November  2014, 8(4): 939-957. doi: 10.3934/ipi.2014.8.939

Stability of the Calderón problem in admissible geometries

1. 

Instituto de Ciencias Matemáticas - CSIC, Nicolás Cabrera 13-15, Campus de Cantoblanco UAM, 28049 Madrid, Spain

2. 

Department of Mathematics and Statistics, University of Helsinki and University of Jyväskylä, P.O. Box 35 FI-40014 Jyväskylä

Received  May 2014 Revised  September 2014 Published  November 2014

In this paper we prove log log type stability estimates for inverse boundary value problems on admissible Riemannian manifolds of dimension $n \geq 3$. The stability estimates correspond to the uniqueness results in [13]. These inverse problems arise naturally when studying the anisotropic Calderón problem.
Citation: 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
References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements,, Appl. Anal., 27 (1988), 153. doi: 10.1080/00036818808839730. Google Scholar

[2]

G. Alessandrini, Open issues of stability for the inverse conductivity problem,, J. Inv. Ill-Posed Probl., 15 (2007), 451. doi: 10.1515/jiip.2007.025. Google Scholar

[3]

G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem,, Adv. Appl. Math., 35 (2005), 207. doi: 10.1016/j.aam.2004.12.002. Google Scholar

[4]

K. Astala, M. Lassas and L. Päivärinta, Calderón's inverse problem for anisotropic conductivity in the plane,, Comm. PDE, 30 (2005), 207. doi: 10.1081/PDE-200044485. Google Scholar

[5]

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

[6]

A. P. Calderón, On an Inverse Boundary Value Problem,, Seminar on Numerical Analysis and its Applications to Continuum Physics, (1980). Google Scholar

[7]

P. Caro, On an inverse problem in electromagnetism with local data: Stability and uniqueness,, Inverse Probl. Imaging, 5 (2011), 297. doi: 10.3934/ipi.2011.5.297. Google Scholar

[8]

P. Caro, A. García and J. M. Reyes, Stability of the Calderón problem for less regular conductivities,, J. Differential Equations, 254 (2013), 469. doi: 10.1016/j.jde.2012.08.018. Google Scholar

[9]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Radon transform with restricted data and applications,, Adv. Math., 267 (2014), 523. doi: 10.1016/j.aim.2014.08.009. Google Scholar

[10]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Calderón problem with partial data,, preprint, (2014). Google Scholar

[11]

A. Clop, D. Faraco and A. Ruiz, Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities,, Inverse Probl. Imaging, 4 (2010), 49. doi: 10.3934/ipi.2010.4.49. Google Scholar

[12]

N.S. Dairbekov, G.P. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field,, Adv. Math., 216 (2007), 535. doi: 10.1016/j.aim.2007.05.014. Google Scholar

[13]

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

[14]

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

[15]

D. Dos Santos Ferreira, Y. Kurylev, M. Lassas and M. Salo, The Calderón problem in transversally anisotropic geometries,, J. Eur. Math. Soc, (). Google Scholar

[16]

B. Frigyik, P. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights,, J. Geom. Anal., 18 (2008), 89. doi: 10.1007/s12220-007-9007-6. Google Scholar

[17]

B. Haberman and D. Tataru, Uniqueness in Calderon's problem with Lipschitz conductivities,, Duke Math. J., 162 (2013), 497. doi: 10.1215/00127094-2019591. Google Scholar

[18]

H. Heck and J.-N. Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data,, Inverse Problems, 22 (2006), 1787. doi: 10.1088/0266-5611/22/5/015. Google Scholar

[19]

H. Heck and J.-N. Wang, Optimal stability estimate of the inverse boundary value problem by partial measurements,, preprint, (2007). Google Scholar

[20]

H. Kang and K. Yun, Boundary determination of conductivities and Riemannian metrics via local Dirichlet-to-Neumann operator,, SIAM J. Math. Anal., 34 (2003), 719. doi: 10.1137/S0036141001395042. Google Scholar

[21]

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

[22]

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

[23]

K. Knudsen, M. Lassas, J. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem,, Inverse Problems and Imaging, 3 (2009), 599. doi: 10.3934/ipi.2009.3.599. Google Scholar

[24]

J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements,, Commun. Pure Appl. Math., 42 (1989), 1097. doi: 10.1002/cpa.3160420804. Google Scholar

[25]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation,, Inverse Problems, 17 (2001), 1435. doi: 10.1088/0266-5611/17/5/313. Google Scholar

[26]

A. Nachman, Reconstructions from boundary measurements,, Ann. Math., 128 (1988), 531. doi: 10.2307/1971435. Google Scholar

[27]

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

[28]

M. Salo, The Calderón problem on Riemannian manifolds,, Chapter in Inverse Problems and Applications: Inside Out II (ed. G. Uhlmann), (2013), 167. Google Scholar

[29]

M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces,, J. Diff. Geom., 88 (2011), 161. Google Scholar

[30]

V. A. Sharafutdinov, Ray transform on Riemannian manifolds. Eight lectures on integral geometry,, , (). Google Scholar

[31]

P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity,, Duke Math. J., 123 (2004), 445. doi: 10.1215/S0012-7094-04-12332-2. Google Scholar

[32]

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

[33]

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

[34]

S. Vessella, A continuous dependence result in the analytic continuation problem,, Forum Math., 11 (1999), 695. doi: 10.1515/form.1999.020. Google Scholar

show all references

References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements,, Appl. Anal., 27 (1988), 153. doi: 10.1080/00036818808839730. Google Scholar

[2]

G. Alessandrini, Open issues of stability for the inverse conductivity problem,, J. Inv. Ill-Posed Probl., 15 (2007), 451. doi: 10.1515/jiip.2007.025. Google Scholar

[3]

G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem,, Adv. Appl. Math., 35 (2005), 207. doi: 10.1016/j.aam.2004.12.002. Google Scholar

[4]

K. Astala, M. Lassas and L. Päivärinta, Calderón's inverse problem for anisotropic conductivity in the plane,, Comm. PDE, 30 (2005), 207. doi: 10.1081/PDE-200044485. Google Scholar

[5]

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

[6]

A. P. Calderón, On an Inverse Boundary Value Problem,, Seminar on Numerical Analysis and its Applications to Continuum Physics, (1980). Google Scholar

[7]

P. Caro, On an inverse problem in electromagnetism with local data: Stability and uniqueness,, Inverse Probl. Imaging, 5 (2011), 297. doi: 10.3934/ipi.2011.5.297. Google Scholar

[8]

P. Caro, A. García and J. M. Reyes, Stability of the Calderón problem for less regular conductivities,, J. Differential Equations, 254 (2013), 469. doi: 10.1016/j.jde.2012.08.018. Google Scholar

[9]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Radon transform with restricted data and applications,, Adv. Math., 267 (2014), 523. doi: 10.1016/j.aim.2014.08.009. Google Scholar

[10]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Calderón problem with partial data,, preprint, (2014). Google Scholar

[11]

A. Clop, D. Faraco and A. Ruiz, Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities,, Inverse Probl. Imaging, 4 (2010), 49. doi: 10.3934/ipi.2010.4.49. Google Scholar

[12]

N.S. Dairbekov, G.P. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field,, Adv. Math., 216 (2007), 535. doi: 10.1016/j.aim.2007.05.014. Google Scholar

[13]

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

[14]

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

[15]

D. Dos Santos Ferreira, Y. Kurylev, M. Lassas and M. Salo, The Calderón problem in transversally anisotropic geometries,, J. Eur. Math. Soc, (). Google Scholar

[16]

B. Frigyik, P. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights,, J. Geom. Anal., 18 (2008), 89. doi: 10.1007/s12220-007-9007-6. Google Scholar

[17]

B. Haberman and D. Tataru, Uniqueness in Calderon's problem with Lipschitz conductivities,, Duke Math. J., 162 (2013), 497. doi: 10.1215/00127094-2019591. Google Scholar

[18]

H. Heck and J.-N. Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data,, Inverse Problems, 22 (2006), 1787. doi: 10.1088/0266-5611/22/5/015. Google Scholar

[19]

H. Heck and J.-N. Wang, Optimal stability estimate of the inverse boundary value problem by partial measurements,, preprint, (2007). Google Scholar

[20]

H. Kang and K. Yun, Boundary determination of conductivities and Riemannian metrics via local Dirichlet-to-Neumann operator,, SIAM J. Math. Anal., 34 (2003), 719. doi: 10.1137/S0036141001395042. Google Scholar

[21]

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

[22]

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

[23]

K. Knudsen, M. Lassas, J. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem,, Inverse Problems and Imaging, 3 (2009), 599. doi: 10.3934/ipi.2009.3.599. Google Scholar

[24]

J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements,, Commun. Pure Appl. Math., 42 (1989), 1097. doi: 10.1002/cpa.3160420804. Google Scholar

[25]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation,, Inverse Problems, 17 (2001), 1435. doi: 10.1088/0266-5611/17/5/313. Google Scholar

[26]

A. Nachman, Reconstructions from boundary measurements,, Ann. Math., 128 (1988), 531. doi: 10.2307/1971435. Google Scholar

[27]

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

[28]

M. Salo, The Calderón problem on Riemannian manifolds,, Chapter in Inverse Problems and Applications: Inside Out II (ed. G. Uhlmann), (2013), 167. Google Scholar

[29]

M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces,, J. Diff. Geom., 88 (2011), 161. Google Scholar

[30]

V. A. Sharafutdinov, Ray transform on Riemannian manifolds. Eight lectures on integral geometry,, , (). Google Scholar

[31]

P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity,, Duke Math. J., 123 (2004), 445. doi: 10.1215/S0012-7094-04-12332-2. Google Scholar

[32]

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

[33]

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

[34]

S. Vessella, A continuous dependence result in the analytic continuation problem,, Forum Math., 11 (1999), 695. doi: 10.1515/form.1999.020. Google Scholar

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