The Calderón problem with partial data in two dimensions

Imanuvilov, Oleg; Uhlmann, Gunther; Yamamoto, Masahiro

doi:10.1090/S0894-0347-10-00656-9

The Calderón problem with partial data in two dimensions
HTML articles powered by AMS MathViewer

by Oleg Yu. Imanuvilov, Gunther Uhlmann and Masahiro Yamamoto;

J. Amer. Math. Soc. 23 (2010), 655-691

DOI: https://doi.org/10.1090/S0894-0347-10-00656-9

Published electronically: February 16, 2010

PDF | Request permission

Abstract:

We prove for a two-dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary uniquely determines the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can uniquely determine the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results.

References

V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal control, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1987. Translated from the Russian by V. M. Volosov. MR 924574, DOI 10.1007/978-1-4615-7551-1
Kari Astala and Lassi Päivärinta, Calderón’s inverse conductivity problem in the plane, Ann. of Math. (2) 163 (2006), no. 1, 265–299. MR 2195135, DOI 10.4007/annals.2006.163.265
Kari Astala, Lassi Päivärinta, and Matti Lassas, Calderón’s inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations 30 (2005), no. 1-3, 207–224. MR 2131051, DOI 10.1081/PDE-200044485
Albrecht Böttcher and Bernd Silbermann, Analysis of Toeplitz operators, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006. Prepared jointly with Alexei Karlovich. MR 2223704
Russell M. Brown and Rodolfo H. Torres, Uniqueness in the inverse conductivity problem for conductivities with $3/2$ derivatives in $L^p,\ p>2n$, J. Fourier Anal. Appl. 9 (2003), no. 6, 563–574. MR 2026763, DOI 10.1007/s00041-003-0902-3
Russell M. Brown and Gunther A. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations 22 (1997), no. 5-6, 1009–1027. MR 1452176, DOI 10.1080/03605309708821292
A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl. 16 (2008), no. 1, 19–33. MR 2387648, DOI 10.1515/jiip.2008.002
Alexander L. Bukhgeim and Gunther Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Differential Equations 27 (2002), no. 3-4, 653–668. MR 1900557, DOI 10.1081/PDE-120002868
Alberto-P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980) Soc. Brasil. Mat., Rio de Janeiro, 1980, pp. 65–73. MR 590275
Jin Cheng and Masahiro Yamamoto, Determination of two convection coefficients from Dirichlet to Neumann map in the two-dimensional case, SIAM J. Math. Anal. 35 (2004), no. 6, 1371–1393. MR 2083783, DOI 10.1137/S0036141003422497
David Dos Santos Ferreira, Carlos E. Kenig, Johannes Sjöstrand, and Gunther Uhlmann, Determining a magnetic Schrödinger operator from partial Cauchy data, Comm. Math. Phys. 271 (2007), no. 2, 467–488. MR 2287913, DOI 10.1007/s00220-006-0151-9
David Dos Santos Ferreira, Carlos E. Kenig, Mikko Salo, and Gunther Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math. 178 (2009), no. 1, 119–171. MR 2534094, DOI 10.1007/s00222-009-0196-4
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845, DOI 10.1090/gsm/019
Allan Greenleaf, Matti Lassas, and Gunther Uhlmann, The Calderón problem for conormal potentials. I. Global uniqueness and reconstruction, Comm. Pure Appl. Math. 56 (2003), no. 3, 328–352. MR 1941812, DOI 10.1002/cpa.10061
Horst Heck and Jenn-Nan Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data, Inverse Problems 22 (2006), no. 5, 1787–1796. MR 2261266, DOI 10.1088/0266-5611/22/5/015
Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
Victor Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Probl. Imaging 1 (2007), no. 1, 95–105. MR 2262748, DOI 10.3934/ipi.2007.1.95
Hyeonbae Kang and Gunther Uhlmann, Inverse problems for the Pauli Hamiltonian in two dimensions, J. Fourier Anal. Appl. 10 (2004), no. 2, 201–215. MR 2054308, DOI 10.1007/s00041-004-8011-5
Carlos E. Kenig, Johannes Sjöstrand, and Gunther Uhlmann, The Calderón problem with partial data, Ann. of Math. (2) 165 (2007), no. 2, 567–591. MR 2299741, DOI 10.4007/annals.2007.165.567
Kim Knudsen, The Calderón problem with partial data for less smooth conductivities, Comm. Partial Differential Equations 31 (2006), no. 1-3, 57–71. MR 2209749, DOI 10.1080/03605300500361610
Kim Knudsen and Mikko Salo, Determining nonsmooth first order terms from partial boundary measurements, Inverse Probl. Imaging 1 (2007), no. 2, 349–369. MR 2282273, DOI 10.3934/ipi.2007.1.349
R. V. Kohn and M. Vogelius, Determining conductivity by boundary measurements. II. Interior results, Comm. Pure Appl. Math. 38 (1985), no. 5, 643–667. MR 803253, DOI 10.1002/cpa.3160380513
Robert V. Kohn and Michael Vogelius, Identification of an unknown conductivity by means of measurements at the boundary, Inverse problems (New York, 1983) SIAM-AMS Proc., vol. 14, Amer. Math. Soc., Providence, RI, 1984, pp. 113–123. MR 773707, DOI 10.1002/cpa.3160370302
Adrian I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2) 143 (1996), no. 1, 71–96. MR 1370758, DOI 10.2307/2118653
Lassi Päivärinta, Alexander Panchenko, and Gunther Uhlmann, Complex geometrical optics solutions for Lipschitz conductivities, Rev. Mat. Iberoamericana 19 (2003), no. 1, 57–72. MR 1993415, DOI 10.4171/RMI/338
Ziqi Sun and Gunther Uhlmann, Anisotropic inverse problems in two dimensions, Inverse Problems 19 (2003), no. 5, 1001–1010. MR 2024685, DOI 10.1088/0266-5611/19/5/301
John Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math. 43 (1990), no. 2, 201–232. MR 1038142, DOI 10.1002/cpa.3160430203
John Sylvester and Gunther Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), no. 1, 153–169. MR 873380, DOI 10.2307/1971291
John Sylvester and Gunther Uhlmann, Inverse boundary value problems at the boundary—continuous dependence, Comm. Pure Appl. Math. 41 (1988), no. 2, 197–219. MR 924684, DOI 10.1002/cpa.3160410205
Leo Tzou, Stability estimates for coefficients of magnetic Schrödinger equation from full and partial boundary measurements, Comm. Partial Differential Equations 33 (2008), no. 10-12, 1911–1952. MR 2475324, DOI 10.1080/03605300802402674
Gunther Uhlmann, Commentary on Calderón’s paper (29), on an inverse boundary value problem, Selected papers of Alberto P. Calderón, Amer. Math. Soc., Providence, RI, 2008, pp. 623–636. MR 2435340
I. N. Vekua, Generalized analytic functions, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Company, Inc., Reading, MA, 1962. MR 150320