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

A partial data result for less regular conductivities in admissible geometries

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.
Citation: Casey Rodriguez. A partial data result for less regular conductivities in admissible geometries. Inverse Problems and 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.

[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.

[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.

[4]

P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, preprint, arXiv:1411.8001.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

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.

[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.

[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.

[4]

P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, preprint, arXiv:1411.8001.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

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