# American Institute of Mathematical Sciences

June  2017, 11(3): 455-476. doi: 10.3934/ipi.2017021

## Reconstruction in the partial data Calderón problem on admissible manifolds

 Department of Mathematics, Northeastern University, Boston, MA 02115, USA

Received  April 2016 Revised  February 2017 Published  April 2017

We consider the problem of developing a method to reconstruct a potential $q$ from the partial data Dirichlet-to-Neumann map for the Schrödinger equation $(-Δ_g+q)u=0$ on a fixed admissible manifold $(M,g)$. If the part of the boundary that is inaccessible for measurements satisfies a flatness condition in one direction, then we reconstruct the local attenuated geodesic ray transform of the one-dimensional Fourier transform of the potential $q$. This allows us to reconstruct $q$ locally, if the local (unattenuated) geodesic ray transform is constructively invertible. We also reconstruct $q$ globally, if $M$ satisfies certain concavity condition and if the global geodesic ray transform can be inverted constructively. These are reconstruction procedures for the corresponding uniqueness results given by Kenig and Salo [7]. Moreover, the global reconstruction extends and improves the constructive proof of Nachman and Street [14] in the Euclidean setting. We derive a certain boundary integral equation which involves the given partial data and describes the traces of complex geometrical optics solutions. For construction of complex geometrical optics solutions, following [14] and improving their arguments, we use a certain family of Green's functions for the Laplace-Beltrami operator and the corresponding single layer potentials. The constructive inversion problem for local or global geodesic ray transforms is one of the major topics of interest in integral geometry.

Citation: Yernat M. Assylbekov. Reconstruction in the partial data Calderón problem on admissible manifolds. Inverse Problems and Imaging, 2017, 11 (3) : 455-476. doi: 10.3934/ipi.2017021
##### References:
 [1] A. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Diff. Eq., 27 (2002), 653-668.  doi: 10.1081/PDE-120002868. [2] A. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasil. Mat., (Rio de Janeiro), (1980), 65-73. [3] D. Dos Santos Ferreira, C. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-17.  doi: 10.1007/s00222-009-0196-4. [4] D. Dos Santos Ferreira, C. Kenig and M. Salo, Determining an unbounded potential from Cauchy data in admissible geometries, Comm. PDE, 38 (2013), 50-68.  doi: 10.1080/03605302.2012.736911. [5] V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Probl. Imaging, 1 (2007), 95-105.  doi: 10.3934/ipi.2007.1.95. [6] C. Kenig and M. Salo, Recent progress in the Calderón problem with partial data, Contemp. Math., 615 (2014), 193-222.  doi: 10.1090/conm/615/12245. [7] C. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications, Analysis & PDE, 6 (2013), 2003-2048.  doi: 10.2140/apde.2013.6.2003. [8] C. Kenig, M. Salo and G. Uhlmann, Reconstructions from boundary measurements on admissible manifolds, Inverse Probl. Imaging, 5 (2011), 859-877.  doi: 10.3934/ipi.2011.5.859. [9] C. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math.(2), 165 (2007), 567-591.  doi: 10.4007/annals.2007.165.567. [10] V. Krishnan, On the inversion formulas of Pestov and Uhlmann for the geodesic ray transform, J. Inv. Ill-Posed Problems, 18 (2010), 401-408.  doi: 10.1515/JIIP.2010.017. [11] J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097-1112.  doi: 10.1002/cpa.3160420804. [12] J. -L. Lions and E. Magenes, Problémes Aux Limites Non Homogénes et Applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968. [13] A. Nachman, Reconstructions from boundary measurements, Ann. Math., 128 (1988), 531-576.  doi: 10.2307/1971435. [14] A. Nachman and B. Street, Reconstruction in the Calderón problem with partial data, Comm. PDE, 35 (2010), 375-390.  doi: 10.1080/03605300903296322. [15] R. G. Novikov, Multidimensional inverse spectral problem for the equation $-Δψ+(v(x)-Eu(x))ψ=0$, Funct. Anal. Appl., 22 (1988), 263-272.  doi: 10.1007/BF01077418. [16] L. Pestov and G. Uhlmann, On the Characterization of the Range and Inversion Formulas for the Geodesic X-Ray Transform, International Math. Research Notices, 80 (2004), 4331-4347.  doi: 10.1155/S1073792804142116. [17] M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces, J. Diff. Geom., 88 (2011), 161-187.  doi: 10.4310/jdg/1317758872. [18] V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994. doi: 10.1515/9783110900095. [19] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math.(2), 125 (1987), 153-169.  doi: 10.2307/1971291. [20] M. E. Taylor, Partial Differential Equations I. Basic Theory, Applied Mathematical Sciences, 115. Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8. [21] G. Uhlmann, Inverse problems: Seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.  doi: 10.1007/s13373-014-0051-9. [22] G. Uhlmann and A. Vasy, The inverse problem for the local geodesic ray transform, Inventiones Mathematicae, 205 (2016), 83-120.  doi: 10.1007/s00222-015-0631-7.

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##### References:
 [1] A. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Diff. Eq., 27 (2002), 653-668.  doi: 10.1081/PDE-120002868. [2] A. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasil. Mat., (Rio de Janeiro), (1980), 65-73. [3] D. Dos Santos Ferreira, C. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-17.  doi: 10.1007/s00222-009-0196-4. [4] D. Dos Santos Ferreira, C. Kenig and M. Salo, Determining an unbounded potential from Cauchy data in admissible geometries, Comm. PDE, 38 (2013), 50-68.  doi: 10.1080/03605302.2012.736911. [5] V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Probl. Imaging, 1 (2007), 95-105.  doi: 10.3934/ipi.2007.1.95. [6] C. Kenig and M. Salo, Recent progress in the Calderón problem with partial data, Contemp. Math., 615 (2014), 193-222.  doi: 10.1090/conm/615/12245. [7] C. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications, Analysis & PDE, 6 (2013), 2003-2048.  doi: 10.2140/apde.2013.6.2003. [8] C. Kenig, M. Salo and G. Uhlmann, Reconstructions from boundary measurements on admissible manifolds, Inverse Probl. Imaging, 5 (2011), 859-877.  doi: 10.3934/ipi.2011.5.859. [9] C. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math.(2), 165 (2007), 567-591.  doi: 10.4007/annals.2007.165.567. [10] V. Krishnan, On the inversion formulas of Pestov and Uhlmann for the geodesic ray transform, J. Inv. Ill-Posed Problems, 18 (2010), 401-408.  doi: 10.1515/JIIP.2010.017. [11] J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097-1112.  doi: 10.1002/cpa.3160420804. [12] J. -L. Lions and E. Magenes, Problémes Aux Limites Non Homogénes et Applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968. [13] A. Nachman, Reconstructions from boundary measurements, Ann. Math., 128 (1988), 531-576.  doi: 10.2307/1971435. [14] A. Nachman and B. Street, Reconstruction in the Calderón problem with partial data, Comm. PDE, 35 (2010), 375-390.  doi: 10.1080/03605300903296322. [15] R. G. Novikov, Multidimensional inverse spectral problem for the equation $-Δψ+(v(x)-Eu(x))ψ=0$, Funct. Anal. Appl., 22 (1988), 263-272.  doi: 10.1007/BF01077418. [16] L. Pestov and G. Uhlmann, On the Characterization of the Range and Inversion Formulas for the Geodesic X-Ray Transform, International Math. Research Notices, 80 (2004), 4331-4347.  doi: 10.1155/S1073792804142116. [17] M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces, J. Diff. Geom., 88 (2011), 161-187.  doi: 10.4310/jdg/1317758872. [18] V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994. doi: 10.1515/9783110900095. [19] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math.(2), 125 (1987), 153-169.  doi: 10.2307/1971291. [20] M. E. Taylor, Partial Differential Equations I. Basic Theory, Applied Mathematical Sciences, 115. Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8. [21] G. Uhlmann, Inverse problems: Seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.  doi: 10.1007/s13373-014-0051-9. [22] G. Uhlmann and A. Vasy, The inverse problem for the local geodesic ray transform, Inventiones Mathematicae, 205 (2016), 83-120.  doi: 10.1007/s00222-015-0631-7.
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