# Uniqueness and Stability in the Cauchy Problem

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

## Abstract

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.

## Keywords

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.

## References

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.
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.
3. [AlVe]
Alessandrini, G., Vessella, S. Remark on the strong unique continuation property for parabolic equations. Proc. AMS, 132 (2003), 499–501.
4. [AliB]
Alinhac, S., Baouendi, M.S. A nonuniqueness result for operators of principal type. Math. Zeit., 220 (1995), 561–568.
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.
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.
8. [Ci]
Ciarlet, P.G. Mathematical Elasticity. North-Holland, Amsterdam, 1988.
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.
12. [ElI]
Eller, M., Isakov, V. Carleman estimates with two large parameters and applications. Contemp. Math., AMS, 268 (2000), 117–137.
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.
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.
17. [GZ]
Gajevski, H., Zacharias, K. Zur Regularizierung einer Klasse nichtkorrecter Probleme bei Evolutionsgleichungen. J. Math. Anal. Appl., 38 (1972), 784–789.
18. [Hö1]
Hörmander, L. Linear Partial Differential Operators. Springer-Verlag, 1963.
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.
21. [Im]
Imanuvilov, O. On Carleman estimates for hyperbolic equations. Asympt. Anal., 32 (2002), 185–220.
22. [IY2]
Imanuvilov, O., Yamamoto, M. Global uniqueness and stability in determining coefficients of wave equations. Comm. Part. Diff. Equat., 26 (2001), 1409–1425.
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.
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.
26. [Is12]
Isakov, V. Carleman Type Estimates in an Anisotropic Case and Applications. J. Diff. Equat., 105 (1993), 217–239.
27. [Is13]
Isakov, V. On uniqueness in a lateral Cauchy Problem with multiple characteristics. J. Diff. Equat., 133 (1997), 134–147.
28. [Is14]
Isakov, V. On the uniqueness of the continuation for a thermoelasticity system. SIAM J. Math. Anal., 33 (2001), 509–522.
29. [Is17]
Isakov, V. Increased stability in the continuation for the Helmholtz equation with variable coefficient. Contemp. Math. AMS, 426 (2007), 255–269.
30. [Is20]
Isakov, V. Carleman estimates for some anisotropic elasticity systems and applications. Evol. Equat. Control Theory, 1 (2012), 141–154.
31. [Is22]
Isakov, V. Increasing stability for near field from scattering amplitude. Contemp. Math. AMS, 640 (2015), 59–70.
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.
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.
36. [J4]
John, F. Partial Differential Equations. Springer, 1982.
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.
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.
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.
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.
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.
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.
46. [LaTZ1]
Lasiecka, I., Triggiani, R., Zhang, X. Non conservative Wave Equations with Unobserved Neumann B.C. Contemp. Math., AMS, 268 (2000), 227–325.
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.
49. [Lax]
Lax, P. Asymptotic solution of oscillatory initial value problems. Duke Math. J., 24 (1957), 627–646.
50. [LitT]
Littman, W., Taylor, S.W. Smoothing evolution equations and boundary control theory. J.d’ Anal. Math., 59 (1992), 201–215.
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.
53. [Me]
Metivier, G. Counterexamples to Holmgren’s uniqueness for analytic non-linear Cauchy Problem. Inv. Math., 112 (1993), 1–8.
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.
58. [Tat1]
Tataru, D. A-priori estimates of Carleman’s type in domains with boundary. J. Math. Pures Appl., 73 (1994), 355–389.
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.
60. [Tat3]
Tataru, D. Carleman estimates and unique continuation for solutions to boundary value problems. J. Math. Pures Appl., 75 (1996), 367–408.
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