕ+iψ))²L2+ ˜QL1+P L0= 0, (3.21)

QL˜ ₂+P L₁= 0. (3.22)

Therefore, (3.19) is the well known eikonal equation forϕandψcan be written as

(∇ψ)²= (∇ϕ)²,∇ϕ· ∇ψ= 0. (3.23)

Case 1. Logarithmic Phase. For the case of logarithmic phase, we haveϕ(x) = log|x−x0|. So we can find a solution of (3.23) of the form

d_S2

x−x₀

|x−x0|, ω₀

= π

2 −tan⁻¹ ω₀·(x−x₀)

p(x−x0)²−(ω0·(x−x0))²,

see [34] for instance. If we choose ψ as above, (3.20), (3.21) and (3.22) are the transport equations for L0, L1 and L2. Now by the change of coordinates so that x0 = 0,Ω ⊂ {x3 >0}, and w0 = e1, we obtain ϕ+iψ = logz, where z = x1 +i|x⁰| is a complex variable with x⁰ := (x2, x3). Therefore, if we write L0:= (a0, b0)^> then (3.20) gives

[(∇logz)· ∇+∇ · ∇logz]a0= 0, (3.24) [(∇logz)· ∇+∇ · ∇logz]b0+

λ0+µ0

λ₀+ 2µ₀

µ0

1

2(∇logz)·a0= 0. (3.25) The equation (3.24) reduces to the following Cauchy-Riemann equation in thez variable

∂z¯− 1 2(z−z)¯

a0(z, θ) = 0,

whereθ=_|x^x00|.This last equation has the following solution

a₀(z, θ) = (z−z)¯ ⁻¹²(β₁, β₂, β₃)^> (3.26) where (β1, β2, β3)∈S²are arbitrary.³ Replacing a0in (3.25), we obtain smooth solutionb0 in Ω.Similarly, we get smooth solutions L1and L2 of the equations (3.21) and (3.22) on Ω, see [1] for instance. Finally, to derive the estimate of the remainder termR, from Proposition 3.2, we have

e^{τ ϕ}τ⁻²P e^{−τ(ϕ+iψ)}R=−e^{−iτ ψ}τ⁻⁴(P L2), (3.27) with the estimate

kRk_H2

scl(Ω)≤Cτke^{−iτ ψ}τ⁻⁴(P L2)k_L2(Ω)=O(τ⁻³).

Now, to estimate the gradient of R in H²(Ω0) for sub-domain Ω0 ⊂⊂ Ω, we use interior estimate. The equation (3.27) can be rewritten in terms ofr, sas

(−∆r+ 2τ∇(ϕ+iψ)· ∇r+τ(∆(ϕ+iψ))r−_µ^κ2

0r=τ⁻²P a2,

−∆s+ 2τ∇(ϕ+iψ)· ∇s+τ(∆(ϕ+iψ))s+τ_λ^λ0+µ0

0+2µ0µ₀¹²[∇(ϕ+iψ)·r− ∇ ·r]−_λ ^κ2

0+2µ0s=τ⁻²P b₂. Taking the∇on the both sides of the above equations and applying the interior estimate on

−∆(∇r) =−2τ∇(∇(ϕ+iψ)· ∇r)−τ∇((∆(ϕ+iψ))r) + κ²

µ₀∇r+τ⁻²∇(P a₂), and −∆(∇s) =−2τ∇(∇(ϕ+iψ)· ∇s)−τ∇((∆(ϕ+iψ))s)

−τ λ₀+µ₀ λ0+ 2µ0

µ0

1

2[∇(∇(ϕ+iψ)·r)− ∇(∇ ·r)] + κ² λ0+ 2µ0

∇s+τ⁻²∇(P b2), we obtain the estimate for the remainder termk∇RkH²(Ω₀)≤C,whereC >0 is a constant.

Case 2. Linear Case. In the linear case, we chooseϕ:=−x·ρandψ:=−x·ρ^⊥,whereρ, ρ^⊥∈S² with ρ·ρ^⊥= 0.Then (3.18) reduces to

e^−τ(x·ρ)τ⁻²P e^{τ x·(ρ+iρ⊥)}(L0+τ⁻¹L1+τ⁻²L2) =e^{iτ(x·ρ⊥)}[τ⁻¹TρL0+τ⁻²{TρL1+P L0} +τ⁻³{T_ρL₂+P L₁}+τ⁻⁴(P L₂)]

whereTρ:=−i(ρ+iρ^⊥)·D+A1(−iρ+ρ^⊥).We choose L0, L1 andL2 such that

TρL0= 0, (3.28)

T_ρL₁+P L₀= 0, (3.29)

TρL2+P L1= 0. (3.30)

Here the equations (3.28), (3.29), (3.30) are the system of Cauchy-Riemann type. Introducing new variable z= (z1, z2) = (ρ+iρ^⊥)·x, the equation (3.28) becomes

−2∂z¯L0+A1(−iρ+ρ^⊥)L0= 0. (3.31) Now we denoteL0:= (a0, b0)^>, where a0:= (a¹₀, a²₀, a³₀)^> ∈C³ andb0∈C.Therefore the equation (3.31) reduces to

(∂z¯a0= 0

−2∂z¯b0+_λ^λ0+µ0

0+2µ₀µ₀¹²(ρ+iρ^⊥)·a0= 0. (3.32)

3In section 6.1, see the proof of Lemma 6.3, we choose (β1, β2, β3) =ω0, whereω0 is given in (3.12).

In particular, any analytic function satisfies∂z¯a0= 0. Take any non-zero constant vectorρ, thenρsatisfies

∂z¯ρ= 0. Again if we replacea0byρin the second equation of (3.32) then we get a smooth solutionb0on Ω.

Next, we look forL1. The transport equation (3.29) becomes

−2∂¯zL₁+A₁(−iρ+ρ^⊥)L₁=−P L0, (3.33) where we denote L1 := (a1, b1)^>, a1 := (a¹₁, a²₁, a³₁)^> ∈ C³ and b1 ∈ C. Therefore the equation (3.33) reduces to

−2∂z¯

a₁ b₁

+ 0

λ₀+µ₀ λ₀+2µ₀µ

1 2

0(ρ+iρ^⊥)·a1

!

=−P a₀

b₀

. (3.34)

The equation (3.34) is solvable and the solution is a smooth function on Ω since the right hand side is smooth, see for instance [1]. We look forL₂ in a similar way. The transport equation (3.30) becomes

−2∂_¯zL₂+A₁(−iρ+ρ^⊥)L₂=−P L₁ (3.35) and then we obtainL2as a smooth function on Ω, where we denoteL2:= (a2, b2)^>, witha2:= (a¹₂, a²₂, a³₂)^>

∈C³ and b2∈C.Finally, to get the remainder term of the complex geometrical optics solution, we choose R∈C⁴,by Proposition 3.2, such that

e^−τ(x·ρ)τ⁻²P e^{τ x·(ρ+iρ⊥)}R=−e^{iτ(x·ρ⊥)}τ⁻⁴(P L2), (3.36) hencekRk_H2

scl(Ω) ≤Cτke^{iτ(x·ρ⊥)}τ⁻⁴(P L2)k_L2(Ω) =O(τ⁻³). We can write the estimates for the remainder term as







kRk_L2(Ω)≤Cτ⁻³ k∇RkL²(Ω)≤Cτ⁻² k∇∇Rk_L2(Ω)≤Cτ⁻¹.

(3.37)

To derive the estimate inH³(Ω0),Ω0⊂⊂Ω we proceed as in Case 1 using interior regularity estimates.

4. Reconstruction Scheme. In this section, we show how one can reconstruct some features of the obstacle using only one part of the farfield pattern. These features are extracted from the behavior of the indicator functions defined in section 2.3 for large τ. Precisely, we can reconstruct the convex hull of the obstacle if we use CGOs with linear phases and some parts of its non-convex part if we use CGOs with logarithmic phase. Let us introduce the following two functions:

h_D(ρ) := sup

x∈D

x·ρ, (ρ∈S²) (4.1)

and d_D(x₀) := inf

x∈Dlog|x−x₀|, (x₀∈R³\ch(Ω)). (4.2) I. Use of purely pors type CGOs

Recall that the indicator function Iss represents the energy when we usesincident field and the s-part of the farfield data. Similarly, the indicator functionIpp represents the energy when we use thesincident field and thep-part of the farfield data. So for this case we choose the p-part and s-part of the CGO solution discussed in subsection 3.1. We have the following theorem.

Theorem 4.1. (CGOs with linear phase.) Let ρ∈S² and take v to be the p-type CGO of linear phase introduced in section 3.1. Let Ipp(τ, t)be the corresponding indicator functions defined in (2.6). For both the penetrable and the impenetrable cases ⁴, we have the following characterizations ofhD(ρ).

|τ⁻¹I_pp(τ, t)| ≤Ce^−cτ, τ >>1, c, C >0,and in particular, lim

τ→∞|I_pp(τ, t)|= 0 (t > h_D(ρ)), (4.3) lim inf

τ→∞ |τ⁻¹Ipp(τ, hD(ρ))|>0, and precisely, c≤τ⁻¹|Ipp(τ, hD(ρ))| ≤Cτ², τ >>1, c, C >0, (4.4)

|τ⁻¹Ipp(τ, t)| ≥Ce^cτ, τ >>1, c, C >0, and in particular, lim

τ→∞|Ipp(τ, t)|=∞ (t < hD(ρ)). (4.5)

4For the impenetrable case, we further assume thatk² is not the Dirichlet eigenvalue for the Lam´e operator inD. This is needed in section 5.2 where we use the single layer potential to represent the solution of the scattered field. This condition can be avoided by using combined single and double layer potentials, as it is done for the acoustic case [20].

(CGOs with logarithmic phase.) Let x0∈R³\ch(Ω) and set v to be the p-type CGO with logarithmic phase introduced in section 3.1.2. Let Ipp(τ, t) the corresponding indicator functions defined in (2.6). For both the penetrable and the impenetrable cases, we have the following characterizations ofdD(x0).

|τ⁻¹Ipp(τ, t)| ≤Ce^−cτ, τ >>1, c, C >0,and in particular, lim

τ→∞|Ipp(τ, t)|= 0 (t < dD(x0)), (4.6) lim inf

τ→∞ |τ⁻¹Ipp(τ, dD(x0))|>0, and precisely, c≤τ⁻¹|Ipp(τ, dD(x0))| ≤Cτ², τ >>1, c, C >0, (4.7)

|τ⁻¹I_pp(τ, t)| ≥Ce^cτ, τ >>1, c, C >0, and in particular, lim

τ→∞|I_pp(τ, t)|=∞ (t > d_D(x₀)). (4.8) The above estimates are also valid if we replaceI_ppbyI_ssand thep-type CGOs by thes-type CGOs introduced in section 3.1 and section 3.2.

II. Use of CGOs with both non-vanishingp and s parts

Regarding the indicator functionI_sp,which depends ons-incident wave andp-part of the farfield and similarly for I_ps, which depends on p-incident wave and s-part of the farfield, we cannot use the expression of the CGO solution as in (3.2) and (3.5), since the s-part of (3.2) and thep-part of (3.5) are zero. Instead, we use the CGOs we discussed in subsection 3.2. Using these CGOs, we have the following theorem.

Theorem 4.2. (CGOs with linear phases.) Letρ∈S². For both the penetrable and the impenetrable cases⁵, we have the following characterizations ofhD(ρ).

|τ⁻³Isp(τ, t)| ≤Ce^−cτ, τ >>1, c, C >0, and in particular, lim

τ→∞|Isp(τ, t)|= 0 (t > hD(ρ)), (4.9) lim inf

τ→∞ |τ⁻³Isp(τ, hD(ρ))|>0, and precisely, c≤τ⁻³|Isp(τ, hD(ρ))| ≤Cτ², τ >>1, c, C >0, (4.10)

|τ⁻³I_sp(τ, t)| ≥Ce^cτ, τ >>1, c, C >0, and in particular, lim

τ→∞|Isp(τ, t)|=∞ (t < h_D(ρ)).(4.11) (CGOs with logarithmic phases.) Let x₀ ∈ R³\ch(Ω). For both the penetrable and the impenetrable cases, we have the following characterizations of d_D(x₀).

|τ⁻³I_sp(τ, t)| ≤Ce^−cτ, τ >>1, c, C >0, and in particular, lim

τ→∞|I_sp(τ, t)|= 0 (t < d_D(x₀)),(4.12) lim inf

τ→∞ |τ⁻³Isp(τ, dD(x0))|>0, and precisely, c≤τ⁻³|Isp(τ, dD(x0))| ≤Cτ², τ >>1, c, C >0,(4.13)

|τ⁻³Isp(τ, t)| ≥Ce^cτ, τ >>1, c, C >0, and in particular, lim

τ→∞|Isp(τ, t)|=∞ (t > dD(x0)).(4.14) The above estimates are valid if we replace Isp byIps.

From the above two theorems, we see that, in case of linear phase for a fixed directionρ(accordingly, in case of logarithmic phase for a fixed directionx0), the behavior of the indicator functionIij, whereij=pp, ss, sp or ps, changes drastically in terms of τ: exponentially decaying if t > hD(ρ) (accordingly t < dD(x0) for the logarithmic phase), polynomially behaving ift=hD(ρ) (accordingly t=dD(x0) for logarithmic phase) and exponentially growing if t < hD(ρ) (accordingly t > dD(x0) for the logarithmic phase). Using this property of the indicator functions we can reconstruct the support functionhD(ρ), ρ∈S² (accordingly the distance dD(x0), x0 ∈R³\ch(Ω), for the logarithmic phase) from the farfield measurements. Finally, from this support function for linear phase, we can reconstruct the convex hull of D and from the distance function for the logarithmic phase, we can, in addition to the convex hull, reconstruct parts of the non-convex part of the obstacle D. Let us finish this section by rephrasing the formulas in Theorem 4.1 and Theorem 4.2 as follows: h_D(ρ)−t = lim_τ→∞^log|I_2τ^ij(τ,t)|, for ij = pp, ss, sp, ps, when we use CGOs with linear phase and t−d_D(x₀) = lim_τ→∞^log|I_2τ^ij(τ,t)|, for ij =pp, ss, sp, ps, when we use CGOs with logarithmic phase. The formulas are easily deduced from (4.4), (4.7), (4.10) and (4.13) and the following identities

Iij(τ, t) =e^{2τ(hD(ρ)−t)}Iij(τ, hD(ρ)), (4.15) I_ij(τ, t) =e^{2τ(t−dD(x0))}I_ij(τ, d_D(x₀)). (4.16)

5Same comments as for Theorem 4.1.

5. Justification of the reconstruction schemes. In this section, we prove the above two theorems using all the CGOs for both the penetrable and impenetrable obstacle cases. For that we only focus the four points (4.4), (4.7), (4.10) and (4.13) since we have (4.15) and (4.16). In addition, the lower estimates in (4.4), (4.7), (4.10) and (4.13) are the most difficult part since the upper bounds are easily obtain using the Cauchy-Schwartz inequality, the well posedness of the forward problems and the upper estimate of the H¹-norms of the CGOs given in section 6. So, we mainly focus in our proofs on the lower estimates.

5.1. The penetrable obstacle case. We considerwas an incident field andu^s(w) the scattered field, therefore the total field ˜w=w+u^s(w) satisfies the following problem

(∇ ·(σ( ˜w)) +κ²w˜= 0, in R³

u^s(w) satisfies the Kupradze radiation condition, (5.1) recalling thatσ( ˜w) =λ(∇ ·w)I˜ 3+µ(∇w˜+ (∇w)˜ ^>).The incident CGO field satisfies

∇ ·(σ₀(w)) +κ²w= 0 in Ω, (5.2)

whereσ0(v) =λ0(∇ ·v)I3+ 2µ0(v) for any displacement fieldv. Accordingly, we will useσD(v) to denote σ(v)−σ0(v), i.e. σD(v) =λD(∇ ·v)I3+ 2µD(v). Note that for a matrixA= (aij), we use |A| to denote (P

i,j|aij|²)¹². For any matricesA= (aij) andB= (bij) we define the product as follows

A·B^>:=tr(AB) (5.3)

wheretr(A) is the trace of the matrixA. Also frequently we will use the following basic identity:

σ(u)·(∇v)^>=σ(v)·(∇u)^> (5.4)

and Betti’s identity Z

Ω

(∇ ·σ(u))·vdx=− Z

Ω

σ(u)·(∇v)^>dx+ Z

∂Ω

(σ(u)·ν)·vds(x). (5.5) Lemma 5.1. We have the following identity

tr(σ(u)∇u) = 3λ+ 2µ

3 |∇ ·u|²+ 2µ|(u)−∇ ·u 3 I3|².

Proof. DefineSym∇u:= ¹₂(∇u+ (∇u)^>) =(u). For anyα, β and a matrix A, we have the following identity, see [16],

α|tr(A)|²+ 2β|SymA|²=3α+ 2β

3 |tr(A)|²+ 2β|SymA−tr(A)

3 I₃|². (5.6) SubstitutingA=∇u, α=λ, β=µin (5.6), we obtain

tr(σ(u)∇u) =λ|∇ ·u|²+ 2µ|(u)|²

= 3λ+ 2µ

3 |∇ ·u|²+ 2µ|(u)−∇ ·u 3 I₃|². Letv andwbe two incident waves. We setI(v, w) :=R

D[u^s(v)·(σ(w)·ν)−v·(σ(u^s(w))·ν)]ds(x).Hence from (2.2), (2.3), (2.4) and (2.5), we have

Iss(τ, t) =I(vs, vs), Ipp(τ, t) =I(vp, vp), Ips(τ, t) =I(vp, vs) and Isp(τ, t) =I(vs, vp). (5.7)