Advertisement

Analysis and Mathematical Physics

, Volume 9, Issue 4, pp 2171–2199 | Cite as

A Hölder stability estimate for inverse problems for the ultrahyperbolic Schrödinger equation

  • Fikret GölgeleyenEmail author
  • Özlem Kaytmaz
Article
  • 40 Downloads

Abstract

In this article, we first establish a global Carleman estimate for an ultrahyperbolic Schrödinger equation. Next, we prove Hölder stability for the inverse problem of determining a coefficient or a source term in the equation by some lateral boundary data.

Keywords

Ultrahyperbolic Schrödinger equation Inverse problem Stability Carleman estimate 

Mathematics Subject Classification

35R30 35B35 35Q40 

Notes

Acknowledgements

The authors would like to thank the anonymous referee for his/her very valuable comments and suggestions.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.

References

  1. 1.
    Ablowitz, M.J., Haberman, R.: Nonlinear evolution equations in two and three dimensions. Phys. Rev. Lett. 35, 1185 (1975)Google Scholar
  2. 2.
    Amirov, A.K.: Integral Geometry and Inverse Problems for Kinetic Equations. VSP, Utrecht (2001)Google Scholar
  3. 3.
    Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control and stabilization from the boundary. SIAM J. Control Optim. 30, 1024–1065 (1992)Google Scholar
  4. 4.
    Baudouin, L., Puel, J.P.: Uniqueness and stability in an inverse problem for the Schrödinger equation. Inverse Probl. 18, 1537 (2002)Google Scholar
  5. 5.
    Bellassoued, M., Choulli, M.: Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by arbitrary boundary observation. J. Math. Pures Appl. 91, 233–255 (2009)Google Scholar
  6. 6.
    Bellassoued, M., Yamamoto, M.: Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation. J. Math. Pures Appl. 85, 193–224 (2006)Google Scholar
  7. 7.
    Bukhgeim, A.L., Klibanov, M.V.: Global uniqueness of a class of inverse problems. Dokl. Akad. Nauk SSSR 260, 269–272 (1981)Google Scholar
  8. 8.
    Calderón, A.P.: Uniqueness in the Cauchy problem for partial differential equations. Am. J. Math. 80, 16–36 (1958)Google Scholar
  9. 9.
    Carleman, T.: Sur un probleme, d’unicité pour les systemes d’équations aux derivées partiellesa deux variables independentes. Ark. Mat. Astr. Fys. 2, 1–9 (1939)Google Scholar
  10. 10.
    Cristofol, M., Soccorsi, E.: Stability estimate in an inverse problem for non-autonomous magnetic Schrödinger equations. Appl. Anal. 90, 1499–1520 (2011)Google Scholar
  11. 11.
    Davey, A., Stewartson, K.: On three-dimensional packets of surface waves. Proc. R. Soc. Lond. A Math. Phys. Sci 338, 101–110 (2011)Google Scholar
  12. 12.
    Djordjevic, V.D., Redekopp, L.G.: On two-dimensional packets of capillary-gravity waves. J. Fluid Mech. 79, 703–714 (1977)Google Scholar
  13. 13.
    Escauriaza, L., Kenig, C.E., Ponce, G., Vega, L.: Unique continuation for Schrödinger evolutions, with applications to profiles of concentration and traveling waves. Commun. Math. Phys. 305, 487–512 (2011)Google Scholar
  14. 14.
    Gölgeleyen, F., Yamamoto, M.: Stability of inverse problems for ultrahyperbolic equations. Chin. Ann. Math. B 35, 527–556 (2014)Google Scholar
  15. 15.
    Hörmander, L.: Linear Partial Differential Operators. Springer, Berlin (1963)Google Scholar
  16. 16.
    Ichinose, W.: A note on the Cauchy problem for Schrödinger type equations on the Riemannian manifold. Math. Japon. 35, 205–213 (1990)Google Scholar
  17. 17.
    Imanuvilov, O.Y., Yamamoto, M.: Global Lipschitz stability in an inverse hyperbolic problem by interior observations. Inverse Probl. 17, 717–728 (2001)Google Scholar
  18. 18.
    Imanuvilov, O.Y., Yamamoto, M.: Global uniqueness and stability in determining coefficients of wave equations. Commun. Partial. Differ. Equ. 26, 1409–1425 (2001)Google Scholar
  19. 19.
    Isakov, V., Yamamoto, M.: Carleman estimate with the Neumann boundary condition and its applications to the observability inequality and inverse hyperbolic problems. Contemp. Math. 268, 191–226 (2000)Google Scholar
  20. 20.
    Kenig, C.E.: Carleman estimates, uniform Sobolev inequalities for second-order differential operators, and unique continuation theorems. Proc. Int. Congr. Math. 1, 948–960 (1986)Google Scholar
  21. 21.
    Kenig, C.E., Ponce, G., Vega, L.: Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations. Invent. Math. 134, 489–545 (1998)Google Scholar
  22. 22.
    Kian, Y., Phan, Q.S., Soccorsi, E.: Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains. J. Math. Anal. Appl. 426, 194–210 (2015)Google Scholar
  23. 23.
    Klibanov, M.V., Yamamoto, M.: Lipschitz stability of an inverse problem for an acoustic equation. Appl. Anal. 85, 515–538 (2006)Google Scholar
  24. 24.
    Konopelchenko, B.G., Matkarimov, B.T.: On the inverse scattering transform for the Ishimori equation. Phys. Lett. A 135, 183–189 (1989)Google Scholar
  25. 25.
    Liu, S., Triggiani, R.: Boundary control and boundary inverse theory for non-homogeneous second-order hyperbolic equations: a common Carleman estimates approach. In: HCDTE Lecture Notes. Part I: Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations. Appl. Math. 6, 227–343 (2013)Google Scholar
  26. 26.
    Mercado, A., Osses, A., Rosier, L.: Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights. Inverse Probl. 24, 015017 (2008)Google Scholar
  27. 27.
    Puel, J.P., Yamamoto, M.: On a global estimate in a linear inverse hyperbolic problem. Inverse Probl. 12, 995 (1996)Google Scholar
  28. 28.
    Sulem, C., Sulem, P.L.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer, New York (1999)Google Scholar
  29. 29.
    Triggiani, R., Zhang, Z.: Global uniqueness and stability in determining the electric potential coefficient of an inverse problem for Schrödinger equations on Riemannian manifolds. J. Inverse Ill-Posed Probl. 23, 587–609 (2015)Google Scholar
  30. 30.
    Yamamoto, M.: Carleman estimates for parabolic equations and applications. Inverse Probl. 25, 123013 (2009)Google Scholar
  31. 31.
    Yuan, G., Yamamoto, M.: Carleman estimates for the Schrödinger equation and applications to an inverse problem and an observability inequality. Chin. Ann. Math. B 31, 555–578 (2010)Google Scholar
  32. 32.
    Zakharov, V.E., Kuznetsov, E.A.: Multi-scale expansions in the theory of systems integrable by the inverse scattering transform. Phys. D 18, 455–463 (1986)Google Scholar
  33. 33.
    Zakharov, V.E., Schulman, E.I.: Degenerative dispersion laws, motion invariants and kinetic equations. Phys. D 1, 192–202 (1980)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsZonguldak Bulent Ecevit UniversityZonguldakTurkey

Personalised recommendations