Scattering Problems and Stationary Waves

  • Victor Isakov
Part of the Applied Mathematical Sciences book series (AMS, volume 127)


The stationary incoming wave u with the wave number k is a solution to the perturbed Helmholtz equation (scattering by medium) \(Au - k^{2}u = 0\,\mathrm{in}\,\mathbb{R}^{3}\)


Helmholtz Equation Stability Estimate Inverse Scattering Harmonic Measure Inverse Scattering Problem 
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  1. [BLT]
    Bao, G., Lin, F., Triki, F. An inverse source problem with multiple frequency data. C. R. Acad. Sci. Paris, A 349 (2011), 855–859.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [CIL]
    Cheng, J., Isakov, V., Lu, S. Increasing stability in the inverse source prolem with many frequencies. J. Diff. Equat., 260 (2016), 4786–4804.CrossRefzbMATHGoogle Scholar
  3. [CheY3]
    Cheng, J., Yamamoto, M. Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves. Inverse Problems, 19 (2003), 1361–1385.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [CoK]
    Colton, D., Kirsch, A. A simple method for solving inverse scattering problems in the resonance region. Inverse Problems, 12 (1996), 383–395.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [CoKr]
    Colton, D., Kress, R. Inverse Acoustic and Electromagnetic Scattering Theory. Appl. Math. Sc., 93, Springer Verlag, 2013.Google Scholar
  6. [DR]
    Di Cristo, M., Rondi, L. Examples of exponential instability for inverse inclusion and scattering problems. Inverse Problems, 19 (2003), 685–701.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [EV]
    Eller, M., Valdivia, N. Acoustic source identification using multiple frequency information. Inverse Problems, 25 (2009), 115005.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [ER2]
    Eskin, G., Ralston, J. Inverse Backscattering Problem in Three Dimensions. Comm. Math. Phys., 124 (1989), 169–215.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [ER3]
    Eskin, G., Ralston, J. Inverse Backscattering in Two Dimensions. Comm. Math. Phys., 138 (1991), 451–486.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [HaH]
    Hähner, P., Hohage, T. New stability estimates for the inverse acoustic inhomogeneous medium problem and applications. SIAM J. Math. Anal., 62 (2001), 670–685.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [HeN]
    Henkin, G.M., Novikov, R.G. A multidimensional inverse problem in quantum and acoustic scattering. Inverse problems, 4 (1988), 103–121.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Is4]
    Isakov, V. Inverse Source Problems. Math. Surveys and Monographs Series, Vol. 34, AMS, Providence, R.I., 1990.Google Scholar
  13. [Is5]
    Isakov, V. On uniqueness in the inverse scattering problem. Comm. Part. Diff. Equat. 15 (1990), 1565–1581.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [Is8]
    Isakov, V. Stability estimates for obstacles in inverse scattering. J. Comp. Appl. Math., 42 (1991), 79–89.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [Is9]
    Isakov, V. New stability results for soft obstacles in inverse scattering. Inverse Problems, 9 (1993), 535–543.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [Is22]
    Isakov, V. Increasing stability for near field from scattering amplitude. Contemp. Math. AMS, 640 (2015), 59–70.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [IsN]
    Isakov, V., Nachman, A. Global Uniqueness for a two-dimensional elliptic inverse problem. Trans. AMS, 347 (1995), 3375–3391.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [KirK]
    Kirsch, A., Kress, R. Uniqueness in Inverse Obstacle Scattering. Inverse Problems 9, (1993), 285–299.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [Kl]
    Klibanov, M.V. Phaseless inverse scattering problems in three dimensions. SIAM J. Appl. Math., 74 (2014), 392–410.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [LaxP1]
    Lax, P., Phillips, R. The scattering of sound waves by an obstacle. Comm. Pure Appl. Math., 30 (1977), 195–233.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [LaxP2]
    Lax, P., Phillips, R. Scattering Theory. Academic Press, 1989.zbMATHGoogle Scholar
  22. [MaT]
    Majda, A., Taylor, M. Inverse Scattering Problems for transparent obstacles, electromagnetic waves and hyperbolic systems. Comm. Part. Diff. Equat., 2 (4) (1977), 395–438.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [Ma]
    Mandache, N. Exponential instability in an inverse problem for the Schrödinger equation. Inverse Problems, 17 (2001), 1435–1444.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [McL]
    McLean, W. Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, 2000.zbMATHGoogle Scholar
  25. [Mo]
    Morawetz, C. Notes on Time Decay and Scattering For Some Hyperbolic Problems. Reg. Conf. Series in Appl. Math. SIAM, Philadelphia, 1975.Google Scholar
  26. [N1]
    Nachman, A. Reconstruction from boundary measurements. Ann. Math., 128 (1988), 531–577.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [N2]
    Nachman, A. Global uniqueness for a two dimensional inverse boundary value problem. Ann. Math., 142 (1995), 71–96.MathSciNetzbMATHGoogle Scholar
  28. [No1]
    Novikov, R.G. Multidimensional inverse spectral problem for the equation \(-\mathrm{\Delta }\psi + (v(x) - Eu(x))\psi = 0\). Funct. Anal. and Appl., 22 (1988), 11–23.MathSciNetGoogle Scholar
  29. [PS]
    Päivärinta, L., Sylvester, J. Transmission eigenvalues. SIAM J. Math. Anal., 40 (2008), 783–753.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [RS]
    Rondi, L., Sini, M. Stable determination of a scattered wave from its far field pattern: the high frequency asymptotics. Arch. Rat. Mech. Anal., 218 (2015), 1–54.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [St]
    Stefanov, P. Inverse Scattering Problem for Moving Obstacles. Math. Z., 207 (1991), 461–481.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [SyU1]
    Sylvester, J., Uhlmann, G. Global Uniqueness Theorem for an Inverse Boundary Problem. Ann. Math., 125 (1987), 153–169.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [Tay]
    Taylor, M. Partial Differential Equations. Springer, 1996.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Victor Isakov
    • 1
  1. 1.Department of Mathematics and StatisticsWichita State UniversityWichitaUSA

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