Recovery of time dependent coefficients from boundary data for hyperbolic equations
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Recovery of time dependent coefficients from boundary data for hyperbolic equations © 2021 European Mathematical Society. Published by EMS Press. Published version Feizmohammadi, Ali; Ilmavirta, Joonas; Kian, Yavar; Oksanen, Lauri Feizmohammadi, A., Ilmavirta, J., Kian, Y., & Oksanen, L. (2021). Recovery of time dependent coefficients from boundary data for hyperbolic equations. Journal of Spectral Theory, 11(3), 1107-1143. https://doi.org/10.4171/jst/367 2021
J. Spectr. Theory 11 (2021), 1107–1143 DOI 10.4171/JST/367 © 2021 European Mathematical Society Published by EMS Press This work is licensed under a CC BY 4.0 license. Recovery of time-dependent coefficients from boundary data for hyperbolic equations Ali Feizmohammadi,1Joonas Ilmavirta,2Yavar Kian,3and Lauri Oksanen4 Abstract. We study uniqueness of the recovery of a time-dependent magnetic vectorvalued potential and an electric scalar-valued potential on a Riemannian manifold from the knowledge of the Dirichlet-to-Neumann map of a hyperbolic equation. The Cauchy data is observed on time-like parts of the space-time boundary and uniqueness is proved up to the natural gauge for the problem. The proof is based on Gaussian beams and inversion of the light ray transform on Lorentzian manifolds under the assumptions that the Lorentzian manifold is a product of a Riemannian manifold with a time interval and that the geodesic ray transform is invertible on the Riemannian manifold. Mathematics Subject Classification (2020). 35R30. Keywords. Dirichlet-to-Neumann map, Gaussian beam, inverse problems, light ray transform, magnetic potential. Contents 1 Introduction................................1108 2 Preliminaries ...............................1114 3 Gaussian Beam solutions . . . . . . . . . . . . . . . . . . . . . . . . . 1116 4 Reduction to the light ray transform . . . . . . . . . . . . . . . . . . . 1131 5 Inversion of the light ray transforms . . . . . . . . . . . . . . . . . . . 1135 References...................................1140 1Ali Feizmohammadi was supported by EPSRC grant EP/P01593X/1. 2Joonas Ilmavirta was supported by the Academy of Finland (decision 295853). 3Yavar Kian was partially supported by the Agence Nationale de la Recherche grant ANR17-CE40-0029. 4Lauri Oksanen was supported by the EPSRC grants EP/P01593X/1 and EP/R002207/1.
1108 A. Feizmohammadi, J. Ilmavirta, Y. Kian, and L. Oksanen 1. Introduction 1.1. Statement of the problem. Let .M;Ng/ be a 1Cndimensional Lorentzian manifold with boundary. Throughout this paper, we will assume that .M;Ng/ has a global product structure, that is to say MDŒ0; T M,NgD dt2Cg, where T > 0 and .M; g/ denotes a smooth compact connected Riemannian manifold of dimension n>2, with smooth boundary @M. We assume that g2C6.MISym2M/, where Sym2Mdenotes the bundle of symmetric twotensors over M. We denote by Ngthe Laplace–Beltrami operator given by NgWD div NgrNg, where div Ng(resp., rNg) denotes the divergence (resp., gradient) operator on .M;Ng/. In local coordinates .x0WD t; x1; : : : ; xn/and for each u2C2.M/we have NguD n X i;j D0 jNgj[email protected]=2 Ngij @xju/; where Ng1WD .Ngij /06i;j 6nand jNgj WD jdet Ngj: As NgD dt2Cg, we have NgD @2 tCg, where gis defined analogously. Consider a complex-valued scalar function q.t; x/ (electric potential) and a complex-valued one-form A(magnetic potential), such that the following regularity assumptions are satisfied q2C.M/and A2C1.MITM/: (1.1) In local coordinates, the one-form Acan be expressed as A.t; x/ Db.t; x/ dt C n X iD1 !j.t; x/ dxjDb.t; x/ dt C!.t; x/; (1.2) where !is a time-dependent one-form on .M; g/. Given Aand qas above, We consider the initial boundary value problem (IBVP) 8 ˆ ˆ < ˆ ˆ : NguCArNguCqu D0on M; uDhon .0; T / @M; u.0; /D0; @tu.0; /D0on M; (1.3)
Recovery of time-dependent coefficients 1109 with non-homogeneous Dirichlet data h2H1 0..0; T / @M /. Note that in local coordinates, ArNguD b @tuCPn i;j D1gij !i@ju. We introduce N, the outward unit normal vector to .0; T /@M , and define the hyperbolic Dirichlet-to-Neumann (DN in short) map, given by ƒA;qWh7! @Nu.AN/u 2ˇˇˇ.0;T /@M where usolves problem (1.3). This is well-defined as equation (1.3) admits a unique solution u2C.0; T IH1.M // \C1.0; T IL2.M //; with @Nuj.0;T /@M 2 L2..0; T /@M /. This follows from [29, Theorem 2.1] as explained in Section 2.1. The goal of this paper is to study the unique recovery of the complex valued coefficients Aand qgiven ƒA;q, up to the natural obstructions discussed in the next section. 1.2. Natural obstructions. The first obstruction concerns the recovery of the magnetic potential A. Indeed, for jD1; 2, fix .Aj; qj/defined as above and assume that there exists .t; x/ 2C2.M/with j.0;T /@M D0, such that A1DA2C2N d ; q1Dq2CNg A2rNg hrNg ; rNg iNg;(1.4) where N ddenotes the exterior derivative on M, that acts on f2C1.M/through N df D@tf dt Cdf with ddenoting the exterior derivative on M, and h;iNg denotes the inner product on .M;Ng/. Then, for ujthe solution of (1.3) with ADAj,qDqj,jD1; 2, it holds that u1De u2and, using the fact that j.0;T /@M D0, we obtain @Nu1.A1N/u1 2D@Nu2C.@N /u2.A1N/u2 2D@Nu2.A2N/u2 2: This proves that ƒA1;q1DƒA2;q2, but A1¤A2as soon as does not vanish identically. In other words, the DN map ƒA;q is invariant with respect to the gauge transformation given by (1.4) and the best we can expect is the recovery of the coefficients Aand qfrom ƒA;q modulo the gauge invariance (1.4). The second obstruction to our problem is due to finite speed of propagation for the wave equation. Let us define the set DD ¹.t; x/ 2Mjdist.x; @M/ < t < T dist.x; @M /º: Due to domain of dependence arguments (see [25, Section 1.1] and [26, Section 1.1]), it is not possible to recover the restriction of any of the coefficients A and qon the set MnDfrom ƒA;q. Therefore, for our problem, the best we can expect, is to recover the coefficients modulo the gauge invariance above, on the set D.
1110 A. Feizmohammadi, J. Ilmavirta, Y. Kian, and L. Oksanen 1.3. Main result. Before stating the main result, we need to recall the definition of the geodesic ray transform on the (spatial) Riemannian manifold M. For each .y; v/ 2SM, with SM denoting the unit tangent bundle of M, let .Iy; v/ denote the unit speed geodesic starting at point y, in the direction v, that is, rg PP.Iy; v/ D0; .0Iy; v/ Dy; P.0Iy; v/ Dv: For any .y; v/ 2SM, we define exit.y; v/ through exit.y; v/ WD inf ¹t > 0 j.tIy; v/ 2@M; P.tIy; v/ …T.tIy;v/@Mº: We now define @SM WD ¹.y; v/ 2SM jy2@M; hv; .y/ig< 0; exit.y; v/ < 1º;(1.5) where denotes the outward normal unit vector on @M. Henceforth, for the sake of brevity, we use the term maximal geodesic to refer to the geodesics .Iy; v/ (or ./in short) with .y; v/ 2@SM, over their maximal interval of definition in Mint , that is the interval IWD .0; exit/. Definition 1.1. Let .y; v/ 2@SM and let .Iy; v/WI!M. We define the geodesic ray transform of .f; ˛/ 2C.M/ C.MITM/, as follows: I.f; ˛/ WD ZI Œf ..t// C˛..t// P.t/ dt: We also need to recall the definition of the solenoidal component, ˛s, of a one-form ˛with local representation ˛DPn kD1˛kdxk. Let ı˛ WD n X i;j D1 1 [email protected] ˛j/ denote the divergence operator on Msending one-forms to functions. Given any ˛2L2.MITM/, it can be uniquely decomposed as ˛D˛sCd ; (1.6) where ı˛sD0and 2H1 0.M/ solves g Dı˛ in the weak sense on M. This is called the Helmholtz decomposition (see for example [41]). We will be working with C1.M/ one-forms. In this case, one can immediately see that since ı˛ 2 C.M/ Ln.M/, elliptic regularity implies that 2W2;n.M / C1.M /. We will use this observation later in the paper to derive the smoothness properties for the gauge. With these notations, we can now state the main geometric assumption on .M; g/.
Recovery of time-dependent coefficients 1111 Hypothesis 1.2. Let .M; g/ be a compact connected Riemannian manifold with smooth boundary. We say that the geodesic ray transform is injective on Mwith respect to functions f2C.M / and one-forms ˛2C.M ITM /, if the following holds: I.Iy;v/.f; ˛/ D0for all .y; v/ 2@SM H) f0and ˛s0: According to Theorems 3 and 4 in [36], Hypothesis 1.2 will be fulfilled if M is simple. This condition can also be fulfilled by a non-simple manifold. We refer to Section 2.2 for a more detailed discussion about this aspect. Finally, let us introduce the set EDwhere we recover the coefficients. For any x2M, we define Dg.x/ WD sup ¹exit.y; v/ j.y; v/ 2@SM; x is in .Iy; v/º and let Dg.M/ WD sup ¹Dg.x/ jx2Mº: For T > 2Dg.M/, we define EWD ¹.t; x/ 2MjDg.x/ < t < T Dg.x/º: Theorem 1.3. Let g2C6.M ISym2M /,A1;A22C1.MITM/and q1; q22 C.M/. Assume that supp .A1A2/E,supp .q1q2/E, and A1.t; x/ DA2.t; x/ for all .t; x/ 2.0; T / @M: If Hypothesis 1.2 holds, then ƒA1;q1DƒA2;q2implies that there exists 2 C2.M/with j.0;T /@M D0, such that (1.4)holds. The proof of this theorem relies in part on injectivity of the so-called light ray transforms of vector valued functions. Recall that a curve ˇin Mis a null geodesic (or a light ray), if rNg P ˇP ˇD0and hP ˇ; P ˇiNgD0. Given the product structure of M, we can parameterize maximal null geodesics through ˇ.t/ D.Q tCt; .t//; t 2I where is a unit speed maximal geodesic in Mand Q t2R. We define the light ray transforms Lqand LAof q2C.M/and A2C.MITM/as follows: Lq.ˇ/ WD ZI q.ˇ.t// dt and LA.ˇ/ WD Zˇ A: We have the following proposition, that will be proved in Section 5.
1112 A. Feizmohammadi, J. Ilmavirta, Y. Kian, and L. Oksanen Proposition 1.4. Let .q; A/2C.M/C1.MITM/be such that supp q; supp AE. If Hypothesis 1.2 holds, then we have the following statements: (i) if Lq.ˇ/ D0for all maximal null geodesics ˇin M, then q0on M; (ii) if LA.ˇ/ D0for all maximal null geodesics ˇin M, then there exists 2C2.M/vanishing on @Msuch that AN d on M. 1.4. Previous literature. Results related to the recovery of coefficients for hyperbolic equations can in general be divided into two categories of timeindependent and time-dependent coefficients. Starting with the seminal works [4,6], there is an extensive literature related to the recovery of time-independent coefficients for hyperbolic equations. These results usually rely on the Boundary Control method, developed in [4,6] and a time sharp unique continuation theorem [40], which provide the building blocks of very general results. We refer the reader to [27] for an introduction to the method and to [28] for an example of a state of the art result in this direction. We also refer to [5,21] for review. The unique continuation theorem in [40] fails if the dependence of the coefficients on time is non-analytic and therefore extension of these results for general time dependent coefficients is not possible (see e.g. [1,2]). We refer the reader to [17] for a uniqueness result, when the dependence of the coefficients on time is real analytic. Starting with [12], methods based on Carleman estimates have also been quite fruitful in deriving uniqueness results for time-independent coefficients of hyperbolic equations. Contrary to the Boundary Control method, where the best known stability estimates are double logarithmic [11], these methods tend to give strong stability estimates. We also mention [30] where Boundary Control method is combined with complex geometric optics and stronger estimates are obtained for low frequencies. In the time-dependent category, most of the results are obtained for the reconstruction of the zeroth order term, q, and are based on a use of geometric optic solutions for the wave equation. Let us mention that this approach has also been used in the time-independent category [8,9,23,37] to obtain strong stability estimates although they suffer from considerably stronger geometric assumptions compared to the Boundary Control method. Methods based on geometric optics were used in the context of recovery of time-dependent coefficients starting with [35]. Among the literature of results in this direction, we refer to [10,19,24,32,33]. The leading coefficients for the wave equation in all these results are constant. Uniqueness of zeroth order coefficient qfor a variable coefficient wave equation was considered in [26] where the recovery of the potential was based on inversion of geodesic
Recovery of time-dependent coefficients 1113 ray transform for scalar functions. It should be noted that even in the case where AD0, the result in this paper is a significant improvement of [26], since there .M; g/ was assumed to be simple. Approaches based on global geometric optic solutions fail if the Riemannian manifold .M; g/ is not simple (see Section 3.5). This motivates the use of Gaussian beams in the current paper. Gaussian beams were introduced in [3,34] and they were first used in the context of inverse problems in [7,20]. We refer the reader to [21] for a thorough presentation in the case of a wave equation with a smooth metric, a smooth electric potential and no magnetic potential. This paper is concerned with the reconstruction of time-dependent vector valued coefficients for the wave equation under weak geometrical assumptions on the spatial manifold .M; g/ and weaker regularity assumptions on the coefficients. We use Gaussian beams to reduce the inverse problem to the inversion of the light ray transform of the unknown coefficients. The closest previous work to this reduction is [39], where the authors study the problem of recovery of the geometry along with a time-dependent magnetic potential Aand an electric potential qin a Lorentzian manifold from a micro-local formulation of a Cauchy data set on the boundary. It is shown that if Ng; A; q belong to some Ck, with ksufficiently large, then this Cauchy data set uniquely determines the scattering relation of Ngalong with light ray transforms of A; q. Their approach is based on the study of Fourier Integral Operators and propagation of singularities. The inversion of the light ray transform on a general Lorentzian manifold is left as an open problem. Our Gaussian beam construction makes the reduction to the light ray transform more explicit in terms of the smoothness required. We also succeed in the inversion of the light lay transform, in the sense of Proposition 1.4. Our inversion method for the light ray transform was inspired in part by techniques developed in the context of the Calderón problem [13]. 1.5. Outline of the paper. This paper is organized as follows. In Section 2, we discuss the forward problem (1.3) and also discuss the Hypothesis 1.2. In Section 3, we present the Gaussian beam construction near a null geodesic in M. In Section 4, we show the reduction step from the knowledge of the Dirichletto-Neumann map ƒA;q, to the knowledge of the light ray transforms of A; q and conclude that Theorem 1.3 follows from Proposition 1.4. Finally, Section 5is concerned with the proof of Proposition 1.4.
1114 A. Feizmohammadi, J. Ilmavirta, Y. Kian, and L. Oksanen 2. Preliminaries 2.1. Direct problem. Let Xu WD ArNguCqu; where Aand qsatisfy (1.1). We consider the wave equation 8 ˆ ˆ < ˆ ˆ : NguCXu DFin M, ujx2@M Dhon .0; T / @M, ujtD0Du0; @tujtD0Du1on M. (2.1) It is classical that uis in the energy space C.Œ0; T IH1.M// \C1.Œ0; T IL2.M // (2.2) when hD0,F2L2.M/,u02H1 0.M/ and u12L2.M/. The wave equation 8 ˆ ˆ < ˆ ˆ : NgvDFin M, vjx2@M Dhon .0; T / @M, vjtD0Du0; @tvjtD0Du1on M, (2.3) was considered in [29]. It was shown there that if Fand u1are as above, and u02H1.M/ and h2H1..0; T / @M / satisfy the compatibility condition hjtD0Du0jx2@M ;(2.4) then the solution vis the energy space (2.2), and @Nvjx2@M 2L2..0; T / @M /: Let us now set uDvwwhere vis the solution of (2.3) with F,u0,u1and has above, and wis the solution of (2.1) with FDXv 2L2.M/,u0D0,u1D0 and hD0. Then usatisfies (2.1) with the same F,u0,u1and has in (2.3) for v. As both vand ware in (2.2), so is u. But then NguDFXu 2L2.M/and @ujx2@M 2L2..0; T / @M /. It is straightforward to turn this regularity result to the corresponding estimate kukC.Œ0;T IH1 0.M //\C1.Œ0;T IL2.M // Ck@ukL2..0;T /@M / C.kFkL2.M/CkhkH1..0;T /@M / Cku0kH1.M / Cku1kL2.M //; (2.5) for solutions uof (2.1) under the compatibility condition (2.4). We write If .t/ DRt 0f .s/ds, and show now that usatisfies the estimate kukL2.M/CkIF kL2.M/;(2.6)
Recovery of time-dependent coefficients 1121 We will also require that the following transport equation holds along the null geodesic ˇ: .TAv0/.s; 0; : : : ; 0/ D0for all s2.a0; b0/: (3.11) 3.3. Construction of the phase. We begin by solving equation (3.10). For mD0, we obtain the equation n X k;lD0Ngkl @' @zk @' @zlˇˇˇˇD0for 16i; j 6n: Recalling that NgjˇD2dz0dz1C.dz2/2C:::C.dzn/2, this reduces to 2@0'@1'C n X kD2 .@k'/2D0: (3.12) Similarly, for mD1, we obtain (recall that for all i; j; k we have @igjkjˇD0) n X k;lD0Ngkl @2 k˛'@l'ˇˇˇˇD0for 16˛6n: (3.13) Recalling the definition of the phase function 'from equation (3.8), it is clear that equations (3.12) and (3.13) will be satisfied if we set '0D0and '1Dr. Next we consider the case mD2in equation (3.10) and write '2.s; z0/WD X 16i;j 6n Hij .s/zizj; where Hij DHji is a symmetric matrix. Let us impose the auxiliary condition that =H.s/ > 0 for s2.a0; b0/: (3.14) This assumption will lead to a Gaussian decay away from the null geodesic ˇ, but we will also provide a geometric motivation behind this assumption in the next section. We require @2 @zi@zjn X k;lD0Ngkl @' @zk @' @zlˇˇˇˇD0for all 16i; j 6n: This is equivalent to .@2 ij Ngkl @k' @l'C2Ngkl @3 kij ' @l'C2Ngkl @2 ki ' @2 lj 'C4@iNgkl @2 jk' @l'/jˇD0:
1122 A. Feizmohammadi, J. Ilmavirta, Y. Kian, and L. Oksanen which again simplifies to @2 ij Ng11 C2Ng10@3 0ij 'C2 n X kD2 @2 ki ' @2 kj 'ˇˇˇˇD0: We therefore obtain the following Riccati type ODE: d ds HCHCH CDD0; s 2.a0; b0/; H.s/DH0;with =H0> 0; (3.15) where Cis the matrix defined through 8 ˆ ˆ < ˆ ˆ : C11 D0; Cii D2for 26i6n; Cij D0otherwise, (3.16) and DD.Dij /nnWD 1 4.@2 ij Ng11jˇ/nnfor all i; j 2 ¹1; : : :; nº. Note that since g is C4smooth in the Fermi coordinates, we have D2C2.Œa0; b0ICnn/. We now recall two lemmas. For the proofs, we refer the reader to [21, Lemma 8, Section 8] and [21, Lemma 10, Section 8] respectively. Lemma 3.2. The Riccati equation (3.15) has a unique solution. The solution H is symmetric and =.H.s// > 0 for all s2.a0; b0/. We have H.s/ DZ.s/Y.s/1 where Z.t/ and Y.t/ solve the following system of first order linear ODEs: d ds YDCZ; Y.s/DI; (3.17) d ds ZD DY; Z.s/DH0:(3.18) In addition, Y.s/ is non-degenerate for all s2Œa0; b0. Lemma 3.3. The following identity is satisfied: det.=.H.s// jdet.Y.s//j2Ddet.=.H0//: Let us make some remarks about the regularity of the solutions Y.s/; H.s/. Since Cis a constant matrix, the matrix Y.s/ also satisfies d2 ds2YD CDY; Y.s/DI; P Y.s/DCH0(3.19) since D2C2.Œa0; b0IC2n/, we immediately deduce that Y2C4.Œa0; b0ICnn/.
Recovery of time-dependent coefficients 1123 Now considering the ODE for the function Z.s/, we deduce that we have Z2 C3.Œa0; b0ICnn/. Finally, since H.s/ DZ.s/Y 1.s/ and since Y.s/ is nonsingular on Œa0; b0, we conclude that H2C3.Œa0; b0ICnn/and '2C3.V/(3.20) in the Fermi coordinates. 3.4. Construction of the amplitude. Let us study the transport equation (3.11). First observe that in Fermi coordinates . Ng'/jˇD N X i;j D0Ngij @2 ij 'jˇD n X jD2 @2 jj 'jˇDTr.CH/: Thus equation (3.11) simplifies to [email protected] / A.s; 0/ P ˇ/v0D0 s 2Œa0; b0; (3.21) where AP ˇWD A@sD hA; dz1iNg:We proceed to prove that v0.s/ Ddet.Y.s//1 2e1 2.Rs sA.;0/ P ˇ d/ s2Œa0; b0(3.22) satisfies equation (3.21). Indeed, this follows immediately from the observation Tr.C.s/H.s// DTr.C.s/Z.s/Y.s/1/ DTr dY ds .s/Y.s/1 Dd ds log.det.Y.s///; where we have used the fact that dY ds .s/ DC.s/Z.s/. Clearly v02C2.Œa0; b0/, which together with the definition of vimplies that v2C2.V/. This concludes the construction of the amplitude function and also the construction of udefined by (3.7) which we refer to as an approximate Gaussian beam. 3.5. Geometrical interpretations. We will briefly discuss some geometrical aspects of the approximate Gaussian beam construction. In particular, we will discuss explicitly, how conjugate points on .M;Ng/ manifest themselves in the vector valued function Y.s/ constructed above. First we note the following lemma. Lemma 3.4. Let ˇbe a null geodesic as above and let us consider the Fermi coordinates near ˇ. We have the following identity: @2Ng11 @zi@zjˇˇˇˇD 2R0i0j jˇ;
1124 A. Feizmohammadi, J. Ilmavirta, Y. Kian, and L. Oksanen for all indices i; j 2 ¹1; : : : ; nº, where R denotes the curvature tensor for the Lorentzian manifold .y M;Ng/. Proof. First note that by Bianchi identities the expression is clearly symmetric with respect to indices i; j . We let rNgand N i jk denote the Levi-Civita connection and the Christoffel symbol for .M;Ng/ respectively. Recall that Ngij denotes the inverse of the matrix Ngij and since NgjˇD2dz0dz1C.dz2/2C C .dzn/2it follows that @2Ng11 @zi@zjˇˇˇˇD @2Ng00 @zi@zjˇˇˇˇ: Since Œ@i; @jD0, we have R0i0j jˇD hrNg @0rNg @i@0; @jiNgjˇhrNg @irNg @0@0; @jiNgjˇ: Using the definition of the Levi-Civita connection, we have rNg @0rNg @i@0D n X kD0 .@0N k i0/@kC n X k;mD0 N k i0 N m 0k@m;(3.23) and rNg @irNg @0@0D n X kD0 .@iN k 00/@kC n X k;mD0 N k 00 N m ik@m:(3.24) Recall that @igjkjˇD0. This implies that N i jkjˇD0but since @0also denotes the tangent vector to ˇ, we observe additionally that @0i jkjˇD0which implies that (3.23) vanishes along ˇ. Hence, R0i0j jˇD Dn X kD0 @iN k 00@k; @jENgˇˇˇˇ D n X kD0Ngjk@iN k 00ˇˇˇˇ D1 2n X k;mD0Ngjk Ngkm@2 im Ng00ˇˇˇˇ D1 2n X mD0 ıjm@2 im Ng00ˇˇˇˇ D1 2@2 ij Ng00jˇ D 1 2@2 ij Ng11jˇ:
Recovery of time-dependent coefficients 1125 We can next use the above lemma in conjunction with the product structure of the Lorentzian manifold Mto derive the following corollary. Corollary 3.5. For any null geodesic ˇas above, we have @2Ng11 @z1@ziˇˇˇˇD0; for all indices i2 ¹1; : : : ; nº. This corollary can be used to simplify the Riccati equation (3.15) further. Indeed, Corollary 3.5 implies that D1i D0for i2 ¹1; : : :; nºand since C1j D Cj1 D0for all j2 ¹1; : : : ; nº, we can simply take H11 DsCc0for any constant c0with =.c0/ > 0,H1j DHj1 D0for all j > 1 and take HiC1;j C1WD z Hi;j for all i; j 2 ¹1; : : : ; n1ºwhere z His a symmetric .n1/.n1/ matrix satisfying d ds z HC2z H2Cz DD0; z H.s/Dz H0;(3.25) with z Dij DDiC1;j C1for all i; j 2 ¹1; : : : ; n1º. This observation also simplifies the construction of the matrix Y. Indeed, we can take Y11 Dc0and Y1j DYj1 D0 for all j2 ¹1; : : : ; nº. Note that a key ingredient in the construction of Gaussian beams is the requirement that the matrix valued function Y.s/ is non-singular. This is indeed guaranteed in the above construction as a consequence of choosing =.H0/ > 0. We will briefly discuss what happens when one pursues real valued solutions to this linear system. Recall that Y.s/ satisfies equation (3.19). This of course implies that the columns of the matrix Yshould also satisfy the same ODE. Let Vbe one of the columns of Ywith representation VDPn jD1Vj.s/ @ @zj. Using the definition of the matrix Dand Lemma 3.4, we deduce that d2 ds2ViDPn jD1R0i0j Vj. Rearranging the indices and using the Bianchi identities, we obtain (recall that on the null geodesic, @0DP ˇ): D2 ds2VCR.V; P ˇ/ P ˇD0: This is the well-known Jacobi equation along ˇ. We therefore see that the columns of Yare variation fields of some variation of ˇthrough null geodesics. In particular, based on this geometric characterization of Y, one can deduce that if there exists a point ˇ.s/ on the interval Œa0; b0that is conjugate to ˇ.a0/, then any real valued solution Y.s/ to (3.19) will always become singular at that point (see for example [15, Section 5.5]). Therefore a global geometric optic construction with a real valued phase function can not be achieved in the presence of conjugate points on .M;Ng/.
1126 A. Feizmohammadi, J. Ilmavirta, Y. Kian, and L. Oksanen 3.6. Construction of the remainder terms. With the WKB construction complete, we now return to the task of constructing solutions u1; u2to (3.1), concentrating on a null geodesic ˇ2D. In particular, we will construct the remainder terms in equations (3.2)-(3.3). We consider the differential operators LA1;q1and L A2;q2(formal adjoint of LA2;q2with respect to the real L2inner product). We can use the previous discussion to obtain two families of approximate solutions given by ei'v1and ei'Nv2to these differential operators. Indeed, let F1; D LA1;q1.ei'v1/; F2; D L A2;q2.ei N'Nv2/: Applying equation (3.9), we obtain F1; D ei'Œ2.S'/v1iTA1v1CLA1;q1v1; F2; D ei N'Œ2.S'/ Nv2CiTN A2v2CL A2;q2Nv2: (3.26) The phase function '2C3.V/is chosen exactly as in Section 3.3 and adapting equation (3.11) to this case, we make the following ansatz for v1; v2: viDn 4vi;0.s/jz0j ıfor iD1; 2; such the functions vi;0.s/ satisfy the following transport equations: 2@sv1;0 C.Tr .CH/ A1P ˇ/v1;0 D0; s 2Œa0; b0; 2@sv2;0 C.Tr .CH/ CN A2P ˇ/v2;0 D0; s 2Œa0; b0: (3.27) Using (3.22), we have v1;0.s/ Ddet.Y.s//1 2e1 2.Rs s.A1P ˇ/.;0/ d/; v2;0.s/ Ddet.Y.s//1 2e1 2.Rs s.N A2P ˇ/.;0/ d/: (3.28) Note that F1;; F2; are compactly supported in a small tubular region around the null geodesic where the Fermi coordinates are well defined. Also recall from the previous discussions that v1; v22C2.V/in the Fermi coordinates. Next we define the expression Rj;,jD1; 2, as the solution of the following IBVP 8 ˆ ˆ < ˆ ˆ : LA1;q1R1; DF1;; .t; x/ 2M; R1;.0; x/ D0; @tR1;.0; x/ D0; x 2M; R1;.t; x/ D0; .t; x/ 2.0; T / @M; (3.29) 8 ˆ ˆ < ˆ ˆ : L A2;q2R2; DF2;; .t; x/ 2M; R2;.T; x/ D0; @tR2;.T; x/ D0; x 2M; R2;.t; x/ D0; .t; x/ 2.0; T / @M: (3.30)
Recovery of time-dependent coefficients 1127 The energy estimate (2.5) in Section 2implies that equations (3.29) and (3.30) admit unique solutions Rj; 2C.Œ0; T IH1 0.M// \C1.Œ0; T IL2.M // j D1; 2; with the estimates kRj;kH1.M/6CkFj;kL2.M/:(3.31) We claim that Rj;,jD1; 2, satisfy the following decay property lim !C1.kRj;kL2.M/C1kRj;kH1.M//D0; (3.32) and showing this completes the construction of the solutions u1; u2of (3.1). Note that for jD1; 2, using (3.10) and (3.11), we have the following bounds: kvjkC2.M/6C n 4; jTAjvjj6Cn 4jz0jjz0j ı; jS'j6Cjz0j2N.jz0j/for z2V; (3.33) where C > 0 only depending on the geometry and kA1kC1;kA2kC1. Here, N denotes a continuous function depending on the geometry .y M;Ng/ and such that N.0/ D0. Note that equation (3.14) implies that jei'j D jei N'j6eDjz0j2;(3.34) with D > 0 independent of and only depending on the geometry. In particular, these estimates imply that kn 4ei'k2 L2.V/.ZV n 2eDjz0j22jz0j ıdz DO.1/; kn 4.S'/ei'k2 L2.V/.ZV jz0j4n 2N2.jz0j/eDjz0j22jz0j ıdz Do.2/; k.TAjvj/ei'k2 L2.V/.ZV n 2jz0j2eDjz0j22jz0j ıdz DO.1/; (3.35) Combining these bounds with (3.26), we find Fj;L2.M/Do./; j D1; 2; (3.36) and using the estimate (3.31), we deduce that lim !C1 1Rj;H1.M/6Clim !C1 1Fj;L2.M/D0; j D1; 2:
1128 A. Feizmohammadi, J. Ilmavirta, Y. Kian, and L. Oksanen Therefore, in order to prove the bound (3.32), it only remains to prove that lim !C1 kRj;kL2.M/D0; j D1; 2: (3.37) Let us begin by stating the following two lemmas that will simplify the proof of the estimate (3.37). Lemma 3.6. Let '; Vbe as above. Then, @t'does not vanish in V. Proof. Since we are considering the neighborhood V, we may use the Fermi coordinate system .s; z0/. Recall that in this coordinate system '.z/ Dz1CHij .z0/zizjDrCHij .s/zizj where i; j 2 ¹1; : : : ; nº. Therefore, p2@t'D@s'@r'D 1CX 16i;j 6n P Hij .s/zizjC2 n X iD1 Hi1.s/zi; which implies that for jz0j D pjz1j2CCjznj2< ı sufficiently small we have j@t'j>1 2:(3.38) This proves the lemma. Lemma 3.7. Let 'be as above and suppose that f2C.V/with supp fV. The following estimate holds: lim !1 n 4 t Z0 f .; /ei'.;/dL2.M/D0: (3.39) Proof. Note that since fis supported near the null geodesic ˇ, we may use the Fermi coordinate system .s; z0/. Define WRnC1!Rthrough .x/ D1nC1 4. 1 4jxj/; with defined as in (3.8) and D kkL1.Rn/. We define the smooth functions fDf.s; z0/2V: It is clear that fis supported near the null geodesic ˇand lim !1 kffkC.M/D0; kfkWk;1.M/6Ckk 4for all k2N:(3.40)
Recovery of time-dependent coefficients 1129 We write n 4 t Z0 f .; /ei'.;/dL2.M/ 6I1CI2; where I1Dn 4 t Z0 f.; /ei'.;/dL2.M/ ; and I2Dn 4 t Z0 .f f/.; /ei'.;/dL2.M/ : For I1, we integrate by part in time using ei' Di @t'@tei' and use Lemma 3.6 together with the fact that the set Vdoes not intersect the sets ¹0ºMand ¹TºM to obtain I16n 4 @t'fei'L2.M/Cn 41 t Z0 @f @'ei' dL2.M/ .3 4; where we are using the first bound in (3.35) together with equation (3.40) to bound the first term by 1and the second term by 3 4. For the term I2, we can use the Cauchy–Schwarz inequality along with inequalities (3.35) and equation (3.40) to obtain I26Tkn 4.f f/ei'kL2.M/ 6TkffkC.M/kn 4ei'kL2.V/! 0as ! 1: Proof of Estimate (3.37).The result for R1; and R2; being similar, we will only give a proof for R1;. Using the energy estimate (2.6) in Section 2, together with the fact that R1; solves equation (3.30), it suffices to prove that lim !1 kF;kL2.M/D0; where F;.t; x/ WD t Z0 F1;.; x/ d: We have F;.t; x/ DI1.t; x/ CI2.t; x/ CI3.t; x/; (3.41)
1130 A. Feizmohammadi, J. Ilmavirta, Y. Kian, and L. Oksanen where I1.t; x/ D t Z0 ei'Œ2.S'/.; x/v1.; x/ d; I2.t; x/ D t Z0 ei'ŒiTA1v1.; x/ d; I3.t; x/ D t Z0 ei'Œ.LA1;q1/v1.; x/ d: We proceed to bound each of the above integrals using the Fermi coordinates .s; z0/ around ˇ. This can be done since each of the above integrands is supported in a small tubular neighborhood of ˇ. For the first integral, I1, we apply integration by parts to obtain I1.t; x/ D t Z0 i2 @'.@ei'/.S'/.; x/v1.; x/ d Diei' @t'.S'/.t; x/v1.t; x/ C t Z0 ei'hi@hS' @'v1ii.; x/ d Diei' @t'.S'/v1Ci t Z0 ei' @2 ' .@'/2.S'/v1dt i t Z0 ei' @Œ.S'/v1 @'dt WD S1.t; x/ CS2.t; x/ CS3.t; x/: (3.42) Using (3.35), we deduce that kS1kL2.M/Do.1/. For S2.t; x/, we first use the Cauchy–Schwarz inequality to observe that kS2kL2.M/6T kei' @2 t' .@t'/2.S'/v1kL2.V/: Recalling that '2C3.V/and applying the bound (3.35) again, we deduce that kS2kL2.M/Do.1/. For S3, we integrate by parts again to obtain S3D ei' @tŒ.S'/v1 .@t'/2C t Z0 ei'@h@..S'/v1/ .@'/2id WD S4CS5: Using equation (3.10), it is clear that [email protected]'/j.jz0jN.jz0j/and j@2 t.S'/j.N.jz0j/ on the set V. Using this observation, together with bounds analogous to (3.35),
Recovery of time-dependent coefficients 1137 Proof. We proceed by induction. The claim is true for kD0as discussed above. Suppose it holds for all m < k. For mDk, we have @k b LA.0; / D k X jD0k jZI .it/kjŒ@j O b.0; .t// C@j O!.0; .t// P.t/ dt D0: Now using the induction hypothesis, we can simplify this expression to obtain ZI Œ@k O b.0; .t// C@k O!.0; .t// P.t/ dt D k1 X jD0k jZI .it/kjŒ.ij / j1Cd jP.t/ dt: Now using the fact that RItkjd jP.t/ dt D RI.k j /tkj1 jdt, we can simplify the right hand side of the above equation to k1 X jD0k jZI Œ.it/kj.ij / j1.it/kj1i.k j / j dt D k1 X jD0k jZI Œ.it/kj.ij / j1 dtC k X jD1k jZI Œ.ij /.it/kj j1 dt Dik ZI k1..t// dt: Hence we have ZI Œ@k O b.0; .t// ik k1..t// C@k O!.0; .t// P.t/ dt D0: Using Hypothesis 1.2 together with smoothness properties of the Helmholtz decomposition in Section 1.3, we conclude that there exists k2C1.M/ vanishing on @M such that @k O!.0; x/ Dd k.x/; and @k O b.0; x/ Dik k1.x/: Since d kD@k O!.0; x/ 2C1.MITM/ and k2C1.M/, we conclude that k2C2.M/: This completes the induction argument.
1138 A. Feizmohammadi, J. Ilmavirta, Y. Kian, and L. Oksanen Proof of statement (ii) in Proposition 1.4.Let us define .t; x/ WD t Z0 b.s; x/ ds D t Z 1 b.s; x/ ds; (5.4) where we are using the fact that supp AM. Note that since Ahas compact support, equation (5.3) implies .t; x/ D .T; x/ D T Z0 b.s; x/ ds DO b.0; x/ D0for t>T. (5.5) Similarly, .t; x/ D0for t60. Again, we let O .; x/ denote the Fourier transform of in the time variable. Note that since is compactly supported in time, O .; x/ is analytic with respect to . Let us define the coefficients ¹Q kº1 kD0 through O .; x/ D 1 X kD0 Q k.x/ kŠ k: We claim that Q k.x/ D k.x/ holds for all k>0and all x2M. To see this note that by definition @t Db. Hence i O ./ DO b./. Now differentiating this expression kC1times and evaluating at D0, we deduce that iQ kDiO .k/.0/ Di1 kC1O b.kC1/.0/ Di k; where we used Lemma 5.1 in the last step. Thus, we have O .; x/ D 1 X kD0 k.x/ kŠ k: Since !.t; x/ is also compactly supported in t, Lemma 5.1 implies that O!.; x/ D 1 X kD0 @k O!.0; x/ kŠ kD 1 X kD0 d k.x/ kŠ k:(5.6) Note that for every fixed x2M, the following estimate holds: j@k O!.0; x/j6k@k O!.; x/kL1.R/. T Z0 jtjkj!.t; x/jdt;
Recovery of time-dependent coefficients 1139 which implies that the infinite series (5.6) is uniformly convergent with respect to x2M, and therefore we can write O!.; x/ D 1 X kD0 d k.x/ kŠ kDd1 X kD0 k.x/ kŠ kDdO .; x/: Hence, !Dd , and subsequently ADb dt C!D.@t / dt Cd DN d : (5.7) Note that j.0;T /@M D0as kj@M D0for all k>0. We will now show that 2C2.M/. Indeed, it is clear from equation (5.4) that 2C1.M/. But then since N d DA2C1.MITM/we may conclude that 2C2.M/. 5.1. Remarks. With the proof of Theorem 1.3 complete, let us state a few remarks. We start with the auxiliary condition that the coefficients A; q should be known on the set MnE. This is merely an artifact of the light ray inversion method, as the Gaussian beam construction shows that the light ray transforms of Aand qcan be obtained on the set D. As the ratio Dg.M/=T grows, the set Egrows and the coefficients are therefore recovered in a large set that is closer in size to the optimal set D. The assumption that Dg.M/ < 1in particular implies that the manifold .M; g/ should be non-trapping. To illustrate this remark, consider MDŒ0; 1 S1with S1denoting the unit circle and note that in this case Dg.M / D 1. Regarding the smoothness of the metric g, note that we only require the metric in Fermi coordinates to be C4smooth. This however, can only be guaranteed if the metric is a priori known to be C6smooth, as there could be some loss of regularity in the angular directions of the exponential map of a Riemannian manifold. One can in fact use a mollification of the phase function 'to improve the result to g2C4.MISym2M/. The key here is that there is no loss of regularity in the direction tangent to the null geodesic in Fermi coordinates. We believe that the result could be extended to g2C2.M ISym2M/, but this will require mollification of the metric, g. Improvements beyond the C2-smoothness for the metric gcould be much harder.
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Recovery of time-dependent coefficients 1143 [42] G. Uhlmann and A. Vasy, The inverse problem for the local geodesic ray transform. Invent. Math. 205 (2016), no. 1, 83–120. MR 3514959 Zbl 1350.53098 Received October 15, 2019 Ali Feizmohammadi, The Fields Institute for Research in Mathematical Sciences, Toronto, Ontario M5T 3J1, Canada e-mail: afeizmoh@fields.utoronto.ca Joonas Ilmavirta, Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), 40014 Jyväskylä, Finland e-mail: joonas.ilmavirta@jyu.fi Yavar Kian, Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France e-mail: [email protected] Lauri Oksanen, Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FI-00014 Helsinki, Finland e-mail: lauri.oksanen@helsinki.fi