# American Institute of Mathematical Sciences

November  2011, 5(4): 859-877. doi: 10.3934/ipi.2011.5.859

## Reconstructions from boundary measurements on admissible manifolds

 1 Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637-1514, United States 2 Department of Mathematics and Statistics, University of Helsinki and University of Jyväskylä, PO Box 68, 00014 Helsinki, Finland 3 Department of Mathematics, University of Washington and, Department of Mathematics, University of California, Irvine, CA 92697-3875, United States

Received  November 2010 Revised  August 2011 Published  November 2011

We prove that a potential $q$ can be reconstructed from the Dirichlet-to-Neumann map for the Schrödinger operator $-\Delta_g + q$ in a fixed admissible $3$-dimensional Riemannian manifold $(M,g)$. We also show that an admissible metric $g$ in a fixed conformal class can be constructed from the Dirichlet-to-Neumann map for $\Delta_g$. This is a constructive version of earlier uniqueness results by Dos Santos Ferreira et al. [10] on admissible manifolds, and extends the reconstruction procedure of Nachman [31] in Euclidean space. The main points are the derivation of a boundary integral equation characterizing the boundary values of complex geometrical optics solutions, and the development of associated layer potentials adapted to a cylindrical geometry.
Citation: Carlos E. Kenig, Mikko Salo, Gunther Uhlmann. Reconstructions from boundary measurements on admissible manifolds. Inverse Problems & Imaging, 2011, 5 (4) : 859-877. doi: 10.3934/ipi.2011.5.859
##### References:
 [1] K. Astala, M. Lassas and L. Päivärinta, Calderón's inverse problem for anisotropic conductivity in the plane,, Comm. PDE, 30 (2005), 207. doi: 10.1081/PDE-200044485. Google Scholar [2] K. Astala and L. Päivärinta, A boundary integral equation for Calderón's inverse conductivity problem,, Collect. Math., 2006 (): 127. Google Scholar [3] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane,, Ann. of Math. (2), 163 (2006), 265. doi: 10.4007/annals.2006.163.265. Google Scholar [4] T. Aubin, "Some Nonlinear Problems in Riemannian Geometry,", Springer Monographs in Mathematics, (1998). Google Scholar [5] M. Belishev and Y. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method),, Comm. PDE, 17 (1992), 767. doi: 10.1080/03605309208820863. Google Scholar [6] A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case,, J. Inverse Ill-posed Probl., 16 (2008), 19. doi: 10.1515/jiip.2008.002. Google Scholar [7] A. P. Calderón, On an inverse boundary value problem,, in, (1980), 65. Google Scholar [8] F. Delbary, P. C. Hansen and K. Knudsen, A direct numerical reconstruction algorithm for the 3D Calderón problem,, in, 290 (2011). doi: 10.1088/1742-6596/290/1/012003. Google Scholar [9] D. Dos Santos Ferreira, C. E. Kenig and M. Salo, Determining an unbounded potential from Cauchy data in admissible geometries,, preprint, (2011). Google Scholar [10] D. Dos Santos Ferreira, C. E. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems,, Invent. Math., 178 (2009), 119. doi: 10.1007/s00222-009-0196-4. Google Scholar [11] L. D. Faddeev, Increasing solutions of the Schrödinger equation,, Sov. Phys. Dokl., 10 (1966), 1033. Google Scholar [12] B. Frigyik, P. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights,, J. Geom. Anal., 18 (2008), 89. doi: 10.1007/s12220-007-9007-6. Google Scholar [13] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998). Google Scholar [14] C. Guillarmou and A. Sá Barreto, Inverse problems for Einstein manifolds,, Inverse Probl. Imaging, 3 (2009), 1. doi: 10.3934/ipi.2009.3.1. Google Scholar [15] C. Guillarmou and L. Tzou, Calderón inverse problem with partial data on Riemann surfaces,, Duke Math. J., 158 (2011), 83. doi: 10.1215/00127094-1276310. Google Scholar [16] C. Guillarmou and L. Tzou, Identification of a connection from Cauchy data on a Riemann surface with boundary,, Geom. Funct. Anal., 21 (2011), 393. doi: 10.1007/s00039-011-0110-2. Google Scholar [17] G. M. Henkin and V. Michel, On the explicit reconstruction of a Riemann surface from its Dirichlet-Neumann operator,, Geom. Funct. Anal., 17 (2007), 116. doi: 10.1007/s00039-006-0590-7. Google Scholar [18] G. M. Henkin and V. Michel, Inverse conductivity problem on Riemann surfaces,, J. Geom. Anal., 18 (2008), 1033. doi: 10.1007/s12220-008-9035-x. Google Scholar [19] G. M. Khenkin and R. G. Novikov, The $\overline{\partial}$ -equation in the multidimensional inverse scattering problem,, Uspekhi Mat. Nauk, 42 (1987), 93. Google Scholar [20] G. M. Henkin and R. G. Novikov, On the reconstruction of conductivity of bordered two-dimensional surface in $\mathbbR^3$ from electrical currents measurements on its boundary,, J. Geom. Anal., 21 (2011), 543. doi: 10.1007/s12220-010-9158-8. Google Scholar [21] G. Henkina and M. Santacesaria, On an inverse problem for anisotropic conductivity in the plane,, Inverse Problems, 26 (2010). Google Scholar [22] G. Henkin and M. Santacesaria, Gel'fand-Calderón's inverse problem for anisotropic conductivities on bordered surfaces in $\mathbbR^3$,, IMRN (to appear), (). Google Scholar [23] A. Katchalov and Y. Kurylëv, Incomplete spectral data and the reconstruction of a Riemannian manifold,, J. Inverse Ill-Posed Probl., 1 (1993), 141. doi: 10.1515/jiip.1993.1.2.141. Google Scholar [24] A. Katchalov and Y. Kurylev, Multidimensional inverse problem with incomplete boundary spectral data,, Comm. PDE, 23 (1998), 55. doi: 10.1080/03605309808821338. Google Scholar [25] A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems,", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123 (2001). Google Scholar [26] C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations,, Duke Math. J., 157 (2011), 369. doi: 10.1215/00127094-1272903. Google Scholar [27] M. Lassas, M. Taylor and G. Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary,, Comm. Anal. Geom., 11 (2003), 207. Google Scholar [28] M. Lassas and G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map,, Ann. Sci. École Norm. Sup. (4), 34 (2001), 771. Google Scholar [29] J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements,, Comm. Pure Appl. Math., 42 (1989), 1097. doi: 10.1002/cpa.3160420804. Google Scholar [30] W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,", Cambridge University Press, (2000). Google Scholar [31] A. Nachman, Reconstructions from boundary measurements,, Ann. Math. (2), 128 (1988), 531. doi: 10.2307/1971435. Google Scholar [32] A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math. (2), 143 (1996), 71. doi: 10.2307/2118653. Google Scholar [33] A. Nachman and B. Street, Reconstruction in the Calderón problem with partial data,, Comm. PDE, 35 (2010), 375. doi: 10.1080/03605300903296322. Google Scholar [34] R. G. Novikov, Multidimensional inverse spectral problem for the equation $-\Delta \psi + (v(x) - E u(x))\psi = 0$,, Funct. Anal. Appl., 22 (1988), 263. doi: 10.1007/BF01077418. Google Scholar [35] R. G. Novikov, An effectivization of the global reconstruction in the Gel'fand-Calderón inverse problem in three dimensions,, in, 494 (2009), 161. Google Scholar [36] P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics,, Duke Math. J., 70 (1993), 617. doi: 10.1215/S0012-7094-93-07014-7. Google Scholar [37] M. Salo, Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field,, Comm. PDE, 31 (2006), 1639. doi: 10.1080/03605300500530420. Google Scholar [38] M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces,, J. Diff. Geom., 88 (2011), 161. Google Scholar [39] V. Sharafutdinov, "Integral Geometry of Tensor Fields,", Inverse and Ill-Posed Problems Series, (1994). Google Scholar [40] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math. (2), 125 (1987), 153. doi: 10.2307/1971291. Google Scholar [41] M. E. Taylor, "Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials,", Mathematical Surveys and Monographs, 81 (2000). Google Scholar

show all references

##### References:
 [1] K. Astala, M. Lassas and L. Päivärinta, Calderón's inverse problem for anisotropic conductivity in the plane,, Comm. PDE, 30 (2005), 207. doi: 10.1081/PDE-200044485. Google Scholar [2] K. Astala and L. Päivärinta, A boundary integral equation for Calderón's inverse conductivity problem,, Collect. Math., 2006 (): 127. Google Scholar [3] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane,, Ann. of Math. (2), 163 (2006), 265. doi: 10.4007/annals.2006.163.265. Google Scholar [4] T. Aubin, "Some Nonlinear Problems in Riemannian Geometry,", Springer Monographs in Mathematics, (1998). Google Scholar [5] M. Belishev and Y. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method),, Comm. PDE, 17 (1992), 767. doi: 10.1080/03605309208820863. Google Scholar [6] A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case,, J. Inverse Ill-posed Probl., 16 (2008), 19. doi: 10.1515/jiip.2008.002. Google Scholar [7] A. P. Calderón, On an inverse boundary value problem,, in, (1980), 65. Google Scholar [8] F. Delbary, P. C. Hansen and K. Knudsen, A direct numerical reconstruction algorithm for the 3D Calderón problem,, in, 290 (2011). doi: 10.1088/1742-6596/290/1/012003. Google Scholar [9] D. Dos Santos Ferreira, C. E. Kenig and M. Salo, Determining an unbounded potential from Cauchy data in admissible geometries,, preprint, (2011). Google Scholar [10] D. Dos Santos Ferreira, C. E. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems,, Invent. Math., 178 (2009), 119. doi: 10.1007/s00222-009-0196-4. Google Scholar [11] L. D. Faddeev, Increasing solutions of the Schrödinger equation,, Sov. Phys. Dokl., 10 (1966), 1033. Google Scholar [12] B. Frigyik, P. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights,, J. Geom. Anal., 18 (2008), 89. doi: 10.1007/s12220-007-9007-6. Google Scholar [13] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998). Google Scholar [14] C. Guillarmou and A. Sá Barreto, Inverse problems for Einstein manifolds,, Inverse Probl. Imaging, 3 (2009), 1. doi: 10.3934/ipi.2009.3.1. Google Scholar [15] C. Guillarmou and L. Tzou, Calderón inverse problem with partial data on Riemann surfaces,, Duke Math. J., 158 (2011), 83. doi: 10.1215/00127094-1276310. Google Scholar [16] C. Guillarmou and L. Tzou, Identification of a connection from Cauchy data on a Riemann surface with boundary,, Geom. Funct. Anal., 21 (2011), 393. doi: 10.1007/s00039-011-0110-2. Google Scholar [17] G. M. Henkin and V. Michel, On the explicit reconstruction of a Riemann surface from its Dirichlet-Neumann operator,, Geom. Funct. Anal., 17 (2007), 116. doi: 10.1007/s00039-006-0590-7. Google Scholar [18] G. M. Henkin and V. Michel, Inverse conductivity problem on Riemann surfaces,, J. Geom. Anal., 18 (2008), 1033. doi: 10.1007/s12220-008-9035-x. Google Scholar [19] G. M. Khenkin and R. G. Novikov, The $\overline{\partial}$ -equation in the multidimensional inverse scattering problem,, Uspekhi Mat. Nauk, 42 (1987), 93. Google Scholar [20] G. M. Henkin and R. G. Novikov, On the reconstruction of conductivity of bordered two-dimensional surface in $\mathbbR^3$ from electrical currents measurements on its boundary,, J. Geom. Anal., 21 (2011), 543. doi: 10.1007/s12220-010-9158-8. Google Scholar [21] G. Henkina and M. Santacesaria, On an inverse problem for anisotropic conductivity in the plane,, Inverse Problems, 26 (2010). Google Scholar [22] G. Henkin and M. Santacesaria, Gel'fand-Calderón's inverse problem for anisotropic conductivities on bordered surfaces in $\mathbbR^3$,, IMRN (to appear), (). Google Scholar [23] A. Katchalov and Y. Kurylëv, Incomplete spectral data and the reconstruction of a Riemannian manifold,, J. Inverse Ill-Posed Probl., 1 (1993), 141. doi: 10.1515/jiip.1993.1.2.141. Google Scholar [24] A. Katchalov and Y. Kurylev, Multidimensional inverse problem with incomplete boundary spectral data,, Comm. PDE, 23 (1998), 55. doi: 10.1080/03605309808821338. Google Scholar [25] A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems,", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123 (2001). Google Scholar [26] C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations,, Duke Math. J., 157 (2011), 369. doi: 10.1215/00127094-1272903. Google Scholar [27] M. Lassas, M. Taylor and G. Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary,, Comm. Anal. Geom., 11 (2003), 207. Google Scholar [28] M. Lassas and G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map,, Ann. Sci. École Norm. Sup. (4), 34 (2001), 771. Google Scholar [29] J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements,, Comm. Pure Appl. Math., 42 (1989), 1097. doi: 10.1002/cpa.3160420804. Google Scholar [30] W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,", Cambridge University Press, (2000). Google Scholar [31] A. Nachman, Reconstructions from boundary measurements,, Ann. Math. (2), 128 (1988), 531. doi: 10.2307/1971435. Google Scholar [32] A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math. (2), 143 (1996), 71. doi: 10.2307/2118653. Google Scholar [33] A. Nachman and B. Street, Reconstruction in the Calderón problem with partial data,, Comm. PDE, 35 (2010), 375. doi: 10.1080/03605300903296322. Google Scholar [34] R. G. Novikov, Multidimensional inverse spectral problem for the equation $-\Delta \psi + (v(x) - E u(x))\psi = 0$,, Funct. Anal. Appl., 22 (1988), 263. doi: 10.1007/BF01077418. Google Scholar [35] R. G. Novikov, An effectivization of the global reconstruction in the Gel'fand-Calderón inverse problem in three dimensions,, in, 494 (2009), 161. Google Scholar [36] P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics,, Duke Math. J., 70 (1993), 617. doi: 10.1215/S0012-7094-93-07014-7. Google Scholar [37] M. Salo, Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field,, Comm. PDE, 31 (2006), 1639. doi: 10.1080/03605300500530420. Google Scholar [38] M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces,, J. Diff. Geom., 88 (2011), 161. Google Scholar [39] V. Sharafutdinov, "Integral Geometry of Tensor Fields,", Inverse and Ill-Posed Problems Series, (1994). Google Scholar [40] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math. (2), 125 (1987), 153. doi: 10.2307/1971291. Google Scholar [41] M. E. Taylor, "Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials,", Mathematical Surveys and Monographs, 81 (2000). Google Scholar
 [1] Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems & Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469 [2] Naoufel Ben Abdallah, Yongyong Cai, Francois Castella, Florian Méhats. Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential. Kinetic & Related Models, 2011, 4 (4) : 831-856. doi: 10.3934/krm.2011.4.831 [3] Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations & Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431 [4] Binhua Feng, Dun Zhao. On the Cauchy problem for the XFEL Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4171-4186. doi: 10.3934/dcdsb.2018131 [5] Chuang Zheng. Inverse problems for the fourth order Schrödinger equation on a finite domain. Mathematical Control & Related Fields, 2015, 5 (1) : 177-189. doi: 10.3934/mcrf.2015.5.177 [6] Jean-Claude Saut, Jun-Ichi Segata. Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 219-239. doi: 10.3934/dcds.2019009 [7] Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102 [8] Gökçe Dİlek Küçük, Gabil Yagub, Ercan Çelİk. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 503-512. doi: 10.3934/dcdss.2019033 [9] Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems & Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183 [10] Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074 [11] Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689 [12] Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control & Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161 [13] D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563 [14] Frank Wusterhausen. Schrödinger equation with noise on the boundary. Conference Publications, 2013, 2013 (special) : 791-796. doi: 10.3934/proc.2013.2013.791 [15] Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations & Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007 [16] Victor Isakov. On uniqueness in the inverse conductivity problem with local data. Inverse Problems & Imaging, 2007, 1 (1) : 95-105. doi: 10.3934/ipi.2007.1.95 [17] Luca Rondi. On the regularization of the inverse conductivity problem with discontinuous conductivities. Inverse Problems & Imaging, 2008, 2 (3) : 397-409. doi: 10.3934/ipi.2008.2.397 [18] Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems & Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959 [19] Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169 [20] Ru-Yu Lai. Global uniqueness for an inverse problem for the magnetic Schrödinger operator. Inverse Problems & Imaging, 2011, 5 (1) : 59-73. doi: 10.3934/ipi.2011.5.59

2018 Impact Factor: 1.469