The Calderón problem with partial data

Johannes Sjöstrand[1]

  • [1] CMLS, École Polytechnique, F-91128 Palaiseau cedex (UMR 7640, CNRS)

Journées Équations aux dérivées partielles (2004)

  • page 1-9
  • ISSN: 0752-0360

Abstract

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We describe a joint work with C.E. Kenig and G. Uhlmann [9] where we improve an earlier result by Bukhgeim and Uhlmann [1], by showing that in dimension n ≥ 3 , the knowledge of the Cauchy data for the Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of [1] but use a richer set of solutions to the Dirichlet problem.

How to cite

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Sjöstrand, Johannes. "The Calderón problem with partial data." Journées Équations aux dérivées partielles (2004): 1-9. <http://eudml.org/doc/10601>.

@article{Sjöstrand2004,
abstract = {We describe a joint work with C.E. Kenig and G. Uhlmann [9] where we improve an earlier result by Bukhgeim and Uhlmann [1], by showing that in dimension $n\ge 3$, the knowledge of the Cauchy data for the Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of [1] but use a richer set of solutions to the Dirichlet problem.},
affiliation = {CMLS, École Polytechnique, F-91128 Palaiseau cedex (UMR 7640, CNRS)},
author = {Sjöstrand, Johannes},
journal = {Journées Équations aux dérivées partielles},
keywords = {Dirichlet to Neumann map; Carleman estimates; analytic microlocal analysis},
language = {eng},
month = {6},
pages = {1-9},
publisher = {Groupement de recherche 2434 du CNRS},
title = {The Calderón problem with partial data},
url = {http://eudml.org/doc/10601},
year = {2004},
}

TY - JOUR
AU - Sjöstrand, Johannes
TI - The Calderón problem with partial data
JO - Journées Équations aux dérivées partielles
DA - 2004/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 9
AB - We describe a joint work with C.E. Kenig and G. Uhlmann [9] where we improve an earlier result by Bukhgeim and Uhlmann [1], by showing that in dimension $n\ge 3$, the knowledge of the Cauchy data for the Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of [1] but use a richer set of solutions to the Dirichlet problem.
LA - eng
KW - Dirichlet to Neumann map; Carleman estimates; analytic microlocal analysis
UR - http://eudml.org/doc/10601
ER -

References

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  1. A. L. Bukhgeim, G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. PDE, 27(3,4)(2002), 653–668. Zbl0998.35063MR1900557
  2. N. Burq, Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonances au voisinage du réel, Acta Math. 180(1)(1998), 1–29. Zbl0918.35081MR1618254
  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), 65-73. MR590275
  4. M. Dimassi, J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268. Cambridge University Press, Cambridge, 1999. Zbl0926.35002MR1735654
  5. J.J. Duistermaat, L. Hörmander, Fourier integral operators II, Acta Mathematica 128(1972), 183-269. Zbl0232.47055MR388464
  6. A. Greenleaf and G. Uhlmann, Local uniqueness for the Dirichlet-to-Neumann map via the two-plane transform, Duke Math. J. 108(2001), 599-617. Zbl1013.35085MR1838663
  7. L. Hörmander, Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math. 24(1971), 671–704. Zbl0226.35019MR294849
  8. H. Isozaki and G. Uhlmann, Hyperbolic geometry and the Dirichlet-to-Neumann map, Advances in Math., to appear. Zbl1062.35172
  9. C.E. Kenig, J. Sjöstrand, G. Uhlmann, The Calderón problem with partial data, preprint http://xxx.lanl.gov/abs/math.AP/0405486 . Zbl1127.35079
  10. G. Lebeau, L. Robbiano, Contrôle exact de l’équation de la chaleur, Comm. P.D.E. 20(1-2)(1995), 335–356. Zbl0819.35071MR1312710
  11. R. Novikov, Multidimensional inverse spectral problems for the equation - Δ ψ + ( v ( x ) - E u ( x ) ) ψ = 0 , Funkt. An. Ego Pril. 22(4)(1988), 11–12, and Funct. An. and its Appl., 22(4)(1988), 263–272. Zbl0689.35098MR976992
  12. M. Sato, T. Kawai, M. Kashiwara, Microfunctions and pseudodifferential equations, Springer Lecture Notes in Math. 287 Zbl0277.46039MR420735
  13. J. Sjöstrand, Singularités analytiques microlocales, Astérisque 95 (1982). Zbl0524.35007MR699623
  14. J. Sylvester, G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125(1987), 153–169. Zbl0625.35078MR873380
  15. G. Uhlmann, Developments in inverse problems since Calderón’s foundational paper, Chapter 19 in “Harmonic Analysis and Partial Differential Equations", University of Chicago Press (1999), 295-345, edited by M. Christ, C. Kenig and C. Sadosky. Zbl0963.35203MR1743870

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