November  2011, 5(4): 859-877. doi: 10.3934/ipi.2011.5.859

Reconstructions from boundary measurements on admissible manifolds

1. 

Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637-1514, United States

2. 

Department of Mathematics and Statistics, University of Helsinki and University of Jyväskylä, PO Box 68, 00014 Helsinki, Finland

3. 

Department of Mathematics, University of Washington and, Department of Mathematics, University of California, Irvine, CA 92697-3875, United States

Received  November 2010 Revised  August 2011 Published  November 2011

We prove that a potential $q$ can be reconstructed from the Dirichlet-to-Neumann map for the Schrödinger operator $-\Delta_g + q$ in a fixed admissible $3$-dimensional Riemannian manifold $(M,g)$. We also show that an admissible metric $g$ in a fixed conformal class can be constructed from the Dirichlet-to-Neumann map for $\Delta_g$. This is a constructive version of earlier uniqueness results by Dos Santos Ferreira et al. [10] on admissible manifolds, and extends the reconstruction procedure of Nachman [31] in Euclidean space. The main points are the derivation of a boundary integral equation characterizing the boundary values of complex geometrical optics solutions, and the development of associated layer potentials adapted to a cylindrical geometry.
Citation: Carlos E. Kenig, Mikko Salo, Gunther Uhlmann. Reconstructions from boundary measurements on admissible manifolds. Inverse Problems & Imaging, 2011, 5 (4) : 859-877. doi: 10.3934/ipi.2011.5.859
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show all references

References:
[1]

Comm. PDE, 30 (2005), 207-224. doi: 10.1081/PDE-200044485.  Google Scholar

[2]

K. Astala and L. Päivärinta, A boundary integral equation for Calderón's inverse conductivity problem,, Collect. Math., 2006 (): 127.   Google Scholar

[3]

Ann. of Math. (2), 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.  Google Scholar

[4]

Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.  Google Scholar

[5]

Comm. PDE, 17 (1992), 767-804. doi: 10.1080/03605309208820863.  Google Scholar

[6]

J. Inverse Ill-posed Probl., 16 (2008), 19-33. doi: 10.1515/jiip.2008.002.  Google Scholar

[7]

in "Seminar on Numerical Analysis and its Applications to Continuum Physics" (Rio de Janeiro, 1980), 65-73, Soc. Brasileira de Matemática, Río de Janeiro, 1980.  Google Scholar

[8]

in "Proc. International Conference on Inverse Problems," 2010, Hong Kong, Journal of Physics: Conference Series, 290 (2011), 012003. doi: 10.1088/1742-6596/290/1/012003.  Google Scholar

[9]

preprint, 2011, arXiv:1104.0232. Google Scholar

[10]

Invent. Math., 178 (2009), 119-171. doi: 10.1007/s00222-009-0196-4.  Google Scholar

[11]

Sov. Phys. Dokl., 10 (1966), 1033-1035. Google Scholar

[12]

J. Geom. Anal., 18 (2008), 89-108. doi: 10.1007/s12220-007-9007-6.  Google Scholar

[13]

Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[14]

Inverse Probl. Imaging, 3 (2009), 1-15. doi: 10.3934/ipi.2009.3.1.  Google Scholar

[15]

Duke Math. J., 158 (2011), 83-120. doi: 10.1215/00127094-1276310.  Google Scholar

[16]

Geom. Funct. Anal., 21 (2011), 393-418. doi: 10.1007/s00039-011-0110-2.  Google Scholar

[17]

Geom. Funct. Anal., 17 (2007), 116-155. doi: 10.1007/s00039-006-0590-7.  Google Scholar

[18]

J. Geom. Anal., 18 (2008), 1033-1052. doi: 10.1007/s12220-008-9035-x.  Google Scholar

[19]

Uspekhi Mat. Nauk, 42 (1987), 93-152, 255.  Google Scholar

[20]

J. Geom. Anal., 21 (2011), 543-587. doi: 10.1007/s12220-010-9158-8.  Google Scholar

[21]

Inverse Problems, 26 (2010), 095011, 18 pp.  Google Scholar

[22]

G. Henkin and M. Santacesaria, Gel'fand-Calderón's inverse problem for anisotropic conductivities on bordered surfaces in $\mathbbR^3$,, IMRN (to appear), ().   Google Scholar

[23]

J. Inverse Ill-Posed Probl., 1 (1993), 141-153. doi: 10.1515/jiip.1993.1.2.141.  Google Scholar

[24]

Comm. PDE, 23 (1998), 55-95. doi: 10.1080/03605309808821338.  Google Scholar

[25]

Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall/CRC, Boca Raton, FL, 2001.  Google Scholar

[26]

Duke Math. J., 157 (2011), 369-419. doi: 10.1215/00127094-1272903.  Google Scholar

[27]

Comm. Anal. Geom., 11 (2003), 207-221.  Google Scholar

[28]

Ann. Sci. École Norm. Sup. (4), 34 (2001), 771-787.  Google Scholar

[29]

Comm. Pure Appl. Math., 42 (1989), 1097-1112. doi: 10.1002/cpa.3160420804.  Google Scholar

[30]

Cambridge University Press, Cambridge, 2000.  Google Scholar

[31]

Ann. Math. (2), 128 (1988), 531-576. doi: 10.2307/1971435.  Google Scholar

[32]

Ann. of Math. (2), 143 (1996), 71-96. doi: 10.2307/2118653.  Google Scholar

[33]

Comm. PDE, 35 (2010), 375-390. doi: 10.1080/03605300903296322.  Google Scholar

[34]

Funct. Anal. Appl., 22 (1988), 263-272. doi: 10.1007/BF01077418.  Google Scholar

[35]

in "Imaging Microstructures," 161-184, Contemp. Math., 494, Amer. Math. Soc., Providence, RI, 2009.  Google Scholar

[36]

Duke Math. J., 70 (1993), 617-653. doi: 10.1215/S0012-7094-93-07014-7.  Google Scholar

[37]

Comm. PDE, 31 (2006), 1639-1666. doi: 10.1080/03605300500530420.  Google Scholar

[38]

J. Diff. Geom., 88 (2011), 161-187. Google Scholar

[39]

Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994.  Google Scholar

[40]

Ann. of Math. (2), 125 (1987), 153-169. doi: 10.2307/1971291.  Google Scholar

[41]

Mathematical Surveys and Monographs, 81, American Mathematical Society, Providence, RI, 2000.  Google Scholar

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