On Recovery of a Bounded Elastic Body by Electromagnetic Far-Field Measurements
OON RECOVERY OF A BOUNDED ELASTIC BODY BYELECTROMAGNETIC FAR-FIELD MEASUREMENTS
TIELEI ZHU ∗ , JIAQING YANG † , AND
BO ZHANG ‡ Abstract.
This paper is concerned with the problem of scattering of a time-harmonic electro-magnetic field by a three-dimensional elastic body. General transmission conditions are consideredto model the interaction between the electromagnetic field and the elastic body on the interface byassuming Voigt’s model. The existence of a unique solution of the interaction problem is proved in anappropriate Sobolev space by employing a variational method together with the classical Fredholmalternative. The inverse problem is then considered, which is to recover the elastic body by thescattered wave-field. It is shown that the shape and location of the elastic body can be uniquely de-termined by the fixed energy magnetic (or electric) far-field measurements corresponding to incidentplane waves with all polarizations.
Key words.
Interaction, Maxwell’s equations, Navier equation, well-posedness, inverse scatter-ing, uniqueness.
AMS subject classifications.
1. Introduction.
The interaction of different physical fields has received consid-erable attention due to the rapidly increasing use of composite materials. Therefore,it is significant to develop the related mathematical model and analysis by physicalprocess. The physical kinematic and dynamic relations are described by the corre-sponding partial differential equations (PDEs) with certain boundary-transmissionconditions. Generally, it is difficult to find an appropriate interaction condition con-nected with different physical fields on the interface.For time-harmonic acoustic wave scattering by a solid body, many work has beendone on the mathematical analysis of the interaction problem (see, e.g., [6, 9]). Re-cently, the corresponding inverse problems have also been studied mathematically andnumerically of detecting an elastic body via the measurement of the acoustic scatteredwave field. We refer the reader to [7, 11–13] for detailed discussions. In particular,it is shown in [12] that a uniqueness result was first proved in recovering an elasticbody by the acoustic far-field measurements. The proof was then simplified by Monkand Selgas [11] by using the technique of H¨ahner in [5] for the case of a penetrable,anisotropic obstacle. However, the analysis in [11] relies on the H -regularity estimateof solutions of the scattering problem, and thus the proposed method remains com-plicated. Very recently, a much simpler proof was introduced by Qu et al. [13], whichis motivated by the previous work of the last two authors [14] for inverse acoustic andelectromagnetic scattering by a penetrable obstacle, and can be extended to deal withother more general cases.In this paper we consider the problem of scattering of a time-harmonic electro-magnetic field by a three-dimensional elastic body. Assume by Voigt’s model that theinteraction is allowed only through the boundary of the body, which means that themodel problem can be described by the Maxwell and Navier equations coupled with a ∗ School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China( [email protected] ) † School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China( [email protected] ) ‡ NCMIS, LSEC and Academy of Mathematics and Systems Sciences, Chinese Academy of Sci-ences, Beijing 100190, China and School of Mathematical Sciences, University of Chinese Academyof Sciences, Beijing 100049, China ( [email protected] )1 a r X i v : . [ m a t h . A P ] A ug T. Zhu, J. Yang and B. Zhang suitable transmission condition on the interface between the elastic and electromag-netic medium. It was shown [2] that an interaction model was first introduced byCakoni and Hsiao with possible interface conditions for the coupled electromagneticand elastic fields, where the uniqueness result and equivalent integral equations andnon-local variational formulations have been established for the model. Applying theframework in [2], Gatica et al. [4] proved the existence of a unique solution of theinteraction problem by using a variational method. The result was later extended byBernardo et al. [1] to a different function space for the elastic field, based on a similaridea to [4].Different from [1] and [4], we study in this paper the interaction problem withgeneral interface transmission conditions which could model more physical situationsin applications. An equivalent non-symmetric variational formulation is then obtainedby using Green’s formulas so that the existence of a unique solution to the problemcan follow from the classical Fredholm alternative with a suitable Helmholtz-typedecomposition of the electromagnetic field. Compared with the forward problem,the inverse problem of determining the elastic body is more challenging due to thecomplication of the interaction model. To the best of our knowledge, no uniquenessresult is available for this problem in the literature. Inspired by our previous work [14]where a novel technique was introduced for showing uniqueness in determining anacoustic or electromagnetic penetrable obstacle, we aim to develop a novel and simpletechnique to prove the unique recovery of the elastic body by the electromagneticfar-field measurement at a fixed frequency. The proposed method is mainly basedon constructing a well-posed system of PDEs for the coupled Maxwell and Navierequations in a small domain near the interface in conjunction with a uniform a prioriestimate in the H (curl , · ) × H ( · ) norm of solutions to the interaction problem whenthe incident electromagnetic fields are induced by a family of electric dipoles with aweak singularity.The remaining part of the paper is organized as follows. In Section 2, we formulatethe interaction scattering problem by collecting some useful functions spaces, traceoperators and related properties. In Section 3, we show the existence of a uniquesolution to the interaction problem by the variational method with aid of a suitableHelmholtz-type decomposition. In Section 4, a global uniqueness theorem is provedfor the associated inverse problem of determining the elastic body from the magneticor electric far-field measurements at a fixed frequency.
2. The model problem.
In this section, we first introduce some basic notationsand function spaces used throughout this paper and then present the mathematicalformulation of the model problem.
For a complex number z ∈ C , its conjugate and modulusare denoted by z and | z | , respectively. For x, y ∈ C define x · y = (cid:80) j =1 x j y j . Let D ∈ R be a bounded domain with a C -boundary ∂D and let ν be the unit outwardnormal to ∂D . For s ∈ R denote by H s ( D ) and H s ( ∂D ) the standard scalar Sobolevspaces defined on D and ∂D , respectively, with L ( · ) := H ( · ). We also need thefollowing vector function spaces defined on D and ∂D : H s ( D ) := [ H s ( D )] , s ∈ R , H s ( ∂D ) := [ H s ( ∂D )] , s ∈ R , H st ( ∂D ) := { µ ∈ H s ( ∂D ) : µ · ν = 0 } , s ∈ R ,H (curl , D ) := { B ∈ L ( D ) : curl B ∈ L ( D ) } . ecovery of elastic body by electromagnetic far-field measurements s ∈ R , (cid:104) H − s , H s (cid:105) and (cid:104) H − s , H s (cid:105) denote the duality product under theextension of the L -bilinear form (cid:104) u , v (cid:105) := (cid:90) D u · v d x, u , v ∈ L ( D ) . By [10] the tangential trace spaces of H (curl , D ) can be characterized as H − / ( ∂D ) := { µ ∈ H − / t ( ∂D ) : Div ∂D µ ∈ H − / ( ∂D ) } ,H − / ( ∂D ) := { µ ∈ H − / t ( ∂D ) : Curl ∂D µ ∈ H − / ( ∂D ) } , where Div ∂D and Curl ∂D denote the surface divergence and surface curl with respectto the boundary ∂D , respectively. For convenience, we also use ∇ ∂D · to denoteDiv ∂D , which is the surface gradient defined by ∇ ∂D f := ( ν × ∇ f ) × ν for a smoothfunction f . For a smooth vector function u ∈ [ C ( D )] , we introduce the tangentialtrace mapping γ t and the tangential projection operator γ T by γ t u := ν × u on ∂D,γ T u := ν × ( u × ν ) on ∂D, which can be extended as bounded and surjective operators from H (curl , D ) into H − / ( ∂D ) and H − / ( ∂D ), respectively.Further, the duality product between H − / ( ∂D ) and H − / ( ∂D ) is defined as (cid:104) ϕ, ψ (cid:105) H − / × H − / := (cid:90) D [curl v · w − curl w · v ] d x for v , w ∈ H (curl , D ) satisfying γ t v = ϕ and γ T w = ψ . In addition, introduce thetangential trace spaces of H ( D ): H / ⊥ ( ∂D ) := γ t ( H ( D )) and H / || ( ∂D ) := γ T ( H ( D )) , and denote by H − / ⊥ ( ∂D ) and H − / || ( ∂D ) the dual space of H / ⊥ ( ∂D ) and H / || ( ∂D ), respectively. Clearly, H − / ( ∂D ) is a closed subspace of H − / || ( ∂D ),which can be understood in the sense that (cid:104) ϕ, ψ (cid:105) H − / × H / || := (cid:90) D [curl w · v − curl v · w ] d x for w ∈ H (curl , D ) and v ∈ H ( D ) satisfying γ t w = ϕ ∈ H − / ( ∂D ) and γ T v = ψ ∈ H / || ( ∂D ). In this subsection, we formulate themathematical formulation of the problem of scattering of a time-harmonic electromag-netic wave by an elastic body in R . As seen in Figure 2.1, the elastic body is describedby a bounded domain D with a smooth boundary of C -class, which is assumed to beinhomogeneous and anisotropic with the stiffness tensor C := ( C ijkl ( x )) i,j,k,l =1 , where C ijkl ∈ L ∞ ( D ) ( i, j, k, l = 1 , , ρ ( x ) ∈ L ∞ ( D ). The background T. Zhu, J. Yang and B. Zhang
D ∂DD c ( E i , H i ) ( E s , H s )( E s , H s )( E s , H s ) u Fig. 2.1 . Interaction between electromagnetic wave and a bounded elastic body medium outside of D is assumed to be homogeneous and isotropic with constantelectric permittivity ε ∈ R + and magnetic permeability µ ∈ R + .Consider a pair of electromagnetic waves of the form E i ( x, d, p ) = − κ curl ( pe i κx · d ) , H i ( x, d, p ) = curl ( pe i κx · d ) , x ∈ R (2.1)which is incident on D from the unbounded exterior domain D c := R \ D , where ω ∈ R + is the wave frequency, d ∈ S is the direction of wave propagation, p ∈ R is the polarization vector and κ = ω √ ε µ is the wave number. Then the elasticdeformation occurs due to the physical property of the elastic body. Following Voigt’smodel, we can assume that the electromagnetic wave does not penetrate the elasticbody so that the interaction occurs only on the interface. Under the above physicalassumption, the elastic field u satisfies the Navier equation ∇ · ( C : ∇ u ) + ρω u = 0 in D, (2.2)where C : ∇ u is defined as C : ∇ u := ( C : ∇ u ) ij = (cid:88) k,l =1 C ijkl ∂u k ∂x l and ∇ · ( C : ∇ u ) is defined as ∇ · ( C : ∇ u ) = (cid:88) j,k,l =1 ∂∂x j ( C ijkl ∂u k ∂x l ) . Moreover, the stiffness tensor C in (2.1) satisfies the symmetry condition C ijkl = C klij = C jikl = C ijlk and the Legendre elliptic condition (cid:88) i,j,k,l =1 C ijkl ( x ) a ij a kl ≥ c (cid:88) i,j =1 | a ij | , a ij = a ji for some positive constant c > ecovery of elastic body by electromagnetic far-field measurements D c , the electromagnetic field ( E , H ), which is the sumof the incident field ( E i , H i ) and the scattered field ( E s , H s ), satisfies the Maxwellequations curl E − i κ H = 0 , curl H + i κ E = 0 in D c , (2.3)with the scattered field ( E s , H s ) satisfying the Silver-M¨uller radiation conditionlim | x |→∞ | x | ( E s × ˆ x + H s ) = 0 , (2.4)where ˆ x = x/ | x | ∈ S .On the interface, the electromagnetic and elastic fields are assumed to be coupledby the general transmission conditions (cf. [2]): T u − b ν × H s = b ν × H i on ∂D, (2.5) ν × u − b ν × E s = b ν × E i on ∂D, (2.6)where b , b ∈ C , b b (cid:54) = 0, ν is the unit outward normal to ∂D and T is defined as( T u ) i = (cid:88) j,k,l =1 ν j C ijkl ∂u k ∂x l , i = 1 , , . We refer the reader to [2, 4] for detailed discussions on different choices of b and b .By the radiation condition (2.4), it is well-known that the scattered field has theasymptotic behavior H s ( x, d, p ) = e i κ | x | | x | H s ∞ (ˆ x, d, p ) + O (cid:18) | x | (cid:19) , as | x | → ∞ E s ( x, d, p ) = e i κ | x | | x | E s ∞ (ˆ x, d, p ) + O (cid:18) | x | (cid:19) , as | x | → ∞ , where E s ∞ (ˆ x, d, p ) and H s ∞ (ˆ x, d, p ) denote the electric and magnetic far-field patterns,respectively, which are analytic in ˆ x ∈ S and d ∈ S , respectively, and satisfy therelations (cf. [3]): H s ∞ (ˆ x, d, p ) = ˆ x × E s ∞ (ˆ x, d, p ) , ˆ x · E s ∞ (ˆ x, d, p ) = 0 , ˆ x · H s ∞ (ˆ x, d, p ) = 0 . (2.7)
3. The well-posedness of the interaction problem.
In this section, weprove the well-posedness of the interaction problem (2.1)-(2.5), employing a varia-tion method. Under the transmission conditions (2.5) and (2.6), the existence ofa unique solution can be obtained by showing the variational formulation to be ofFredholm with index 0 in an appropriate Sobolev space.
As known for the fluid-solid interaction prob-lem, non-uniqueness may exist for certain frequencies which are called Jones frequen-cies. Similarly, there may exist pathological frequencies in the interaction between anelectromagnetic wave and an elastic body so that a nontrivial solution exists for thehomogeneous problem corresponding to the problem (2.1)-(2.5). Thus, introduce thehomogeneous problem ∇ · ( C : ∇ u ) + ρω u = 0 in D, (3.1) ν × u = 0 on ∂D, (3.2) T u = 0 on ∂D (3.3) T. Zhu, J. Yang and B. Zhang and let the set P ( ω ) be consisting of the frequency ω ∈ R such that (3.1) has anontrivial solution. Then we have the following result on uniqueness of solutions tothe problem (2.1)-(2.5). Theorem 3.1.
Assume that ρ, κ, ω ∈ R and ω / ∈ P ( ω ) . If Re( b b ) = 0 , then thescattering problem (2 . − (2 . has at most one solution.Proof . Let E i = H i = 0. Then it is enough to prove that E s = H s = 0. UsingGreen’s formula and the transmission conditions (2.5) and (2.6), we have (cid:90) ∂D ν × E s · H s d s = − b b (cid:90) ∂D T u · u d s = − b b (cid:90) D [ E ( u , u ) − ρω | u | ] d x, where E ( u , v ) := (cid:88) i,j,k,l =1 C ijkl ∂u k ∂x l ∂v i ∂x j for u , v ∈ H ( D ). ThusRe (cid:90) ∂D ν × E s · H s d s = − Re 1 b b (cid:90) D [ E ( u , u ) − ρω | u | ] d x. By this, Rellich’s lemma (see [3]) and the fact that Re( b b ) = 0, we conclude that E s = H s = 0. Then the elastic field u satisfies (3.1), yielding u = 0 since ω / ∈ P ( ω ). We now prove the existence of solutions of thescattering problem (2.1)-(2.5), employing a variational method. To this end, we elim-inate the electric field E and consider the boundary value problem for ( H , u ): ∇ · ( C : ∇ u ) + ρω u = 0 in D, curl curl H − κ H = 0 in D c ,T u − b ν × H = f on ∂D,ν × u + b i κ ν × curl H = f on ∂D, lim r →∞ r (ˆ x × curl H + i κ H ) = 0 r = | x | , (3.4)where f ∈ H − / ( ∂D ) and f ∈ H − / ( ∂D ). Note that the scattering problem(2.1)-(2.5) can be viewed as a special case of the problem (3.4) with f := b ν × H i and f := ( b / i κ ) ν × curl H i .We now reduce the problem (3.4) into one in the bounded domain B R := { x ∈ R : | x | ≤ R } with R large enough. To this end, we introduce the Calder´on mapping G e : H − / ( S R ) → H − / ( S R ) defined by G e ( λ ) := 1i κ ˆ x × curl (cid:101) w on S R := ∂B R (3.5)for λ ∈ H − / ( S R ), where (cid:101) w satisfies the problem curl curl (cid:101) w − κ (cid:101) w = 0 in R \ B R , ˆ x × (cid:101) w = λ on S R , lim r →∞ r (ˆ x × curl (cid:101) w + i κ (cid:101) w ) = 0 r = | x | . ecovery of elastic body by electromagnetic far-field measurements G e has the following properties which were proved in [10]. Lemma 3.2.
Let (cid:101) G e be defined as G e in (3 . with κ = i . Then (a) G e + i κ (cid:101) G e is compact from H − / ( S R ) to H − / ( S R ) ; (b) (cid:104) (cid:101) G e λ, λ × ˆ x (cid:105) H − / × H − / < for any λ ∈ H − / ( S R ) with λ (cid:54) = 0 ; (c) G e can be split as G e λ := G e λ + G e λ for λ ∈ H − / Div ( S R ) such that (c.1) The map H (cid:55)→ G e (ˆ x × H ) is compact from X into H − / Div ( S R ) , where X is defined in (3.9) below; (c.2) i κ (cid:104) G e (ˆ x × H ) , γ T H (cid:105) H − / × H − / ≥ for H ∈ H (curl , B R ) . With the aid of the Calder´on map G e , the problem (3.4) can be equivalentlyreduced to the boundary value problem ∇ · ( C : ∇ u ) + ρω u = 0 in D, curl curl H − κ H = 0 in B R \ D,T u − b ν × H = f on ∂D,ν × u + b i κ ν × curl H = f on ∂D, ˆ x × curl H = i κG e (ˆ x × H ) on S R , (3.6)where f ∈ H − / ( ∂D ) and f ∈ H − / ( ∂D ).Multiplying the first equation of (3.6) with v ∈ H ( D ) and using integration byparts together with the third equation of (3.6) yield (cid:90) D ( E ( u , v ) − ρω u · v ) d x − b (cid:104) γ t H , γ T v (cid:105) H − × H − = (cid:104) f , v (cid:105) H − × H or equivalently − i κb b (cid:90) D ( E ( u , v ) − ρω u · v ) d x + i κb (cid:104) γ t H , γ T v (cid:105) H − × H − = − i κb b (cid:104) f , v (cid:105) H − × H for all v ∈ H ( D ). Multiplying the second equation of (3.6) by − w ∈ H (curl , B R \ D )and utilizing the fourth and fifth equations of (3.6) give − (cid:90) B R \ D (curl H · curl w − κ H · w ) d x − i κ (cid:104) G e (ˆ x × H ) , γ T w (cid:105) H − × H − + i κb (cid:104) γ t w , γ T u (cid:105) H − × H − = − i κb (cid:104) f , γ T w (cid:105) H − × H − . Adding the above two equations together and letting X = H (curl , B R \ D ) and Q = H ( D ), we obtain the variational formulation of (3.6): find ( u , H ) ∈ Q × X such that A (( u , H ) , ( v , w )) = F ( v , w ) ∀ ( v , w ) ∈ Q × X, (3.7) T. Zhu, J. Yang and B. Zhang where A (( u , H ) , ( v , w )) := − i κb b (cid:90) D ( E ( u , v ) − ρω u · v ) d x + i κb (cid:104) γ t H , γ T v (cid:105) H − × H − − (cid:90) B R \ D (curl H · curl w − κ H · w ) d x − i κ (cid:104) G e (ˆ x × H ) , γ T w (cid:105) H − × H − + i κb (cid:104) γ t w , γ T u (cid:105) H − × H − , F (( v , w )) := − i κb b (cid:104) f , v (cid:105) H − × H − i κb (cid:104) f , γ T w (cid:105) H − × H − . We now split the sesquilinear form A (( · , · ) , ( · , · )) on ( Q × X ) × ( Q × X ) into two parts: A (( u , H ) , ( v , w )) := A (( u , H ) , ( v , w )) + K (( u , H ) , ( v , w )) (3.8)with A (( · , · ) , ( · , · )) and K (( · , · ) , ( · , · )) defined as follows: A (( u , H ) , ( v , w )) := − i κb b (cid:90) D ( E ( u , v ) + M u · v ) d x + i κb (cid:104) γ t H , γ T v (cid:105) H − × H − − (cid:90) B R \ D (curl H · curl w − κ H · w ) d x − i κ (cid:104) G e (ˆ x × H ) , γ T w (cid:105) H − × H − + i κb (cid:104) γ t w , γ T u (cid:105) H − × H − , K (( u , H ) , ( v , w )) := i κb b (cid:90) D ( ρω + M ) u · v d x for all ( u , H ) , ( v , w ) ∈ Q × X , where M > M >ω (cid:107) ρ (cid:107) L ∞ . Further, the sesquilinear form A (( · , · ) , ( · , · )) can be written as A (( u , H ) , ( v , w )) = A ( u , v ) − A ( H , w ) + i κb (cid:104) γ t H , γ T v (cid:105) H − × H − + i κb (cid:104) γ t w , γ T u (cid:105) H − × H − for all ( u , H ) , ( v , w ) ∈ Q × X , where A ( H , w ) := (cid:90) B R \ D (curl H · curl w − κ H · w ) d x + i κ (cid:104) G e ( ν × H ) , γ T w (cid:105) H − × H − , A ( u , v ) := − i κb b (cid:90) D ( E ( u , v ) + M u · v ) d x. Note that A ( · , · ) corresponds to the magnetic field H and A ( · , · ) corresponds to theelastic field u . It is easy to see that A ( · , · ) is coercive in H ( D ) × H ( D ). However,it is difficult to directly analyze A ( · , · ) in H (curl , B R \ D ) × H (curl , B R \ D ) sincethe imbedding map of H (curl , B R \ D ) (cid:44) → L ( B R \ D ) is not compact. To overcome ecovery of elastic body by electromagnetic far-field measurements A ( · , · ). To this end,define the scalar space S := ψ ∈ H ( B R \ D ) : (cid:90) S R ψ d s = 0 , which is clearly a closed linear subspace of H ( B R \ D ) and thus a Hilbert space.Then the sesquilinear form A ( · , · ) can be rewritten on S as A ( ∇ φ, ∇ ψ ) := a ( φ, ψ ) + b ( φ, ψ ) for all φ, ψ ∈ S, where a ( · , · ) and b ( · , · ) are defined as a ( φ, ψ ) := − κ (cid:104)∇ φ, ∇ ψ (cid:105) + κ (cid:104) (cid:101) G e (ˆ x × ∇ φ ) , ∇ S R ψ (cid:105) H − × H − ,b ( φ, ψ ) := i κ (cid:104) ( G e + i κ (cid:101) G e )(ˆ x × ∇ φ ) , ∇ S R ψ (cid:105) H − × H − . We have the following result which was proved in [10].
Lemma 3.3. a ( · , · ) is bounded and elliptic on S × S , and there exists a compactoperator K on S such that b ( φ, ψ ) = a ( K φ, ψ ) for all φ, ψ ∈ S . Further, I + K isan isomorphism on S . In order to analyze A ( · , · ) on X × X , we introduce the following subspace of X : X := { H ∈ X : − κ (cid:104) H , ∇ ψ (cid:105) + i κ (cid:104) G e (ˆ x × H ) , ∇ S R ψ (cid:105) H − × H − = 0 ∀ ψ ∈ S } = { H ∈ X : ∇ · H = 0 in B R \ D, − κ ˆ x · H = i κ ∇ S R · G e (ˆ x × H ) on S R and ν · H = 0 on ∂D } . (3.9)We then have the following Helmholtz-type decomposition for X . Lemma 3.4. ∇ S and X are closed linear subspaces of X , and X = X (cid:76) ∇ S isthe direct sum of ∇ S and X . Further, there exist constants c , c > such that c (cid:107) w + ∇ φ (cid:107) X ≤ (cid:107) w (cid:107) X + (cid:107)∇ φ (cid:107) X ≤ c (cid:107) w + ∇ φ (cid:107) X (3.10) for all w ∈ X and φ ∈ S .Proof . The closeness of ∇ S follows from the property that curl ∇ ψ = 0for ψ ∈ X and the boundedness of the differential operator ∇ from H ( · ) into L ( · ). For a fixed ψ ∈ S , the linear functionals H → (cid:104) H , ∇ ψ (cid:105) and H → (cid:10) G e (ˆ x × H ) , ∇ S R ψ (cid:11) H − × H − are bounded on X , yielding that X is closed.Given H ∈ X , we now construct a function φ ∈ S such that A ( ∇ φ, ∇ ψ ) = A ( H , ∇ ψ ) for all ψ ∈ S. (3.11)From Lemma 3.3 it follows that such φ is well defined and satisfies that (cid:107)∇ φ (cid:107) L ( B R \ D ) ≤ c (cid:107) H (cid:107) H (curl ,B R \ D ) for some constant c >
0. Let w = H − ∇ φ . Then, and by (3.11) and the definitionof A ( · , · ) we deduce that w ∈ X . It remains to show that the intersection ∇ S ∩ X contains only a trivial element. In fact, if there exists φ ∈ S such that ∇ φ ∈ X , then A ( ∇ φ, ∇ ψ ) = 0 for all ψ ∈ S, T. Zhu, J. Yang and B. Zhang implying that φ = 0.Finally, the inequality (3.10) follows from the boundedness of the projection op-erators X (cid:44) → ∇ S and X (cid:44) → X . Lemma 3.5. X is compactly imbedded in L ( B R \ D ) .Proof . Since X is a Hilbert space, it is enough to show that u j → L ( B R \ D )as j → ∞ if { u j } j ∈ N ⊂ X and u j (cid:42) j → ∞ .For each j ∈ N , define v j ∈ H loc (curl , R \ B R ) which satisfies that curl curl v j − κ v j = 0 in R \ B R , ˆ x × v j = ˆ x × u j on S R , lim r →∞ r (ˆ x × curl v j + i κ v j ) = 0 r = | x | . For each j ∈ N , define u ej = (cid:40) u j in B R \ D, v j in R \ B R . Then it is clear that u ej is the extension of u j in the sense of H (curl , · ).Recalling the definition of the space X , one has − κ ˆ x · u j = i κ ∇ S R · G e (ˆ x × u j )on S R , which, combined with the definition of G e and the Maxwell equation for v j ,gives ˆ x · u j = − i κ ∇ S R · G e (ˆ x × u j ) = − i κ ∇ S R · (cid:18) κ ˆ x × curl v j (cid:19) = − κ ∇ S R · (ˆ x × curl v j ) = 1 κ ˆ x · curl curl v j = ˆ x · v j on S R . (3.12)Noting that ∇ · u j = 0 in B R \ D and ∇ · v j = 0 in R \ B R , we conclude from (3.12)that the extended function u ej satisfiesdiv u ej = 0 in R \ D and ν · u ej = 0 on ∂D. Then it follows from Theorem 3.50 of [10] that u ej ∈ H / sloc ( R \ D ) for some s ≥
0. Bythe compactness of the imbedding H / ( B R \ D ) (cid:44) → L ( B R \ D ), there is a subsequenceof { u j } converging to 0 in L ( B R \ D ). This completes the proof.We are now ready to analyze the sesquilinear form A ( · , · ) on X . First, for H , w ∈ X , by Lemma 3.4 there exist H , w ∈ X and φ, ψ ∈ S such that H = H + ∇ φ and w = w + ∇ ψ . Thus, by the definition of X one has A ( H , w ) = A ( H , w ) + A ( ∇ φ, w ) + A ( ∇ φ, ∇ ψ ) . (3.13)We split A ( · , · ) into two parts: A ( H , w ) := a ( H , w ) + b ( H , w ) ∀ H , w ∈ X , where a ( H , w ) := (curl H , curl w ) + ( H , w ) + i κ (cid:104) G e (ˆ x × H ) , γ T w (cid:105) H − × H − ,b ( H , w ) := − ( κ + 1)( H , w ) + i κ (cid:104) G e (ˆ x × H ) , γ T w (cid:105) H − × H − . ecovery of elastic body by electromagnetic far-field measurements a ( · , · ) and b ( · , · ) on X . Lemma 3.6. a ( · , · ) is coercive on X × X , and there exists a compact operator K on X such that b ( H , w ) = a ( K H , w ) for all H , w ∈ X . Further, I + K is an isomorphism on X .Proof . The coerciveness of a ( · . · ) follows easily from the property of G e (see (c.2)in Lemma 3.2). By the Cauchy-Schwartz inequality, it is easy to see that, for eachfixed H ∈ X , b ( H , · ) defines a bounded linear functional on X . Thus, and bythe Lax-Milgram theorem and the coerciveness of a ( · . · ) on X × X , there is anoperator K on X such that b ( H , w ) = a ( K H , w ). The compactness of K follows easily from the property of G e (see Lemma 3.2) and the compact imbedding X (cid:44) → L ( B R \ D ) (see Lemma 3.5).We now prove that I + K is an isomorphism on X . By the Risze-Fredholmtheory, it is enough to show that I + K is injective. Let ( I + K ) w = 0 with w ∈ X .Then w satisfies A ( w , ψ ) = a ( w + K w , ψ ) = 0 for all ψ ∈ X . By the definition of X , we know that A ( w , ∇ φ ) = 0 for all φ ∈ S . This, combinedwith the Helmholtz-type decomposition for Ψ ∈ X , yields A ( w , Ψ) = 0 for all Ψ ∈ X. Therefore, w satisfies the boundary value problem curl curl w − κ w = 0 in B R \ D,ν × curl w = 0 on ∂D, κ ˆ x × curl w = G e (ˆ x × w ) on S R in the distribution sense. By the third equation in the above problem it is seen that w can be extended into R \ B R by considering the exterior problem curl curl v − κ v = 0 in R \ B R , ˆ x × v = ˆ x × w on S R , lim r →∞ r (ˆ x × curl v + i κ v ) = 0 , r = | x | . By the definition of G e it follows that ˆ x × curl w = ˆ x × curl v on S R . Hence, thefunction w e , which is defined by w in B R \ D and by v in R \ B R , satisfies theexterior problem curl curl w e − κ w e = 0 in R \ D,ν × curl w e = 0 on ∂D, lim r →∞ r (ˆ x × curl w e + i κ w e ) = 0 , r = | x | . Using Green’s formula for w e , one has (cid:90) S R ν × w e · κ curl w e d s = 0 . This, together with the Rellich lemma and the unique continuation principle, implies w e ≡ R \ D , and thus w = 0. Therefore, I + K is injective on X , which endsthe proof.2 T. Zhu, J. Yang and B. Zhang
Based on the above analysis and Lemmas 3.3, 3.4, 3.5 and 3.6, we can split A (( u , H ) , ( v , w )) as follows: A (( u , H ) , ( v , w )) := (cid:101) A (( u , H ) , ( v , w )) + K (( u , H ) , ( v , w ))for all u , v ∈ Q and H , w ∈ X with H = H + ∇ φ and w = w + ∇ ψ , H , w ∈ X and φ, ψ ∈ S , where (cid:101) A ( · , · ) and K ( · , · ) are defined as (cid:101) A (( u , H ) , ( v , w )) := A ( u , v ) − a ( φ, ψ ) − a ( H , w ) − A ( ∇ φ, w )+ i κb (cid:104) γ t ∇ φ, γ T v (cid:105) H − × H − + i κb (cid:104) γ t ∇ ψ, γ T u (cid:105) H − × H − , K (( u , H ) , ( v , w )) := − b ( φ, ψ ) − b ( H , w ) + i κb (cid:104) γ t H , γ T v (cid:105) H − × H − + i κb (cid:104) γ t w , γ T u (cid:105) H − × H − . Further, define the sesquilinear form (cid:102) K (( u , H ) , ( v , w )) := K (( u , H ) , ( v , w )) + K (( u , H ) , ( v , w )) . Then (3.7) can be reduced to the problem: find ( u , H ) ∈ Q × X such that (cid:101) A (( u , H ) , ( v , w )) + (cid:102) K (( u , H ) , ( v , w )) = F (( v , w )) (3.14)for all ( v , w ) ∈ Q × X .Let (cid:101) A , (cid:102) K : Q × X (cid:55)→ ( Q × X ) (cid:48) be the linear, bounded operators induced bythe corresponding sesquilinear forms (cid:101) A ( · , · ), (cid:102) K ( · , · ), respectively, with using the Rieszrepresentation lemma in Hilbert spaces. Then we have the following result. Theorem 3.7. If Im( b b ) < , then (cid:101) A + (cid:102) K is of Fredholm type with index .Proof . By the compact imbedding H ( D ) (cid:55)→ L ( D ) we deduce that K ( · , · )is a compact form on H ( · ) × H ( · ). By Lemma 3.2 it is known that b ( · , · ) is acompact form on S × S , and by Lemmas 3.5 and 3.2 it is also known that b ( · , · ) is acompact form on X × X . Further, by a similar argument as in deriving (3.12) (seealso [10]) it follows that, if w ∈ X then w | ∂D ∈ H ( ∂D ). This, combined withthe compact imbedding H ( ∂D ) (cid:55)→ L ( ∂D ), gives that K ( · , · ) is a compact form on( Q × X ) × ( Q × X ). Thus, the operator (cid:102) K is compact from Q × X into ( Q × X ) (cid:48) . Itremains to show that (cid:101) A is an isomorphism from Q × X into ( Q × X ) (cid:48) .Since Im( b b ) <
0, we obtain by using Korn’s inequality thatRe [ A ( u , u )] ≥ C (cid:107) u (cid:107) Q for all u ∈ Q (3.15)for some constant C >
0. Further, by Lemmas 3.2 and 3.3 it can be concluded that − Re [ a ( φ, φ )] ≥ C (cid:107)∇ φ (cid:107) X for all φ ∈ S, (3.16)Re [ a ( H , H )] ≥ C (cid:107) H (cid:107) X for all H ∈ X (3.17)for some constant C >
0. Recalling thati κcb + i κcb = 2i κ Re( b c ) 1 | b | , ecovery of elastic body by electromagnetic far-field measurements (cid:20) i κb (cid:104) γ t ∇ φ, γ T u (cid:105) H − × H − + i κb (cid:104) γ t ∇ φ, γ T u (cid:105) H − × H − (cid:21) = 0 . (3.18)Therefore, by (3.15)-(3.16) and (3.18) we obtain that the real part of (cid:101) A (( u , φ ) , ( v , ψ )) := A ( u , v ) − a ( φ, ψ )+ i κb (cid:104) γ t ∇ φ, γ T v (cid:105) H − × H − + i κb (cid:104) γ t ∇ ψ, γ T u (cid:105) H − × H − is coercive on ( Q × S ) × ( Q × S ). Thus, and by the Lax-Milgram lemma, for eachbounded functional ( f , f ) ∈ Q (cid:48) × ( ∇ S ) (cid:48) there exists a unique element ( (cid:101) u , (cid:101) φ ) ∈ ( Q × S )satisfying that (cid:101) A (( (cid:101) u , (cid:101) φ ) , ( v , ψ )) = f ( v ) + f ( ∇ ψ ) for all ( v , ψ ) ∈ Q × S (3.19)and the estimate (cid:107) (cid:101) u (cid:107) Q + (cid:107)∇ (cid:101) φ (cid:107) X ≤ C ( (cid:107) f (cid:107) Q (cid:48) + (cid:107) f (cid:107) ( ∇ S ) (cid:48) ) (3.20)for some constant C > A ( · , · ) and the Lax-Milgram lemma, foreach f ∈ ( X ) (cid:48) there exists a unique element (cid:102) H ∈ X satisfying that − a ( (cid:102) H , w ) = A ( ∇ (cid:101) φ, w ) + f ( w ) for all w ∈ X (3.21)and the estimate (cid:107) (cid:102) H (cid:107) X ≤ C ( (cid:107)∇ (cid:101) φ (cid:107) X + (cid:107) f (cid:107) ( X ) (cid:48) ) ≤ C ( (cid:107) f (cid:107) Q (cid:48) + (cid:107) f (cid:107) ( ∇ S ) (cid:48) + (cid:107) f (cid:107) ( X ) (cid:48) ) , (3.22)where C > (cid:101) A is an isomorphismfrom Q × X to ( Q × X ) (cid:48) . The proof is thus compete.Using Theorem 3.7, we can easily obtain the following well-posedness result forthe problem (3.6). Theorem 3.8.
Let ω / ∈ P ( ω ) . If Re( b b ) = 0 and Im( b b ) < , then theproblem (3.6) has a unique solution ( u , H ) ∈ Q × X satisfying the estimate (cid:107) u (cid:107) Q + (cid:107) H (cid:107) X ≤ C ( (cid:107) f (cid:107) H − ( ∂D ) + (cid:107) f (cid:107) H − ( ∂D ) ) , where C > is a constant independent of the choice of f and f .
4. Uniqueness of the inverse problem.
In this section, based on the analysisfor the forward scattering problem (2.1)-(2.5), we investigate the inverse problem ofdetermining the elastic body D by the electromagnetic far-field measurements. Weshall show that the shape and location of the elastic body can be uniquely recoveredby the magnetic or electric far-field pattern corresponding to incident plane waveswith all incident directions and polarizations. Motivated by our previous work in [14]for the reduced wave equation and the Maxwell equations, our method is based ona coupled system of PDEs constructed in a sufficiently small domain as well as theuniform a priori estimate in H ( · ) for the elastic field.4 T. Zhu, J. Yang and B. Zhang
In order to study the inverse problem,we introduce the following boundary value problem in a bounded, simply connecteddomain Ω with a Lipschitz continuous boundary ∂ Ω: curl curl H + H = ξ in Ω , ∇ · ( ˆ C : ∇ u ) − u = ξ in Ω ,T u − ˆ b ν × H = h on ∂ Ω ,ν × curl H + i κ ˆ b ν × u = h on ∂ Ω , (4.1)where ˆ C := ( ˆ C ijkl ( x )) i,j,k,l =1 with ˆ C ijkl ∈ L ∞ (Ω) ( i, j, k, l = 1 , , b , ˆ b ∈ C withˆ b ˆ b (cid:54) = 0, ξ , ξ ∈ L (Ω), h ∈ H − / ( ∂ Ω), h ∈ H − / ( ∂ Ω) and ˆ C satisfies thesymmetry condition and the Legendre elliptic condition (see Subsection 2.2). Lemma 4.1. If Im(ˆ b ˆ b ) < , then the problem (4 . has a unique solution ( H , u ) ∈ H (curl , Ω) × H (Ω) such that (cid:107) H (cid:107) H (curl , Ω) + (cid:107) u (cid:107) H (Ω) ≤ C (cid:104) (cid:107) ξ (cid:107) L (Ω) + (cid:107) ξ (cid:107) L (Ω) + (cid:107) h (cid:107) H − / ( ∂ Ω) + (cid:107) h (cid:107) H − / ( ∂ Ω) (cid:105) , where C > is a constant independent of ξ , ξ , h and h .Proof . By using Green’s formula, the problem (4.1) can be reformulated as thevariational problem: find ( H , u ) ∈ H (curl , Ω) × H (Ω) such thatˆ A (( H , u ) , ( w , v )) = ˆ F (( w , v )) for ( w , v ) ∈ H (curl , Ω) × H (Ω) , (4.2)whereˆ A (( H , u ) , ( w , v )) := (cid:90) Ω (curl H · curl w + H · w ) d x + − i κ ˆ b ˆ b (cid:90) Ω ( E ( u , v ) + u · v )) d x + i κ ˆ b (cid:104) γ t w , γ T u (cid:105) H − × H − − − i κ ˆ b (cid:104) γ t H , γ T v (cid:105) H − × H − , ˆ F (( w , v )) := − i κ ˆ b ˆ b (cid:104) h , v (cid:105) H − × H − (cid:104) h , γ T w (cid:105) H − × H − + (cid:90) Ω ( ξ · w + i κ ˆ b ˆ b ξ · v ) d x. By the Cauchy-Schwarz inequality and the trace theorem, there exists a constant
C > | ˆ A (( H , u ) , ( w , v )) | ≤ C (cid:107) ( H , u ) (cid:107) H (curl , Ω) × H (Ω) (cid:107) ( w , v ) (cid:107) H (curl , Ω) × H (Ω) , | ˆ F (( w , v )) | ≤ C (cid:2) (cid:107) ξ (cid:107) L (Ω) + (cid:107) ξ (cid:107) L (Ω) + (cid:107) h (cid:107) H − ( ∂ Ω) + (cid:107) h (cid:107) H − ( ∂ Ω) (cid:21) (cid:107) ( w , v ) (cid:107) H (curl , Ω) × H (Ω) , which implies that both ˆ A ( · , · ) and ˆ F ( · ) are bounded in H (curl , Ω) × H (Ω). ecovery of elastic body by electromagnetic far-field measurements w , v ) := ( H , u ) in (4.2). Then it follows thatˆ A (( H , u ) , ( H , u )) = (cid:90) Ω (curl H · curl H + | H | + − i κ ˆ b ˆ b ( E ( u , u ) + | u | )) d x + i κ ˆ b (cid:104) γ t H , γ T u (cid:105) H − × H − − − i κ ˆ b (cid:104) γ t H , γ T u (cid:105) H − × H − . Thus, Re (cid:104) ˆ A (( H , u ) , ( H , u )) (cid:105) ≥ c (cid:104) (cid:107) H (cid:107) H (curl , Ω) + (cid:107) u (cid:107) H (Ω) (cid:105) for some constant c >
0. The required result then follows from the Lax-Milgramlemma.
Assume that D and (cid:101) D are two elastic bodies corresponding to with the electromagnetic far-field patterns( E s ∞ (ˆ x, d, p ), H s ∞ (ˆ x, d, p )) and ( (cid:101) E s ∞ (ˆ x, d, p ), (cid:102) H s ∞ (ˆ x, d, p )), generated by the incidentplane waves given in (2.1) with the incident direction d ∈ S and the polarizationvector p ∈ R . Theorem 4.2. If H s ∞ (ˆ x, d, p ) = (cid:102) H s ∞ (ˆ x, d, p ) for all ˆ x, d ∈ S and p ∈ R , then D = (cid:101) D .Proof . Suppose D (cid:54) = (cid:101) D . Then there would exist z ∗ ∈ ∂D \ ∂ (cid:101) D and a small ball B centered at z ∗ such that z j := z ∗ + δj ν ( z ∗ ) ∈ B, for j = 1 , , · · · for small enough δ > B ∩ (cid:101) D = ∅ ; see Figure 4.1 for the geometric description. e D D z ∗ z j = z ∗ + δj ν ( z ∗ ) D B Fig. 4.1 . Two different elastic bodies
Consider the scattering problem (3.4) with the boundary data f and f inducedby the electric dipoles H i ( x, z j , q ) = curl ( q Φ( x, z j )) / (cid:107) curl ( q Φ( x, z j )) (cid:107) L ( ∂D ) , (4.3) E i ( x, z j , q ) = − κ curl H i ( x, z j , q ) (4.4)for q ∈ R , where Φ( · , · ) is the fundamental solution to the three-dimensionalHelmhlotz equation given byΦ( x, z ) := 14 π e i κ | x − z | | x − z | , x (cid:54) = z. T. Zhu, J. Yang and B. Zhang
By Theorem 3.8 we know that, for each j ∈ N the problem (3.4) has a uniquesolution ( E s ( · , z j , q ) , H s ( · , z j , q ) , u ( · , z j , q )) ∈ H loc (curl , R \ D ) × H loc (curl , R \ D ) × H ( D ) with respect to the elastic body D and a unique solution( (cid:101) E s ( · , z j , q ) , (cid:102) H s ( · , z j , q ) , (cid:101) u ( · , z j , q )) ∈ H loc (curl , R \ (cid:101) D ) × H loc (curl , R \ (cid:101) D ) × H ( (cid:101) D )with respect to the elastic body (cid:101) D . Define the total electromagnetic fields as follows: E ( · , z j , q ) := E i ( x, z j , q ) + E s ( · , z j , q ) in R \ D, H ( · , z j , q ) := H i ( x, z j , q ) + H s ( · , z j , q ) in R \ D, (cid:101) E ( · , z j , q ) := (cid:101) E i ( x, z j , q ) + (cid:101) E s ( · , z j , q ) in R \ (cid:101) D, (cid:102) H ( · , z j , q ) := (cid:102) H i ( x, z j , q ) + (cid:102) H s ( · , z j , q ) in R \ (cid:101) D. We now prove that the following mixed reciprocity relation holds for the scatteringsolutions of the problem (3.4) associated with the incident plane wave given in (2.1)and the electric dipoles given in (4.3) and (4.4):1 c j πp · E s ∞ ( − d, z j , q ) = q · E s ( z j , d, p ) , (4.5)1 c j πp · (cid:101) E s ∞ ( − d, z j , q ) = q · (cid:101) E s ( z j , d, p ) , (4.6)where c j := 1 / (cid:107) curl ( q Φ( x, z j )) (cid:107) L ( ∂D ) , E s ∞ ( − d, z j , q ) and (cid:101) E s ∞ ( − d, z j , q ) are the elec-tric far-field patterns corresponding to D and (cid:101) D , respectively. We only prove (4.5)since (4.6) can be shown similarly. First, use the vector Gauss divergence theoremand the radiation condition (2.4) to obtain that for each p, q ∈ R , (cid:90) ∂D ( ν × E i ( · , z j , q ) · H i ( · , d, p ) + ν × H i ( · , z j , q ) · E i ( · , d, p )) d s = 0 , (cid:90) ∂D ( ν × E s ( · , z j , q ) · H s ( · , d, p ) + ν × H s ( · , z j , q ) · E s ( · , d, p )) d s = 0 . Next, by the Stratton-Chu formula (cf. [3]) we get4 πp · E s ∞ ( − d, z j , q ) = (cid:90) ∂D ( ν × E se ( · , z j , q ) · H i ( · , d, p ) + ν × H s ( · , z j , q ) · E i ( · , d, p )) d s,q · E s ( z j , d, p ) = 1 c j (cid:90) ∂D ( ν × E s ( · , d, p ) · H i ( · , z j , q ) + ν × H s ( · , d, p ) · E i ( · , z j , q )) d s. Combining the above four equations with the transmission conditions yields4 πp · E s ∞ ( − d, z j , q ) − c j q · E s ( z j , d, p )= (cid:90) ∂D [ ν × E ( · , z j , q ) · H ( · , d, p ) + ν × H ( · , z j , q ) · E ( · , d, p )] d s = − b b (cid:90) ∂D [ u ( · , z j , q ) · T u ( · , d, p ) − u ( · , d, p ) · T u ( · , z j , q )] d s = − b b (cid:90) D [ u ( · , z j , q ) · ( ∇ · ( C : ∇ u ( · , d, p ))) − u ( · , d, p ) · ( ∇ · ( C : ∇ u ( · , z j , q )))] d x = 0 , ecovery of elastic body by electromagnetic far-field measurements H s ∞ (ˆ x, d, p ) = (cid:102) H s ∞ (ˆ x, d, p ) for all ˆ x, d ∈ S and p ∈ R , we obtain by (2.7)and Rellich’s lemma that for each p ∈ R , E s ( x, d, p ) = (cid:101) E s ( x, d, p ) , H s ( x, d, p ) = (cid:102) H s ( x, d, p ) x ∈ G , where G denotes the unbounded component of R \ ( D ∪ (cid:101) D ). This, together with themixed reciprocity relations (4.5) and (4.6) and Rellich’s lemma again, implies that foreach q ∈ R , E s ( x, z j , q ) = (cid:101) E s ( x, z j , q ) , H s ( x, z j , q ) = (cid:102) H s ( x, z j , q ) x ∈ G . (4.7)We now prove the uniform boundedness in an appropriate Sobolev space of both H s ( · , z j , q ) and u ( · , z j , q ) as j → ∞ . To this end, define the functionˆ H j ( x ) := H s ( x, z j , q ) − curl [ q Φ( x, y j )] / (cid:107) curl [ q Φ( x, z j )] (cid:107) L ( ∂D ) , x ∈ R \ D, (4.8)where y j := z ∗ − ( δ/j ) ν ( z ∗ ) ∈ D for j ∈ N . It is easy to verify that ( u ( · , z j , q ) , ˆ H j ( · ))satisfies the scattering problem curl curl ˆ H j − κ ˆ H j = 0 in D c , ∇ · ( C : ∇ u ( · , z j , q )) + ρω u ( · , z j , q ) = 0 in D,T u ( · , z j , q )) − b ν × ˆ H j = f j on ∂D,ν × curl ˆ H j + i κb ν × u ( · , z j , q )) = i κ f j on ∂D, lim r →∞ r (ˆ x × curl ˆ H j + i κ ˆ H j ) = 0 , r = | x | , (4.9)where the data f j and f j are given by f j ( x ) := ν × H i ( x, z j , q ) + ν × curl [ q Φ( x, y j )] (cid:107) curl [ q Φ( x, z j )] (cid:107) L ( ∂D ) = ν × curl [ q Φ( x, z j )] (cid:107) curl [ q Φ( x, z j )] (cid:107) L ( ∂D ) + ν × curl [ q Φ( x, y j )] (cid:107) curl [ q Φ( x, z j )] (cid:107) L ( ∂D ) , f j ( x ) := ν × E i ( x, z j , q ) − κ ν × curl curl [ q Φ( x, y j )] (cid:107) curl [ q Φ( x, z j )] (cid:107) L ( ∂D ) = − κ (cid:18) ν × curl curl [ q Φ( x, z j )] (cid:107) curl [ q Φ( x, z j )] (cid:107) L ( ∂D ) + ν × curl curl [ q Φ( x, y j )] (cid:107) curl [ q Φ( x, z j )] (cid:107) L ( ∂D ) (cid:19) . Noting that Div ( ν × f ) = − ν · curl f and curl curl f = ( − ∆ + ∇ div ) f for a smoothfunction f , one immediately has (cid:107) f j (cid:107) L ( ∂D ) + (cid:107) Div f j (cid:107) L ( ∂D ) ≤ C (4.10)uniformly for all j ∈ N , where C > z j and y j , and on taking q := ν ( z ∗ ), we can further prove that f j isuniformly bounded in L ( ∂D ) for all j ∈ N , that is, (cid:107) f j (cid:107) L ( ∂D ) ≤ C (4.11)8 T. Zhu, J. Yang and B. Zhang for some fixed constant C >
0. In fact, a direct calculation shows that1 (cid:107) curl [ q Φ( x, z j )] (cid:107) L ( ∂D ) { ν × curl [ q Φ( x, z j )] + ν × curl [ q Φ( x, y j )] } = ν × grad div [ q Φ( x, z j ) + q Φ( x, y j )] + κ ν × [ q Φ( x, z j ) + q Φ( x, y j )] (cid:107) curl [ q Φ( x, z j )] (cid:107) L ( ∂D ) = ν × grad grad[Φ( x, z j ) + Φ( x, y j )] q (cid:107) curl [ q Φ( x, z j )] (cid:107) L ( ∂D ) + κ ν × q [Φ( x, z j ) + Φ( x, y j )] (cid:107) curl [ q Φ( x, z j )] (cid:107) L ( ∂D ) =: I j + II j . (4.12)It is easy to see that II j ∈ L ( ∂D ) is uniformly bounded for j ∈ N since the funda-mental solution Φ( · , · ) is weakly singular. To estimate I j , without loss of generality,we may take z ∗ = (0 , , T and ν ( z ∗ ) = (0 , , T . Since q = ν ( z ∗ ), we have I j = 1 (cid:107) curl ( q Φ( x, z j )) (cid:107) L ( ∂D ) ν ( x ) × ∂ Φ( x, z j ) + ∂ Φ( x, y j ) ∂ Φ( x, z j ) + ∂ Φ( x, y j ) ∂ Φ( x, z j ) + ∂ Φ( x, y j ) = 1 (cid:107) curl ( q Φ( x, z j )) (cid:107) L ( ∂D ) ν ( z ∗ ) × ∂ Φ( x, z j ) + ∂ Φ( x, y j ) ∂ Φ( x, z j ) + ∂ Φ( x, y j ) ∂ Φ( x, z j ) + ∂ Φ( x, y j ) + 1 (cid:107) curl ( q Φ( x, z j )) (cid:107) L ( ∂D ) ( ν ( x ) − ν ( z ∗ )) × ∂ Φ( x, z j ) + ∂ Φ( x, y j ) ∂ Φ( x, z j ) + ∂ Φ( x, y j ) ∂ Φ( x, z j ) + ∂ Φ( x, y j ) = 1 (cid:107) curl ( q Φ( x, z j )) (cid:107) L ( ∂D ) − ∂ Φ( x, z j ) − ∂ Φ( x, y j ) ∂ Φ( x, z j ) + ∂ Φ( x, y j )0 + 1 (cid:107) curl ( q Φ( x, z j )) (cid:107) L ( ∂D ) ( ν ( x ) − ν ( z ∗ )) × ∂ Φ( x, z j ) + ∂ Φ( x, y j ) ∂ Φ( x, z j ) + ∂ Φ( x, y j ) ∂ Φ( x, z j ) + ∂ Φ( x, y j ) =: I (1) j + I (2) j . Let x = ( x , x , x ) T and z = ( z (1) , z (2) , z (3) ) T . Then a direct calculation gives4 π ∂ (cid:96) Φ( x, z ) = − κ ( x − z (3) )( x (cid:96) − z ( (cid:96) ) ) e i κ | x − z | | x − z | − κ ( x − z (3) )( x j − z ( (cid:96) ) ) e i κ | x − z | | x − z | + 3 ( x − z (3) )( x (cid:96) − z ( (cid:96) ) ) e i κ | x − z | | x − z | for (cid:96) = 1 , , (4.13)and4 π ∂ (cid:96) Φ( x, z ) = − κ ( x − z (3) ) e i κ | x − z | | x − z | + i κ e i κ | x − z | | x − z | − κ ( x − z (3) ) e i κ | x − z | | x − z | − e i κ | x − z | | x − z | + 3 ( x − z (3) ) e i κ | x − z | | x − z | for (cid:96) = 3 . (4.14)Since ∂D is C smooth, we have the unit normal vector function ν ( x ) ∈ C ( ∂D ), andthus | ν ( x ) − ν ( z ∗ ) | = O ( | x − z ∗ | ) for all x ∈ ∂D . This, combined with (4.13), (4.14) ecovery of elastic body by electromagnetic far-field measurements (cid:107) curl ( q Φ( · , z j )) (cid:107) L ( ∂D ) = (cid:90) ∂D | grad Φ( x, z j ) × ν ( z ∗ ) | d s ∼ = (cid:90) ∂D | x − z j | d s + O (1) , (4.15)implies that I (2) j ∈ L ( ∂D ) is uniformly bounded for all j ∈ N . It remains to showthat I (1) j ∈ L ( ∂D ) is uniformly bounded for all j ∈ N . From the definition of I (1) j and the equality (4.13), it is sufficient to prove this fact for the first component of I (1) j . In view of (4.13) and (4.15), we only need to show that the sequence1 (cid:107) curl [ q Φ( x, z j )] (cid:107) L ( ∂D ) (cid:32) ( x − z (3) j ) x e i κ | x − z j | | x − z j | + ( x − y (3) j ) x e i κ | x − y j | | x − y j | (cid:33) (4.16)is uniformly bounded in L ( ∂D ) for all j ∈ N .For x = ( x , x , x ) T , y = ( y (1) , y (2) , y (3) ) T and z = ( z (1) , z (2) , z (3) ) T , by a directcalculation we have( x − z (3) ) x e i κ | x − z | | x − z | + ( x − y (3) ) x e i κ | x − y | | x − y | = ( x − z (3) ) x | x − z | (cid:18) e i κ | x − z | | x − z | − e i κ | x − y | | x − y | (cid:19) + ( x − z (3) + x − y (3) ) x e i κ | x − y | | x − y | + (cid:18) ( x − z (3) ) x e i κ | x − y | | x − z | | x − y | + ( x − z (3) ) x e i κ | x − y | | x − z || x − y | (cid:19) (cid:18) | x − z | − | x − y | (cid:19) + (cid:18) ( x − z (3) ) x e i κ | x − y | | x − y | + ( x − z (3) ) x e i κ | x − y | | x − z | | x − y | (cid:19) (cid:18) | x − z | − | x − y | (cid:19) (4.17)It follows from [8] that there exists a constant C > (cid:12)(cid:12)(cid:12)(cid:12) | x − z j | − | x − y j | (cid:12)(cid:12)(cid:12)(cid:12) ≤ C , | Φ( x, z j ) − Φ( x, y j ) | ≤ C , x ∈ ∂D. (4.18)Recalling z ∗ = (0 , , T and ν ( z ∗ ) = (0 , , T , we deduce by Taylor’s expansion thatthere exists a constant C > | x | ≤ C ( x + x ) for all x ∈ ∂D. (4.19)Inserting (4.18) and (4.19) into (4.17) and using (4.15), we obtain that the sequencein (4.16) is uniformly bounded in L ( ∂D ) for all j ∈ N , which means that I (1) j isuniformly bounded in L ( ∂D ) for all j ∈ N . Thus, and by (4.12), the inequality(4.11) holds. By (4.10), (4.11) and Theorem 3.8 for the problem (4.9), we obtain theuniform estimate (cid:107) ˆ H j ( · ) (cid:107) H (curl ,B R \ D ) + (cid:107) u ( · , z j , q ) (cid:107) H ( D ) ≤ C ( (cid:107) f j (cid:107) H − / ( ∂D ) + (cid:107) f j (cid:107) H − / ( ∂D ) ) ≤ C , (4.20)where C , C > j ∈ N .0 T. Zhu, J. Yang and B. Zhang
Since ∂D ∈ C and D (cid:54) = D , we can choose a small subdomain of D , denoted by D with a C -smooth boundary ∂D , satisfying that D ⊂ D \ (cid:101) D and the intersection B ∩ ∂D contains an open segment of ∂D . In D , we can construct the followingboundary value problem curl curl F ,j + F ,j = ξ ,j in D , ∇ · ( C : ∇ E ,j ) − E ,j = ξ ,j in D ,T E ,j − b ν × F ,j = h ,j on ∂D ,ν × curl F ,j + i κb ν × E ,j = h ,j on ∂D , (4.21)with ξ ,j , ξ ,j , h ,j and h ,j defined as follows: ξ ,j : = ( κ + 1) (cid:102) H ( · , z j , q ) ,ξ ,j : = ( ρω + 1) u ( · , z j , q ) , h ,j : = T u ( · , z j , q ) − b ν × (cid:102) H ( · , z j , q ) , h ,j : = ν × curl (cid:102) H ( · , z j , q ) + i κb ν × u ( · , z j , q ) . By Theorem 3.8 it can be verified that all data are well-defined in the related Sobolevspaces for each fixed j ∈ N . By Lemma 4.1, the problem (4.21) is uniquely solvablewith the estimate (cid:107) F ,j (cid:107) H (curl ,D ) + (cid:107) E ,j (cid:107) H ( D ) ≤ C (cid:2) (cid:107) ξ ,j (cid:107) L ( D ) + (cid:107) ξ ,j (cid:107) L ( D ) + (cid:107) h ,j (cid:107) H − ( ∂D ) + (cid:107) h ,j (cid:107) H − ( ∂D ) (cid:21) , (4.22)where C > j ∈ N . In fact, it first follows from (4.20) that ξ ,j is uniformlybounded in L ( D ) for all j ∈ N . Due to the positive distance between D and (cid:101) D ,it can be concluded by the well-posedness of the problem (3.4) associated with theelastic body (cid:101) D and the uniform boundedness of H i ( · , z j , q ) in L ( D ) that ξ ,j isuniformly bounded in L ( D ) for all j ∈ N . By recalling the equality (4.7) and thetransmission conditions (2.5) and (2.6), it is deduced that h ,j = 0 , h ,j = 0 on ∂D ∩ ∂D for all j ∈ N . Thus we only need to show that h ,j and h ,j are uniformly boundedin H − / (Γ) and H − / (Γ), respectively, where Γ := ∂D \ B ε ( z ∗ ) and B ε ( z ∗ ) is aball with sufficiently small radius ε > B ε ( z ∗ ) ⊂ D . Since (cid:102) H ( · , z j , q ) = H i ( · , z j , q ) + (cid:102) H s ( · , z j , q ), and by (4.20), we deduce that (cid:107) u ( · , z j , q ) (cid:107) H ( D \ B ε ( z ∗ )) + (cid:107) (cid:102) H ( · , z j , q ) (cid:107) H (curl ,D \ B ε ( z ∗ )) ≤ C for a some constant C >
0, whence the uniform boundedness of (cid:107) h ,j (cid:107) H − / ( ∂D ) and (cid:107) h ,j (cid:107) H − / ( ∂D ) follows from the trace theorems.It is easy to verify that ( F ,j , E ,j ) := ( (cid:102) H ( · , z j , q ) , u ( · , z j , q )) is the unique solutionto the problem (4.21). Then, by (4.22) we have (cid:107) (cid:102) H ( · , z j , q ) (cid:107) H (curl ,D ) ≤ C ∀ j ∈ N (4.23) ecovery of elastic body by electromagnetic far-field measurements C > j ∈ N . On the other hand, due to thepositive distance between z ∗ and (cid:101) D , we have (cid:107) (cid:102) H s ( · , z j , q ) (cid:107) H (curl ,D ) ≤ C ∀ j ∈ N (4.24)for some constant C > j ∈ N . From (4.24) it follows that (cid:107) (cid:102) H ( · , z j , q ) (cid:107) H (curl ,D ) = (cid:107) H i ( · , z j , q ) + (cid:102) H s ( · , z j , q ) (cid:107) H (curl ,D ) ≥ (cid:107) H i ( · , z j , q ) (cid:107) H (curl ,D ) − (cid:107) (cid:102) H s ( · , z j , q ) (cid:107) H (curl ,D ) ≥ (cid:107) H i ( · , z j , q ) (cid:107) H (curl ,D ) − C . By [14, Theorem 3.8] it is seen that the right-hand side of the above inequality goes toinfinity as j → ∞ , which contradicts to the inequality (4.23), meaning that D = (cid:101) D .The proof is thus complete. Acknowledgements.
This work is partially supported by the NNSF of ChinaGrants No. 11771349, 91730306, 91630309 and Fundamental Research Funds for theCentral Universities of China.
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