# Microlocal analysis and inverse problems Lecture 4 : Uniqueness results in admissible geometries

Size: px
Start display at page:

## Transcription

1 Microlocal analysis and inverse problems Lecture 4 : Uniqueness results in admissible geometries David Dos Santos Ferreira LAGA Université de Paris 13 Wednesday May 18 Instituto de Ciencias Matemáticas, Madrid David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 1 / 24

2 Outline Introduction 1 Introduction 2 Complex Geometrical Optics 3 Attenuated X-ray transform 4 Uniqueness of unbounded potentials David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 2 / 24

3 Introduction Introduction It is time now to state some precise uniqueness results. So far we have seen that: 1 In order to have Carleman estimates with opposite weights, one has to work with limiting Carleman weights. This is in order to comply with Hörmander s (necessary) criterium of solvability for non-selfadjoint operators. 2 On Riemannian manifolds, the existence of LCW is a limiting condition. It implies that manifolds have to be locally conformal to a product. 3 For reasons related to the global solvability of the transport equation, we will ask that the manifolds under scope be globally conformal to a product. 4 In fact, we need more conditions: for reasons related to the global solvability of the eikonal eqution, we ask the cutlocus of the manifold to be empty. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 3 / 24

4 Introduction Introduction It is time now to state some precise uniqueness results. So far we have seen that: 1 In order to have Carleman estimates with opposite weights, one has to work with limiting Carleman weights. This is in order to comply with Hörmander s (necessary) criterium of solvability for non-selfadjoint operators. 2 On Riemannian manifolds, the existence of LCW is a limiting condition. It implies that manifolds have to be locally conformal to a product. 3 For reasons related to the global solvability of the transport equation, we will ask that the manifolds under scope be globally conformal to a product. 4 In fact, we need more conditions: for reasons related to the global solvability of the eikonal eqution, we ask the cutlocus of the manifold to be empty. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 3 / 24

5 Introduction Introduction It is time now to state some precise uniqueness results. So far we have seen that: 1 In order to have Carleman estimates with opposite weights, one has to work with limiting Carleman weights. This is in order to comply with Hörmander s (necessary) criterium of solvability for non-selfadjoint operators. 2 On Riemannian manifolds, the existence of LCW is a limiting condition. It implies that manifolds have to be locally conformal to a product. 3 For reasons related to the global solvability of the transport equation, we will ask that the manifolds under scope be globally conformal to a product. 4 In fact, we need more conditions: for reasons related to the global solvability of the eikonal eqution, we ask the cutlocus of the manifold to be empty. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 3 / 24

6 Introduction Introduction It is time now to state some precise uniqueness results. So far we have seen that: 1 In order to have Carleman estimates with opposite weights, one has to work with limiting Carleman weights. This is in order to comply with Hörmander s (necessary) criterium of solvability for non-selfadjoint operators. 2 On Riemannian manifolds, the existence of LCW is a limiting condition. It implies that manifolds have to be locally conformal to a product. 3 For reasons related to the global solvability of the transport equation, we will ask that the manifolds under scope be globally conformal to a product. 4 In fact, we need more conditions: for reasons related to the global solvability of the eikonal eqution, we ask the cutlocus of the manifold to be empty. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 3 / 24

7 Introduction Admissible geometries The former remarks justify the introduction of: Definition A compact Riemannian manifold (M, g), with dimension n 3 and with boundary M, is called admissible if M R M 0 for some (n 1)-dimensional simple manifold (M 0, g 0 ), and if g = c(e g 0 ) where e is the Euclidean metric on R and c is a smooth positive function on M. Definition Here, a compact manifold (M 0, g 0 ) with boundary is simple if for any p M 0 the exponential map exp p with its maximal domain of definition is a diffeomorphism onto M 0, and if M 0 is strictly convex (that is, the second fundamental form of M 0 M 0 is positive definite). David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 4 / 24

8 Introduction Admissible geometries The former remarks justify the introduction of: Definition A compact Riemannian manifold (M, g), with dimension n 3 and with boundary M, is called admissible if M R M 0 for some (n 1)-dimensional simple manifold (M 0, g 0 ), and if g = c(e g 0 ) where e is the Euclidean metric on R and c is a smooth positive function on M. Definition Here, a compact manifold (M 0, g 0 ) with boundary is simple if for any p M 0 the exponential map exp p with its maximal domain of definition is a diffeomorphism onto M 0, and if M 0 is strictly convex (that is, the second fundamental form of M 0 M 0 is positive definite). David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 4 / 24

9 Introduction Admissible geometries The former remarks justify the introduction of: Definition A compact Riemannian manifold (M, g), with dimension n 3 and with boundary M, is called admissible if M R M 0 for some (n 1)-dimensional simple manifold (M 0, g 0 ), and if g = c(e g 0 ) where e is the Euclidean metric on R and c is a smooth positive function on M. Definition Here, a compact manifold (M 0, g 0 ) with boundary is simple if for any p M 0 the exponential map exp p with its maximal domain of definition is a diffeomorphism onto M 0, and if M 0 is strictly convex (that is, the second fundamental form of M 0 M 0 is positive definite). David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 4 / 24

10 Introduction Main results (smooth potentials) Theorem Let (M, g) be admissible, and let q 1 and q 2 be two smooth functions on M. If Λ g,q1 = Λ g,q2, then q 1 = q 2. (In fact, we have results for anisotropic magnetic Schrödinger operators). Theorem Let (M, g 1 ) and (M, g 2 ) be two admissible Riemannian manifolds in the same conformal class. If Λ g1 = Λ g2, then g 1 = g 2. The above theorems were proved in a joint paper with Carlos Kenig, Mikko Salo and Gunther Uhlmann. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 5 / 24

11 Introduction Main results (smooth potentials) Theorem Let (M, g) be admissible, and let q 1 and q 2 be two smooth functions on M. If Λ g,q1 = Λ g,q2, then q 1 = q 2. (In fact, we have results for anisotropic magnetic Schrödinger operators). Theorem Let (M, g 1 ) and (M, g 2 ) be two admissible Riemannian manifolds in the same conformal class. If Λ g1 = Λ g2, then g 1 = g 2. The above theorems were proved in a joint paper with Carlos Kenig, Mikko Salo and Gunther Uhlmann. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 5 / 24

12 Introduction Main results (unbounded potentials) Theorem Let (M, g) be admissible and let q 1, q 2 be complex functions in L n/2 (M). If Λ g,q1 = Λ g,q2, then q 1 = q 2. The above theorem was proved in a joint paper with Carlos Kenig and Mikko Salo. From the point of view of unique continuation, the regularity L n/2 is optimal. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 6 / 24

13 Introduction Main results (unbounded potentials) Theorem Let (M, g) be admissible and let q 1, q 2 be complex functions in L n/2 (M). If Λ g,q1 = Λ g,q2, then q 1 = q 2. The above theorem was proved in a joint paper with Carlos Kenig and Mikko Salo. From the point of view of unique continuation, the regularity L n/2 is optimal. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 6 / 24

14 Outline Complex Geometrical Optics 1 Introduction 2 Complex Geometrical Optics 3 Attenuated X-ray transform 4 Uniqueness of unbounded potentials David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 7 / 24

15 Complex Geometrical Optics Reminder of the formal WKB expansion Conjugated operator : P ϕ = e ϕ/h h 2 g e ϕ/h P ϕ = P ϕ Principal symbol : p ϕ = ξ 2 dϕ 2 + 2i ξ, dϕ We have g (e 1 h (ϕ+iψ) a) = e 1 h ϕ P ϕ (e i h ψ a) = e 1 h (h (ϕ+iψ) 0 p ϕ (x, dψ) [ + 2h (grad g ϕ + igrad g ψ)a + 1 ] 2 g(ϕ + iψ)a ) + h 2 g a. Eikonal equation: p ϕ (x, dψ) = 0 Transport equation: (grad g ϕ + igrad g ψ)a g(ϕ + iψ)a = 0 David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 8 / 24

16 Complex Geometrical Optics Reminder of the formal WKB expansion Conjugated operator : P ϕ = e ϕ/h h 2 g e ϕ/h P ϕ = P ϕ Principal symbol : p ϕ = ξ 2 dϕ 2 + 2i ξ, dϕ We have g (e 1 h (ϕ+iψ) a) = e 1 h ϕ P ϕ (e i h ψ a) = e 1 h (h (ϕ+iψ) 0 p ϕ (x, dψ) [ + 2h (grad g ϕ + igrad g ψ)a + 1 ] 2 g(ϕ + iψ)a ) + h 2 g a. Eikonal equation: p ϕ (x, dψ) = 0 Transport equation: (grad g ϕ + igrad g ψ)a g(ϕ + iψ)a = 0 David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 8 / 24

17 Complex Geometrical Optics Reminder of the formal WKB expansion Conjugated operator : P ϕ = e ϕ/h h 2 g e ϕ/h P ϕ = P ϕ Principal symbol : p ϕ = ξ 2 dϕ 2 + 2i ξ, dϕ We have g (e 1 h (ϕ+iψ) a) = e 1 h ϕ P ϕ (e i h ψ a) = e 1 h (h (ϕ+iψ) 0 p ϕ (x, dψ) [ + 2h (grad g ϕ + igrad g ψ)a + 1 ] 2 g(ϕ + iψ)a ) + h 2 g a. Eikonal equation: p ϕ (x, dψ) = 0 Transport equation: (grad g ϕ + igrad g ψ)a g(ϕ + iψ)a = 0 David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 8 / 24

18 Complex Geometrical Optics Reminder of the formal WKB expansion Conjugated operator : P ϕ = e ϕ/h h 2 g e ϕ/h P ϕ = P ϕ Principal symbol : p ϕ = ξ 2 dϕ 2 + 2i ξ, dϕ We have g (e 1 h (ϕ+iψ) a) = e 1 h ϕ P ϕ (e i h ψ a) = e 1 h (h (ϕ+iψ) 0 p ϕ (x, dψ) [ + 2h (grad g ϕ + igrad g ψ)a + 1 ] 2 g(ϕ + iψ)a ) + h 2 g a. Eikonal equation: p ϕ (x, dψ) = 0 Transport equation: (grad g ϕ + igrad g ψ)a g(ϕ + iψ)a = 0 David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 8 / 24

19 Complex Geometrical Optics Complex geometrical optics (eikonal equation) We suppose that the metric has the form ( ) 1 0 g(x) = c(x) 0 g 0 (x ) and we choose ϕ(x) = x 1. We have dϕ = dx 1 and grad g ϕ = c 1 x1. Eikonal equation: dψ 2 g = dϕ 2 g = c 1 dϕ, dψ g = c 1 x1 ψ We choose ψ to be independent of x 1, and solution to dψ g0 = 1. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 9 / 24

20 Complex Geometrical Optics Complex geometrical optics (eikonal equation) We suppose that the metric has the form ( ) 1 0 g(x) = c(x) 0 g 0 (x ) and we choose ϕ(x) = x 1. We have dϕ = dx 1 and grad g ϕ = c 1 x1. Eikonal equation: dψ 2 g = dϕ 2 g = c 1 dϕ, dψ g = c 1 x1 ψ We choose ψ to be independent of x 1, and solution to dψ g0 = 1. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 9 / 24

21 Complex Geometrical Optics Complex geometrical optics (eikonal equation) Eikonal equation: dψ 2 g 0 = 1 In simple manifolds, it is easy to give explicit solutions of this eikonal equation ψ(x) = d g0 (x, ω 0 ), ω 0 M \ M where d g0 is the geodesical distance ( M is a simple extension of M). We have grad g0 ψ = (dψ) = exp 1 ω 0 (x ) ψ(x ) In fact, one can use geodesical polar coordinates x = exp ω0 (rθ), r = d g0 (x, ω 0 ) > 0, θ S ω0 M. In those coordinates, the metric reads g 0 = dr 2 + h 0 (r) David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 10 / 24

22 Complex Geometrical Optics Complex geometrical optics (eikonal equation) Eikonal equation: dψ 2 g 0 = 1 In simple manifolds, it is easy to give explicit solutions of this eikonal equation ψ(x) = d g0 (x, ω 0 ), ω 0 M \ M where d g0 is the geodesical distance ( M is a simple extension of M). We have grad g0 ψ = (dψ) = exp 1 ω 0 (x ) ψ(x ) In fact, one can use geodesical polar coordinates x = exp ω0 (rθ), r = d g0 (x, ω 0 ) > 0, θ S ω0 M. In those coordinates, the metric reads g 0 = dr 2 + h 0 (r) David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 10 / 24

23 Complex Geometrical Optics Complex geometrical optics (eikonal equation) Eikonal equation: dψ 2 g 0 = 1 In simple manifolds, it is easy to give explicit solutions of this eikonal equation ψ(x) = d g0 (x, ω 0 ), ω 0 M \ M where d g0 is the geodesical distance ( M is a simple extension of M). We have grad g0 ψ = (dψ) = exp 1 ω 0 (x ) ψ(x ) In fact, one can use geodesical polar coordinates x = exp ω0 (rθ), r = d g0 (x, ω 0 ) > 0, θ S ω0 M. In those coordinates, the metric reads g 0 = dr 2 + h 0 (r) David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 10 / 24

24 Complex Geometrical Optics Complex geometrical optics (transport equation) Transport equation: c 1 x1 a + igrad g ψa g(ϕ + iψ)a = 0 In polar coordinates grad g0 ψ = r, L gradg0 ψ = r, g0 ψ = 1 ( 1/2 ψ ) g 0 1/2 g 0 = 1 r r 2 r log g 0 hence the transport equation reads and can easily be solved where h is a holomorphic function. x1 a + r a r log g 0 a = 0 a = h(x 1 + ir) g 0 1/4 b(θ) David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 11 / 24

25 Complex Geometrical Optics Complex geometrical optics (transport equation) Transport equation: c 1 x1 a + igrad g ψa g(ϕ + iψ)a = 0 In polar coordinates grad g0 ψ = r, L gradg0 ψ = r, g0 ψ = 1 ( 1/2 ψ ) g 0 1/2 g 0 = 1 r r 2 r log g 0 hence the transport equation reads and can easily be solved where h is a holomorphic function. x1 a + r a r log g 0 a = 0 a = h(x 1 + ir) g 0 1/4 b(θ) David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 11 / 24

26 Complex Geometrical Optics Complex geometrical optics (L p case) Proposition Assume that q L n/2 (M). Let ω M 0 \ M 0 be a fixed point, let λ R be fixed, and let b C (S n 2 ) be a function. Write x = (x 1, r, θ) where (r, θ) are polar normal coordinates with center ω in ( M 0, g 0 ). For τ sufficiently large outside a countable set, there exists u 0 H 1 (M) satisfying where r 0 satisfies ( g + q)u = 0 in M, u = e τx 1 (e iτr g 1/4 e iλ(x 1+ir) b(θ) + r) τ r L 2 (M) + r H 1 (M) + r 2n 1. L n 2 (M) David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 12 / 24

27 Outline Attenuated X-ray transform 1 Introduction 2 Complex Geometrical Optics 3 Attenuated X-ray transform 4 Uniqueness of unbounded potentials David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 13 / 24

28 Attenuated X-ray transform Simple manifolds Definition A compact manifold (M 0, g 0 ) with boundary is simple if for any p M 0 the exponential map exp p with its maximal domain of definition is a diffeomorphism onto M 0, and if M 0 is strictly convex (that is, the second fundamental form of M 0 M 0 is positive definite). 1 Simple manifolds are non-trapping. 2 Simple manifolds are diffeomorphic to a ball. 3 A hemisphere is not simple. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 14 / 24

29 Attenuated X-ray transform Simple manifolds Definition A compact manifold (M 0, g 0 ) with boundary is simple if for any p M 0 the exponential map exp p with its maximal domain of definition is a diffeomorphism onto M 0, and if M 0 is strictly convex (that is, the second fundamental form of M 0 M 0 is positive definite). 1 Simple manifolds are non-trapping. 2 Simple manifolds are diffeomorphic to a ball. 3 A hemisphere is not simple. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 14 / 24

30 Attenuated X-ray transform Simple manifolds Definition A compact manifold (M 0, g 0 ) with boundary is simple if for any p M 0 the exponential map exp p with its maximal domain of definition is a diffeomorphism onto M 0, and if M 0 is strictly convex (that is, the second fundamental form of M 0 M 0 is positive definite). 1 Simple manifolds are non-trapping. 2 Simple manifolds are diffeomorphic to a ball. 3 A hemisphere is not simple. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 14 / 24

31 Attenuated X-ray transform Simple manifolds Definition A compact manifold (M 0, g 0 ) with boundary is simple if for any p M 0 the exponential map exp p with its maximal domain of definition is a diffeomorphism onto M 0, and if M 0 is strictly convex (that is, the second fundamental form of M 0 M 0 is positive definite). 1 Simple manifolds are non-trapping. 2 Simple manifolds are diffeomorphic to a ball. 3 A hemisphere is not simple. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 14 / 24

32 Attenuated X-ray transform Attenuated X-ray transform The unit sphere bundle : SM 0 = { S x, S x = (x, ξ) Tx M 0 ; ξ g = 1 }. x M 0 Boundary: (SM 0 ) = {(x, ξ) SM 0 ; x M 0 } union of inward and outward pointing vectors: ± (SM 0 ) = { (x, ξ) SM 0 ; ± ξ, ν 0 }. Denote by t γ(t, x, ξ) the unit speed geodesic starting at x in direction ξ, and let τ(x, ξ) be the time when this geodesic exits M 0. Godesic ray transform with constant attenuation λ: T λ f(x, ξ) = τ(x,ξ) 0 f(γ(t, x, ξ))e λt dt, (x, ξ) + (SM 0 ). David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 15 / 24

33 Attenuated X-ray transform Attenuated X-ray transform The unit sphere bundle : SM 0 = { S x, S x = (x, ξ) Tx M 0 ; ξ g = 1 }. x M 0 Boundary: (SM 0 ) = {(x, ξ) SM 0 ; x M 0 } union of inward and outward pointing vectors: ± (SM 0 ) = { (x, ξ) SM 0 ; ± ξ, ν 0 }. Denote by t γ(t, x, ξ) the unit speed geodesic starting at x in direction ξ, and let τ(x, ξ) be the time when this geodesic exits M 0. Godesic ray transform with constant attenuation λ: T λ f(x, ξ) = τ(x,ξ) 0 f(γ(t, x, ξ))e λt dt, (x, ξ) + (SM 0 ). David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 15 / 24

34 Attenuated X-ray transform Attenuated X-ray transform The unit sphere bundle : SM 0 = { S x, S x = (x, ξ) Tx M 0 ; ξ g = 1 }. x M 0 Boundary: (SM 0 ) = {(x, ξ) SM 0 ; x M 0 } union of inward and outward pointing vectors: ± (SM 0 ) = { (x, ξ) SM 0 ; ± ξ, ν 0 }. Denote by t γ(t, x, ξ) the unit speed geodesic starting at x in direction ξ, and let τ(x, ξ) be the time when this geodesic exits M 0. Godesic ray transform with constant attenuation λ: T λ f(x, ξ) = τ(x,ξ) 0 f(γ(t, x, ξ))e λt dt, (x, ξ) + (SM 0 ). David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 15 / 24

35 Attenuated X-ray transform Injectivity of the attenuated X-ray transform Proposition Let (M 0, g 0 ) be a simple manifold. There exists ε > 0 such that if λ is a real number with λ < ε and if f C (M), then the condition T λ f(x, ξ) = 0 for all (x, ξ) + (SM 0 ) implies that f = 0. This was known when λ = 0. The proof in the attenuated case uses perturbation arguments. What about nonsmooth functions? David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 16 / 24

36 Attenuated X-ray transform Injectivity of the attenuated X-ray transform Proposition Let (M 0, g 0 ) be a simple manifold. There exists ε > 0 such that if λ is a real number with λ < ε and if f C (M), then the condition T λ f(x, ξ) = 0 for all (x, ξ) + (SM 0 ) implies that f = 0. This was known when λ = 0. The proof in the attenuated case uses perturbation arguments. What about nonsmooth functions? David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 16 / 24

37 Attenuated X-ray transform Injectivity of the attenuated X-ray transform Proposition Let (M 0, g 0 ) be a simple manifold. There exists ε > 0 such that if λ is a real number with λ < ε and if f C (M), then the condition T λ f(x, ξ) = 0 for all (x, ξ) + (SM 0 ) implies that f = 0. This was known when λ = 0. The proof in the attenuated case uses perturbation arguments. What about nonsmooth functions? David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 16 / 24

38 Normal operator Attenuated X-ray transform Notations: µ(x, ξ) = ξ, ν(x) and dn is the volume form on N. Scalar product: (h, h) L 2 µ ( + (SM 0 )) = h hµ d( (SM 0 )) Adjoint of the ray transform: + (SM 0 ) T λ h(x) = S x e λτ(x, ξ) h(ϕ τ(x, ξ) (x, ξ)), ds x (ξ), x M 0. where ϕ t (x, ξ) = (γ(t, x, ξ), γ(t, x, ξ)) is the geodesic flow. Lemma T λ T λ is a self-adjoint elliptic pseudodifferential operator of order 1 in M int 0. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 17 / 24

39 Normal operator Attenuated X-ray transform Notations: µ(x, ξ) = ξ, ν(x) and dn is the volume form on N. Scalar product: (h, h) L 2 µ ( + (SM 0 )) = h hµ d( (SM 0 )) Adjoint of the ray transform: + (SM 0 ) T λ h(x) = S x e λτ(x, ξ) h(ϕ τ(x, ξ) (x, ξ)), ds x (ξ), x M 0. where ϕ t (x, ξ) = (γ(t, x, ξ), γ(t, x, ξ)) is the geodesic flow. Lemma T λ T λ is a self-adjoint elliptic pseudodifferential operator of order 1 in M int 0. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 17 / 24

40 Normal operator Attenuated X-ray transform Notations: µ(x, ξ) = ξ, ν(x) and dn is the volume form on N. Scalar product: (h, h) L 2 µ ( + (SM 0 )) = h hµ d( (SM 0 )) Adjoint of the ray transform: + (SM 0 ) T λ h(x) = S x e λτ(x, ξ) h(ϕ τ(x, ξ) (x, ξ)), ds x (ξ), x M 0. where ϕ t (x, ξ) = (γ(t, x, ξ), γ(t, x, ξ)) is the geodesic flow. Lemma T λ T λ is a self-adjoint elliptic pseudodifferential operator of order 1 in M int 0. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 17 / 24

41 Normal operator Attenuated X-ray transform Notations: µ(x, ξ) = ξ, ν(x) and dn is the volume form on N. Scalar product: (h, h) L 2 µ ( + (SM 0 )) = h hµ d( (SM 0 )) Adjoint of the ray transform: + (SM 0 ) T λ h(x) = S x e λτ(x, ξ) h(ϕ τ(x, ξ) (x, ξ)), ds x (ξ), x M 0. where ϕ t (x, ξ) = (γ(t, x, ξ), γ(t, x, ξ)) is the geodesic flow. Lemma T λ T λ is a self-adjoint elliptic pseudodifferential operator of order 1 in M int 0. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 17 / 24

42 Attenuated X-ray transform Injectivity of the attenuated X-ray transform (non-smooth case) Lemma Let (M 0, g 0 ) be an (n 1)-dimensional simple manifold, and let f L 1 (M 0 ). Consider the integrals S n 2 τ(ω,θ) 0 f(r, θ)e λr b(θ) dr dθ where (r, θ) are polar normal coordinates in (M 0, g 0 ) centered at some ω M 0, and τ(ω, θ) is the time when the geodesic r (r, θ) exits M 0. If λ is sufficiently small, and if these integrals vanish for all ω M 0 and all b C (S n 2 ), then f = 0. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 18 / 24

43 Attenuated X-ray transform Using Elliptic regularity Preliminary step: extend (M 0, g 0 ) to a slightly larger simple manifold and f by zero. In this way we can assume that f is compactly supported in M int 0. Let b also depend on ω and change notations to write τ(x,ξ) e λt f(γ(t, x, ξ))b(x, ξ) dt ds x (ξ) = 0. S x 0 Next we make the choice b(x, ξ) = h(x, ξ)µ(x, ξ) and integrate the last identity over M 0 to obtain τ(x,ξ) e λt f(γ(t, x, ξ))h(x, ξ)µ dt d( (SM 0 )) = 0. + (SM 0 ) 0 By adjunction, we get f(x)tλ h(x) dv (x) = 0 M 0 for all h C 0 (( +(SM 0 )) int ). David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 19 / 24

44 Attenuated X-ray transform Using Elliptic regularity Preliminary step: extend (M 0, g 0 ) to a slightly larger simple manifold and f by zero. In this way we can assume that f is compactly supported in M int 0. Let b also depend on ω and change notations to write τ(x,ξ) e λt f(γ(t, x, ξ))b(x, ξ) dt ds x (ξ) = 0. S x 0 Next we make the choice b(x, ξ) = h(x, ξ)µ(x, ξ) and integrate the last identity over M 0 to obtain τ(x,ξ) e λt f(γ(t, x, ξ))h(x, ξ)µ dt d( (SM 0 )) = 0. + (SM 0 ) 0 By adjunction, we get f(x)tλ h(x) dv (x) = 0 M 0 for all h C 0 (( +(SM 0 )) int ). David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 19 / 24

45 Attenuated X-ray transform Using Elliptic regularity Preliminary step: extend (M 0, g 0 ) to a slightly larger simple manifold and f by zero. In this way we can assume that f is compactly supported in M int 0. Let b also depend on ω and change notations to write τ(x,ξ) e λt f(γ(t, x, ξ))b(x, ξ) dt ds x (ξ) = 0. S x 0 Next we make the choice b(x, ξ) = h(x, ξ)µ(x, ξ) and integrate the last identity over M 0 to obtain τ(x,ξ) e λt f(γ(t, x, ξ))h(x, ξ)µ dt d( (SM 0 )) = 0. + (SM 0 ) 0 By adjunction, we get f(x)tλ h(x) dv (x) = 0 M 0 for all h C 0 (( +(SM 0 )) int ). David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 19 / 24

46 Attenuated X-ray transform Using Elliptic regularity Choose h = T λ ϕ for ϕ C0 Since T λ T λ is self-adjoint, we have (M int 0 ) so that M 0 f(x)t λ T λϕ(x) dv (x) = 0. M 0 (T λ T λf(x))ϕ(x) dv (x) = 0 for all test functions ϕ, so T λ T λf = 0. By ellipticity, since f was compactly supported in M0 int, it follows that f C0 (M int 0 ). One can now use the injectivity result for f smooth to conclude that f = 0. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 20 / 24

47 Attenuated X-ray transform Using Elliptic regularity Choose h = T λ ϕ for ϕ C0 Since T λ T λ is self-adjoint, we have (M int 0 ) so that M 0 f(x)t λ T λϕ(x) dv (x) = 0. M 0 (T λ T λf(x))ϕ(x) dv (x) = 0 for all test functions ϕ, so T λ T λf = 0. By ellipticity, since f was compactly supported in M0 int, it follows that f C0 (M int 0 ). One can now use the injectivity result for f smooth to conclude that f = 0. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 20 / 24

48 Outline Uniqueness of unbounded potentials 1 Introduction 2 Complex Geometrical Optics 3 Attenuated X-ray transform 4 Uniqueness of unbounded potentials David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 21 / 24

49 Using CGOs Uniqueness of unbounded potentials Starting point: If q = q 1 q 2 where M qu 1 u 2 dv g = 0 with b C (S n 2 ) and u 1 = e τ(x 1+ir) ( g 1/4 e iλ(x 1+ir) b(θ) + r 1 ), u 2 = e τ(x 1+ir) ( g 1/4 + r 2 ). r j L 2n n 2 (M) = O(1), r j L 2 (M) = o(1) as τ. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 22 / 24

50 Using CGOs Uniqueness of unbounded potentials Noting that dv g = g 1/2 dx 1 dr dθ, we obtain that qe iλ(x1+ir) b(θ) dx 1 dr dθ = q(a 1 r 2 + a 2 r 1 + r 1 r 2 ) dv M The RHS converges to 0 as τ. Taking the limit as τ, we obtain that 0 M S n 2 q(x 1, r, θ)e iλ(x 1+ir) b(θ) dx 1 dr dθ = 0. This statement is true for all choices of ω M 0 \ M 0, for all real numbers λ, and for all functions b C (S n 2 ). Hence f λ (r, θ)e λr b(θ) dr dθ = 0 S n 2 where f λ L 1 (M 0 ) is the function given by 0 f λ (r, θ) = e iλx 1 q(x 1, r, θ) dx 1. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 23 / 24

51 Using CGOs Uniqueness of unbounded potentials Noting that dv g = g 1/2 dx 1 dr dθ, we obtain that qe iλ(x1+ir) b(θ) dx 1 dr dθ = q(a 1 r 2 + a 2 r 1 + r 1 r 2 ) dv M The RHS converges to 0 as τ. Taking the limit as τ, we obtain that 0 M S n 2 q(x 1, r, θ)e iλ(x 1+ir) b(θ) dx 1 dr dθ = 0. This statement is true for all choices of ω M 0 \ M 0, for all real numbers λ, and for all functions b C (S n 2 ). Hence f λ (r, θ)e λr b(θ) dr dθ = 0 S n 2 where f λ L 1 (M 0 ) is the function given by 0 f λ (r, θ) = e iλx 1 q(x 1, r, θ) dx 1. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 23 / 24

52 Using CGOs Uniqueness of unbounded potentials Noting that dv g = g 1/2 dx 1 dr dθ, we obtain that qe iλ(x1+ir) b(θ) dx 1 dr dθ = q(a 1 r 2 + a 2 r 1 + r 1 r 2 ) dv M The RHS converges to 0 as τ. Taking the limit as τ, we obtain that 0 M S n 2 q(x 1, r, θ)e iλ(x 1+ir) b(θ) dx 1 dr dθ = 0. This statement is true for all choices of ω M 0 \ M 0, for all real numbers λ, and for all functions b C (S n 2 ). Hence f λ (r, θ)e λr b(θ) dr dθ = 0 S n 2 where f λ L 1 (M 0 ) is the function given by 0 f λ (r, θ) = e iλx 1 q(x 1, r, θ) dx 1. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 23 / 24

53 End of the proof Uniqueness of unbounded potentials If λ is sufficiently small, it follows that f λ = 0. Since q(, r, θ) is a compactly supported function in L 1 (R) for a.e. (r, θ), the Paley-Wiener theorem shows that q(, r, θ) = 0 for such (r, θ). Consequently q 1 = q 2. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 24 / 24

54 End of the proof Uniqueness of unbounded potentials If λ is sufficiently small, it follows that f λ = 0. Since q(, r, θ) is a compactly supported function in L 1 (R) for a.e. (r, θ), the Paley-Wiener theorem shows that q(, r, θ) = 0 for such (r, θ). Consequently q 1 = q 2. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 24 / 24

55 End of the proof Uniqueness of unbounded potentials If λ is sufficiently small, it follows that f λ = 0. Since q(, r, θ) is a compactly supported function in L 1 (R) for a.e. (r, θ), the Paley-Wiener theorem shows that q(, r, θ) = 0 for such (r, θ). Consequently q 1 = q 2. David Dos Santos Ferreira (LAGA) Inverse Problems 4 ICMAT 24 / 24

### Microlocal analysis and inverse problems Lecture 3 : Carleman estimates

Microlocal analysis and inverse problems ecture 3 : Carleman estimates David Dos Santos Ferreira AGA Université de Paris 13 Monday May 16 Instituto de Ciencias Matemáticas, Madrid David Dos Santos Ferreira

### THE CALDERÓN PROBLEM AND NORMAL FORMS

THE CALDERÓN PROBLEM AND NORMAL FORMS MIKKO SALO Abstract. We outline an approach to the inverse problem of Calderón that highlights the role of microlocal normal forms and propagation of singularities

### An inverse source problem in optical molecular imaging

An inverse source problem in optical molecular imaging Plamen Stefanov 1 Gunther Uhlmann 2 1 2 University of Washington Formulation Direct Problem Singular Operators Inverse Problem Proof Conclusion Figure:

### Hyperbolic inverse problems and exact controllability

Hyperbolic inverse problems and exact controllability Lauri Oksanen University College London An inverse initial source problem Let M R n be a compact domain with smooth strictly convex boundary, and let

### THE ATTENUATED RAY TRANSFORM ON SIMPLE SURFACES

THE ATTENUATED RAY TRANSFORM ON SIMPLE SURFACES MIKKO SALO AND GUNTHER UHLMANN Abstract. We show that the attenuated geodesic ray transform on two dimensional simple surfaces is injective. Moreover we

### DETERMINING A MAGNETIC SCHRÖDINGER OPERATOR FROM PARTIAL CAUCHY DATA

DETERMINING A MAGNETIC SCHRÖDINGER OPERATOR FROM PARTIAL CAUCHY DATA DAVID DOS SANTOS FERREIRA, CARLOS E. KENIG, JOHANNES SJÖSTRAND, GUNTHER UHLMANN Abstract. In this paper we show, in dimension n 3, that

### DETERMINING A FIRST ORDER PERTURBATION OF THE BIHARMONIC OPERATOR BY PARTIAL BOUNDARY MEASUREMENTS

DETERMINING A FIRST ORDER PERTURBATION OF THE BIHARMONIC OPERATOR BY PARTIAL BOUNDARY MEASUREMENTS KATSIARYNA KRUPCHYK, MATTI LASSAS, AND GUNTHER UHLMANN Abstract. We consider an operator 2 + A(x) D +

### Recent progress on the explicit inversion of geodesic X-ray transforms

Recent progress on the explicit inversion of geodesic X-ray transforms François Monard Department of Mathematics, University of Washington. Geometric Analysis and PDE seminar University of Cambridge, May

### COMPLEX SPHERICAL WAVES AND INVERSE PROBLEMS IN UNBOUNDED DOMAINS

COMPLEX SPHERICAL WAVES AND INVERSE PROBLEMS IN UNBOUNDED DOMAINS MIKKO SALO AND JENN-NAN WANG Abstract. This work is motivated by the inverse conductivity problem of identifying an embedded object in

### INVERSE BOUNDARY VALUE PROBLEMS FOR THE MAGNETIC SCHRÖDINGER EQUATION

INVERSE BOUNDARY VALUE PROBLEMS FOR THE MAGNETIC SCHRÖDINGER EQUATION MIKKO SALO Abstract. We survey recent results on inverse boundary value problems for the magnetic Schrödinger equation. 1. Introduction

### THE ATTENUATED RAY TRANSFORM FOR CONNECTIONS AND HIGGS FIELDS

THE ATTENUATED RAY TRANSFORM FOR CONNECTIONS AND HIGGS FIELDS GABRIEL P. PATERNAIN, MIKKO SALO, AND GUNTHER UHLMANN Abstract. We show that for a simple surface with boundary the attenuated ray transform

### A SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY

A SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY PLAMEN STEFANOV 1. Introduction Let (M, g) be a compact Riemannian manifold with boundary. The geodesic ray transform I of symmetric 2-tensor fields f is

### THE CALDERÓN PROBLEM FOR CONNECTIONS. Mihajlo Cekić

THE CALDERÓN PROBLEM FOR CONNECTIONS Mihajlo Cekić Trinity College and Department of Pure Mathematics and Mathematical Statistics University of Cambridge The dissertation is submitted for the degree of

### Pseudo-Poincaré Inequalities and Applications to Sobolev Inequalities

Pseudo-Poincaré Inequalities and Applications to Sobolev Inequalities Laurent Saloff-Coste Abstract Most smoothing procedures are via averaging. Pseudo-Poincaré inequalities give a basic L p -norm control

### Before you begin read these instructions carefully.

MATHEMATICAL TRIPOS Part IB Thursday, 6 June, 2013 9:00 am to 12:00 pm PAPER 3 Before you begin read these instructions carefully. Each question in Section II carries twice the number of marks of each

### Recovery of anisotropic metrics from travel times

Purdue University The Lens Rigidity and the Boundary Rigidity Problems Let M be a bounded domain with boundary. Let g be a Riemannian metric on M. Define the scattering relation σ and the length (travel

### A few words about the MTW tensor

A few words about the Ma-Trudinger-Wang tensor Université Nice - Sophia Antipolis & Institut Universitaire de France Salah Baouendi Memorial Conference (Tunis, March 2014) The Ma-Trudinger-Wang tensor

### A local estimate from Radon transform and stability of Inverse EIT with partial data

A local estimate from Radon transform and stability of Inverse EIT with partial data Alberto Ruiz Universidad Autónoma de Madrid U. California, Irvine.June 2012 w/ P. Caro (U. Helsinki) and D. Dos Santos

### Inverse problems for hyperbolic PDEs

Inverse problems for hyperbolic PDEs Lauri Oksanen University College London Example: inverse problem for the wave equation Let c be a smooth function on Ω R n and consider the wave equation t 2 u c 2

### The Karcher Mean of Points on SO n

The Karcher Mean of Points on SO n Knut Hüper joint work with Jonathan Manton (Univ. Melbourne) Knut.Hueper@nicta.com.au National ICT Australia Ltd. CESAME LLN, 15/7/04 p.1/25 Contents Introduction CESAME

### Wave equation on manifolds and finite speed of propagation

Wave equation on manifolds and finite speed of propagation Ethan Y. Jaffe Let M be a Riemannian manifold (without boundary), and let be the (negative of) the Laplace-Beltrami operator. In this note, we

### LOCAL LENS RIGIDITY WITH INCOMPLETE DATA FOR A CLASS OF NON-SIMPLE RIEMANNIAN MANIFOLDS

LOCAL LENS RIGIDITY WITH INCOMPLETE DATA FOR A CLASS OF NON-SIMPLE RIEMANNIAN MANIFOLDS PLAMEN STEFANOV AND GUNTHER UHLMANN Abstract. Let σ be the scattering relation on a compact Riemannian manifold M

### On stable inversion of the attenuated Radon transform with half data Jan Boman. We shall consider weighted Radon transforms of the form

On stable inversion of the attenuated Radon transform with half data Jan Boman We shall consider weighted Radon transforms of the form R ρ f(l) = f(x)ρ(x, L)ds, L where ρ is a given smooth, positive weight

### LECTURE NOTES ON GEOMETRIC OPTICS

LECTURE NOTES ON GEOMETRIC OPTICS PLAMEN STEFANOV We want to solve the wave equation 1. The wave equation (1.1) ( 2 t c 2 g )u = 0, u t=0 = f 1, u t t=0 = f 2, in R t R n x, where g is a Riemannian metric

### APPLICATIONS OF DIFFERENTIABILITY IN R n.

APPLICATIONS OF DIFFERENTIABILITY IN R n. MATANIA BEN-ARTZI April 2015 Functions here are defined on a subset T R n and take values in R m, where m can be smaller, equal or greater than n. The (open) ball

### MICROLOCAL ANALYSIS METHODS

MICROLOCAL ANALYSIS METHODS PLAMEN STEFANOV One of the fundamental ideas of classical analysis is a thorough study of functions near a point, i.e., locally. Microlocal analysis, loosely speaking, is analysis

### Transport Continuity Property

On Riemannian manifolds satisfying the Transport Continuity Property Université de Nice - Sophia Antipolis (Joint work with A. Figalli and C. Villani) I. Statement of the problem Optimal transport on Riemannian

### Travel Time Tomography and Tensor Tomography, I

Travel Time Tomography and Tensor Tomography, I Plamen Stefanov Purdue University Mini Course, MSRI 2009 Plamen Stefanov (Purdue University ) Travel Time Tomography and Tensor Tomography, I 1 / 17 Alternative

### 1. Geometry of the unit tangent bundle

1 1. Geometry of the unit tangent bundle The main reference for this section is [8]. In the following, we consider (M, g) an n-dimensional smooth manifold endowed with a Riemannian metric g. 1.1. Notations

### LECTURE 15: COMPLETENESS AND CONVEXITY

LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other

### Deforming conformal metrics with negative Bakry-Émery Ricci Tensor on manifolds with boundary

Deforming conformal metrics with negative Bakry-Émery Ricci Tensor on manifolds with boundary Weimin Sheng (Joint with Li-Xia Yuan) Zhejiang University IMS, NUS, 8-12 Dec 2014 1 / 50 Outline 1 Prescribing

### THERMOACOUSTIC TOMOGRAPHY ARISING IN BRAIN IMAGING

THERMOACOUSTIC TOMOGRAPHY ARISING IN BRAIN IMAGING PLAMEN STEFANOV AND GUNTHER UHLMANN Abstract. We study the mathematical model of thermoacoustic and photoacoustic tomography when the sound speed has

### THE BOUNDARY RIGIDITY PROBLEM IN THE PRESENCE OF A MAGNETIC FIELD

THE BOUNDARY RIGIDITY PROBLEM IN THE PRESENCE OF A MAGNETIC FIELD NURLAN S. DAIRBEKOV, GABRIEL P. PATERNAIN, PLAMEN STEFANOV, AND GUNTHER UHLMANN Abstract. For a compact Riemannian manifold with boundary,

### On L p resolvent and Carleman estimates on compacts manifolds

On L p resolvent and Carleman estimates on compacts manifolds David Dos Santos Ferreira Institut Élie Cartan Université de Lorraine Three days on Analysis and PDEs ICMAT, Tuesday June 3rd Collaborators

### Numerical Methods for geodesic X-ray transforms and applications to open theoretical questions

Numerical Methods for geodesic X-ray transforms and applications to open theoretical questions François Monard Department of Mathematics, University of Washington. Nov. 13, 2014 UW Numerical Analysis Research

### The oblique derivative problem for general elliptic systems in Lipschitz domains

M. MITREA The oblique derivative problem for general elliptic systems in Lipschitz domains Let M be a smooth, oriented, connected, compact, boundaryless manifold of real dimension m, and let T M and T

### fy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f))

1. Basic algebra of vector fields Let V be a finite dimensional vector space over R. Recall that V = {L : V R} is defined to be the set of all linear maps to R. V is isomorphic to V, but there is no canonical

### PICARD S THEOREM STEFAN FRIEDL

PICARD S THEOREM STEFAN FRIEDL Abstract. We give a summary for the proof of Picard s Theorem. The proof is for the most part an excerpt of [F]. 1. Introduction Definition. Let U C be an open subset. A

### Complex geometrical optics solutions for Lipschitz conductivities

Rev. Mat. Iberoamericana 19 (2003), 57 72 Complex geometrical optics solutions for Lipschitz conductivities Lassi Päivärinta, Alexander Panchenko and Gunther Uhlmann Abstract We prove the existence of

### Local semiconvexity of Kantorovich potentials on non-compact manifolds

Local semiconvexity of Kantorovich potentials on non-compact manifolds Alessio Figalli, Nicola Gigli Abstract We prove that any Kantorovich potential for the cost function c = d / on a Riemannian manifold

### GABRIEL P. PATERNAIN, MIKKO SALO, AND GUNTHER UHLMANN

ON THE RANGE OF THE ATTENUATED RAY TRANSFORM FOR UNITARY CONNECTIONS GABRIEL P. PATERNAIN, MIKKO SALO, AND GUNTHER UHLMANN Abstract. We describe the range of the attenuated ray transform of a unitary connection

### The Schrödinger propagator for scattering metrics

The Schrödinger propagator for scattering metrics Andrew Hassell (Australian National University) joint work with Jared Wunsch (Northwestern) MSRI, May 5-9, 2003 http://arxiv.org/math.ap/0301341 1 Schrödinger

### KUIPER S THEOREM ON CONFORMALLY FLAT MANIFOLDS

KUIPER S THEOREM ON CONFORMALLY FLAT MANIFOLDS RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA HOWARD@MATH.SC.EDU 1. Introduction These are notes to that show

### arxiv: v1 [math.dg] 24 Feb 2017

GEODESIC X-RAY TOMOGRAPHY FOR PIECEWISE CONSTANT FUNCTIONS ON NONTRAPPING MANIFOLDS JOONAS ILMAVIRTA, JERE LEHTONEN, AND MIKKO SALO arxiv:1702.07622v1 [math.dg] 24 Feb 2017 Abstract. We show that on a

### The X-ray transform for a non-abelian connection in two dimensions

The X-ray transform for a non-abelian connection in two dimensions David Finch Department of Mathematics Oregon State University Corvallis, OR, 97331, USA Gunther Uhlmann Department of Mathematics University

### CALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =

CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.

### YERNAT M. ASSYLBEKOV AND PLAMEN STEFANOV

SHARP STABILITY ESTIMATE FOR THE GEODESIC RAY TRANSFORM YERNAT M. ASSYLBEKOV AND PLAMEN STEFANOV Abstract. We prove a sharp L 2 H stability estimate for the geodesic X-ray transform of tensor fields of

### THE LINEARIZED CALDERÓN PROBLEM IN TRANSVERSALLY ANISOTROPIC GEOMETRIES

THE LINEARIZED CALDERÓN PROBLEM IN TRANSVERSALLY ANISOTROPIC GEOMETRIES DAVID DOS SANTOS FERREIRA, YAROSLAV KURYLEV, MATTI LASSAS, TONY LIIMATAINEN, AND MIKKO SALO Abstract. In this article we study the

### ASYMPTOTIC BEHAVIOR OF GENERALIZED EIGENFUNCTIONS IN N-BODY SCATTERING

ASYMPTOTIC BEHAVIOR OF GENERALIZED EIGENFUNCTIONS IN N-BODY SCATTERING ANDRAS VASY Abstract. In this paper an asymptotic expansion is proved for locally (at infinity) outgoing functions on asymptotically

### Rigidity and Non-rigidity Results on the Sphere

Rigidity and Non-rigidity Results on the Sphere Fengbo Hang Xiaodong Wang Department of Mathematics Michigan State University Oct., 00 1 Introduction It is a simple consequence of the maximum principle

### Microlocal Analysis : a short introduction

Microlocal Analysis : a short introduction Plamen Stefanov Purdue University Mini Course, Fields Institute, 2012 Plamen Stefanov (Purdue University ) Microlocal Analysis : a short introduction 1 / 25 Introduction

### A new class of pseudodifferential operators with mixed homogenities

A new class of pseudodifferential operators with mixed homogenities Po-Lam Yung University of Oxford Jan 20, 2014 Introduction Given a smooth distribution of hyperplanes on R N (or more generally on a

### THE GEODESIC RAY TRANSFORM ON RIEMANNIAN SURFACES WITH CONJUGATE POINTS

THE GEODESIC RAY TRANSFORM ON RIEMANNIAN SURFACES WITH CONJUGATE POINTS FRANÇOIS MONARD, PLAMEN STEFANOV, AND GUNTHER UHLMANN Abstract. We study the geodesic X-ray transform X on compact Riemannian surfaces

### Analysis in weighted spaces : preliminary version

Analysis in weighted spaces : preliminary version Frank Pacard To cite this version: Frank Pacard. Analysis in weighted spaces : preliminary version. 3rd cycle. Téhéran (Iran, 2006, pp.75.

### Optimal Transportation. Nonlinear Partial Differential Equations

Optimal Transportation and Nonlinear Partial Differential Equations Neil S. Trudinger Centre of Mathematics and its Applications Australian National University 26th Brazilian Mathematical Colloquium 2007

### arxiv: v2 [math.dg] 26 Feb 2017

LOCAL AND GLOBAL BOUNDARY RIGIDITY AND THE GEODESIC X-RAY TRANSFORM IN THE NORMAL GAUGE PLAMEN STEFANOV, GUNTHER UHLMANN AND ANDRÁS VASY arxiv:170203638v2 [mathdg] 26 Feb 2017 Abstract In this paper we

### Some topics in sub-riemannian geometry

Some topics in sub-riemannian geometry Luca Rizzi CNRS, Institut Fourier Mathematical Colloquium Universität Bern - December 19 2016 Sub-Riemannian geometry Known under many names: Carnot-Carathéodory

### LOCAL AND GLOBAL BOUNDARY RIGIDITY AND THE GEODESIC X-RAY TRANSFORM IN THE NORMAL GAUGE

LOCAL AND GLOBAL BOUNDARY RIGIDITY AND THE GEODESIC X-RAY TRANSFORM IN THE NORMAL GAUGE PLAMEN STEFANOV, GUNTHER UHLMANN AND ANDRÁS VASY Abstract In this paper we analyze the local and global boundary

### Reduction of Homogeneous Riemannian structures

Geometric Structures in Mathematical Physics, 2011 Reduction of Homogeneous Riemannian structures M. Castrillón López 1 Ignacio Luján 2 1 ICMAT (CSIC-UAM-UC3M-UCM) Universidad Complutense de Madrid 2 Universidad

### Inégalités de dispersion via le semi-groupe de la chaleur

Inégalités de dispersion via le semi-groupe de la chaleur Valentin Samoyeau, Advisor: Frédéric Bernicot. Laboratoire de Mathématiques Jean Leray, Université de Nantes January 28, 2016 1 Introduction Schrödinger

### Nonlinear stabilization via a linear observability

via a linear observability Kaïs Ammari Department of Mathematics University of Monastir Joint work with Fathia Alabau-Boussouira Collocated feedback stabilization Outline 1 Introduction and main result

### 1 First and second variational formulas for area

1 First and second variational formulas for area In this chapter, we will derive the first and second variational formulas for the area of a submanifold. This will be useful in our later discussion on

### arxiv: v2 [math.ap] 13 Sep 2015

THE X-RAY TRANSFORM FOR CONNECTIONS IN NEGATIVE CURVATURE COLIN GUILLARMOU, GABRIEL P. PATERNAIN, MIKKO SALO, AND GUNTHER UHLMANN arxiv:1502.04720v2 [math.ap] 13 Sep 2015 Abstract. We consider integral

### Chap. 1. Some Differential Geometric Tools

Chap. 1. Some Differential Geometric Tools 1. Manifold, Diffeomorphism 1.1. The Implicit Function Theorem ϕ : U R n R n p (0 p < n), of class C k (k 1) x 0 U such that ϕ(x 0 ) = 0 rank Dϕ(x) = n p x U

### Topological properties of Z p and Q p and Euclidean models

Topological properties of Z p and Q p and Euclidean models Samuel Trautwein, Esther Röder, Giorgio Barozzi November 3, 20 Topology of Q p vs Topology of R Both R and Q p are normed fields and complete

### Microlocal Methods in X-ray Tomography

Microlocal Methods in X-ray Tomography Plamen Stefanov Purdue University Lecture I: Euclidean X-ray tomography Mini Course, Fields Institute, 2012 Plamen Stefanov (Purdue University ) Microlocal Methods

### satisfying the following condition: If T : V V is any linear map, then µ(x 1,,X n )= det T µ(x 1,,X n ).

ensities Although differential forms are natural objects to integrate on manifolds, and are essential for use in Stoke s theorem, they have the disadvantage of requiring oriented manifolds in order for

### BOUNDARY RIGIDITY AND STABILITY FOR GENERIC SIMPLE METRICS

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0894-0347(XX)0000-0 BOUNDARY RIGIDITY AND STABILITY FOR GENERIC SIMPLE METRICS PLAMEN STEFANOV AND GUNTHER UHLMANN 1. Introduction

### Finite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product

Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )

### Asymptotics of generalized eigenfunctions on manifold with Euclidean and/or hyperbolic ends

Asymptotics of generalized eigenfunctions on manifold with Euclidean and/or hyperbolic ends Kenichi ITO (University of Tokyo) joint work with Erik SKIBSTED (Aarhus University) 3 July 2018 Example: Free

### Transversality. Abhishek Khetan. December 13, Basics 1. 2 The Transversality Theorem 1. 3 Transversality and Homotopy 2

Transversality Abhishek Khetan December 13, 2017 Contents 1 Basics 1 2 The Transversality Theorem 1 3 Transversality and Homotopy 2 4 Intersection Number Mod 2 4 5 Degree Mod 2 4 1 Basics Definition. Let

### EXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018

EXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018 While these notes are under construction, I expect there will be many typos. The main reference for this is volume 1 of Hörmander, The analysis of liner

### LECTURE 16: CONJUGATE AND CUT POINTS

LECTURE 16: CONJUGATE AND CUT POINTS 1. Conjugate Points Let (M, g) be Riemannian and γ : [a, b] M a geodesic. Then by definition, exp p ((t a) γ(a)) = γ(t). We know that exp p is a diffeomorphism near

### INTRINSIC MEAN ON MANIFOLDS. Abhishek Bhattacharya Project Advisor: Dr.Rabi Bhattacharya

INTRINSIC MEAN ON MANIFOLDS Abhishek Bhattacharya Project Advisor: Dr.Rabi Bhattacharya 1 Overview Properties of Intrinsic mean on Riemannian manifolds have been presented. The results have been applied

### Quasi-conformal minimal Lagrangian diffeomorphisms of the

Quasi-conformal minimal Lagrangian diffeomorphisms of the hyperbolic plane (joint work with J.M. Schlenker) January 21, 2010 Quasi-symmetric homeomorphism of a circle A homeomorphism φ : S 1 S 1 is quasi-symmetric

### On the exponential map on Riemannian polyhedra by Monica Alice Aprodu. Abstract

Bull. Math. Soc. Sci. Math. Roumanie Tome 60 (108) No. 3, 2017, 233 238 On the exponential map on Riemannian polyhedra by Monica Alice Aprodu Abstract We prove that Riemannian polyhedra admit explicit

### Hamiltonian flows, cotangent lifts, and momentum maps

Hamiltonian flows, cotangent lifts, and momentum maps Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto April 3, 2014 1 Symplectic manifolds Let (M, ω) and (N, η) be symplectic

### Math Topology II: Smooth Manifolds. Spring Homework 2 Solution Submit solutions to the following problems:

Math 132 - Topology II: Smooth Manifolds. Spring 2017. Homework 2 Solution Submit solutions to the following problems: 1. Let H = {a + bi + cj + dk (a, b, c, d) R 4 }, where i 2 = j 2 = k 2 = 1, ij = k,

### A local estimate from Radon transform and stability of Inverse EIT with partial data

A local estimate from Radon transform and stability of Inverse EIT with partial data Alberto Ruiz Universidad Autónoma de Madrid ge Joint work with P. Caro (U. Helsinki) and D. Dos Santos Ferreira (Paris

### MORERA THEOREMS VIA MICROLOCAL ANALYSIS. Josip Globevnik and Eric Todd Quinto

MORERA THEOREMS VIA MICROLOCAL ANALYSIS Josip Globevnik and Eric Todd Quinto Abstract. We prove Morera theorems for curves in the plane using microlocal analysis. The key is that microlocal smoothness

### Vector fields Lecture 2

Vector fields Lecture 2 Let U be an open subset of R n and v a vector field on U. We ll say that v is complete if, for every p U, there exists an integral curve, γ : R U with γ(0) = p, i.e., for every

### THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS

THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS OGNJEN MILATOVIC Abstract. We consider H V = M +V, where (M, g) is a Riemannian manifold (not necessarily

### 1 )(y 0) {1}. Thus, the total count of points in (F 1 (y)) is equal to deg y0

1. Classification of 1-manifolds Theorem 1.1. Let M be a connected 1 manifold. Then M is diffeomorphic either to [0, 1], [0, 1), (0, 1), or S 1. We know that none of these four manifolds are not diffeomorphic

### Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on

Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on manifolds. Author: Ognjen Milatovic Department Address: Department

### The inverse conductivity problem with power densities in dimension n 2

The inverse conductivity problem with power densities in dimension n 2 François Monard Guillaume Bal Dept. of Applied Physics and Applied Mathematics, Columbia University. June 19th, 2012 UC Irvine Conference

### Notes on quotients and group actions

Notes on quotients and group actions Erik van den Ban Fall 2006 1 Quotients Let X be a topological space, and R an equivalence relation on X. The set of equivalence classes for this relation is denoted

### Distances, volumes, and integration

Distances, volumes, and integration Scribe: Aric Bartle 1 Local Shape of a Surface A question that we may ask ourselves is what significance does the second fundamental form play in the geometric characteristics

### Differential Geometry Exercises

Differential Geometry Exercises Isaac Chavel Spring 2006 Jordan curve theorem We think of a regular C 2 simply closed path in the plane as a C 2 imbedding of the circle ω : S 1 R 2. Theorem. Given the

### Virasoro hair on locally AdS 3 geometries

Virasoro hair on locally AdS 3 geometries Kavli Institute for Theoretical Physics China Institute of Theoretical Physics ICTS (USTC) arxiv: 1603.05272, M. M. Sheikh-Jabbari and H. Y Motivation Introduction

### ERRATUM TO AFFINE MANIFOLDS, SYZ GEOMETRY AND THE Y VERTEX

ERRATUM TO AFFINE MANIFOLDS, SYZ GEOMETRY AND THE Y VERTEX JOHN LOFTIN, SHING-TUNG YAU, AND ERIC ZASLOW 1. Main result The purpose of this erratum is to correct an error in the proof of the main result

### Visibility estimates in Euclidean and hyperbolic germ-grain models and line tessellations

Visibility estimates in Euclidean and hyperbolic germ-grain models and line tessellations Pierre Calka Lille, Stochastic Geometry Workshop, 30 March 2011 Outline Visibility in the vacancy of the Boolean

### An Overview of Mathematical General Relativity

An Overview of Mathematical General Relativity José Natário (Instituto Superior Técnico) Geometria em Lisboa, 8 March 2005 Outline Lorentzian manifolds Einstein s equation The Schwarzschild solution Initial

### Exercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1

Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines

### Surfaces JWR. February 13, 2014

Surfaces JWR February 13, 214 These notes summarize the key points in the second chapter of Differential Geometry of Curves and Surfaces by Manfredo P. do Carmo. I wrote them to assure that the terminology

### Inversions of ray transforms on simple surfaces

Inversions of ray transforms on simple surfaces François Monard Department of Mathematics, University of Washington. June 09, 2015 Institut Henri Poincaré - Program on Inverse problems 1 / 42 Outline 1

### SYMPLECTIC GEOMETRY: LECTURE 5

SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The

### C 1 regularity of solutions of the Monge-Ampère equation for optimal transport in dimension two

C 1 regularity of solutions of the Monge-Ampère equation for optimal transport in dimension two Alessio Figalli, Grégoire Loeper Abstract We prove C 1 regularity of c-convex weak Alexandrov solutions of

### arxiv: v2 [math.ap] 8 Sep 2008

CARLEMAN ESTIMATES AND INVERSE PROBLEMS FOR DIRAC OPERATORS MIKKO SALO AND LEO TZOU arxiv:0709.2282v2 [math.ap] 8 Sep 2008 Abstract. We consider limiting Carleman weights for Dirac operators and prove