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

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 & 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. 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. 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), 65. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1137/S0036141096301038. Google Scholar
[17]	G. Uhlmann, Electrical impedance tomography and Calderón's problem,, Inverse Problems, 25 (2009). 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). doi: 10.1088/0266-5611/28/10/105008. Google Scholar

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References:

[1]	K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane,, Ann. of Math., 163 (2006), 265. 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. 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), 65. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1137/S0036141096301038. Google Scholar
[17]	G. Uhlmann, Electrical impedance tomography and Calderón's problem,, Inverse Problems, 25 (2009). 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). doi: 10.1088/0266-5611/28/10/105008. Google Scholar

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