American Institute of Mathematical Sciences

Advanced
February  2016, 10(1): 247-262. doi: 10.3934/ipi.2016.10.247

A partial data result for less regular conductivities in admissible geometries

Casey Rodriguez 1,

1. 

University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, United States

Received  December 2014 Revised  August 2015 Published  February 2016

We consider the Calderón problem with partial data in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that measuring the Dirichlet--to--Neumann map on roughly half of the boundary determines a conductivity that has essentially 3/2 derivatives. As a corollary, we strengthen a partial data result due to Kenig, Sjöstrand, and Uhlmann.
Keywords: Calderón problem, partial data., rough conductivities.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Casey Rodriguez. A partial data result for less regular conductivities in admissible geometries. Inverse Problems & Imaging, 2016, 10 (1) : 247-262. doi: 10.3934/ipi.2016.10.247
References:
[1]

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

[2]

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

[3]

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

[4]

P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, preprint,, , ().   Google Scholar

[5]

D. Dos Santos Ferreira, C. E. Kenig and M. Salo, Determining an unbounded potential from Cauchy data in admissible geometries, Comm. Partial Differential Equations, 38 (2013), 50-68. doi: 10.1080/03605302.2012.736911.  Google Scholar

[6]

D. Dos Santos Ferreira, C. E. Kenig, M. 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

[7]

B. Haberman and D. Tataru, Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Math. J., 162 (2013), 496-516. doi: 10.1215/00127094-2019591.  Google Scholar

[8]

B. Haberman, Uniqueness in Calderón's problems for conductivities with unbounded gradient, Comm. Math. Phys., 340 (2015), 639-659. doi: 10.1007/s00220-015-2460-3.  Google Scholar

[9]

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

[10]

C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations, Duke Math. J., 157 (2011), 369-419. doi: 10.1215/00127094-1272903.  Google Scholar

[11]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math., 165 (2007), 567-591. doi: 10.4007/annals.2007.165.567.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.  Google Scholar

[16]

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

[17]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.  Google Scholar

[18]

G. Zhang, Uniqueness in the Calderón problem with partial data for less smooth conductivities, Inverse Problems, 28 (2012), 105008, 18pp. doi: 10.1088/0266-5611/28/10/105008.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, preprint,, , ().   Google Scholar

[5]

D. Dos Santos Ferreira, C. E. Kenig and M. Salo, Determining an unbounded potential from Cauchy data in admissible geometries, Comm. Partial Differential Equations, 38 (2013), 50-68. doi: 10.1080/03605302.2012.736911.  Google Scholar

[6]

D. Dos Santos Ferreira, C. E. Kenig, M. 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

[7]

B. Haberman and D. Tataru, Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Math. J., 162 (2013), 496-516. doi: 10.1215/00127094-2019591.  Google Scholar

[8]

B. Haberman, Uniqueness in Calderón's problems for conductivities with unbounded gradient, Comm. Math. Phys., 340 (2015), 639-659. doi: 10.1007/s00220-015-2460-3.  Google Scholar

[9]

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

[10]

C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations, Duke Math. J., 157 (2011), 369-419. doi: 10.1215/00127094-1272903.  Google Scholar

[11]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math., 165 (2007), 567-591. doi: 10.4007/annals.2007.165.567.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.  Google Scholar

[16]

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

[17]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.  Google Scholar

[18]

G. Zhang, Uniqueness in the Calderón problem with partial data for less smooth conductivities, Inverse Problems, 28 (2012), 105008, 18pp. doi: 10.1088/0266-5611/28/10/105008.  Google Scholar

[1]

Yernat M. Assylbekov. Reconstruction in the partial data Calderón problem on admissible manifolds. Inverse Problems & Imaging, 2017, 11 (3) : 455-476. doi: 10.3934/ipi.2017021
[2]

Matteo Santacesaria. Note on Calderón's inverse problem for measurable conductivities. Inverse Problems & Imaging, 2019, 13 (1) : 149-157. doi: 10.3934/ipi.2019008
[3]

Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems & Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49
[4]

María Ángeles García-Ferrero, Angkana Rüland, Wiktoria Zatoń. Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021049
[5]

Petteri Piiroinen, Martin Simon. Probabilistic interpretation of the Calderón problem. Inverse Problems & Imaging, 2017, 11 (3) : 553-575. doi: 10.3934/ipi.2017026
[6]

Angkana Rüland, Eva Sincich. Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data. Inverse Problems & Imaging, 2019, 13 (5) : 1023-1044. doi: 10.3934/ipi.2019046
[7]

Kim Knudsen, Jennifer L. Mueller. The born approximation and Calderón's method for reconstruction of conductivities in 3-D. Conference Publications, 2011, 2011 (Special) : 844-853. doi: 10.3934/proc.2011.2011.844
[8]

Pedro Caro, Mikko Salo. Stability of the Calderón problem in admissible geometries. Inverse Problems & Imaging, 2014, 8 (4) : 939-957. doi: 10.3934/ipi.2014.8.939
[9]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems & Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117
[10]

Yang Han. On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 233-242. doi: 10.3934/dcds.2005.12.233
[11]

Fabrice Delbary, Kim Knudsen. Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem. Inverse Problems & Imaging, 2014, 8 (4) : 991-1012. doi: 10.3934/ipi.2014.8.991
[12]

Sarah J. Hamilton, David Isaacson, Ville Kolehmainen, Peter A. Muller, Jussi Toivanen, Patrick F. Bray. 3D Electrical Impedance Tomography reconstructions from simulated electrode data using direct inversion $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ and Calderón methods. Inverse Problems & Imaging, 2021, 15 (5) : 1135-1169. doi: 10.3934/ipi.2021032
[13]

Suman Kumar Sahoo, Manmohan Vashisth. A partial data inverse problem for the convection-diffusion equation. Inverse Problems & Imaging, 2020, 14 (1) : 53-75. doi: 10.3934/ipi.2019063
[14]

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
[15]

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
[16]

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
[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]

Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091
[19]

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
[20]

Yavar Kian, Morgan Morancey, Lauri Oksanen. Application of the boundary control method to partial data Borg-Levinson inverse spectral problem. Mathematical Control & Related Fields, 2019, 9 (2) : 289-312. doi: 10.3934/mcrf.2019015

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (107)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy