Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-16T17:29:53.136Z Has data issue: false hasContentIssue false
  • English
  • Français

GLOBAL UNIQUENESS FOR THE CALDERÓN PROBLEM WITH LIPSCHITZ CONDUCTIVITIES

Part of: Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  13 January 2016

PEDRO CARO  and
KEITH M. ROGERS
PEDRO CARO
Affiliation:
BCAM - Basque Center for Applied Mathematics, 48009 Bilbao, Spain; [email protected] Ikerbasque, Basque Foundation for Science, 48011 Bilbao, Spain
KEITH M. ROGERS
Affiliation:
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, 28049 Madrid, Spain; [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove uniqueness for the Calderón problem with Lipschitz conductivities in higher dimensions. Combined with the recent work of Haberman, who treated the three- and four-dimensional cases, this confirms a conjecture of Uhlmann. Our proof builds on the work of Sylvester and Uhlmann, Brown, and Haberman and Tataru who proved uniqueness for $C^{1}$ -conductivities and Lipschitz conductivities sufficiently close to the identity.

MSC classification

Primary: 35R30: Inverse problems
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 4 , 2016 , e2
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

References

Alessandrini, G., ‘Singular solutions of elliptic equations and the determination of conductivity by boundary measurements’, J. Differential Equations 84 (1990), 252–272.Google Scholar
Astala, K., Lassas, M. and Päivärinta, L., ‘The borderlines of invisibility and visibility for Calderon’s inverse problem’, Anal. PDE, to appear.Google Scholar
Astala, K. and Päivärinta, L., ‘Calderón’s inverse conductivity problem in the plane’, Ann. of Math. (2) 163 (2006), 265–299.CrossRefGoogle Scholar
Brown, R. M., ‘Global uniqueness in the impedance-imaging problem for less regular conductivities’, SIAM J. Math. Anal. 27 (1996), 1049–1056.Google Scholar
Brown, R. M., ‘Recovering the conductivity at the boundary from the Dirichlet to Neumann map: a pointwise result’, J. Inverse Ill-Posed Probl. 9 (2001), 567–574.Google Scholar
Brown, R. M. and Torres, R. H., ‘Uniqueness in the inverse conductivity problem for conductivities with 3∕2 derivatives in L p , p > 2n ’, J. Fourier Anal. Appl. 9 (2003), 563–574.Google Scholar
Calderón, A. P., ‘On an inverse boundary value problem’, Comput. Appl. Math. 25 (1980), 133–138.Google Scholar
Caro, P., García, A. and Reyes, J. M., ‘Stability of the Calderón problem for less regular conductivities’, J. Differential Equations 254 (2013), 469–492.Google Scholar
Dos Santos Ferreira, D., Kenig, C. E., Sjöstrand, J. and Uhlmann, G., ‘Determining a magnetic Schrödinger operator from partial Cauchy data’, Comm. Math. Phys. 271 (2007), 467–488.CrossRefGoogle Scholar
García, A. and Zhang, G., ‘Reconstruction from boundary measurements for less regular conductivities’, Preprint, 2012, arXiv:1212.0727.Google Scholar
Greenleaf, A., Kurylev, Y., Lassas, M. and Uhlmann, G., ‘Invisibility and inverse problems’, Bull. Amer. Math. Soc. (N.S.) 46(1) (2009), 55–97.Google Scholar
Greenleaf, A., Lassas, M. and Uhlmann, G., ‘The Calderón problem for conormal potentials. I. Global uniqueness and reconstruction’, Comm. Pure Appl. Math. 56(3) (2003), 328–352.CrossRefGoogle Scholar
Haberman, B., ‘Uniqueness in Calderón’s problem for conductivities with unbounded gradient’, Comm. Math. Phys. 340(2) (2015), 639–659.Google Scholar
Haberman, B. and Tataru, D., ‘Uniqueness in Calderón’s problem with Lipschitz conductivities’, Duke Math. J. 162 (2013), 497–516.Google Scholar
Jerison, D. and Kenig, C. E., ‘The inhomogeneous Dirichlet problem in Lipschitz domains’, J. Funct. Anal. 130 (1995), 161–219.Google Scholar
Kenig, C. E., Sjöstrand, J. and Uhlmann, G., ‘The Calderón problem with partial data’, Ann. of Math. (2) 165(2) (2007), 567–591.Google Scholar
Kohn, R. and Vogelius, M., ‘Determining conductivity by boundary measurements’, Comm. Pure Appl. Math. 37 (1984), 289–298.CrossRefGoogle Scholar
Krupchyk, K. and Uhlmann, G., ‘Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential’, Comm. Math. Phys. 327 (2014), 993–1009.Google Scholar
Nachman, A., Sylvester, J. and Uhlmann, G., ‘An n-dimensional Borg–Levinson theorem’, Comm. Math. Phys. 115(4) (1988), 595–605.Google Scholar
Päivärinta, L., Panchenko, A. and Uhlmann, G., ‘Complex geometrical optics solutions for Lipschitz conductivities’, Rev. Mat. Iberoam. 19(1) (2003), 57–72.Google Scholar
Pliś, A., ‘On non-uniqueness in Cauchy problem for an elliptic second order differential equation’, Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 11 (1963), 95–100.Google Scholar
Stein, E. M., Singular Integrals and Differentiability Properties of Functions (Princeton University Press, New Jersey, 1970).Google Scholar
Sylvester, J. and Uhlmann, G., ‘A uniqueness theorem for an inverse boundary value problem in electrical prospection’, Comm. Pure Appl. Math. 39 (1986), 91–112.Google Scholar
Sylvester, J. and Uhlmann, G., ‘A global uniqueness theorem for an inverse boundary value problem’, Ann. of Math. (2) 125 (1987), 153–169.CrossRefGoogle Scholar
Sylvester, J. and Uhlmann, G., ‘Inverse boundary value problems at the boundary-continuous dependence’, Comm. Pure Appl. Math. 41 (1988), 197–219.CrossRefGoogle Scholar
Tataru, D., ‘The X 𝜃 s spaces and unique continuation for solutions to the semilinear wave equation’, Comm. Partial Differential Equations 21 (1996), 841–887.Google Scholar
Uhlmann, G., ‘Inverse boundary value problems for partial differential equations’, inProceedings of the International Congress of Mathematicians, Doc. Math., Vol. III (Berlin, 1998), 77–86.Google Scholar
Wolff, T. H., ‘Recent work on sharp estimates in second-order elliptic unique continuation problems’, J. Geom. Anal. 3(6) (1993), 621–650.CrossRefGoogle Scholar