A partial data result for less regular conductivities in admissible geometries

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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

Abstract
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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 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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