Uniqueness and Stability in the Cauchy Problem

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


In this chapter we formulate and in many cases prove results on the uniqueness and stability of solutions of the Cauchy problem for general partial differential equations. One of the basic tools is Carleman-type estimates. In Section 3.1 we describe the results for a simplest problem of this kind (the backward parabolic equation), where a choice of the weight function in Carleman estimates is obvious, and the method is equivalent to that of the logarithmic convexity. In Section 3.2 we formulate general conditional Carleman estimates and their simplifications for second-order equations, and we apply the results to the general Cauchy problem and give numerous counterexamples showing that the assumptions of positive results are quite sharp.


Carleman Estimates Backward Parabolic Equation Lateral Cauchy Problem Strongly Pseudoconvex Tataru 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [AN]
    Agmon, S., Nirenberg, L. Lower Bounds and Uniqueness Theorems for Solutions of Differential Equations in a Hilbert Space. Comm. Pure Appl. Math., 20 (1967), 207–229.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [AlBGHV]
    Aleksandrov, A., Bourgain, J., Giesecke, M., Havin, V.P., Vymenetz, Yu. Uniqueness and free interpolation for logarithmic potentials and the Cauchy problem for the Laplace equation in \(\mathbb{R}^{2}\). Geometric and Functional Analysis, 5 (1995), 529–571.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [AlVe]
    Alessandrini, G., Vessella, S. Remark on the strong unique continuation property for parabolic equations. Proc. AMS, 132 (2003), 499–501.CrossRefzbMATHGoogle Scholar
  4. [AliB]
    Alinhac, S., Baouendi, M.S. A nonuniqueness result for operators of principal type. Math. Zeit., 220 (1995), 561–568.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [BarLR]
    Bardos, C., Lebeau, G., Rauch, J. Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundaries. SIAM J. Control and Optimization, 30 (1992), 1024–1065.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BoW]
    Bourgain, J., Wolff, T. A remark on gradients of harmonic functions in dimension 3. Coll. Math. LX/LXI (1990), 253–260.Google Scholar
  7. [Ca]
    Carleman, T. Sur un probleme d’unicite pour les systemes d’equations aux derivees partielles a deux variables independentes. Ark. Mat. Astr. Fys., 26B (1939), 1–9.MathSciNetzbMATHGoogle Scholar
  8. [Ci]
    Ciarlet, P.G. Mathematical Elasticity. North-Holland, Amsterdam, 1988.zbMATHGoogle Scholar
  9. [CoV]
    Colombo, F., Vespri, V. Generations of analytic semigroups in W k, p(Ω) and \(C^{k}(\mathrm{\bar{\varOmega }})\). Differential and Integral Equations 9 (1996), 421–436.Google Scholar
  10. [CoKr]
    Colton, D., Kress, R. Inverse Acoustic and Electromagnetic Scattering Theory. Appl. Math. Sc., 93, Springer Verlag, 2013.Google Scholar
  11. [CouH]
    Courant, R., Hilbert, D. Methods of Mathematical Physics. Vol. II. Wiley, 1962.zbMATHGoogle Scholar
  12. [ElI]
    Eller, M., Isakov, V. Carleman estimates with two large parameters and applications. Contemp. Math., AMS, 268 (2000), 117–137.zbMATHGoogle Scholar
  13. [EINT]
    Eller, M., Isakov, V., Nakamura, G., Tataru, D. Uniqueness and stability in the Cauchy problem for Maxwell’s and elasticity systems. College de France Seminar, 14, “Studies in Math. Appl.”, Vol. 31, North-Holland, Elsevier Science, 2002, 329–349.Google Scholar
  14. [ElLT]
    Eller, M., Lasiecka, I., Triggiani, R. Simulteneous exact/approximate boundary controllability of thermoelastic plates with variable thermal coefficients and moment controls. J. Math. Anal. Appl., 251 (2000), 452–478.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [Fe]
    Fernandez, F. J. Unique continuation for Parabolic Operators. II Comm. Part Diff. Equat, 28 (2003), 1597–1604.Google Scholar
  16. [Fri]
    Friedrichs, K.O. Symmetric Positive Linear Differential Equations. Comm. Pure Appl. Math., 11 (1958), 333–418.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [GZ]
    Gajevski, H., Zacharias, K. Zur Regularizierung einer Klasse nichtkorrecter Probleme bei Evolutionsgleichungen. J. Math. Anal. Appl., 38 (1972), 784–789.MathSciNetCrossRefGoogle Scholar
  18. [Hö1]
    Hörmander, L. Linear Partial Differential Operators. Springer-Verlag, 1963.CrossRefzbMATHGoogle Scholar
  19. [Hö2]
    Hörmander, L. The Analysis of Linear Partial Differential Operators. Springer Verlag, New York, 1983–1985.Google Scholar
  20. [HrI]
    Hrycak, T., Isakov, V. Increased stability in the continuation of solutions of the Helmholtz equation. Inverse Problems, 20 (2004), 697–712.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [Im]
    Imanuvilov, O. On Carleman estimates for hyperbolic equations. Asympt. Anal., 32 (2002), 185–220.MathSciNetzbMATHGoogle Scholar
  22. [IY2]
    Imanuvilov, O., Yamamoto, M. Global uniqueness and stability in determining coefficients of wave equations. Comm. Part. Diff. Equat., 26 (2001), 1409–1425.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [IIY]
    Imanuvilov, O., Isakov, V., Yamamoto, M. An inverse problem for the dynamical Lame system with two sets of boundary data. Comm. Pure Appl. Math., 56 (2003), 1366–1382.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [Is1]
    Isakov, V. A Nonhyperbolic Cauchy Problem for □ b □ c and its Applications to Elasticity Theory. Comm. Pure Appl. Math. 39 (1986), 474–769.Google Scholar
  25. [Is2]
    Isakov, V. Uniqueness of continuation across a time-like hyperplane and related inverse problems for hyperbolic equations. Comm. Part. Diff. Equat. 14 (1989), 465–478.MathSciNetzbMATHGoogle Scholar
  26. [Is12]
    Isakov, V. Carleman Type Estimates in an Anisotropic Case and Applications. J. Diff. Equat., 105 (1993), 217–239.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [Is13]
    Isakov, V. On uniqueness in a lateral Cauchy Problem with multiple characteristics. J. Diff. Equat., 133 (1997), 134–147.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Is14]
    Isakov, V. On the uniqueness of the continuation for a thermoelasticity system. SIAM J. Math. Anal., 33 (2001), 509–522.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [Is17]
    Isakov, V. Increased stability in the continuation for the Helmholtz equation with variable coefficient. Contemp. Math. AMS, 426 (2007), 255–269.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [Is20]
    Isakov, V. Carleman estimates for some anisotropic elasticity systems and applications. Evol. Equat. Control Theory, 1 (2012), 141–154.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [Is22]
    Isakov, V. Increasing stability for near field from scattering amplitude. Contemp. Math. AMS, 640 (2015), 59–70.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [Is23]
    Isakov, V. Carleman estimates with two large parameters for an anisotropic system of elasticity. Rend. Ist. Mat. Univ. Trieste, 48 (2016), 261–283.Google Scholar
  33. [IsKi]
    Isakov, V., Kim, N. Weak Carleman estimates with two large parameters for second order equations and applications. Discr. Cont. Dyn. Syst.,A, 27 (2010), 799–827.Google Scholar
  34. [J1]
    John, F. Numerical Solution of the heat equation for preceding time. Ann. Mat. Pura Appl., 40 (1955), 129–142.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [J2]
    John, F. Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Comm. Pure Appl. Math., 13 (1960), 551–585.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [J4]
    John, F. Partial Differential Equations. Springer, 1982.CrossRefzbMATHGoogle Scholar
  37. [Ki]
    Kim, N. Uniqueness and Hölder stability of continuation for the linear thermoelasticity system with residual tress. Evol. Equat. Control Theory, 2 (2013), 679–693.CrossRefzbMATHGoogle Scholar
  38. [KlM]
    Klibanov, M.V., Malinskii, J. Newton-Kantorovich method for 3-dimensional inverse scattering problem and stability of the hyperbolic Cauchy problem with time independent data. Inverse Problems, 7 (1991), 577–595.MathSciNetCrossRefGoogle Scholar
  39. [KnV]
    Knabner, P., Vessella, S. Stabilization of ill-posed Cauchy problems for parabolic equations. Ann. Mat. Pura Appl., (4)149 (1987), 393–409.Google Scholar
  40. [KT1]
    Koch, H., Tataru, D. Carleman estimates and unique continuation for second order elliptic equations with non smooth coefficients. Comm. Pure Appl. Math., 54 (2001), 339–360.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [KT2]
    Koch, H., Tataru, D. Carleman estimates and unique continuation for second order parabolic equations with non smooth coefficients. Comm. Part. Diff. Equat., 34 (2009), 305–366.CrossRefzbMATHGoogle Scholar
  42. [Kre]
    Krein, S.G. Linear Differential Equations in Banach Space. Transl. Math. Monogr., 29, AMS, 1971.Google Scholar
  43. [KuG]
    Kumano-Go, H. On a example of non-uniqueness of solutions of the Cauchy problem for the wave equation. Proc. Japan Acad., 39 (1963), 578–582.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [LSU]
    Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N. Linear and quasilinear equations of parabolic type. Transl. Math. Monogr., 23, AMS, Providence, R.I., 1968.Google Scholar
  45. [LaTY]
    Lasiecka, I., Triggiani, R., Yao, P.F. Inverse/observability estimates for second order hyperbolic equations with variable coefficients. J. Math. Anal. Appl., 235 (1999), 13–57.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [LaTZ1]
    Lasiecka, I., Triggiani, R., Zhang, X. Non conservative Wave Equations with Unobserved Neumann B.C. Contemp. Math., AMS, 268 (2000), 227–325.zbMATHGoogle Scholar
  47. [LaTZ2]
    Lasiecka, I., Triggiani, R., Zhang, X. Global Uniqueness, Observability, and Stability of Non conservative Schrödinger Equations Via Carleman Estimates. Part II: L 2-estimates. J. Inv. Ill-Posed Problems, 11 (2003), 1–39.Google Scholar
  48. [LL]
    Lattes, R., Lions, J.L. Methodes de quasi-reversibilité et applications. Dunod, Paris, 1967.zbMATHGoogle Scholar
  49. [Lax]
    Lax, P. Asymptotic solution of oscillatory initial value problems. Duke Math. J., 24 (1957), 627–646.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [LitT]
    Littman, W., Taylor, S.W. Smoothing evolution equations and boundary control theory. J.d’ Anal. Math., 59 (1992), 201–215.MathSciNetzbMATHGoogle Scholar
  51. [Lo]
    Lop Fat Ho. Observabilité frontierè de l’équation des ondes. C.R. Acad. Sc. Paris, t. 302, Ser. I, #12 (1986), 443–446.Google Scholar
  52. [MT]
    Magnanini, R., Talenti, G. On complex-valued solutions to a 2D eikonal equation. SIAM J. Math. Anal., 34 (2003), 805–835.MathSciNetCrossRefzbMATHGoogle Scholar
  53. [Me]
    Metivier, G. Counterexamples to Holmgren’s uniqueness for analytic non-linear Cauchy Problem. Inv. Math., 112 (1993), 1–8.CrossRefGoogle Scholar
  54. [Ni]
    Nirenberg, L. Lectures on linear partial differential equations. Conf. Board in the Math. Sci Regional Conference 17, AMS, Providence, R.I., 1973.Google Scholar
  55. [Pl1]
    Plis̆, A. A smooth linear elliptic differential equation without any solution in a sphere. Comm. Pure Appl. Math., 14 (1961), 599–617.Google Scholar
  56. [Pl2]
    Plis̆, A. On non-uniqueness in Cauchy problem for an elliptic second order equation. Bull. Acad. Polon. Sci. Ser. Sc. Math. Astr. Phys., 11 (1963), 95–100.Google Scholar
  57. [Ro]
    Robbiano, L. Theorème d’Unicité Adapté au Contrôle des Solutions des Problèmes Hyperboliques. Comm. Part. Diff. Equat., 16 (1991), 789–801.CrossRefGoogle Scholar
  58. [Tat1]
    Tataru, D. A-priori estimates of Carleman’s type in domains with boundary. J. Math. Pures Appl., 73 (1994), 355–389.MathSciNetzbMATHGoogle Scholar
  59. [Tat2]
    Tataru, D. Unique continuation for solutions to PDE’s; between Hormander’s Theorem and Holmgren’s Theorem. Comm. Part. Diff. Equat., 20 (1995), 855–884.CrossRefzbMATHGoogle Scholar
  60. [Tat3]
    Tataru, D. Carleman estimates and unique continuation for solutions to boundary value problems. J. Math. Pures Appl., 75 (1996), 367–408.MathSciNetzbMATHGoogle Scholar
  61. [TrY]
    Triggiani, R., Yao, P.F. Carleman Estimates with No Lower-Order Terms for General Riemann Wave Equations. Global Uniqueness and Observability in One Shot. Appl. Math. Optim., 46 (2002), 331–375.Google Scholar
  62. [W]
    Watanabe, K. Sur l’unicité retrograde dans les problèmes mixtes paraboliques. Case de dimension 1. Math. Soc. Japan, 42 (1990), 377–386.Google Scholar
  63. [Wo]
    Wolff, T. Counterexamples with harmonic gradients in \(\mathbb{R}^{3}\). Essays on Fourier analysis, Princeton Math. Ser. 42, Princeton Univ. Press, Princeton, N.J., 1995.Google Scholar
  64. [Z]
    Zuily, C. Uniqueness and Non uniqueness in the Cauchy Problem. Progress in Mathematics 33, Birkhäuser, Boston, 1983.Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

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

Personalised recommendations