June  2018, 12(3): 801-830. doi: 10.3934/ipi.2018034

An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data

Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bangalore, India

Received  September 2016 Revised  January 2018 Published  March 2018

We consider the inverse problem of recovering the magnetic and potential term of a magnetic Schrödinger operator on certain compact Riemannian manifolds with boundary from partial Dirichlet and Neumann data on suitable subsets of the boundary. The uniqueness proof relies on proving a suitable Carleman estimate for functions which vanish only on a part of boundary and constructing complex geometric optics solutions which vanish on a part of the boundary.

Citation: Sombuddha Bhattacharyya. An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data. Inverse Problems & Imaging, 2018, 12 (3) : 801-830. doi: 10.3934/ipi.2018034
References:
[1]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math. (2), 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265. Google Scholar

[2]

A. L. Bukhgeim, Recovering a potential from cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33. Google Scholar

[3]

A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial cauchy data, Communications in Partial Differential Equations, 27 (2002), 653-668. doi: 10.1081/PDE-120002868. Google Scholar

[4]

A. P. Calderón, On an inverse boundary value problem, In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pages 65-73. Soc. Brasil. Mat., Rio de Janeiro, 1980. Google Scholar

[5]

F. J. Chung, Partial data for the neumann-dirichlet magnetic schrödinger inverse problem, Inverse Probl. Imaging, 8 (2014), 959-989. doi: 10.3934/ipi.2014.8.959. Google Scholar

[6]

F. J. Chung, A partial data result for the magnetic schrödinger inverse problem, Anal. PDE, 7 (2014), 117-157. doi: 10.2140/apde.2014.7.117. Google Scholar

[7]

F. J. Chung, Partial data for the neumann-to-dirichlet map, J. Fourier Anal. Appl., 21 (2015), 628-665. doi: 10.1007/s00041-014-9379-5. Google Scholar

[8]

F. J. Chung, M. Salo and L. Tzou, Partial data inverse problems for the hodge laplacian, Anal. PDE, 10 (2017), 43-93, https://arXiv.org/abs/1310.4616 doi: 10.2140/apde.2017.10.43. Google Scholar

[9]

D. D. S. FerreiraC. E. KenigM. Salo and G. Uhlmann, Limiting carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-171. doi: 10.1007/s00222-009-0196-4. Google Scholar

[10]

D. D. S. FerreiraC. E. KenigJ. Sjöstrand and G. Uhlmann, Determining a magnetic schrödinger operator from partial cauchy data, Comm. Math. Phys., 271 (2007), 467-488. doi: 10.1007/s00220-006-0151-9. Google Scholar

[11]

L. Hörmander, Remarks on Holmgren's uniqueness theorem, Ann. Inst. Fourier, 43 (1993), 1223-1251. doi: 10.5802/aif.1371. Google Scholar

[12]

O. Y. ImanuvilovG. Uhlmann and M. Yamamoto, Determination of second-order elliptic operators in two dimensions from partial Cauchy data, Proc. Natl. Acad. Sci. USA, 108 (2011), 467-472. doi: 10.1073/pnas.1011681107. Google Scholar

[13]

O. Y. ImanuvilovG. Uhlmann and M. Yamamoto, The Calderón problem with partial data in two dimensions, J. Amer. Math. Soc., 23 (2010), 655-691. doi: 10.1090/S0894-0347-10-00656-9. Google Scholar

[14]

O. Y. Imanuvilov, G. Uhlmann and M. Yamamoto, Inverse boundary value problem by measuring Dirichlet data and Neumann data on disjoint sets, Inverse Problems, 27 (2011), 085007, 26pp. Google Scholar

[15]

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. Google Scholar

[16]

C. E. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications, Anal. PDE, 6 (2013), 2003-2048. doi: 10.2140/apde.2013.6.2003. Google Scholar

[17]

C. E. KenigJ. 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. Google Scholar

[18]

K. Knudsen and M. Salo, Determining nonsmooth first order terms from partial boundary measurements, Inverse Probl. Imaging, 1 (2007), 349-369. doi: 10.3934/ipi.2007.1.349. Google Scholar

[19]

K. Knudsen, The Calderón problem with partial data for less smooth conductivities, Comm. Partial Differential Equations, 31 (2006), 57-71. doi: 10.1080/03605300500361610. Google Scholar

[20]

K. Krupchyk and G. Uhlmann, Inverse problems for magnetic schrödinger operators in transversally anisotropic geometries, ArXiv https://arXiv.org/abs/1702.07974Google Scholar

[21]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), 71-96. doi: 10.2307/2118653. Google Scholar

[22]

G. NakamuraZ. Sun and G. Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann., 303 (1995), 377-388. doi: 10.1007/BF01460996. Google Scholar

[23]

M. Salo, Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field, Comm. Partial Differential Equations, 31 (2006), 1639-1666. doi: 10.1080/03605300500530420. Google Scholar

[24]

M. Salo and L. Tzou, Carleman estimates and inverse problems for dirac operators, Math. Ann., 344 (2009), 161-184. doi: 10.1007/s00208-008-0301-9. Google Scholar

[25]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-posed Problems Series. VSP, Utrecht, Utrecht, the Netherlands, 1994. Google Scholar

[26]

Z. Sun, An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc., 338 (1993), 953-969. Google Scholar

[27]

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. Google Scholar

[28]

C. F. Tolmasky, Exponentially growing solutions for nonsmooth first-order perturbations of the Laplacian, SIAM J. Math. Anal., 29 (1998), 116-133 (electronic). doi: 10.1137/S0036141096301038. Google Scholar

show all references

References:
[1]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math. (2), 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265. Google Scholar

[2]

A. L. Bukhgeim, Recovering a potential from cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33. Google Scholar

[3]

A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial cauchy data, Communications in Partial Differential Equations, 27 (2002), 653-668. doi: 10.1081/PDE-120002868. Google Scholar

[4]

A. P. Calderón, On an inverse boundary value problem, In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pages 65-73. Soc. Brasil. Mat., Rio de Janeiro, 1980. Google Scholar

[5]

F. J. Chung, Partial data for the neumann-dirichlet magnetic schrödinger inverse problem, Inverse Probl. Imaging, 8 (2014), 959-989. doi: 10.3934/ipi.2014.8.959. Google Scholar

[6]

F. J. Chung, A partial data result for the magnetic schrödinger inverse problem, Anal. PDE, 7 (2014), 117-157. doi: 10.2140/apde.2014.7.117. Google Scholar

[7]

F. J. Chung, Partial data for the neumann-to-dirichlet map, J. Fourier Anal. Appl., 21 (2015), 628-665. doi: 10.1007/s00041-014-9379-5. Google Scholar

[8]

F. J. Chung, M. Salo and L. Tzou, Partial data inverse problems for the hodge laplacian, Anal. PDE, 10 (2017), 43-93, https://arXiv.org/abs/1310.4616 doi: 10.2140/apde.2017.10.43. Google Scholar

[9]

D. D. S. FerreiraC. E. KenigM. Salo and G. Uhlmann, Limiting carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-171. doi: 10.1007/s00222-009-0196-4. Google Scholar

[10]

D. D. S. FerreiraC. E. KenigJ. Sjöstrand and G. Uhlmann, Determining a magnetic schrödinger operator from partial cauchy data, Comm. Math. Phys., 271 (2007), 467-488. doi: 10.1007/s00220-006-0151-9. Google Scholar

[11]

L. Hörmander, Remarks on Holmgren's uniqueness theorem, Ann. Inst. Fourier, 43 (1993), 1223-1251. doi: 10.5802/aif.1371. Google Scholar

[12]

O. Y. ImanuvilovG. Uhlmann and M. Yamamoto, Determination of second-order elliptic operators in two dimensions from partial Cauchy data, Proc. Natl. Acad. Sci. USA, 108 (2011), 467-472. doi: 10.1073/pnas.1011681107. Google Scholar

[13]

O. Y. ImanuvilovG. Uhlmann and M. Yamamoto, The Calderón problem with partial data in two dimensions, J. Amer. Math. Soc., 23 (2010), 655-691. doi: 10.1090/S0894-0347-10-00656-9. Google Scholar

[14]

O. Y. Imanuvilov, G. Uhlmann and M. Yamamoto, Inverse boundary value problem by measuring Dirichlet data and Neumann data on disjoint sets, Inverse Problems, 27 (2011), 085007, 26pp. Google Scholar

[15]

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. Google Scholar

[16]

C. E. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications, Anal. PDE, 6 (2013), 2003-2048. doi: 10.2140/apde.2013.6.2003. Google Scholar

[17]

C. E. KenigJ. 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. Google Scholar

[18]

K. Knudsen and M. Salo, Determining nonsmooth first order terms from partial boundary measurements, Inverse Probl. Imaging, 1 (2007), 349-369. doi: 10.3934/ipi.2007.1.349. Google Scholar

[19]

K. Knudsen, The Calderón problem with partial data for less smooth conductivities, Comm. Partial Differential Equations, 31 (2006), 57-71. doi: 10.1080/03605300500361610. Google Scholar

[20]

K. Krupchyk and G. Uhlmann, Inverse problems for magnetic schrödinger operators in transversally anisotropic geometries, ArXiv https://arXiv.org/abs/1702.07974Google Scholar

[21]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), 71-96. doi: 10.2307/2118653. Google Scholar

[22]

G. NakamuraZ. Sun and G. Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann., 303 (1995), 377-388. doi: 10.1007/BF01460996. Google Scholar

[23]

M. Salo, Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field, Comm. Partial Differential Equations, 31 (2006), 1639-1666. doi: 10.1080/03605300500530420. Google Scholar

[24]

M. Salo and L. Tzou, Carleman estimates and inverse problems for dirac operators, Math. Ann., 344 (2009), 161-184. doi: 10.1007/s00208-008-0301-9. Google Scholar

[25]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-posed Problems Series. VSP, Utrecht, Utrecht, the Netherlands, 1994. Google Scholar

[26]

Z. Sun, An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc., 338 (1993), 953-969. Google Scholar

[27]

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. Google Scholar

[28]

C. F. Tolmasky, Exponentially growing solutions for nonsmooth first-order perturbations of the Laplacian, SIAM J. Math. Anal., 29 (1998), 116-133 (electronic). doi: 10.1137/S0036141096301038. Google Scholar

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