Uniqueness for the inverse boundary value problem of piecewise homogeneous anisotropic elasticity in the time domain
aa r X i v : . [ m a t h . A P ] M a r Uniqueness for the inverse boundary valueproblem of piecewise homogeneous anisotropicelasticity in the time domain
C˘at˘alin I. Cˆarstea ∗ Gen Nakamura † Lauri Oksanen ‡ Abstract
We consider the inverse boundary value problem of recoveringpiecewise homogeneous elastic tensor and piecewise homogeneous massdensity from a localized lateral Dirichlet-to-Neumann or Neumann-to-Dirichlet map for the elasticity equation in the space-time domain.We derive uniqueness for identifying these tensor and density on alldomains of homogeneity that may be reached from the part of theboundary where the measurements are taken by a chain of subdo-mains whose successive interfaces contain a curved portion.
Keywords.
Inverse boundary value problem; uniqueness; anisotropicelasticity.
MSC(2000):
Inverse boundary value problems are concerned with the determination of thephysical properties (represented as various coefficients of a model equation)of an object from measurements taken on the boundary. The mathematicalinvestigation of such problems already has a history spanning nearly fourdecades, going back to [10]. In this context, uniqueness refers to the propertycertain equations might have that if two sets of coefficients produce identicalsets of boundary data, then the coefficients must also be identical. ∗ School of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, P.R.China;email: [email protected] † Hokkaido University, Sapporo 060-0808, Japan; email: [email protected] ‡ University College London, London, UK; email: [email protected] we obtain uniqueness closer to the part of the boundary where he measurements are taken for shorter observation times and further awayfrom that part of the boundary for longer times .We would like to remark on the importance of showing uniqueness inthe inverse boundary value problem for the local Neumann-to-Dirichlet map.The vibroseis exploration technique in the reflection seismology is used to in-vestigate the underground structure of the Earth. The measurements takenusing this technique correspond almost precisely to this kind of map, and notto a local Dirichlet-to-Neumann map (see [3]). Furthermore, the geologicalstructure of the Earth is isotropic or transversally isotropic layered in itsshallow part, and in its further depth part, it is close to piecewise homoge-neous and can contain regions with more complicated anisotropic elasticitythan transversally isotropic elasticity.The rest of this paper is organized as follows. In what remains of Section1, we formulate our inverse problem and give our main results on the uniqueidentification of piecewise homogeneous density and elasticity tensor. Thefirst main results is the unique identification for the case when the geometryof the homogeneous pieces is known. The second main result is in the casewhen the geometry of this pieces is unknown. In Section 2 we briefly review aboundary determination result that has been obtained previously in [11], [13].In Section 3 we discuss interior determination results for both Dirichlet-to-Neumann and Neumann-to-Dirichlet maps. Then in Section 4, by combiningthese the results of the previous two sections, we complete the proofs of ourmain results, in the second case making use of the theory of subanalytic sets.Section A which is an appendix provides a brief summary of results on thetheory of subanalytic sets which are used in the proof of the second mainresult. Let Ω ⊂ R be an open bounded connected domain, with Lipschitz boundary.For a T > T = (0 , T ) × Ω an Γ = ∂ Ω.An elastic tensor C = C ( x ) = ( C ijkl ( x )) i,j,k,l =1 , , , x = ( x , x , x ) ∈ Ω,is defined by real valued functions C ijkl ( x ) which satisfy the symmetries C ijkl ( x ) = C ijlk ( x ) , C ijkl ( x ) = C klij ( x ) , x ∈ Ω , i, j, k, l ∈ { , , } , (1)and the strong convexity condition, i.e. there exists λ > ǫ = ( ǫ ij ), ǫ : ( C : ǫ ) = X i,j,k,l =1 C ijkl ( x ) ǫ ij ǫ kl ≥ λ ( ǫ : ǫ ) , x ∈ Ω , (2)4here ǫ : η is the inner product of matrices ǫ and η = ( η ij ) defined by ǫ : η = P i,j =1 ǫ ij η ij , and C : η is a matrix whose ( i, j ) component ( C : η ) ij is defined as ( C : η ) ij = P k,l =1 C ijkl ( x ) η kl . The density of mass function isgiven as a function ρ ( x ) > λ , x ∈ Ω, which will be simply called density .In this paper we will make the further assumption that there are a finitenumber of open, connected, Lipschtiz subdomains D α , α ∈ A , that is sub-domains with Lipschitz boundaries, such that ¯Ω = ∪ α ∈ A ¯ D α , D α ∩ D β = ∅ if α = β , and C , ρ are homogeneous in each D α .For a function u : Ω → R denoting the diplacement we write( L C u ) i = X j,k,l =1 ∂ j ( C ijkl ∂ k u l ) , (3)where ∂ j = ∂ x j and for u : Ω T → R we write( P ρ, C u ) i = ρ∂ t u i − ( L C u ) i . (4)Let ν be the unit outer normal to ∂ Ω. We will denote the traction at ∂ Ω by( ∂ C u ) i := [( C : Du ) ν ] i = X j,k,l =1 ν j C ijkl ∂ k u l . (5) We want to consider in this paper the equations ( P ρ, C u = 0 in Ω T , ( u, ∂ t u ) | t =0 = 0 , (6)with Dirichlet or Neumann boundary conditions. At one point in the ar-gument we will however need a result with slightly more general boundaryconditions, namely with both Dirichlet and Neumann data given on comple-mentary parts of the boundary. This kind of problems have been consideredin the case of piezoelectric equations in [1]. Here we will give the restrictionof their result to the case of hyperbolic elasticity equations.We need to introduce a few function spaces. Suppose Σ is an open subsetof Γ. Let H ± / co (Σ) = (cid:8) f ∈ H ± / (Γ) : supp g ⊂ Σ (cid:9) . (7)with the restriction of the norm of H ± / (Γ), and H ± / (Σ) = (cid:8) f | Σ : f ∈ H ± / (Γ) (cid:9) , (8)5ith the norm || f || H ± / (Σ) = inf ˜ f, ˜ f | Σ = f || ˜ f || H ± / (Γ) . (9)Suppose Γ u , Γ s ⊂ Γ are disjoint connected open sets such that ∂ Γ u = ∂ Γ s is a Lipschitz curve. We write H u (Ω) = (cid:8) u ∈ H (Ω) : u | Γ u = 0 (cid:9) and ( H u (Ω)) ′ for its dual space . (10)Let f ∈ H (0 , T ; H / (Γ u )), g ∈ H (0 , T ; H − / (Γ s )) be boundary data, F ∈ H (0 , T ; ( H u (Ω)) ′ ) be a source term, and u ∈ H (Ω), u ∈ L (Ω) beinitial data satisfying the compatibility condition f (0 , · ) = u | Γ u . (11)Then we have the following theorem. Theorem 1.1.
There exists a unique u ∈ L ∞ (0 , T ; H (Ω)) with ∂ t u ∈ L ∞ (0 , T ; L (Ω)) , ∂ t u ∈ L ∞ (0 , T ; ( H u (Ω)) ′ ) such that P ρ, C u = F in Ω T ,u = f on (0 , T ) × Γ u ,∂ C u = g on (0 , T ) × Γ s u | t =0 = u , ∂ t u | t =0 = u , (12) and || u || L ∞ (0 ,T ; H (Ω)) + || ∂ t u || L ∞ (0 ,T ; L (Ω)) + || ∂ t u || L ∞ (0 ,T ;( H u (Ω)) ′ ) ≤ C ( || u || H (Ω; R ) + || u || L (Ω) + || F || H (0 ,T ;( H u (Ω)) ′ ) + || f || H (0 ,T ; H / (Γ u )) + || g || H (0 ,T ; H − / (Γ s )) ) , (13) where the constant C depends on Ω , T , λ , || C || L ∞ (Ω) , || ρ || L ∞ (Ω) . Remark 1.1.
One reason why the source term F has higher regularity intime than is usually assumed is the fact that it is more singular in space. Wewill in fact at one point in the proof need to solve the equation with a sourceterm that would have exactly this regularity. The Dirichlet data is required tohave three derivatives in time because, in the course of the proof of the result,it is necessary to convert the given equation to one with zero Dirichlet data,but with an additional source term, which has to have H in time regularity.
6e can define the local Neumann-to-Dirichlet and Dirichlet-to-Neumannmaps, which contain the information that may be collected by applying var-ious tractions to Σ and measuring displacements on Σ, or by producing dis-placements and measuring tractions on Σ as follows. The local Neumann-to-Dirichlet map (abbreviated as “ND map”)Φ T, Σ ρ, C : H (0 , T ; H − / co (Σ)) → H (0 , T ; H / (Σ)) (14)is defined by Φ T, Σ ρ, C g = u | (0 ,T ) × Σ , with (0 , T ) × Σ = (0 , T ) × Σ , (15)where u solves P ρ, C u = 0 in Ω T ,∂ C u | (0 ,T ) × Σ = g, ( u, ∂ t u ) | t =0 = 0 . (16)The local Dirichlet-to-Neumann map (abbreviated as “DN map”)Λ T, Σ ρ, C : H (0 , T ; H / co (Σ)) → H (0 , T ; H − / (Σ)) (17)is defined by Λ T, Σ ρ, C f = ∂ C u | (0 ,T ) × Σ , (18)where u solves P ρ, C u = 0 in Ω T ,u | (0 ,T ) × Σ = f, ( u, ∂ t u ) | t =0 = 0 . (19) If Σ ⊂ ∂D α is open, we will say that Σ is curved if it is C and { ν ( x ) : x ∈ Σ } ⊂ S contains the image of a non-constant continuous curve.Suppose C ( I ) , ρ ( I ) I = 1 ,
2, are two elastic tensors and two densitieson Ω, all of which are homogeneous in common Lipschitz subdomains D α .Since each boundary ∂D α of D α can have the discontinuity of the densityand elastic tensor, we also call each ∂D α interface. We will consider a chain D α i , i = 1 , . . . , N of these subdomains (which we will abbreviate as D i )and nonempty open surfaces Σ i ⊂ ∂D i such that Σ = Σ ⊂ Γ ∩ ∂D , andΣ i +1 ⊂ ¯ D i ∩ ¯ D i +1 , i = 1 , . . . , N − Theorem 1.2.
Let C (1) , C (2) , ρ (1) , ρ (2) be as above, assume that each Σ i , i = 1 , . . . , N − is curved in the sense given above, and ∂ ( ∂ Ω ∩ ∂D ) , ∂ ( ∂D i ∩ D i +1 ) , i = 1 , . . . , N − are Lipschitz curves, then there exist times < T < · · · < T N < ∞ , such that if Λ T k , Σ ρ (1) , C (1) = Λ T k , Σ ρ (2) , C (2) , (20) or if Φ T k , Σ ρ (1) , C (1) = Φ T k , Σ ρ (2) , C (2) , (21) then ρ (1) | D i = ρ (2) | D i , C (1) | D i = C (2) | D i , i = 1 , . . . , k. (22) Remark 1.2.
The condition that ∂ ( ∂ Ω ∩ ∂D ) , ∂ ( ∂D i ∩ ∂D i +1 ) , i = 1 , . . . , N − are Lipschitz curves is in fact not necessary. Given a chain of domains thatsatisfies the other assumptions, we can pick a smooth curve that connects apoint in Σ to a point inside D N and crosses each interface ∂D i transversely.The intersection of the original chain of domains with an appropriately cho-sen tubular neighborhood of this curve would satisfy this extra condition. Remark 1.3.
Note that in the case when ρ | D i , C | D i are known for some i ,then we do not need to assume that Γ i is curved. This would permit our resultto apply, for example, to measurements of elastic waves taken on the surfaceof the Earth, which is locally very close to flat, as long as the properties of thetop layer of the ground are known by other means. If the underground regionswhose elastic properties are unknown can be reached by passing through anumber of interfaces that do have curved portions, then the result still holds. Suppose C ( I ) , ρ ( I ) I = 1 ,
2, are two elastic tensors and two densities on Ω,all of which are homogeneous on Lipschitz subdomains D ( I ) α . Suppose furtherthat there is a “region of interest” R ⊂ Ω, Σ ⊂ ∂R , such that any D ( I ) α ∩ R is sub-analytic. Theorem 1.3.
Let C (1) , C (2) , ρ (1) , ρ (2) , be as above, and assume that Σ iscurved and that any boundary ( ∂D ( I ) α ∩ R ) \ ∂R is curved on all its smoothcomponents. Then there exists a time < T < ∞ so that if Λ T, Σ ρ (1) , C (1) = Λ T, Σ ρ (2) , C (2) , (23) or if Φ T, Σ ρ (1) , C (1) = Φ T, Σ ρ (2) , C (2) , (24) then ρ (1) | R = ρ (2) | R , C (1) | R = C (2) | R . (25) For a definition and summary of properties of sub-analytic sets, see Appendix A Boundary determination
In this section we follow [13] to show that the DN map Λ T, Σ ρ, C , or the ND mapΦ T, Σ ρ, C , determines the values of ρ | Σ and C | Σ . More precisely we will show thefollowing. Proposition 2.1 (see [13, Theorem 5.4]) . If Σ is curved, < T < ∞ ,and Λ T, Σ ρ (1) , C (1) = Λ T, Σ ρ (2) , C (2) or Φ T, Σ ρ (1) , C (1) = Φ T, Σ ρ (2) , C (2) , then ρ (1) | D = ρ (2) | D and C (1) | D = C (2) | D .Proof. We will sketch the argument in the DN map and the ND map casesseparately.
The DN map case:
This is proved in [13]. Here we will repeat enoughof the argument to give the reader an idea of how it works, but we will notreproduce it in full. The main idea is to use a finite-time Laplace transformin order to convert the hyperbolic elasticity boundary value problem to an el-liptic boundary value problem. The boundary determination will then followfrom the results on the elliptic case proven in [11], [13].Suppose C and ρ are of the kind we are considering. Let ψ ∈ H / co (Σ)and let u be a solution of P ρ, C u = 0 in Ω T ,u | (0 ,T ) × Γ = t ψ, ( u, ∂ t u ) | t =0 = 0 . (26)Also for φ ∈ H / co (Σ), we will consider the elliptic boundary value problem ( ρv − h L C v = 0 in Ω ,v | Γ = φ (27)depending on a parameter h >
0. Let˜Λ h, Σ ρ, C ( φ ) = h∂ C v | Σ (28)be the associated DN map.We are interested in comparing v ( · , h ) to the finite-time Laplace transform( L T u )( · , h ) := T Z u ( t, · ) e − th d t, (29)9here the Dirichlet data for v is chosen so that φ = ψ T Z t e − th d t. (30)Let u ∈ H (Ω) be the solution of ( L C u = 0 in Ω ,u | Γ = ψ, (31)and define u so that u ( t, x ) = t u ( x ) + u ( t, x ) , ( t, x ) ∈ Ω T . (32)Then u should satisfy P ρ, C u = − ρu in Ω T u | (0 ,T ) × Γ = 0 , ( u , ∂ t u ) | t =0 = 0 . (33)It is known that by the Korn inequality, there exists a unique solution suchthat u ∈ C ([0 , T ]; H (Ω)) ∩ C ([0 , T ]; L (Ω)) (34)and || u ( t ) || H (Ω) + || ∂ t u ( t ) || L (Ω) ≤ C || ρu || L (Ω T ) ≤ C || ψ || H / (Ω) , t ∈ [0 , T ] (35)(see [27]).Let r ( · , h ) = v ( · , h ) − T Z u ( t, · ) e − th d t. (36)An elementary computation shows that r satisfies ( h − ρr − L C r = e − Th [ ∂ t u ( T ) + h − u ( T ) + ρu ( T h − + 2 T )] in Ω ,r | Γ = 0 . (37)Let ˜ T = max(1 , T ). By the standard elliptic estimates it follows that || r || H (Ω) ≤ Ch − ˜ T e − Th || ψ || H / co (Σ) , (38)10or 0 < h <
1, and with a constant
C > T or h .From (30) it follows that t φ = χψ , where χ ( t, T, h ) = t T Z s e − sh d s − . (39)It is easy to see that there exists h > < h < h , then χ ( t, T, h ) < CT h − , ≤ t ≤ T, (40)where the constant C > T or h . We can concludethat || ˜Λ h, Σ ρ, C φ − h L T Λ T, Σ ρ, C ( χφ ) || H − / (Σ) ≤ C ˜ T h − e − Th || φ || H / co (Σ) , (41)or || ˜Λ h, Σ ρ, C − h L T Λ T, Σ ρ, C χ || H / co (Σ) → H − / (Σ) ≤ C ˜ T h − e − Th , (42)where the constant C > T or h .Considering ˜Λ h, Σ ρ,C as a semiclassical pseudodifferential operator with thesmall parameter h , it follows that its full symbol can be obtained from Λ T, Σ ρ,C (but not necessarily the operator itself).It is shown in [13] (see their Theorem 4.2) that the principal symbol of˜Λ h, Σ ρ,C determines Γ( x, h ) , x ⊥ ν , where Γ( x, h ) is the fundamental solution of ρ − h L C associated to the pair ρ | D and C | D whose Fourier transform withrespect to x with x ⊥ ν is bounded as h →
0. See also [20], [26], for similarresults. Once having this, it can be shown that by using Σ is curved, we canrecover ρ | D and C | D . This is shown in [13, Apendix B]. The ND map case:
The method of proof is almost entirely analogous tothe DN map case. We will give only a brief sketch of its argument.Let ψ ∈ H − / co (Σ) and let u be a solution of P ρ, C u = 0 in Ω T ,∂ C u | (0 ,T ) × Γ = t ψ, ( u, ∂ t u ) | t =0 = 0 . (43)For φ ∈ H − / co (Σ) consider the elliptic boundary value problem ( ρv − h L C v = 0 in Ω ,∂ C v | Γ = φ (44)11epending on a parameter h >
0. Let˜Φ h, Σ ρ, C φ = h − v | Σ (45)be the associated ND map.We will choose the Neumann data φ as φ = ψ T Z t e − th d t (46)and let u ∈ H (Ω) be the solution of ( L C u = 0 in Ω ,∂ C u | Γ = ψ. (47)Likewise before for the DN map case, we define u so that u ( t, x ) = t u ( x ) + u ( t, x ) , ( t, x ) ∈ Ω T . (48)Then u should satisfy P ρ, C u = − ρu in Ω T ∂ C u | (0 ,T ) × Γ = 0 , ( u , ∂ t u ) | t =0 = 0 . (49)By the Korn inequality, this equation has a unique solution such that u ∈ C ([0 , T ]; H (Ω)) ∩ C ([0 , T ]; L (Ω)) (50)and || u ( t ) || H (Ω) + || ∂ t u ( t ) || L (Ω) ≤ C || ρu || L (Ω T ) ≤ C || ψ || H − / (Ω) , t ∈ [0 , T ] (51)(see [27]).Now let r ( · , h ) = v ( · , h ) − T Z u ( t, · ) e − th d t. (52)It satisfies ( h − ρr − L C z = e − Th [ ∂ t u ( T ) + h − u ( T ) + ρu ( T h − + 2 T )] in Ω ,∂ C r | Γ = 0 . (53)12et ˜ T = max(1 , T ). By the standard elliptic estimates it follows that || r || H (Ω) ≤ Ch − ˜ T e − Th || ψ || H − / co (Σ) , (54)for 0 < h < C independent on T or h .Similarly to the DN case, we can conclude that || ˜Φ h, Σ ρ, C φ − h − L T Φ T, Σ ρ, C ( χφ ) || H / (Σ) ≤ C ˜ T h − e − Th || φ || H − / co (Σ) , (55)or || ˜Φ h, Σ ρ, C − h − L T Φ T, Σ ρ, C χ || H − / co (Σ) → H / (Σ) ≤ C ˜ T h − e − Th , (56)where the constant C is independent of T or h .As above, we can obtain the symbol of ˜Φ h, Σ ρ, C from Φ T, Σ ρ, C . The principalsymbol of ˜Λ h, Σ ρ, C is the inverse of the symbol of ˜Φ h, Σ ρ, C . We can therefore concludeas above that the local ND map determines the elastic tensor and density atthe boundary. For the purposes of this section, Ω will be a domain in R , D ⊂ Ω a subdo-main, Σ ⊂ ∂D ∩ ∂ Ω = ∅ . Let Ω = Ω \ D and Σ = ∂ Ω \ ∂ Ω. Suppose C and ρ are homogeneous in D and let Λ T Σ be the DN map for the domain Ω with data on Σ and Φ T Σ be the similarly defined ND map. We will provethe following proposition. Proposition 3.1.
There exists some < δ < ∞ depending on D , C | D , ρ | D such that(i) Λ T, Σ ρ, C determines Λ T − δ Σ ,(ii) Φ T, Σ ρ, C determines Φ T − δ Σ . We need the following result from [14]:
Proposition 3.2.
Suppose C , ρ are homogeneous in D . There is a (non-Riemannian) metric N on T D , determined by C | D and ρ | D , such that if P ρ, C w = 0 in D × (0 , T ) , ( w, ∂ C w ) | (0 ,T ) × Σ = 0 , then w ( T / , x ) = 0 for any x ∈ D such that d N ( x, Σ) < T / . Here N is a family of norms N x on R ≡ T x D , x ∈ Ω, which induces adistance on D by d ( x, y ) = inf γ Z N γ ( t ) ( γ ′ ( t )) d t, (57)13here the infimum is taken over curves γ ∈ C ([0 , D ) such that γ (0) = x and γ (1) = y . The distance to the boundary is defined in the usual way as d N ( x, Σ) = inf y ∈ Σ d N ( x, y ) . (58)Now let H − (Ω) be the dual space of H (Ω), and for F ∈ H ((0 , T ); H − (Ω))let u satisfy P ρ, C u = F in Ω T ,u | (0 ,T ) × Γ = 0 , ( u, ∂ t u ) | t =0 = 0 . (59)Then u has the estimate || u || X D ≤ C || F || H ((0 ,T ); H − (Ω)) , (60)with || u || X D = || u || L ∞ ((0 ,T ); H (Ω)) + || ∂ t u || L ∞ ((0 ,T ); L (Ω)) . (61)We write G Dρ, C ( F ) = u, G Dρ, C : H ((0 , T ); H − (Ω)) → X D . (62)Also, if u satisfies P ρ, C u = F in Ω T ,∂ C u | (0 ,T ) × Γ = 0 , ( u, ∂ t u ) | t =0 = 0 , (63)then || u || X N ≤ C || F || H ((0 ,T ); H − (Ω)) , (64)with || u || X N = || u || L ∞ ((0 ,T ); H (Ω)) + || ∂ t u || L ∞ ((0 ,T ); L (Ω)) . (65)We write G Nρ, C ( F ) = u, G Nρ, C : H ((0 , T ); H − (Ω)) → X N . (66)Now let δ = sup x ∈ D d N ( x, Σ). Then we have the following lemma.
Lemma 3.1.
For any F ∈ C ∞ (Ω T ) , supp F ⊂ (0 , T − δ ) × D , the DN map Λ T, Σ ρ, C , ρ | D , and C | D determine G Dρ, C ( F ) | (0 ,T − δ ) × D . roof. For f ∈ C ∞ ((0 , T ) × Σ) we define S T Σ ( f ) = u | (0 ,T ) × D , where u satisfies P ρ, C u = 0 in Ω T ,u | (0 ,T ) × Γ = f, ( u, ∂ t u ) | t =0 = 0 . (67)Similarly, if F ∈ C ∞ ((0 , T ) × D ) we define S T ( F ) = u | (0 ,T ) × D , where u satisfies (59). The first step in the proof is to show that Λ T, Σ ρ, C , ρ | D , and C | D determine S T − δ Σ . Let f ∈ C ∞ ((0 , T ) × Σ) and u that satisfies (67). Extend u by 0 to( −∞ , × Ω. Let S Tf ,f = { v ∈ C ∞ (( −∞ , T ) × D ) : P ρ, C v = 0 ,v | ( −∞ ,T ) × Σ = f , ∂ C v | ( −∞ ,T ) × Σ = f } , (68)which is determined by f , f , D , Σ, ρ | D , and C | D . Let u ′ ∈ S Tf, Λ T, Σ ρ, C f , and set w = u − u ′ . Then P ρ, C w = 0 on D , and ( w, ∂ C w ) | (0 ,T ) × Σ = 0. For t < T − δ we can apply Proposition 3.2 with the time interval ( t − δ, t + δ ) to concludethat w ( t, x ) = 0 for all x ∈ D . So we can now assume S T − δ Σ is known.Let r be the time reversal operator on (0 , T − δ ). That is ( r ℓ )( t ) = ℓ ( T − δ − t ) , t ∈ (0 , T − δ ) for any function ℓ over (0 , T − δ ). If we write u ∗ = r S T − δ Σ r f, (69) u ∗ satisfies ( P ρ, C u ∗ = 0 in (0 , T − δ ) × D,u ∗ | (0 ,T ) × Γ = f, ( u ∗ , ∂ t u ∗ ) | t = T − δ = 0 . (70)We want to identify the adjoint in L ((0 , T − δ ) × D ) of r S T − δ Σ r . Let F ∈ C ∞ ((0 , T ) × D ) and denote v = G Dρ, C ( F ). Then, using integration by parts, h F, r S T − δ Σ r f i = T − δ Z Z D F ( t, x ) u ∗ ( t, x ) d x d t = T − δ Z Z Ω P ρ, C ( v ) u ∗ d x d t = − T − δ Z Z Σ ∂ C vf. (71)We may then take the map F → ∂ C v | Σ to be known. Extend v as 0 to( −∞ , × Ω and let v ′ ∈ S T − δ ,∂ C v | Σ . Define w = v − v ′ . As above we can We use this two notations as defined here only within the proof of this lemma. w ( t, x ) = 0 for all t < T − δ and x ∈ D . This proves that S T − δ ( F ) is determined by the knowledge of D , the DN map Λ T, Σ ρ, C , ρ | D and C | D . Lemma 3.2.
For any F ∈ C ∞ (Ω T ) , supp F ⊂ (0 , T − δ ) × D , the ND map Φ T, Σ ρ, C , ρ | D , and C | D determine G Nρ, C ( F ) | (0 ,T − δ ) × D .Proof. For g ∈ C ∞ ((0 , T ) × Σ) we define S T Σ ( g ) = u | (0 ,T ) × D , where u satisfies P ρ, C u = 0 in Ω T ,∂ C u | (0 ,T ) × Γ = g, ( u, ∂ t u ) | t =0 = 0 . (72)Similarly, if F ∈ C ∞ ((0 , T ) × D ) we define S T ( F ) = u | (0 ,T ) × D , where u satisfies (63).The first step in the proof is to show that Φ T, Σ ρ, C , ρ | D , and C | D determine S T − δ Σ . Let g ∈ C ∞ ((0 , T ) × Σ) and u that satisfies (72). Extend u by 0 to( −∞ , × Ω. Let u ′ ∈ S T Φ T, Σ ρ, C g,g , and set w = u − u ′ . Then P ρ, C w = 0 on D ,and ( w, ∂ C w ) | (0 ,T ) × Σ = 0. For t < T − δ we can apply Proposition 3.2 withthe time interval ( t − δ, t + δ ) to conclude that w ( t, x ) = 0 for all x ∈ D . Sowe may now assume S T − δ Σ is known.If we write u ∗ = r S T − δ Σ r g, (73) u ∗ satisfies P ρ, C u ∗ = 0 in (0 , T − δ ) × D,∂ C u ∗ | (0 ,T ) × Γ = g, ( u ∗ , ∂ t u ∗ ) | t = T − δ = 0 . (74)We want to identify the adjoint in L ((0 , T − δ ) × D ) of r S T − δ Σ r . Let F ∈ C ∞ ((0 , T ) × D ) and denote v = G Nρ, C ( F ). Then, using integration by parts, h F, r S T − δ Σ r f i = T − δ Z Z D F ( t, x ) u ∗ ( t, x ) d x d t = T − δ Z Z Ω P ρ, C ( v ) u ∗ d x d t = T − δ Z Z Σ v · g. (75)We can then take the map F → v | Σ to be known. Extend v as 0 to ( −∞ , × Ω and let v ′ ∈ S T − δv | Σ , . Define w = v − v ′ . As above we can conclude that w ( t, x ) = 0 for all t < T − δ and x ∈ D . This proves that S T − δ ( F ) isdetermined by the knowledge of D , the ND map Φ T, Σ ρ, C , ρ | D and C | D .16et f ∈ C ((0 , T ); H − / (Σ )), and define T f ∈ C ((0 , T ); H − (Ω)) by h T f , φ i = T Z h f, φ | Σ i , φ ∈ H (Ω) . (76)Define the operator L TD by L TD ( f ) = u | Σ , where P ρ, C u = T f in Ω T ,u | (0 ,T ) × Γ = 0 , ( u, ∂ t u ) | t =0 = 0 , (77)and the operator L TN by L TN ( f ) = u | Σ , where P ρ, C u = T f in Ω T ,∂ C u | (0 ,T ) × Γ = 0 , ( u, ∂ t u ) | t =0 = 0 . (78) Lemma 3.3. L T − δD is determined by the knowledge of D , the DN map Λ T, Σ ρ, C , ρ | D , and C | D . L T − δN is determined by the knowledge of D , the NDmap Φ T, Σ ρ, C , ρ | D , and C | D .Proof. Let f ∈ C ((0 , T ); C ∞ (Σ )). In local (in space) coordinates we can ar-range that Σ is { x = 0 } and D is { x > } . Suppose the spatial support of f is entirely in this coordinate patch. For ǫ > F ǫ ∈ C ((0 , T ); H − (Ω))by F ǫ ( φ )( t ) = Z x =0 f ( t, x ′ ) φ ( x ′ , ǫ ) d x ′ , φ ∈ C ∞ ( D ) . (79)Then | F ǫ ( φ )( t ) − T f ( φ )( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z ≤ x ≤ ǫ f ( t, x ′ ) ∂ φ ( x ′ , x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C || φ || H (Ω) || f || L ∞ ((0 ,T ) × Σ ) ǫ / , (80)and similarly | ∂ t F ǫ ( φ )( t ) − ∂ t T f ( φ )( t ) | ≤ C || φ || H (Ω) || ∂ t f || L ∞ ((0 ,T ) × Σ ) ǫ / . (81)Using a partition of unity argument, we can therefore construct a sequence F n ∈ C ((0 , T ); H − (Ω)), supp F n ⊂ (0 , T ) × D , such that F n → T f in C ((0 , T ); H − (Ω)). 17uppose now that we take two functions f, h ∈ C ((0 , T − δ ); C (Σ ))and construct sequences F n , H n as above. By Lemma 3.1, H n ( G Dρ,C ( F n )) isdetermined by the knowledge of D , the DN map Λ T, Σ ρ, C , ρ | D , and C | D . Passingto the limit we see that so is T h ( G Dρ, C ( T f )) and therefore so is L T − δD . Thesame is true for L T − δN .Let Λ T, +Σ be the DN map for the domain D with data on Σ . Also letΦ T, +Σ be the ND map for the domain D with data on Σ . Lemma 3.4. If (Λ T Σ − Λ T, +Σ ) f = 0 for f ∈ C ∞ ((0 , T ); H / co (Σ )) , then f = 0 .If (Φ T Σ − Φ T, +Σ ) g = 0 for g ∈ C ∞ ((0 , T ); H − / co (Σ )) , then g = 0 .Proof. Suppose (Λ T Σ − Λ T, +Σ ) f = 0. Let u and u + be the solutions in Ω and D respectively with Dirichlet data f on Σ and zero on the rest of theirrespective boundaries. Define ˜ u = uχ Ω + u + χ D . By the assumption, sinceboth Dirichlet and Neumann data accross Σ match, we have that P ρ, C ˜ u = 0in Ω. Since ˜ u has zero initial Cauchy and lateral Dirichlet data, it must bezero.In the case when (Φ T Σ − Φ T, +Σ ) f = 0, the argument is identical. Lemma 3.5.
For f ∈ C ∞ ((0 , T ) × Σ ) , (Λ T Σ − Λ T, +Σ ) L TD f = f. (82) Proof.
Let u be as in (77) and φ ∈ C ∞ ((0 , T ) × Ω). Then h f, φ | Σ i = T Z Z Ω P ρ, C ( u ) φ = T Z Z Ω (cid:0) ∂ t uφ + Du : ( C : Dφ ) (cid:1) (83)and T Z Z Ω Du : ( C : Dφ ) = T Z Z Ω Du : ( C : Dφ ) + T Z Z D Du : ( C : Dφ )= − T Z Z Ω L C ( u ) φ − T Z Z D L C ( u ) φ + h Λ T Σ L TD f, φ | Σ i − h Λ T, +Σ L TD f, φ | Σ i . (84)18 roof of Proposition 3.1 (i). It follows that Λ T − δ Σ − Λ T − δ, +Σ is the inverse of L T − δD , which is then determined by Λ T, Σ ρ, C . The claim (i) follows as Λ T − δ, +Σ is clearly determined by D , C D and ρ | D .The argument for the ND map case is a little bit more involved. Beforestating the lemma for ND maps that is analogous to Lemma 3.5 we introducetwo more notations. If u = G Nρ, C ( T f ), then ∂ C D u | Σ +2 will denote the restrictionof ∂ C u to Σ , taken from the D side of Σ . Similarly, ∂ C Ω2 u | Σ will be therestriction of the same function to Σ , taken from the Ω side of Σ . Notethat here the unit normal vector in all cases points away from Ω and into D . Lemma 3.6.
For f ∈ C ∞ ((0 , T ) × Σ ) , (Φ T Σ − Φ T, +Σ )( ∂ C | D u | Σ +2 + f ) = − Φ T, +Σ ( f ) . (85) Proof.
As in the proof of Lemma 3.5, for φ ∈ C ∞ ((0 , T ) × Ω) h f, φ | Σ i = T Z Z Ω P ρ, C ( u ) φ = T Z Z Ω (cid:0) ∂ t uφ + Du : ( C : Dφ ) (cid:1) , (86)and T Z Z Ω Du : ( C : Dφ ) = T Z Z Ω Du : ( C | Ω : Dφ ) + T Z Z D Du : ( C | D : Dφ )= − T Z Z Ω L C ( u ) φ − T Z Z D L C ( u ) φ + h ∂ C Ω2 u | Σ , φ | Σ i − h ∂ C D u | Σ +2 , φ | Σ i . (87)It follows that ∂ C Ω2 u | Σ = ∂ C D u | Σ +2 + f. (88)Since u ( t, · ) ∈ H (Ω), its Dirichlet data on each side of Σ coincide. ThereforeΦ T Σ ( ∂ C Ω2 u | Σ ) = Φ T, +Σ ( ∂ C D u | Σ +2 ) . (89)The conclusion follows immediately.The surjectivity of the difference Φ T Σ − Φ T, +Σ follows from the next lemma.19 emma 3.7. The local ND map Φ T, +Σ : C ∞ ((0 , T ]; H − / co (Σ )) → C ∞ ((0 , T ]; H / (Σ )) (90) is surjective. First note that since the coefficients of the equation are time independent,taking time derivatives of all orders and applying Theorem 1.1, we have thefollowing corollary with the same notations as in Theorem 1.1.
Corollary 3.1. If f ∈ C ∞ ((0 , T ]; H / (Γ u )) , g ∈ C ∞ ((0 , T ]; H − / (Γ s )) ,then there exists a unique u ∈ C ∞ ((0 , T ]; H (Ω)) such that P ρ, C u = 0 in Ω ,u | (0 ,T ) × Γ u = f,∂ C u | (0 ,T ) × Γ s = g, ( u, ∂ t u ) | t =0 = 0 . (91)In particular, taking Γ u = ∅ and Γ s = Γ, this justifies the spaces inbetween which Φ T, +Σ maps in the statement of the lemma. Proof of Lemma 3.7.
To prove the surjectivity, let f ∈ C ∞ ((0 , T ]; H / (Σ )).We choose Γ u = Σ , Γ s = ∂D \ Σ . Let u ∈ C ∞ ((0 , T ]; H (Ω)) be such that P ρ, C u = 0 in D,u | (0 ,T ) × Γ u = f,∂ C u | (0 ,T ) × Γ s = 0 , ( u, ∂ t u ) | t =0 = 0 . (92)Since Φ T, +Σ ( ∂ C | D u | Σ +2 ) = f , we have our desired conclusion. Proof of Proposition 3.1 (ii).
With the same notation used in Lemma 3.6,for f ∈ C ∞ ((0 , T ); H − / (Σ )) define the operator K : C ∞ ((0 , T ); H − / co (Σ )) → C ∞ ((0 , T ); H − / co (Σ )) , K ( f ) = ∂ C | D u | Σ +2 + f. (93)By Lemmas 3.6 and 3.7, this operator is surjective. It follows from the proofof Lemma 3.3 that K is determined by the knowledge of Ω, D , C | D , ρ | D ,and Φ T, Σ ρ, C . Its right inverse, which we will denote by κ , is then determined bythe same quantities. We have now that Φ T Σ − Φ T, +Σ = − Φ T, +Σ ◦ κ is determinedby the knowledge of Ω, D , C | D , ρ | D , and Φ T, Σ ρ, C . This ends the proof.20 Proofs of Theorem 1.2 and Theorem 1.3
Proof of Theorem 1.2.
The time T can be chosen arbitrarily small. ByProposition 2.1, if Λ T , Σ ρ (1) ,C (1) = Λ T , Σ ρ (2) ,C (2) or Φ T , Σ ρ (1) ,C (1) = Φ T , Σ ρ (2) ,C (2) , then ρ (1) | D = ρ (2) | D and C (1) | D = C (2) | D . Now let δ be as in Proposition 3.1, with D = D . We can choose T = T + 2 δ . If Λ T , Σ ρ (1) ,C (1) = Λ T , Σ ρ (2) ,C (2) or Φ T , Σ ρ (1) ,C (1) =Φ T , Σ ρ (2) ,C (2) , then Λ T , Γ ρ (1) ,C (1) = Λ T , Γ ρ (2) ,C (2) or Φ T , Γ ρ (1) ,C (1) = Φ T , Γ ρ (2) ,C (2) , respectively,where these DN and ND maps are taken relative to Ω \ D . By Proposi-tion 2.1 it follows that ρ (1) | D = ρ (2) | D and C (1) | D = C (2) | D . It is clear thatwe may continue in this way to inductively construct all the times T k . Proof of Theorem 1.3.
It is clear that we only need to prove the result in thecase R = Ω. Notice that ρ ( I ) , C ( I ) are all constant on all the elements of thecommon partition { ˜ D γ } = { D (1) α ∩ D (2) β } . (94)Let P ∈ R and pick a smooth curve ω : [0 , → R so that ω (0) ∈ Σ, ω (1) = P , and which intersects the boundaries of the subdomains ˜ D γ onlyat smooth points and transversally. Let V ǫ be the tubular neighborhood of ω ([0 , ǫ >
0. We can choose ǫ small enough that V ǫ only intersectsthe smooth components of the boundaries of the subdomanins ˜ D γ and ∂V ǫ is transversal to all of them. It then follows that each D ′ γ = ˜ D γ ∩ V ǫ is aLipschitz set.We can label by D ′ , . . . , D ′ N the chain of non-empty sets in { D ′ γ } , in theorder in which the curve ω intersects them. This chain satisfies the conditionsof Theorem 1.2 and we may conclude that there is a time 0 < T P < ∞ suchthat if Λ T P , Σ ρ (1) ,C (1) = Λ T P , Σ ρ (2) ,C (2) or Φ T P , Σ ρ (1) ,C (1) = Φ T P , Σ ρ (2) ,C (2) , then ρ (1) ( P ) = ρ (2) ( P ) and C (1) ( P ) = C (2) ( P ). Choosing T = max P ∈ R T P < ∞ , we have the result. Acknowledgement
This project began while the first author was employed at the Hong KongUniversity of Science and Technology, Jockey Club Institute for AdvancedStudy, and concluded while he was visiting Hokkaido University, partiallysupported by Sichuan University. The second author was partially sup-ported by Grant-in-Aid for Scientific Research (15K21766, 15H05740) of theJapan Society for the Promotion of Science doing the research of this pa-per. The third author was supported by EPSRC grants EP/P01593X/1 andEP/R002207/1. 21
Sub-analytic sets
In this appendix, for the convenience of the reader, we give the definitionand summarize a few of the properties of sub-analytic sets.Let X be a real analytic manifold. A set A ⊂ X is semi-analytic if for any x ∈ A (here A denotes the closure of A ) there exists an open neighborhood U of x in X and finitely many real-analytic functions f ij : U → R , i = 1 , . . . , p , j = 1 , . . . , q , such that A ∩ U = p [ i =1 q \ j =1 { x ∈ U : f ij ( x ) ∗ ij } , (95)where the relations ∗ ij are either “ > ” or “=”. For example, a finite unionof linear or curved polyhedra in R n , whose boundaries are level sets of real-analytic functions, is a semi-analytic set. A good reference for semi-analyticsets is [9].Now we introduce the notion of a subanalytic set, which is just obtainedin the above definition by replacing subsets determined by inequalities withthe ones of images of analytic maps. That is, A is said to be subanalytic if forany x ∈ A there exist an open neighborhood U of x , real analytic compactmanifolds Y i,j , i = 1 , , ≤ j ≤ N and real analytic maps Φ i,j : Y i,j → X such that A ∩ U = N [ j =1 (Φ ,j ( Y ,j ) \ Φ ,j ( Y ,j )) \ U. (96)Reference is made to [9] and [16], where we can find all the required proofs forproperties stated below: A family of subanalytic sets is stable under severalset theoretical operations. Note that, by definition, a semi-analytic subset issubanalytic.1. A finite union and a finite intersection of subanalytic subsets are sub-analytic.2. The closure, interior and complement of a subanalytic subset are againsubanalytic. In particular, its boundary is subanalytic.3. The inverse image of a subanalytic set by an analytic map is subana-lytic. Further, the direct image of a subanalytic set by a proper analyticmap is also subanalytic.The other important properties needed in this paper are the following“finiteness property” and “triangulation theorem” of a subanalytic set.22 emma A.1 (Theorem 3.14 [9]) . Each connected component of a subanalyticset is subanalytic. Furthermore, connected components of a subanalytic setare locally finite, that is, for any compact subset K and a subanalytic subset A , the number of connected components of A intersecting K is finite. In particular, for two relatively compact subanalytic subsets A and B ,the number of connected components of A ∩ B is always finite. Lemma A.2 (Proposition 8.2.5 [16]) . Let X = ⊔ λ ∈ Λ X λ be a locally finitepartition of X by subanalytic subsets. Then there exist a simplicial complex S = ( S, ∆) and a homeomorphism i : | S | → X such that1. for any simplex σ ∈ ∆ , the image ˆ σ := i ( | σ | ) is subanalytic in X andreal analytic smooth at every point in ˆ σ .2. for any simplex σ ∈ ∆ , there exists λ ∈ Λ with i ( | σ | ) ⊂ X λ . References [1] M. Akamatsu and G. Nakamura. Well-posedness of initial-boundaryvalue problems for piezoelectric equations.
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