## Analysis & PDE

• Anal. PDE
• Volume 11, Number 6 (2018), 1381-1414.

### The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds

#### Abstract

We consider the Dirichlet-to-Neumann map $Λ$ on a cylinder-like Lorentzian manifold related to the wave equation related to the metric $g$, the magnetic field $A$ and the potential $q$. We show that we can recover the jet of $g , A , q$ on the boundary from $Λ$ up to a gauge transformation in a stable way. We also show that $Λ$ recovers the following three invariants in a stable way: the lens relation of $g$, and the light ray transforms of $A$ and $q$. Moreover, $Λ$ is an FIO away from the diagonal with a canonical relation given by the lens relation. We present applications for recovery of $A$ and $q$ in a logarithmically stable way in the Minkowski case, and uniqueness with partial data.

#### Article information

Source
Anal. PDE, Volume 11, Number 6 (2018), 1381-1414.

Dates
Revised: 6 January 2018
Accepted: 14 February 2018
First available in Project Euclid: 23 May 2018

https://projecteuclid.org/euclid.apde/1527040815

Digital Object Identifier
doi:10.2140/apde.2018.11.1381

Mathematical Reviews number (MathSciNet)
MR3803714

Zentralblatt MATH identifier
06881246

#### Citation

Stefanov, Plamen; Yang, Yang. The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds. Anal. PDE 11 (2018), no. 6, 1381--1414. doi:10.2140/apde.2018.11.1381. https://projecteuclid.org/euclid.apde/1527040815

#### References

• G. Bao and H. Zhang, “Sensitivity analysis of an inverse problem for the wave equation with caustics”, J. Amer. Math. Soc. 27:4 (2014), 953–981.
• A. K. Begmatov, “A certain inversion problem for the ray transform with incomplete data”, Sibirsk. Mat. Zh. 42:3 (2001), 507–514. In Russian; translated in Siberian Math. J. 42:3 (2001), 428–434.
• M. I. Belishev, “An approach to multidimensional inverse problems for the wave equation”, Dokl. Akad. Nauk SSSR 297:3 (1987), 524–527. In Russian.
• M. I. Belishev, “Recent progress in the boundary control method”, Inverse Problems 23:5 (2007), R1–R67.
• M. I. Belishev and Y. V. Kurylev, “To the reconstruction of a Riemannian manifold via its spectral data (BC-method)”, Comm. Partial Differential Equations 17:5-6 (1992), 767–804.
• M. Bellassoued and I. Ben Aïcha, “Stable determination outside a cloaking region of two time-dependent coefficients in an hyperbolic equation from Dirichlet to Neumann map”, J. Math. Anal. Appl. 449:1 (2017), 46–76.
• M. Bellassoued and D. Dos Santos Ferreira, “Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map”, Inverse Probl. Imaging 5:4 (2011), 745–773.
• I. Ben Aïcha, “Stability estimate for a hyperbolic inverse problem with time-dependent coefficient”, Inverse Problems 31:12 (2015), art. id. 125010.
• J. Boman and E. T. Quinto, “Support theorems for Radon transforms on real analytic line complexes in three-space”, Trans. Amer. Math. Soc. 335:2 (1993), 877–890.
• J. Cooper and W. Strauss, “The leading singularity of a wave reflected by a moving boundary”, J. Differential Equations 52:2 (1984), 175–203.
• N. S. Dairbekov, G. P. Paternain, P. Stefanov, and G. Uhlmann, “The boundary rigidity problem in the presence of a magnetic field”, Adv. Math. 216:2 (2007), 535–609.
• D. Dos Santos Ferreira, C. E. Kenig, M. Salo, and G. Uhlmann, “Limiting Carleman weights and anisotropic inverse problems”, Invent. Math. 178:1 (2009), 119–171.
• J. J. Duistermaat, Fourier integral operators, Progress in Mathematics 130, Birkhäuser, Boston, 1996.
• G. Eskin, “Inverse problems for general second order hyperbolic equations with time-dependent coefficients”, Bull. Math. Sci. 7:2 (2017), 247–307.
• G. Eskin and J. Ralston, “The determination of moving boundaries for hyperbolic equations”, Inverse Problems 26:1 (2010), art. id. 015001.
• A. Greenleaf and G. Uhlmann, “Nonlocal inversion formulas for the X-ray transform”, Duke Math. J. 58:1 (1989), 205–240.
• A. Greenleaf and G. Uhlmann, “Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms”, Ann. Inst. Fourier $($Grenoble$)$ 40:2 (1990), 443–466.
• A. Greenleaf and G. Uhlmann, “Microlocal techniques in integral geometry”, pp. 121–135 in Integral geometry and tomography (Arcata, CA, 1989), edited by E. Grinberg and E. T. Quinto, Contemp. Math. 113, Amer. Math. Soc., Providence, RI, 1990.
• L. Hörmander, The analysis of linear partial differential operators, III: Pseudodifferential operators, Grundlehren der Mathematischen Wissenschaften 274, Springer, 1985.
• L. Hörmander, The analysis of linear partial differential operators, IV: Fourier integral operators, Grundlehren der Mathematischen Wissenschaften 275, Springer, 1985.
• V. Isakov and Z. Q. Sun, “Stability estimates for hyperbolic inverse problems with local boundary data”, Inverse Problems 8:2 (1992), 193–206.
• Y. Kian, “Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data”, SIAM J. Math. Anal. 48:6 (2016), 4021–4046.
• Y. Kian, L. Oksanen, E. Soccorsi, and M. Yamamoto, “Global uniqueness in an inverse problem for time fractional diffusion equations”, J. Differential Equations 264:2 (2018), 1146–1170.
• Y. Kurylev, M. Lassas, and G. Uhlmann, “Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations”, preprint, 2014.
• Y. Kurylev, M. Lassas, and G. Uhlmann, “Inverse problems in spacetime, I: Inverse problems for Einstein equations (extended preprint version)”, preprint, 2014.
• M. Lassas, G. Uhlmann, and Y. Wang, “Inverse problems for semilinear wave equations on Lorentzian manifolds”, preprint, 2016.
• M. Lassas, L. Oksanen, P. Stefanov, and G. Uhlmann, “On the inverse problem of finding cosmic strings and other topological defects”, Comm. Math. Phys. (online publication November 2017).
• C. Montalto, “Stable determination of a simple metric, a covector field and a potential from the hyperbolic Dirichlet-to-Neumann map”, Comm. Partial Differential Equations 39:1 (2014), 120–145.
• R. S. Palais, “Extending diffeomorphisms”, Proc. Amer. Math. Soc. 11 (1960), 274–277.
• A. Z. Petrov, Einstein spaces, Pergamon Press, Oxford, 1969.
• S. RabieniaHaratbar, “Support theorem for the light ray transform on Minkoswki spaces”, preprint, 2017. To appear in Inverse Probl. Imaging.
• A. G. Ramm and Rakesh, “Property $C$ and an inverse problem for a hyperbolic equation”, J. Math. Anal. Appl. 156:1 (1991), 209–219.
• A. G. Ramm and J. Sjöstrand, “An inverse problem of the wave equation”, Math. Z. 206:1 (1991), 119–130.
• R. Salazar, “Determination of time-dependent coefficients for a hyperbolic inverse problem”, Inverse Problems 29:9 (2013), art. id. 095015.
• R. Salazar, “Stability estimate for the relativistic Schrödinger equation with time-dependent vector potentials”, Inverse Problems 30:10 (2014), art. id. 105005.
• P. D. Stefanov, “Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials”, Math. Z. 201:4 (1989), 541–559.
• P. D. Stefanov, “Inverse scattering problem for moving obstacles”, Math. Z. 207:3 (1991), 461–480.
• P. Stefanov, “Support theorems for the light ray transform on analytic Lorentzian manifolds”, Proc. Amer. Math. Soc. 145:3 (2017), 1259–1274.
• P. Stefanov and G. Uhlmann, “Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media”, J. Funct. Anal. 154:2 (1998), 330–358.
• P. Stefanov and G. Uhlmann, “Boundary rigidity and stability for generic simple metrics”, J. Amer. Math. Soc. 18:4 (2005), 975–1003.
• P. Stefanov and G. Uhlmann, “Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map”, Int. Math. Res. Not. 2005:17 (2005), 1047–1061.
• P. Stefanov, G. Uhlmann, and A. Vasy, “On the stable recovery of a metric from the hyperbolic DN map with incomplete data”, Inverse Probl. Imaging 10:4 (2016), 1141–1147.
• Z. Q. Sun, “On continuous dependence for an inverse initial-boundary value problem for the wave equation”, J. Math. Anal. Appl. 150:1 (1990), 188–204.
• D. Tataru, “Unique continuation for solutions to PDE's: between Hörmander's theorem and Holmgren's theorem”, Comm. Partial Differential Equations 20:5-6 (1995), 855–884.
• D. Tataru, “Unique continuation for operators with partially analytic coefficients”, J. Math. Pures Appl. $(9)$ 78:5 (1999), 505–521.
• M. E. Taylor, Pseudodifferential operators, Princeton Mathematical Series 34, Princeton University Press, 1981.
• A. Waters, “Stable determination of X-ray transforms of time dependent potentials from partial boundary data”, Comm. Partial Differential Equations 39:12 (2014), 2169–2197.