Elements of noncommutative geometry in inverse problems on manifolds
aa r X i v : . [ m a t h - ph ] J un Elements of noncommutative geometry ininverse problems on manifolds
M.I.Belishev ∗ and M.N.Demchenko † Abstract
We deal with two dynamical systems associated with a Riemannianmanifold with boundary. The first one is a system governed by thescalar wave equation, the second is governed by the Maxwell equations.Both of the systems are controlled from the boundary. The inverseproblem is to recover the manifold via the relevant measurements atthe boundary (inverse data).We show that the inverse data determine a C*-algebras, whose(topologized) spectra are identical to the manifold. By this, to re-cover the manifold is to determine a proper algebra from the inversedata, find its spectrum, and provide the spectrum with a Riemannianstructure.The paper develops an algebraic version of the boundary controlmethod, which is an approach to inverse problems based on their re-lations to control theory.
About the paper
One of the basic theses of noncommutative geometry is that a topologicalspace can be characterized via an algebra associated with it [8], [11], [16]. Inother words, a space can be encoded into an algebra. As was recognized in [2] ∗ Saint-Petersburg Department of the Steklov Mathematical Institute, Saint-PetersburgState University, Russia; [email protected]. † Saint-Petersburg Department of the Steklov Mathematical Institute; [email protected]. [ A (Ω) of an appropriate Banach algebra A (Ω), thealgebra being determined by the inverse data up to isometric isomorphism.Therefore, one can reconstruct Ω by the scheme: • extract an isometric copy ˜ A (Ω) of A (Ω) from the data • find its spectrum [ ˜ A (Ω) =: ˜Ω, which is homeomorphic to [ A (Ω) by virtueof ˜ A (Ω) isom = A (Ω). Thus, we have ˜Ω hom = Ω • endow ˜Ω with a proper Riemannian structure.As a result, we get a Riemannian manifold ˜Ω isometric to the original Ω byconstruction. It is ˜Ω, which solves the reconstruction problem.Our paper keeps this scheme and extends it to the inverse problem ofelectrodynamics. Content
We deal with a smooth compact Riemannian manifold Ω with boundary.
Eikonals.
We introduce the eikonals , which play the role of main instrumentfor reconstruction. An eikonal τ σ ( · ) = dist ( · , σ ) is a distance function on Ωwith the base σ ⊂ ∂ Ω. The eikonals determine the Riemannian structure onΩ. With each eikonal one associates a self-adjoint operator ˇ τ σ in L (Ω), whichmultiplies functions by τ σ . Its representation via the Spectral Theorem isˇ τ σ = R ∞ s dX sσ , where X sσ is the projection onto the subspace L (Ω s [ σ ]) offunctions supported in the metric neighborhood Ω s [ σ ] ⊂ Ω of σ of radius s .For an oriented 3d-manifold Ω, by analogy with the scalar case, we in-troduce the solenoidal eikonals ε σ = R ∞ s dY sσ , which act in the space C = { curl h | h, curl h ∈ ~L (Ω) } relevant to electrodynamics. Here Y sσ projectsvector-fields onto the subspace of curls supported in Ω s [ σ ]. Algebras.
Eikonals { τ σ | σ ⊂ ∂ Ω } generate the Banach algebra C (Ω) of realcontinuous functions. By the Gelfand theorem, its Gelfand spectrum (the setof characters) [ C (Ω) is homeomorphic to Ω [13], [14].2perator eikonals { ˇ τ σ | σ ⊂ ∂ Ω } generate an operator algebra T , whichis a commutative C*-subalgebra of the bounded operator algebra B ( L (Ω)).The algebras T and C (Ω) are isometrically isomorphic (via ˇ τ σ τ σ ). Bythis, their spectra are homeomorphic, and we have b T hom = [ C (Ω) hom = Ω.Solenoidal eikonals generate an operator algebra E , which is a C*-sub-algebra of B ( C ). In contrast to T , the algebra E is noncommutative . However,the factor-algebra ˙ E = E / K over the ideal of compact operators K ∈ E turns out to be commutative. Moreover, one has ˙ E isom = C (Ω) that implies b ˙ E hom = [ C (Ω) hom = Ω. Inverse problems.
Following [4], we begin with a dynamical system, whichis governed by the scalar wave equation in Ω and controlled from the boundary ∂ Ω. The input output correspondence is realized by a response operator R ,which plays the role of inverse data. A reconstruction (inverse) problem isto recover the manifold Ω via given R .Solving this problem, we construct (via R ) an operator algebra ˜ T isomet-ric to T , find its spectrum ˜Ω := b ˜ T hom = b T hom = Ω, endow it with the Riemannianstructure by the use of images of eikonals, and eventually turn ˜Ω into an iso-metric copy of the original manifold Ω. The copy ˜Ω provides the solution tothe reconstruction problem.In electrodynamics, the corresponding system is governed by the Maxwellequations and also controlled from the boundary. The relevant response op-erator R plays the role of inverse data for the reconstruction problem. Tosolve this problem, we repeat all the steps of the above described proce-dure. The only additional step is the factorization E ˙ E , which eliminatesnoncommutativity. Appendix.
Here the basic lemmas on the eikonals ε σ and algebra E areproven. CommentsWhat is ”to recover a manifold”?
Setting the goal to determine Ω from R , one has to take into account the evident nonuniqueness of such a de-termination. Indeed, if two manifolds Ω and Ω ′ are isometric and have themutual boundary ∂ Ω = ∂ Ω ′ then their boundary inverse data (in particular,the response operators) turn out to be identical. Hence, the correspondenceΩ R in not injective and to recover the original Ω via R is impossible.3rom the physical viewpoint, the inverse data formalize the measure-ments, which the external observer implements at the boundary. The abovementioned nonuniqueness means that the observer is not able to distinguishΩ from Ω ′ in principle. In such a situation, the only reasonable understand-ing of the reconstruction problem is the following: to construct a manifold ˜Ω ,which possesses the prescribed inverse data . It is the above mentioned isomet-ric copy ˜Ω, which satisfies this requirement: we have ˜ R = R by construction. Remark
Reconstruction via algebras is known in Noncommutative Geom-etry: see [8], [11], [16]. However, there is a principle difference: in the men-tioned papers the starting point for reconstruction is the so-called spectraltriple {A , H , D} , which consists of a commutative algebra, a Hilbert space,and a self-adjoint (Dirac-like) operator. So, an algebra is given .In our case, we at first have to extract an algebra from R . Then we dealwith this algebra imposed by inverse data, whereas its ”good” properties arenot guaranteed. For instance, a metric graph is a ”commutative space” butits eikonal algebra T turns out to be strongly noncommutative . The latterleads to difficulties in reconstruction problem, which are not overcome yet.Reconstruction via algebras in inverse problems was originated in [2] anddeveloped in [4]. It represents an algebraic version of the boundary controlmethod , which is an approach to inverse problems based on their relationsto control theory [1], [3]. We hope for further applications of this version toinverse problems of mathematical physics. Acknowledgements
The authors thank B.A.Plamenevskii for kind anduseful consultations. The work is supported by the grants RFBR 11-01-00407A, RFBR 12-01-31446, SPbGU 11.38.63.2012, 6.38.670.2013 and RFGovernment grant 11.G34.31.0026.
We deal with a real smooth compact Riemannian manifold Ω with theboundary Γ, g is the metric tensor, dim Ω = n > A ⊂ Ω, byΩ r [ A ] := { x ∈ Ω | dist ( x, A ) < r } , r > no factorization turns T into a commutative algebra everywhere in the paper, ”smooth” means C ∞ -smooth
4e denote its metric r -neighborhood. Compactness implies diam Ω :=sup { dist ( x, y ) | x, y ∈ Ω } < ∞ andΩ r [ A ] = Ω as r > diam Ω . (2.1) Let us say a subset σ ⊂ Γ to be regular and write σ ∈ R (Γ) if σ is diffeomor-phic to a ”disk” { p ∈ R n − | k p k } .By a (scalar) eikonal we name a distant function of the form τ σ ( x ) := dist ( x, σ ) , x ∈ Ω ( σ ∈ R (Γ)) . The set σ is said to be a base . Eikonals are Lipschitz functions: τ σ ∈ Lip(Ω) ⊂ C (Ω). Moreover, eikonals are smooth almost everywhere and |∇ τ σ ( x ) | = 1 a . a . x ∈ Ω (2.2)holds. Also, note the following simple geometric facts.
Proposition 1.
For any x ∈ Ω there is σ ∈ R (Γ) such that τ σ ( x ) = 0 . Forany different x, x ′ ∈ Ω there is a σ ∈ R (Γ) such that τ σ ( x ) = τ σ ( x ′ ) (i.e.,the eikonals distinguish points of Ω ). The equality σ = { γ ∈ Γ | τ σ ( γ ) = 0 } holds. Copy ˜ΩAs functions on Ω, eikonals are determined by the Riemannian structure ofΩ. The converse is also true in the following sense.Assume that we are given with a topological space ˜Ω, which is home-omorphic to Ω (with the Riemann metric topology) via a homeomorphism η : Ω → ˜Ω; let ˜ τ σ := τ σ ◦ η − . Also, assume that η is unknown but we aregiven with the map R (Γ) ∋ σ ˜ τ σ ∈ C ( ˜Ω) . (2.3)Then one can endow ˜Ω with the Riemannian structure, which turns it intoa manifold isometric to Ω. Roughly speaking, the way is the following .For a fixed point p ∈ ˜Ω one can find its neighborhood ω ⊂ ˜Ω and thesets σ , . . . , σ n ∈ R (Γ) such that the functions x = ˜ τ σ ( · ) , . . . , x n = ˜ τ σ n ( · ) see [5] for detail φ : ω ∋ p
7→ { x k ( p ) } nk =1 ∈ R n . The coordinatesendow ω with tangent spaces. These spaces can be provided with the metrictensor ˜ g = η ∗ g : one can determine its components ˜ g ij from the equations˜ g ij ( x ) ∂ ˜ τ σ ◦ φ − ∂x i ( x ) ∂ ˜ τ σ ◦ φ − ∂x j ( x ) = 1 , x ∈ φ ( ω ) , σ ∈ R (Γ) (2.4)which are just (2.2) written in coordinates. Choosing here σ = σ i , we get˜ g ii = 1. Choosing (a finite number of) additional sets σ , we can determinethe functions ∂ ˜ τ σ ◦ φ − ∂x i and then find all other components ˜ g ij ( x ) by solvingthe system (2.4) with respect to them.So, although the homeomorphism η is unknown, we are able to endow ˜Ωwith the metric tensor ˜ g = η ∗ g , which turns it into a Riemannian manifold( ˜Ω , ˜ g ) isometric to (Ω , g ) by construction.Moreover, there is a natural way to identify the boundaries ˜Γ := ∂ ˜Ω andΓ = ∂ Ω. At first, we can select the boundary points in ˜Ω by˜Γ = [ σ ∈R (Γ) ˜ σ, where ˜ σ := { ˜ γ ∈ ˜Ω | ˜ τ σ (˜ γ ) = 0 } . Then we identify Γ ∋ γ ≡ ˜ γ ∈ ˜Γ if γ ∈ σ implies ˜ γ ∈ ˜ σ for all regular σ containing γ .As a result, we get the manifold ( ˜Ω , ˜ g ) isometric to (Ω , g ), these manifoldshaving the mutual boundary Γ. In what follows we refer to ( ˜Ω , ˜ g ) as acanonical copy of the original manifold Ω (shortly: the copy ˜Ω).The aforesaid is summarized as follows. Proposition 2.
A space ˜Ω along with the map (2 . determine the copy ˜Ω and, hence, determine Ω up to isometry of Riemannian manifolds. Introduce the space H := L (Ω) with the inner product( u, v ) H = Z Ω u ( x ) v ( x ) dx . Let A ⊂ Ω be a measurable subset, χ A ( · ) its indicator (a characteristicfunction). By Hh A i := { χ A y | y ∈ H}
6e denote the subspace of functions supported on A . The (orthogonal) pro-jection X A in H onto Hh A i multiplies functions by χ A , i.e., cuts off functionson A .Let B ( H ) be the normed algebra of bounded operators in H . With ascalar eikonal τ σ one associates an operator ˇ τ σ ∈ B ( H ), which acts in H by(ˇ τ σ y ) ( x ) := τ σ ( x ) y ( x ) , x ∈ Ωand is bounded since Ω is compact. Moreover, one has k ˇ τ σ k = max x ∈ Ω | τ σ ( x ) | = k τ σ k C (Ω) diam Ω . (2.5)With a slight abuse of terms, we also call ˇ τ σ an eikonal .Each eikonal is a self-adjoint positive operator, which is represented bythe Spectral Theorem in the well-known form. Proposition 3.
The representation ˇ τ σ = Z ∞ s dX sσ (2.6) is valid, where the projections X sσ := X Ω s [ σ ] cut off functions on the metricneighborhoods of σ . Note that the integration interval is in fact 0 s k ˇ τ σ k .The eikonals corresponding to different bases do commute. This followsfrom commutation of X sσ and X s ′ σ ′ for all σ, σ ′ ∈ R (Γ) and s, s ′ > Here we introduce an analog of ˇ τ σ relevant to electrodynamics. Now, let dim Ω = 3. Also, let Ω be orientable and endowed with a volume3-form dv . On such a manifold, the intrinsic operations of vector analysis ∧ (vector product), ∇ , div , curl, are well defined on smooth functions andvector fields (sections of the tangent bundle T Ω): see, e.g., [17].7 olenoidal spaces
The class of smooth fields ~C ∞ (Ω) is dense in the space ~ H of square-summablefields with the product ( a, b ) ~ H = Z Ω a ( x ) · b ( x ) dx , where · is the inner product in T Ω x . This space contains the (sub)spaces J := { y ∈ ~ H | div y = 0 in Ω } , C := { curl h ∈ ~ H | h, curl h ∈ ~ H } ⊂ J of solenoidal fields and curls. Note that the smooth classes
J ∩ ~C ∞ (Ω) and C ∩ ~C ∞ (Ω) are dense in J and C respectively.Recall the well-known decompositions ~ H = G ⊕ J = G ⊕ C ⊕ D , (2.7)where G := {∇ q | q ∈ H (Ω) } is the space of potential fields, D := { y ∈ J | curl h = 0 , ν ∧ y = 0 on Γ } is a finite-dimensional subspace of harmonicDirichlet fields [17].For an A ⊂ Ω we denote by ~ Hh A i := { χ A y | y ∈ ~ H} , J h A i := { y ∈ J | supp y ⊂ A } , Ch A i := { curl h | h ∈ ~C ∞ (Ω) , supp h ⊂ A } (the closure in ~ H ) the subspaces of fields supported in A . Eikonals ε σ Fix a σ ∈ R (Γ) and take A = Ω s [ σ ]. Let Y sσ be the projection in C onto thesubspace Ch Ω s [ σ ] i . Note that the action of Y sσ is not reduced to cutting offfields on Ω s [ σ ], it acts in more complicated way (see [3], [5]).By analogy with (2.6), define a solenoidal operator eikonal ε σ := Z ∞ s dY sσ , (2.8)which is an operator in C . We omit a simple proof of the following result.8 roposition 4. The eikonal ε σ is a bounded self-adjoint positive operator,the equalities k ε σ k = k τ σ k C (Ω) ( . ) = k ˇ τ σ k (2.9) being valid. An important fact is that, in contrast to the cutting off projections X sσ ,the projections Y sσ and Y s ′ σ ′ do not commute in general. As a consequence,the eikonals ε σ and ε σ ′ also do not commute .Multiplying a field h ∈ C by a bounded function ϕ , one takes the fieldout of the subspace of curls: ϕh ∈ ~ H but ϕh
6∈ C in general. However, amap h ϕh is a well defined bounded operator from C to ~ H . For instance,understanding ˇ τ σ as an operator, which multiplies vector fields by the scalareikonal τ σ , we have ˇ τ σ ∈ B ( C ; ~ H ).The following result is of crucial character for future application to inverseproblems. By K ( C ; ~ H ) ⊂ B ( C ; ~ H ) we denote the set of compact operators. Lemma 1.
For any σ ⊂ Γ the relation ε σ − ˇ τ σ ∈ K ( C ; ~ H ) holds. In the proof (see Appendix) we use the technique developed in [9].
We begin with minimal information about algebras: for detail see, e.g., [13],[14]. The abbreviations BA and CBA mean a Banach and commutativeBanach algebra respectively. A BA is a (complex or real) Banach space A equipped with themultiplication operation ab satisfying k ab k ≤ k a k k b k a, b ∈ A . We dealwith algebras with the unit e ∈ A : ea = ae = a .A BA A is called commutative if ab = ba for all a, b ∈ A . Example : thealgebra C ( X ) of continuous functions on a topological space X with the norm k a k = sup X | a ( · ) | . The subalgebras of C ( X ) are called function algebras.A CBA is said to be uniform if k a k = k a k holds. All function algebrasare uniform. Let A ′ be the space of linear continuous functionals on a CBA A .A functional δ ∈ A ′ is called multiplicative if δ ( ab ) = δ ( a ) δ ( b ). Example : a9irac measure δ x ∈ C ′ ( X ) : δ x ( a ) = a ( x ) ( x ∈ X ). Each multiplicativefunctional is of the norm 1.The set of multiplicative functionals endowed with ∗ -weak topology (in A ′ ) is called a spectrum of A and denoted by b A . A spectrum is a compactHausdorff space. The
Gelfand transform acts from a CBA A to C ( b A ) by the rule G : a a ( · ) , a ( δ ) := δ ( a ) , δ ∈ b A . It represents A as a function algebra. Thepassage from A to G A ⊂ C ( b A ) is referred to as a geometrization of A . Theorem 1. (I.M.Gelfand) If A is a uniform CBA, then G is an isometricisomorphism from A onto G A , i.e., G ( αa + βb + cd ) = αGa + βGb + Gc Gd and k Ga k C ( b A ) = k a k A holds for all a, b, c, d ∈ A and numbers α, β . If two CBA A and B are isometrically isomorphic (we write A isom = B )via an isometry j , then the dual isometry j ∗ : B ′ → A ′ provides a homeo-morphism of their spectra: j ∗ b B = b A . Also, one has G A isom = G B via the map j ♯ : Ga ( Ga ) ◦ j ∗ . Let A ( X ) ⊂ C ( X ) be a closed function algebra. For each x ∈ X , theDirac measure δ x belongs to \ A ( X ). Therefore, the map x δ x providesa canonical embedding X ⊂ \ A ( X ).If X is a compact Hausdorff space, then the Dirac measures exhaustthe spectrum of C ( X ), whereas the map x δ x provides a canonicalhomeomorphism from X onto \ C ( X ) (we write X hom = \ C ( X )). Also, one has C ( X ) isom = GC ( X ).The trick, which is used in inverse problems for reconstruction of mani-folds, is the following. Assume that we are given with an ”abstract” CBA A , which is known to be isometrically isomorphic to C ( X ), but neither the(compact Hausdorff) space X nor the isometry map is given. Then, by de-termining the spectrum b A , we in fact recover the space X up to a homeomor-phism: X hom = \ C ( X ) hom = b A , whereas C ( X ) isom = GC ( X ) isom = G A does hold.Thus, A provides a homeomorphic copy b A of the space X and a concreteisometric copy C ( b A ) of the algebra C ( X ). A C ∗ -algebra is a BA endowed with an involution ( ∗ ) satisfying ( αa + βb + cd ) ∗ = ¯ αa ∗ + ¯ βb ∗ + d ∗ c ∗ and k a ∗ a k = k a k for all elements a, b, c, d andnumbers α, β . In the real case, we have just ¯ α = α . Example : the algebra10 ( H ) of bounded operators in a Hilbert space H with the operator normand conjugation. Let I be a norm-closed two-side ideal in a C*-algebra A . Then a ∼ b ⇔ a − b ∈ I is an equivalence. The factor A / I is endowed with a C*-structure via the projection π : A → A / I (element a equivalence classof a ). Namely, one sets k πa k := inf {k b k A | b ∈ πa } , απa + βπb + πc πd := π ( αa + βb + cd ) , ( πa ) ∗ := π ( a ∗ ) for elements a, b, c, d ∈ A and numbers α, β .Thus, π is a homomorphism of C*-algebras. T Now let X be our Riemannian manifold Ω, which is definitely a compactHausdorff space. Let C (Ω) be the CBA of real continuous functions on Ω.The eikonals τ σ generate C (Ω) in the following sense. For a Banachalgebra A and a subset S ⊂ A , by ∨ S we denote the minimal norm-closedsubalgebra of A , which contains S . The following fact is a straightforwardconsequence of the separating properties of eikonals (Proposition 1) and theStone-Weierstrass theorem [14]. Proposition 5.
The equality ∨{ τ σ | σ ∈ R (Γ) } = C (Ω) is valid. Recall that H = L (Ω), B ( H ) is the bounded operator algebra, ˇ τ σ ∈ B ( H ) is the multiplication by τ σ (see sec 2.2). Introduce the (sub)algebra T := ∨{ ˇ τ σ | σ ∈ R (Γ) } ⊂ B ( H ) (3.1)generated by scalar operator eikonals. As easily follows from (2.5) and Propo-sition 5, the map C (Ω) ∋ τ σ ˇ τ σ ∈ T , which connects the generators, isextended to an isometric isomorphism of CBA C (Ω) and T . With regard toitems
4, 5 of sec 3.1, the isometry impliesΩ hom = [ C (Ω) hom = b T . (3.2) On reconstruction
Here we prepare a fragment of the procedure, which will be used for solvinginverse problems.Assume that we are given with a Hilbert space ˜ H = U H , where U is aunitary operator. Also assume that we know the map R (Γ) × [0 , T ] ∋ { σ, s } 7→ ˜ X sσ ∈ B ( ˜ H ) ( T > diam Ω) , (3.3)11here ˜ X sσ := U X sσ U ∗ , but the operator U : H → ˜ H is unknown . Show thatthis map determines the manifold Ω up to isometry. Indeed,1. using the map, one can construct the operators τ ′ σ := Z T s d ˜ X sσ = Z T s d [ U X sσ U ∗ ] ( . ) = U ˇ τ σ U ∗
2. determine the algebra ˜ T = ∨{ τ ′ σ | σ ∈ R (Γ) } ⊂ B ( ˜ H ) , which is iso-metric to T ⊂ B ( H ) (via the unknown U )3. applying the Gelfand transform to ˜ T , find its spectrum b ˜ T =: ˜Ω and thefunctions ˜ τ σ := Gτ ′ σ on ˜Ω.Since ˜ T isom = T , one has ˜Ω := b ˜ T hom = b T hom = Ω (see (3.2)). Hence, we geta homeomorphic copy ˜Ω of the original Ω along with the images ˜ τ σ of theoriginal eikonals τ σ on Ω . Thus, we have a version of the map (2.3), whichdetermines the copy ˜Ω (see Proposition 2).Summarizing, we arrive at the following assertion. Proposition 6.
The map (3 . determines the copy ˜Ω and, hence, deter-mines Ω up to isometry of Riemannian manifolds. Moreover, the procedure 1.– 3. provides the copy ˜Ω. E Recall that the eikonals ε σ are introduced on a 3d-manifold Ω by (2.8).An operator (sub)algebra E := ∨{ ε σ | σ ∈ R (Γ) } ⊂ B ( C ) (3.4)is a ”solenoidal” analog of the algebra T defined by (3.1). It is a real algebragenerated by self-adjoint operators. As such, E is a C*-algebra. In contrast in other words, we are given with a representation of the projection family { X sσ } σ ∈R (Γ) in a space ˜ H by construction, ˜ τ σ turns out to be a pull-back function of τ σ via the homeomorphism˜Ω → Ω T , the algebra E is not commutative (see the remark below Proposition4). However, this non-commutativity is weak in the following sense.Let K ⊂ B ( C ) be the ideal of compact operators. Denote K [ E ] := K ∩ E and ˙ E := E / K [ E ]; let π : B ( C ) → B ( C ) / K be the canonical projection.By (3.4), the latter factor-algebra is generated by the equivalence classes ofeikonals: ˙ E := ∨{ πε σ | σ ∈ R (Γ) } . Recall that the eikonals τ σ generate the algebra C (Ω): see Proposition 5. Theorem 2. ˙ E is a commutative C*-algebra. The map C (Ω) ∋ τ σ πε σ ∈ ˙ E ( σ ∈ R (Γ)) , which relates the generators, can be extended to an isometric isomorphismfrom C (Ω) onto ˙ E .Proof. Define a map ˙ π : C (Ω) → B ( C ) / K in the following way. Let Y be the projection on C acting in ~ H . With afunction f ∈ C (Ω) we associate an operator Y [ f ] ∈ B ( C ) acting by Y [ f ] y := Y ( f y ) , y ∈ C . Now, define ˙ π ( f ) := π ( Y [ f ]) . For f ∈ C (Ω) we denote by ˇ f the operator in ~ H , which multiplies fieldsby f . The following two Lemmas are proved in Appendix. Lemma 2.
For any f ∈ C (Ω) we have ˇ f − Y [ f ] ∈ K ( C ; ~ H ) . Lemma 3.
The mapping ˙ π is an injective homomorphism of C*-algebras. To prove Theorem 2 it suffices to show that the map ˙ π is an extensionof the map τ σ πε σ . Toward this end, let us show that ε σ − Y [ τ σ ] ∈ K . Indeed, we have ε σ − Y [ τ σ ] = ε σ − ˇ τ σ + ˇ τ σ − Y [ τ σ ]and, due to Lemmas 1 and 2, there is a sum of two compact operators from K ( C ; ~ H ) in the right hand side. Now Theorem 2 follows from Lemma 3 andthe fact that algebra ˙ E is generated by elements πε σ .13ith regard to items
4, 5 of sec 3.1, the relation C (Ω) isom = ˙ E establishedby Theorem 2 implies Ω hom = [ C (Ω) hom = b ˙ E . (3.5) Remark
Examples, in which factorization eliminates noncommutativity, arewell known. For instance, let X be a compact smooth manifold (withoutboundary) and let A ⊂ B ( L ( X )) be a C*-algebra generated by a certainclass of pseudo-differential operators of order 0. Then the factor-algebra A / K is commutative and isomorphic to the algebra of continuous functions on thecosphere bundle of X (see [15]). On reconstruction
Here we provide an analog of the procedure described in sec 3.2. This analogis relevant to inverse problems of electrodynamics. Recall that Y sσ is theprojection in C onto the subspace Ch Ω s [ σ ] i .Assume that we are given with a Hilbert space ˜ C = U C , where U is aunitary operator. Also assume that we know the map R (Γ) × [0 , T ] ∋ { σ, s } 7→ ˜ Y sσ ∈ B ( ˜ C ) ( T > diam Ω) , (3.6)where ˜ Y sσ := U Y sσ U ∗ , but the operator U : C → ˜ C is unknown . Show thatthis map determines the manifold Ω up to isometry. Indeed,1. using the map, one can construct the operators ε ′ σ := Z T s d ˜ Y sσ = Z T s d [ U Y sσ U ∗ ] ( . ) = U ˇ ε σ U ∗
2. determine the algebra E ′ = ∨{ ε ′ σ | σ ∈ R (Γ) } ⊂ B ( ˜ C ) , which is iso-metric to E ⊂ B ( C ) (via unknown U )3. construct the factor-algebra ˜ E := E ′ / K [ E ′ ] over the compact operatorideal in E ′ . By construction, one has ˜ E isom = E / K [ E ] =: ˙ E .4. applying the Gelfand transform to ˜ E , find its spectrum b ˜ E =: ˜Ω and thefunctions ˜ τ σ := Gπε ′ σ on ˜Ω. 14ince ˜ E isom = ˙ E , one has ˜Ω := b ˜ E hom = b ˙ E hom = Ω(see (3.5)). So, we get a homeomorphic copy ˜Ω of the original Ω along withthe images ˜ τ σ of the original eikonals τ σ on Ω. Thus, we have a version ofthe map (2.3). This map determines the Riemannian structure on ˜Ω, whichturns it into an isometric copy of Ω (see Proposition 2).Summarizing, we arrive at the following. Proposition 7.
The map (3 . determines the copy ˜Ω and, hence, deter-mines Ω up to isometry of Riemannian manifolds. Moreover, the procedure 1.– 4. enables one to construct the copy ˜Ω.This procedure differs from its scalar analog by one additional step that isfactorization.
With the manifold Ω one associates a dynamical system α T of the form u tt − ∆ u = 0 in (Ω \ Γ) × (0 , T ) (4.1) u | t =0 = u t | t =0 = 0 in Ω (4.2) u = f on Γ × [0 , T ] , (4.3)where ∆ is the (scalar) Beltrami–Laplace operator, t = T > f is a boundary control , u = u f ( x, t ) is a solution. For controls of the smoothclass M T := { f ∈ C ∞ (Γ × [0 , T ]) | supp f ⊂ Γ × (0 , T ] } problem (4.1)–(4.3) has a unique classical (smooth) solution u f . Note thatthe condition on supp f means that f vanishes near t = 0.¿From the physical viewpoint, u f can be interpreted as an acoustical wave , which is initiated by the boundary sound source f and propagates intoa domain Ω filled with an inhomogeneous medium.15 ttributes • The space of controls F T := L (Γ × [0 , T ]) is said to be an outer space ofthe system α T . The smooth class M T is dense in F T .The outer space contains the subspaces F T,sσ := { f ∈ F T | supp f ⊂ σ × [ T − s, T ] } , σ ∈ R (Γ) . Such a subspace consists of controls, which are located on σ and switched onwith delay T − s (the value s is an action time). • An inner space of the system is H = L (Ω). The waves u f ( · , t ) are timedependent elements of H . • In the system α T , the input state correspondence is realized by a controloperator W T : F T → H , Dom W T = M T W T f := u f ( · , T ) . A specifics of the system governed by the scalar wave equation (4.1) is that W T is a bounded operator. Therefore one can extend it from M T onto F T by continuity that we assume to be done. • The input output map is represented by a response operator R T : F T →F T , Dom R T = M T , R T f := ∂u f ∂ν (cid:12)(cid:12)(cid:12)(cid:12) Γ × [0 ,T ] , where ν = ν ( γ ) is an outward normal at γ ∈ Γ.The following evident fact was already mentioned in Introduction.
Proposition 8.
If two Riemannian manifolds have the mutual boundaryand are isometric (the isometry being identity at the boundary), then their(acoustical) response operators coincide. In particular, for the manifold Ω and its copy ˜Ω one has R T = ˜ R T for any T > . • A connecting operator C T : F T → F T is defined by C T := ( W T ) ∗ W T . (4.4)By the definition, we have( C T f, g ) F T = ( W T f, W T g ) H = (cid:0) u f ( · , T ) , u g ( · , T ) (cid:1) H , C T connects the Hilbert metrics of the outer and inner spaces. A sig-nificant fact is that the connecting operator is determined by the responseoperator of the system α T through an explicit formula C T = 12 ( S T ) ∗ R T J T S T , (4.5)where the map S T : F T → F T extends the controls from Γ × [0 , T ] toΓ × [0 , T ] as odd functions (of time t ) with respect to t = T ; J T : F T → F T is an integration: ( J T f )( · , t ) = R t f ( · , s ) ds (see [1], [3]). Controllability
The set U sσ := { u f ( · , s ) | f ∈ F Tσ } is said to be reachable (from σ , at themoment t = s ).The operator ∆, which governs the evolution of the system α T , does notdepend on time. By this, a time delay of controls implies the same delay ofthe waves. As a result, one has U sσ = W T F T,sσ , s T .
Problem (4.1)–(4.3) is hyperbolic and the finiteness of domains of influ-ence does hold for its solutions: for the delayed controls one hassupp u f ( · , T ) ⊂ Ω s [ σ ] , f ∈ F T,sσ . (4.6)The latter means that in the system α T the waves propagate with the unitvelocity. As a result, the embedding U sσ ⊂ Hh Ω s [ σ ] i is valid. The charac-ter of this embedding is of principal importance: it turns out to be dense .The following result is based upon the fundamental Holmgren–John–Tataruuniqueness theorem (see [1], [3] for detail). Proposition 9.
For any s > and σ ∈ R (Γ) , the relation U sσ = Hh Ω s [ σ ] i is valid (the closure in H ). In particular, for s = T > diam Ω one has U Tσ = H . In control theory this property is referred to as a local approximate bound-ary controllability of the system α T . It shows that the reachable sets are richenough: any function supported in the neighborhood Ω s [ σ ] can be approx-imated (in H -metric) by a wave u f ( · , T ) by means of the proper choice ofthe control f ∈ F T,sσ . 17y P sσ we denote the projection in H onto the reachable subspace U sσ andcall it a wave projection . Recall that X sσ is the projection in H onto Hh Ω s [ σ ] i ,which cuts off functions onto the neighborhood Ω s [ σ ]. As a consequence ofthe Proposition 9 we obtain P sσ = X sσ , s > , σ ∈ R (Γ) . (4.7) Setup
A dynamical inverse problem (IP) for the system (4.1)–(4.3) is set up asfollows: given for a fixed
T > diam Ω the response operator R T , to recover the mani-fold Ω.A physical meaning of the condition
T > diam Ω is that the waves u f , whichprospect the manifold from the parts σ of its boundary, need big enough timeto fill the whole Ω: see (4.6) and (2.1).As was clarified in Introduction, to recover Ω means to construct (viagiven R T ) a Riemannian manifold, which has the same boundary Γ, andpossesses the response operator, which is equal to R T . Speaking in advance,it will be shown that R T determines the copy ˜Ω. Thus, ˜Ω provides thesolution to the IP. Model
As an operator connecting two Hilbert spaces, the control operator W T : F T → H can be represented in the form of a polar decomposition W T = Φ T | W T | , where | W T | := (cid:2)(cid:0) W T (cid:1) ∗ W T (cid:3) ( . ) = (cid:0) C T (cid:1) and Φ T : | W T | f W T f is an isometry from Ran | W T | ⊂ F T onto Ran W T ⊂H (see, e.g., [7]). In what follows we assume that Φ T is extended by conti-nuity to an isometry from Ran | W T | onto Ran W T .Recall that U sσ := W T F T,sσ are the reachable sets of the system α T and P sσ is the projection in H onto U sσ . 18et us say the (sub)space ˜ H := Ran | W T | ⊂ F T to be a model innerspace , ˜ U sσ := | W T |F T,sσ ⊂ ˜ H a model reachable set . By ˜ P sσ we denote theprojection in ˜ H onto ˜ U sσ and call it a model wave projection .The model and original objects are related through the isometry Φ T . Inparticular, the definitions imply Φ T ˜ P sσ = P sσ Φ T .Now let T > diam Ω, so that Ω T [ σ ] = Ω holds for any σ . By Proposition9, one has Ran W T = H . By this, the isometry Φ T turns out to be a unitaryoperator from ˜ H onto H . Its inverse U := (Φ T ) ∗ maps H onto ˜ H isometricallyand U P sσ = ˜ P sσ U holds.Let ˜ X sσ := U X sσ U ∗ be the image (in ˜ H ) of the cutting off projection. Theproperty (4.7) implies˜ P sσ = ˜ X sσ , s > , σ ∈ R (Γ) . (4.8) Solving IP
It suffices to show that the operator R T determines the copy ˜Ω. The proce-dure is the following.1. Find the connecting operator by (4.5). Determine the operator | W T | = (cid:0) C T (cid:1) and the subspace ˜ H = Ran | W T | ⊂ F T .2. Fix a σ ∈ R (Γ) and s ∈ (0 , T ]. In ˜ H recover the model reachableset ˜ U sσ = | W T |F T,sσ ⊂ ˜ H and determine the corresponding projection˜ P sσ . By (4.8), we get the projection ˜ X sσ . Thus, the map (3.3) is at ourdisposal.3. By Proposition 6, this map determines the copy ˜Ω. Its response oper-ator ˜ R T coincides with the given R T : see Proposition 8.The acoustical IP is solved. Here Ω is a smooth compact oriented Riemannian 3d-manifold.19ropagation of electromagnetic waves in a curved space is described bythe dynamical Maxwell system α T M e t = curl h, h t = − curl e in (Ω \ Γ) × (0 , T ) (4.9) e | t =0 = 0 , h | t =0 = 0 in Ω (4.10) e θ = f on Γ × [0 , T ] , (4.11)where e θ := e − e · ν ν is a tangent component of e at the boundary, f is atime-dependent tangent field on Γ ( boundary control ), e and h are the electricand magnetic components of the solution. For controls of the smooth class M T := n f ∈ ~C ∞ (Γ × [0 , T ]) (cid:12)(cid:12) ν · f = 0 , supp f ⊂ Γ × (0 , T ] o , problem (4.9)-(4.11) has a unique classical smooth solution { e f ( x, t ) , h f ( x, t ) } .Note that the condition on supp f means that f vanishes near t = 0.Since a divergence is an integral of motion of the Maxwell system, onehas div e f ( · , t ) = 0 , div h f ( · , t ) = 0 , t > . Attributes • An outer space of the system α T M is the space F T := n f ∈ ~L (Γ × [0 , T ]) (cid:12)(cid:12) ν · f = 0 o . The smooth class M T is dense in F T .The outer space contains the subspaces F T,sσ := (cid:8) f ∈ F T (cid:12)(cid:12) supp f ⊂ σ × [ T − s, T ] (cid:9) , σ ∈ R (Γ)of controls, which are located on σ and switched on with delay T − s (thevalue s is an action time). • An inner space of the system is the space C ⊕ C . By (4.9), the solutions { e f ( · , t ) , h f ( · , t ) } are time dependent elements of this space. Also, we selectits electric part C ⊕ { } ∋ e f ( · , t ). • The input state correspondence is realized by a control operator W T M : F T → C ⊕ C , Dom W T M = M T , W T M f := { e f ( · , T ) , h f ( · , T ) } . Its electricpart is W T : F T → C , W T : f e f ( · , T ) .
20n contrast to the acoustical (scalar) system, W T M and W T are unbounded(but closable) operators.A reason to select an electric part of the system α T M is that it is the electriccomponent, which is controlled at the boundary: see (4.11). By this, e f and h f are not quite independent. Moreover, for T < inf { r > | Ω r [Γ] = Ω } theoperator W T is injective and, hence, e f ( · , T ) determines h f ( · , T ) [3], [5]. • The input output map of the system α T M is represented by a responseoperator R T : F T → F T , Dom R T = M T ,R T f := ν ∧ h f (cid:12)(cid:12) Γ × [0 ,T ] . The following fact is quite evident.
Proposition 10.
If two Riemannian manifolds have the mutual boundaryand are isometric (the isometry being identity at the boundary), then theirMaxwell response operators coincide. In particular, for the manifold Ω andits canonical copy ˜Ω one has R T = ˜ R T for any T > . • An electric connecting operator C T : F T → F T is introduced via a con-necting form c T , Dom c T = M T × M T , c T [ f, g ] := (cid:0) e f ( · , T ) , e g ( · , T ) (cid:1) C = (cid:0) W T f, W T g (cid:1) C . It is a Hermitian nonnegative bilinear form. As such, it is closable, the closure¯ c T being defined on N T × N T , where N T is a lineal in F T , N T ⊃ M T . Theform ¯ c T determines a unique self-adjoint operator C T by the relation( C T f, g ) F T = ¯ c T [ f, g ] , f ∈ Dom C T , g ∈ N T (see, e.g., [7]). In fact, to close c T is to close W T , and one has N T =Dom ¯ W T = Dom ( C T ) . Hence, the knowledge of c T enables one to extend W T from M T to N T . In what follows this extension (closure) is assumed tobe done and denoted by the same symbol W T . The images W T f for f ∈ N T are regarded as the generalized solutions e f ( · , T ).As a result, one has the relations¯ c T [ f, g ] = (cid:16) ( C T ) f, ( C T ) g (cid:17) F T = (cid:0) W T f, W T g (cid:1) C , f, g ∈ N T . (4.12)A key fact is that the connecting form is determined by the response operatorof the system α T M through an explicit formula c T [ f, g ] = (cid:0) − ( S T ) ∗ R T J T S T f, g (cid:1) F T , f, g ∈ M T , (4.13)21here the map S T : F T → F T extends the controls from Γ × [0 , T ] toΓ × [0 , T ] as odd functions (of time t ) with respect to t = T ; J T : F T → F T is an integration: ( J T f )( · , t ) = R t f ( · , s ) ds (see [3]).Resuming the aforesaid, we can claim that R T determines the operator( C T ) by the scheme R T ( . ) ⇒ c T ⇒ ¯ c T ⇒ C T ⇒ ( C T ) . (4.14) Controllability
The set E sσ := { e f ( · , s ) | f ∈ F Tσ ∩ M T } is said to be reachable (from σ , atthe moment t = s ).The operators curl , which govern the evolution of the system α T M , doesnot depend on time. By this, a time delay of controls implies the same delayof the waves. As a result, one can represent E sσ = W T (cid:2) F T,sσ ∩ M T (cid:3) . The Maxwell system (4.9)–(4.11) obeys the finiteness of domains of in-fluence principle: for the delayed controls one hassupp e f ( · , T ) ⊂ Ω s [ σ ] , f ∈ (cid:2) F T,sσ ∩ M T (cid:3) . (4.15)The latter means that electromagnetic waves propagate with the unit veloc-ity. As a consequence, the embedding E sσ ⊂ Ch Ω s [ σ ] i is valid. Moreover,this embedding is dense . This fact is derived from a vectorial version of theHolmgren–John–Tataru uniqueness theorem (see [3] for detail). Proposition 11.
For any s > and σ ∈ R (Γ) , the relation E sσ = Ch Ω s [ σ ] i isvalid (the closure in C ). In particular, for s = T > diam Ω one has E Tσ = C . This property is interpreted as a local approximate boundary controllabil-ity of the electric subsystem of α T M .By E sσ we denote the projection in C onto the reachable subspace E sσ andcall it a wave projection . Recall that Y sσ is the projection in C onto Ch Ω s [ σ ] i .As a consequence of the Proposition 11 we obtain E sσ = Y sσ , s > , σ ∈ R (Γ) . (4.16)22 .4 IP of electrodynamics Setup
A dynamical inverse problem (IP) for the system (4.9)–(4.11) is set up asfollows: given for a fixed
T > diam Ω the response operator R T , to recover the mani-fold Ω.A physical meaning of the condition
T > diam Ω is the same as in theacoustical case: the electromagnetic waves need big enough time to prospectthe whole Ω: see (4.15) and (2.1).As before, to recover
Ω means to construct (via given R T ) a Riemannianmanifold, which has the same boundary Γ, and possesses the response oper-ator, which is equal to R T . As well as in the scalar case, we will show that R T determines the copy ˜Ω. Thus, ˜Ω will provide the solution to the IP. Model
Representing the (closed) control operator W T : F T → C in the polar decom-position form, one has W T = Ψ T | W T | , where | W T | := (cid:2)(cid:0) W T (cid:1) ∗ W T (cid:3) andΨ T : | W T | f W T f is an isometry from Ran | W T | ⊂ F T onto Ran W T ⊂ C [7]. In what follows Ψ T is assumed to be extended by continuity to an isom-etry from Ran | W T | onto Ran W T . Also note that (4.12) implies | W T | =( C T ) .Recall that E sσ := W T [ F T,sσ ∩ M T ] is an electric reachable set and E sσ isthe (wave) projection in C onto E sσ .Let us say the (sub)space ˜ C := Ran | W T | ⊂ F T to be a model inner space ,˜ E sσ := | W T | (cid:2) F T,sσ ∩ M T (cid:3) ⊂ ˜ C the model reachable sets . By ˜ E sσ we denote theprojection in ˜ C onto ˜ E sσ and call it a model wave projection .The model and original objects are related through the isometry Ψ T . Inparticular, the definitions imply Ψ T ˜ E sσ = E sσ Ψ T .Now, let T > diam Ω. By Proposition 11, one has Ran W T = C . Thereforethe isometry Ψ T turns out to be a unitary operator from ˜ C onto C . Its inverse U := (Ψ T ) ∗ maps C onto ˜ C isometrically and U E sσ = ˜ E sσ U holds.Let ˜ Y sσ := U Y sσ U ∗ . The property (4.16) implies˜ E sσ = ˜ Y sσ , s > , σ ∈ R (Γ) . (4.17)23 olving IP Let us show that the operator R T determines the copy ˜Ω.1. Find the connecting form c T by (4.13). Determine the model controloperator | W T | = (cid:0) C T (cid:1) (see (4.14)) and the model inner space ˜ C =Ran | W T | ⊂ F T .2. Fix a σ ∈ R (Γ) and s ∈ (0 , T ). In ˜ C recover the model reachable set˜ E sσ = | W T | (cid:2) F T,sσ ∩ M T (cid:3) ⊂ ˜ C and determine the corresponding projec-tion ˜ E sσ . By (4.17), we get the projection ˜ Y sσ . Thus, the map (3.6) isat our disposal.3. By Proposition 7, this map determines the copy ˜Ω. Its Maxwell re-sponse operator ˜ R T coincides with the given R T (see Proposition 10).The IP of electrodynamics is solved. • In this paper, the condition
T > diam Ω is imposed for the sake of sim-plicity. It provides the embedding ˇ τ σ C (Ω) ⊂ C (Ω), which is convenient justby technical reasons. However, there is a time-optimal setup of the recon-struction problem, which takes into account a local character of dependenceof the acoustical and Maxwell response operators on a near-boundary partof the manifold. Namely, by the finiteness of the domain of influence, foran arbitrary fixed T > R T is determined by the submanifoldΩ T [Γ] (does not depend on the part Ω \ Ω T [Γ]). Therefore, the natural setupis: given for a fixed T > the operator R T , to recover Ω T [Γ]. In such astronger form the problem is solved in [3] and [6]. • In reconstruction via a spectral triple {A , H , D} (see [8], [16]), the algebraprovides a topological space (that is b A ), whereas the operator D encodesa Riemannian metric on b A . The metric is recovered (via D ) by means ofthe Connes distance formula . In our scheme, the object responsible for themetric is a selected family of generators of the algebra (that is the eikonals). • Dealing with the reconstruction problem for a graph, one can introducethe straightforward analog of the eikonal algebra T . However, this algebraturns out to be noncommutative. By this, we have to deal with its Jacobson pectrum b T , which is the topologized set of the primitive ideals of T [13]. Asthe known examples show, its structure is related with geometry of the graphbut the relation is of rather implicit character. This challenging problemis open yet. An intriguing fact is that in some examples the space b T isnon-Hausdorff. It contains ”clusters”, which are the groups of nonseparablepoints. Presumably, the clusters of b T are related with interior vertices of thegraph. Here we give proof of Lemmas 1, 2, 3.The standard operations on vector fields on the manifold ∇ , div , curl areunderstood in the generalized sense. Here are standard formulas of vectoranalysis: div ( ϕu ) = ∇ ϕ · u + ϕ div u, (5.1)div ( u ∧ v ) = curl u · v − u · curl v, (5.2)curl ( ϕu ) = ∇ ϕ ∧ u + ϕ curl u. (5.3)In (5.1) and (5.3) a function ϕ is Lipschitz; a field u is locally integrable andits divergence is also locally integrable. In (5.2) we may suppose that u or v is Lipschitz, and the other field is locally integrable and has locally integrablecurl . Let the field z ∈ ~ H satisfy curl z ∈ ~ H . Following [12], we say that the field z satisfies the condition z θ | Γ = 0 , (5.4)if for any field v ∈ ~ H , such that curl v ∈ ~ H , we have( z, curl v ) Ω = (curl z, v ) Ω . Here and further in this section ( · , · ) U and k · k U means the inner productand the norm in L ( U ) or ~L ( U ). It can be shown, that due to smoothnessof the boundary Γ it suffices to check this condition only for v ∈ ~C ∞ (Ω).25ntroduce the space F := { u ∈ ~ H : div u ∈ L (Ω) , curl u ∈ ~ H , u θ | Γ = 0 } with the norm k u k F := k u k + k div u k + k curl u k . The following result is valid for an Ω ⊂ R (see [12], section 8.4) and canbe easily generalized on a smooth manifold. Theorem 3.
The embedding of the space F to ~ H is compact. Actually, the stronger fact holds true: the space F coincides with vectorSobolev space ~H (Ω), which is compactly embedded to ~ H . However, Theo-rem 3 will suffice for our purposes. Theorem 3 is used in spectral analysis ofthe Maxwell operator on compact manifolds (see, e.g., [10]).Let us outline the scheme of the proof of Lemma 1. We obtain estimatesfor L -norms of curl and divergence of the difference ˇ τ σ u − ε σ u by L -norm of u ∈ C (inequalities (5.13), (5.15)), and establish the boundary condition (5.4)on Γ for this difference. This means that the field ˇ τ σ u − ε σ u belongs to F withthe corresponding norm estimate, which implies that the operator ˇ τ σ − ε σ restricted to C is compact (by compactness of the embedding F ⊂ ~ H ).In what follows we consider X sσ as the projections in ~ H , which cut offfields on Ω s [ σ ].We will use the following relations, which are valid for any T > Z [0 ,T ] s dX sσ = T X Tσ − Z [0 ,T ] X sσ ds, Z [0 ,T ] s dY sσ = T Y Tσ − Z [0 ,T ] Y sσ ds. Along with (2.6) this implies that for
T > diam Ω we have( ε σ − ˇ τ σ ) y = (cid:18)Z [0 ,T ] ( X sσ − Y sσ ) ds (cid:19) y, y ∈ C . (5.5)To prove Lemma 1 we need to establish a compactness of the operator, whichacts from C to ~ H by K σ := Z T ( X ξσ − Y ξσ ) dξ T > diam Ω). Define a family of operatorsacting from C to ~ H by K sσ := Z s ( X ξσ − Y ξσ ) dξ, s < ∞ . One can easily check the following relation (cid:18)Z s X ξσ dξ y (cid:19) ( x ) = max { s − τ σ ( x ) , } y ( x ) , x ∈ Ω . (5.6) Lemma 4.
Choose σ ⊂ Γ and s > . Let a field β ∈ ~ Hh Ω s [ σ ] i be smoothin Ω s [ σ ] (in particular, smooth on the boundary Ω s [ σ ] ∩ Γ ) and orthogonal to Ch Ω s [ σ ] i . Then for any z ∈ ~C ∞ (Ω) one has ( β, K sσ curl z ) Ω s [ σ ] = ( β, ∇ τ σ ∧ z ) Ω s [ σ ] . Proof.
Let 0 < s ′ < s . By the absolute continuity of Lebesgue integral wehave ( β, K s ′ σ curl z ) Ω s ′ [ σ ] → ( β, K sσ curl z ) Ω s [ σ ] , s ′ → s − . (5.7)As is evident, β is orthogonal to Ch Ω ξ [ σ ] i for ξ s ; therefore( β, K s ′ σ curl z ) Ω s ′ [ σ ] = Z s ′ dξ ( β, ( X ξσ − Y ξσ ) curl z ) Ω ξ [ σ ] = Z s ′ dξ ( β, X ξσ curl z ) Ω ξ [ σ ] ( . ) = ( β, ( s ′ − τ σ ) curl z ) Ω s ′ [ σ ] =(( s ′ − τ σ ) β, curl z ) Ω s ′ [ σ ] . Define a Lipschitz function h in Ω as follows h ( x ) := max { s ′ − τ σ ( x ) , } We have (( s ′ − τ σ ) β, curl z ) Ω s ′ [ σ ] = ( hβ, curl z ) Ω (5.8)(the field hβ is defined in Ω since h vanishes outside of Ω s ′ [ σ ] ⊂ Ω s [ σ ]). Thefield hβ is Lipschitz, as function h is Lipschitz, and the field β is smoothin the neighborhood of supp h , so we can apply a formula of integration byparts to the right hand side in (5.8). Orthogonality of β to Ch Ω s [ σ ] i impliescurl β | Ω s [ σ ] = 0 , β θ | Ω s [ σ ] ∩ Γ = 0 . (5.9)27ue to the second equality we have ( hβ ) θ | Γ = 0. So the integral over Γin integration by parts vanishes. Applying the first equality in (5.9) andformula (5.3), we obtain:( hβ, curl z ) Ω = (curl ( hβ ) , z ) Ω = ( ∇ h ∧ β, z ) Ω = (( −∇ τ σ ) ∧ β, z ) Ω s ′ [ σ ] =( β, ∇ τ σ ∧ z ) Ω s ′ [ σ ] . The latter term tends to ( β, ∇ τ σ ∧ z ) Ω s [ σ ] as s ′ → s . Taking into account(5.7), we obtain the required equality.Note that Lemma 4 holds true if Ω s [ σ ] = Ω. Lemma 5.
Let σ ⊂ Γ . For a field z ∈ ~C ∞ (Ω) we have ( K σ curl z, K σ curl z ) Ω = 2 ( K σ curl z, ∇ τ σ ∧ z ) Ω . (5.10) Proof.
We have( K σ curl z, K σ curl z ) Ω = Z T ds (( X sσ − Y sσ ) curl z, K σ curl z ) Ω = Z T ds Z T dξ (( X sσ − Y sσ ) curl z, ( X ξσ − Y ξσ ) curl z ) Ω =2 Z T ds Z s dξ (( X sσ − Y sσ ) curl z, ( X ξσ − Y ξσ ) curl z ) Ω =2 Z T ds (( X sσ − Y sσ ) curl z, K sσ curl z ) Ω s [ σ ] . (5.11)As is clear, the field β := ( X sσ − Y sσ ) curl z is orthogonal to Ch Ω s [ σ ] i . Moreover,it is smooth in Ω s [ σ ], since it is solenoidal and satisfies (5.9). So we can applyLemma 4 to the integrand:(( X sσ − Y sσ ) curl z, K sσ curl z ) Ω s [ σ ] = (( X sσ − Y sσ ) curl z, ∇ τ σ ∧ z ) Ω s [ σ ] . Substituting this to (5.11), we obtain( K σ curl z, K σ curl z ) Ω = 2 Z T ds (( X sσ − Y sσ ) curl z, ∇ τ σ ∧ z ) Ω s [ σ ] =2 ( K σ curl z, ∇ τ σ ∧ z ) Ω . z ∈ ~C ∞ (Ω), we obtain k K σ curl z k = 2 ( K σ curl z, ∇ τ σ ∧ z ) Ω C k K σ curl z k Ω · k z k Ω . Therefore, k K σ curl z k Ω C k z k Ω . (5.12) Lemma 6.
For any field u ∈ C the relations k curl ( K σ u ) k Ω C k u k Ω (5.13) and ( K σ u ) θ | Γ = 0 (5.14) are valid.Proof. Let z ∈ ~C ∞ (Ω). Operator K σ is self-adjoint by (5.12) and we have | ( K σ u, curl z ) Ω | = | ( u, K σ curl z ) Ω | k u k Ω · k K σ curl z k Ω C k u k Ω · k z k Ω . Since z is arbitrary this estimate implies (5.13). Since z is not necessarilycompactly supported, the equality (5.14) holds true. Lemma 7.
Let σ ⊂ Γ . For any field u ∈ C we have k div ( K σ u ) k Ω C k u k Ω . (5.15) Proof.
By the definition of K σ , for large enough T we have K σ u = (cid:18)Z T X sσ ds (cid:19) u − (cid:18)Z T E sσ ds (cid:19) u. The second term belongs to C and thus is solenoidal in Ω. By (5.6) the firstterm is equal to ( T − τ σ ) u . Then by formula (5.1) we havediv ( K σ u ) = div (( T − τ σ ) u ) = −∇ τ σ ∧ u. This completes the proof.
Proof of Lemma 1.
Suppose u ∈ C . It follows from the estimates (5.13),(5.15) and boundary condition (5.14) that k K σ u k F e C k u k Ω . Then by compactness of the embedding F ⊂ ~ H (Theorem 3) we concludethat K σ ∈ K ( C ; ~ H ). In view of (5.5) this completes the proof.29 .2 Proof of Lemma 2 At first we prove Lemma for f ∈ C ∞ (Ω).Choose a finite open cover { U j } of the support of f such that every setof this cover is C ∞ -diffeomorphic to a ball in case U j ∩ Γ = ∅ or to a semi-ball { x ∈ R : | x | < , x > } otherwise. Choose a partition of unity ζ j ∈ C ∞ ( U j ) such that 0 ζ j , X j ζ j (cid:12)(cid:12)(cid:12) supp f = 1 . It is clear that ˇ f − Y [ f ] = X j ( ˇ ζ j f − Y [ ζ j f ]) , and the functions ζ j f belong to C ∞ ( U j ). Thus, it is necessary to prove theLemma for a function f supported in some open set U C ∞ -diffeomorphic toa ball or a semiball. In this case, for any y ∈ C we have( f y − Y [ f ] y ) | U = ∇ p y , p y ∈ H ( U ) , (5.16)and if the set U intersects with Γ, then the following equality holds true p y | U ∩ Γ = const . This can be easily obtained with the help of the Helmholtz decomposition in U . The function p y in (5.16) is uniquely determined up to additive constant,which can be chosen so that p y | U ∩ Γ = 0 (5.17)if U ∩ Γ = ∅ , and Z U p y dx = 0otherwise. The Friedrichs and Poincar´e inequalities imply that, in the bothcases, there is a constant C such that k p y k U C k∇ p y k U = k f y − Y [ f ] y k U C k ˇ f − Y [ f ] k · k y k . Therefore, the mapping y p y is continuous from C to H ( U ).30ow assume that a sequence y n weakly converges to zero in C . Then thesequence p y n weakly converges to zero in H ( U ), and due to compactness ofthe embedding H ( U ) ⊂ L ( U ) this implies k p y n k U → , n → ∞ . (5.18)Next, we have k f y n − Y [ f ] y n k = ( f y n , f y n − Y [ f ] y n ) Ω = ( f y n , ∇ p y n ) Ω . In the last equality we used (5.16) and the inclusion supp f ⊂ U . Integrat-ing by parts in this inner product, and applying formula (5.1) and equalitydiv y n = 0, we arrive at( f y n , ∇ p y n ) Ω = − Z U ∇ f · y n p y n dx M k y n k Ω · k p y n k U ( M depends only on f ). Integral over ∂U vanishes since f vanishes on ∂U \ Γand in the case U ∩ Γ = ∅ we have (5.17). The right hand side of the latterinequality tends to zero because the norms of y n are bounded and (5.18)takes place. Then, with regard to the result of the previous calculation, weget the relation k f y n − Y [ f ] y n k Ω → , n → ∞ , which shows that the operator ˇ f − Y [ f ] is compact.Now let us consider the case f ∈ C (Ω). The function f can be approxi-mated in C (Ω) by functions f n ∈ C ∞ (Ω). Operators of multiplication by f n tend to the operator of multiplication by f in the operator norm. Hence, theoperator ˇ f − Y [ f ] is compact as a limit of compact operators. Here we prove the following properties:˙ π ( αf + βg ) = α ˙ π ( f ) + β ˙ π ( g ) , ˙ π ( f g ) = ˙ π ( f ) ˙ π ( g ) , k ˙ π ( f ) k = k f k , where f, g ∈ C (Ω), α, β ∈ R . The first and second relations follow fromLemma 2. For example, consider the second one. We show that Y [ f ] Y [ g ] − Y [ f g ] ∈ K . (5.19)31y Lemma 2 we have Y [ f ] Y [ g ] = ( f + K ) Y [ g ] = f Y [ g ] + K = f ( g + K ) + K = f g + e K, where K , K , K, e K ∈ K ( C , ~ H ). Applying Lemma 2 to the function f g , weobtain (5.19).Consider the fourth property. We can restrict ourselves with smooth f since the mapping ˙ π is bounded. The latter follows from the obviousinequality k ˙ π ( f ) k k f k . Let us establish the opposite inequality. We need to show that for any com-pact operator K ∈ K we have k Y [ f ] + K k > k f k . (5.20)Fix a point x ∈ Ω \ Γ such that ∇ f ( x ) = 0 (the case of a constant f istrivial). Choose a sequence of functions ϕ j ∈ C ∞ (Ω \ Γ) such that supp ϕ j shrink to x as j → ∞ . Introduce the fields y j := ∇ f ∧ ∇ ϕ j . Functions ϕ j can be chosen such that every field y j does not vanish identically.Owing to (5.2) we have div y j = 0. Since supp y j tend to x as j → ∞ , forsufficiently large j the fields y j belong to C . Further, we have f y j = f ∇ f ∧ ∇ ϕ j = 12 ∇ ( f ) ∧ ∇ ϕ j , so by (5.2) div ( f y j ) = 0 and for large j the fields f y j also belong to C . Hence Y [ f ] y j = Y ( f y j ) = f y j . (5.21)Consider a normed sequence ˜ y j = y j / k y j k . Obviously, the sequence ˜ y j weakly converges to zero in C . Therefore K ˜ y j → C . With regard to (5.21) this yields k ( Y [ f ] + K ) ˜ y j k = k f ˜ y j + K ˜ y j k → | f ( x ) | , j → ∞ . Since k ˜ y j k = 1 we arrive at the inequality k Y [ f ] + K k > | f ( x ) | . This occursfor all points x , at which f has nonzero gradient. So (5.20) holds true.32 eferences [1] M.I.Belishev. Boundary control in reconstruction of manifolds and met-rics (the BC method). Inverse Problems , 13(5): R1–R45, 1997.[2] M.I.Belishev. The Calderon problem for two-dimensional manifolds bythe BC-method.
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