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Carleman Estimates for Second Order Partial Differential Operators and Applications

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Abstract

In this chapter, we shall establish a fundamental weighted identity for second order partial differential operators, via which the main results (Carleman estimates and applications) in this book and most of other related results in previous references can be deduced. Also, some frequently used notations (throughout this book) will be introduced, and some background for Carleman estimates and two stimulating examples explaining the main idea of these sort of estimates will be presented.

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References

  1. Adams, R.A.: Sobolev Spaces. Academic, Cambridge (1975)

    Google Scholar 

  2. Aronszajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl. 36, 235–249 (1957)

    MathSciNet  MATH  Google Scholar 

  3. Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30, 1024–1065 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baudouin, L., Puel, J.P.: Uniqueness and stability in an inverse problem for the Schrödinger equation. Inverse Probl. 18, 1537–1554 (2002)

    Article  MATH  Google Scholar 

  5. Baudouin, L., de Buhan, M., Ervedoza, S.: Global Carleman estimates for waves and applications. Commun. Part. Differ. Equ. 38, 823–859 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bourgain, J., Kenig, C.: On localization in the continuous Anderson-Bernoulli model in higher dimension. Invent. Math. 161, 389–426 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bukhgeim, A.L., Klibanov, M.V.: Global uniqueness of class of multidimensional inverse problems. Soviet Math. Dokl. 24, 244–247 (1981)

    MATH  Google Scholar 

  9. Calderón, A.P.: Uniqueness in the cauchy problem for partial differential equations. Am. J. Math. 80, 16–36 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cannarsa, P., Martinez, P., Vancostenoble, J.: Global Carleman Estimates for Degenerate Parabolic Operators with Applications. Memoirs of the American Mathematical Society, vol. 239 (2016)

    Google Scholar 

  11. Carleman, T.: Sur un problème d’unicité pour les systèmes d’équations aux dérivées partielles à deux variables indépendantes. Ark. Mat. Astr. Fys. 26 B. 17, 1–9 (1939)

    Google Scholar 

  12. Chen, X.: A strong unique continuation theorem for parabolic equations. Math. Ann. 311, 603–630 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Choulli, M.: Applications of Elliptic Carleman Inequalities to Cauchy and Inverse Problems. Springer Briefs in Mathematics. BCAM Springer Briefs. Springer, Cham (2016)

    Book  MATH  Google Scholar 

  14. Courant, R., Hilbert, D.: Methods of Mathematical Physics II. Wiley-Interscience, New York (1962)

    MATH  Google Scholar 

  15. Donnelly, H., Fefferman, C.: Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math. 93, 161–183 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  16. Doubova, A., Fernández-Cara, E., González-Burgos, M., Zuazua, E.: On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim. 41, 798–819 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Duyckaerts, T., Zhang, X., Zuazua, E.: On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire. 25, 1–41 (2008)

    Google Scholar 

  18. Fernández-Cara, E., Zuazua, E.: Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire. 17, 583–616 (2000)

    Google Scholar 

  19. Fernández-Cara, E.: Null controllability of the semilinear heat equations. ESAIM: Control Optim. Calc. Var. 2, 87–107 (1997)

    Google Scholar 

  20. Fernández-Cara, E., Zuazua, E.: The cost of approximate controllability for heat equations: the linear case. Adv. Differ. Equ. 5, 465–514 (2000)

    MathSciNet  MATH  Google Scholar 

  21. Fernández-Cara, E., Guerrero, S., Imanuvilov, OYu., Puel, J.-P.: Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83, 1501–1542 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fu, X.: A weighted identity for partial differential operators of second order and its applications. C. R. Acad. Sci. Paris Série I(342), 579–584 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fu, X.: Null controllability for the parabolic equation with a complex principal part. J. Funct. Anal. 257, 1333–1354 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fu, X.: Logarithmic decay of hyperbolic equations with arbitrary small boundary damping. Commun. Partial Differ. Equ. 34, 957–975 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fu, X.: Longtime behavior of the hyperbolic equations with an arbitrary internal damping. Z. angew. Math. Phys. 62, 667–680 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fu, X.: Sharp decay rates for the weakly coupled hyperbolic system with one internal damping. SIAM J. Control Optim. 50, 1643–1660 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Fu, X., Yong, J., Zhang, X.: Exact controllability for the multidimensional semilinear hyperbolic equations. SIAM J. Control Optim. 46, 1578–1614 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Fursikov, A.V., Imanuvilov, O.Y.: Controllability of Evolution Equations. Lecture Notes Series, vol. 34. Seoul National University, Seoul, Korea (1996)

    Google Scholar 

  29. Garofalo, N., Lin, F.: Monotonicity properties of variational integrals, \(A_p\) weights and unique continuation. Indiana Univ. Math. J. 35, 245–268 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  30. Holmgren, E.: Über systeme von linearen partiellen differentialgleichungen. Öfversigt af Kongl. Vetenskaps-Academien Förhandlinger 58, 91–103 (1901)

    MATH  Google Scholar 

  31. Hörmander, L.: Linear Partial Differential Operators. Springer, Berlin (1963)

    Book  MATH  Google Scholar 

  32. Hörmander, L.: The Analysis of Linear Partial Differential Operators IV. Fourier Integral Operators. Springer, Berlin (1985)

    MATH  Google Scholar 

  33. Imanuvilov, OYu.: Controllability of the parabolic equations. Sbornik Math. 186, 879–900 (1995)

    Article  MathSciNet  Google Scholar 

  34. Imanuvilov, OYu.: On Carlerman estimates for hyperbolic equations. Asymptot. Anal. 32, 185–220 (2002)

    MathSciNet  MATH  Google Scholar 

  35. Imanuvilov, OYu., Yamamoto, M.: Global Lipschitz stability in an inverse hyperbolic problem by interior observations. Inverse Probl. 17, 717–728 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  36. Imanuvilov, OYu., Yamamoto, M.: Carleman inequalities for parabolic equations in a Sobolev spaces of negative order and exact controllability for semilinear parabolic equations. Publ. RIMS Kyoto Univ. 39, 227–274 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  37. Isakov, V.: Inverse Problems for Partial Differential Equations. Springer, New York (2006)

    MATH  Google Scholar 

  38. Jerison, D., Lebeau, G.: Nodal sets of sums of eigenfunctions. In: Harmonic Analysis and Partial Differential Equations, pp. 223–239. Chicago (1996). Chicago Lectures in Mathematics, University of Chicago Press, Chicago (1999)

    Google Scholar 

  39. Kazemi, M., Klibanov, M.V.: Stability estimates for ill-posed Cauchy problems involving hyperbolic equations and inequalities. Appl. Anal. 50, 93–102 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kenig, C.E.: Carleman estimates, uniform Sobolev inequalities for second-order differential operators, and unique continuation theorems. In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 948–960. Berkeley, California, USA (1986)

    Google Scholar 

  41. Klibanov, M.V.: Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems. J. Inverse Ill-Posed Probl. 21, 477–560 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Klibanov, M.V., Timonov, A.: Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. Inverse and Ill-Posed Problems Series. VSP, Utrecht (2004)

    Book  MATH  Google Scholar 

  43. Lasiecka, I., Triggiani, R., Zhang, X.: Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. I. \(H^1(\Omega )\)-estimates. J. Inverse Ill-Posed Probl. 11, 43–123 (2004)

    Google Scholar 

  44. Lasiecka, I., Triggiani, R., Zhang, X.: Nonconservative wave equations with purely Neumann B. C.: global uniqueness and observability in one shot. Contemp. Math. 268, 227–326 (2000)

    Google Scholar 

  45. Lavrent’ev, M.M., Romanov, V.G., Shishatskiĭ, S.P.: Ill-Posed Problems of Mathematical Physics and Analysis. Translations of Mathematical Monographs, vol. 64. American Mathematical Society, Providence (1986)

    Google Scholar 

  46. Le Rousseau, J., Robbiano, L.: Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces. Invent. Math. 183, 245–336 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  47. Lebeau, G., Robbiano, L.: Contrôle exact de l’équation de la chaleur. Commun. Part. Differ. Equ. 20, 335–356 (1995)

    Article  MATH  Google Scholar 

  48. Lebeau, G., Robbiano, L.: Stabilisation de l’équation des ondes par le bord. Duke Math. J. 86, 465–491 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  49. Li, W., Zhang, X.: Controllability of parabolic and hyperbolic equations: toward a unified theory. In: Control Theory of Partial Differential Equations. Lecture Notes in Pure and Applied Mathematics, vol. 242, pp. 157–174. Chapan & Hall/CRC, Boca Raton (2005)

    Google Scholar 

  50. Lin, F.: A uniqueness theorem for parabolic equations. Commun. Pure Appl. Math. 43, 127–136 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  51. Lions, J.L.: Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes distribués, Tome 1, Contrôlabilité exacte. Recherches en Mathématiques Appliquées. vol. 8. Masson, Paris (1988)

    Google Scholar 

  52. Liu, X., Zhang, X.: Local controllability of multidimensional quasilinear parabolic equations. SIAM J. Control Optim. 50, 2046–2064 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  53. López, A., Zhang, X., Zuazua, E.: Null controllality of the heat equation as singular limit of the exact controllability of dissipative wave equations. J. Math. Pures Appl. 79, 741–808 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  54. Lü, Q.: A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators. ESAIM: Control Optim. Calc. Var. 19, 255–273 (2013)

    Google Scholar 

  55. Lü, Q.: Observability estimate and state observation problems for stochastic hyperbolic equations. Inverse Probl. 29, 095011 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  56. Lü, Q., Yin, Z.: The \(L^\infty \)-null controllability of parabolic equation with equivalued surface boundary conditions. Asymptot. Anal. 83, 355–378 (2013)

    MathSciNet  MATH  Google Scholar 

  57. Mercado, A., Osses, A., Rosier, L.: Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights. Inverse Probl. 24 (2008)

    Google Scholar 

  58. Müller, C.: On the behavior of the solutions of the differential equation \(\Delta U=F(x, U)\) in the neighborhood of a point. Commun. Pure Appl. Math. 7, 505–515 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  59. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. In: Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

    Google Scholar 

  60. Phung, K.D., Zhang, X.: Time reversal focusing of the initial state for Kirchoff plate. SIAM J. Appl. Math. 68, 1535–1556 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  61. Robbiano, L.: Carleman estimates, results on control and stabilization for partial differential equations. In: Proceedings of the International Congress of Mathematicians, vol. IV, pp. 897–919. Seoul, South Korea (2014)

    Google Scholar 

  62. Saut, J.C., Scheurer, B.: Unique continuation for some evolution equations. J. Differ. Equ. 66, 118–139 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  63. Tang, S., Zhang, X.: Null controllability for forward and backward stochastic parabolic equations. SIAM J. Control Optim. 48, 2191–2216 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  64. Tataru, D.: Carleman estimates and unique continuation for solutions to boundary value problems. J. Math. Pures Appl. 75, 367–408 (1996)

    MathSciNet  MATH  Google Scholar 

  65. Tebou, L.: A Carleman estimate based approach for the stabilization of some locally damped semilinear hyperbolic equations. ESAIM: Control Optim. Calc. Var. 14, 561–574 (2008)

    Google Scholar 

  66. Vessella, S.: Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-caring boundaries and optimal stability estimates. Inverse Probl. 24, 023001 (2008)

    Article  MATH  Google Scholar 

  67. Yamabe, H.: A unique continuation theorem of a diffusion equation. Ann. Math. 69, 462–466 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  68. Yamamoto, M.: Carleman estimates for parabolic equations and applications. Inverse Probl. 25, 123013 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  69. Yuan, G., Yamamoto, M.: Lipschitz stability in inverse problems for a Kirchhoff plate equation. Asympt. Anal. 53, 29–60 (2007)

    MathSciNet  MATH  Google Scholar 

  70. Zhang, X.: A unified controllability/observability theory for some stochastic and deterministic partial differential equations. In: Proceedings of the International Congress of Mathematicians, vol. IV, pp. 3008–3034. Hyderabad, India (2010)

    Google Scholar 

  71. Zhang, X.: Exact controllability of the semilinear distributed parameter system and some related problems. PhD thesis, Fudan University, Shanghai, China (1998)

    Google Scholar 

  72. Zhang, X.: Explicit observability estimate for the wave equation with potential and its application. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456, 1101–1115 (2000)

    Google Scholar 

  73. Zhang, X.: Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities. SIAM J. Control Optim. 39, 812–834 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  74. Zhang, X.: Exact controllability of semilinear plate equations. Asymptot. Anal. 27, 95–125 (2001)

    MathSciNet  MATH  Google Scholar 

  75. Zhang, X.: Carleman and observability estimates for stochastic wave equations. SIAM J. Math. Anal. 40, 851–868 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  76. Zuazua, E.: Controllability and observability of partial differential equations: some results and open problems. In: Handbook of Differential Equations: Evolutionary Differential Equations, vol. 3, pp. 527–621. Elsevier Science, Amsterdam (2006)

    Google Scholar 

  77. Zuily, C.: Uniqueness and Non-Uniqueness in the Cauchy Problem. Birkhäuser, Boston (1983)

    Book  MATH  Google Scholar 

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  1. School of Mathematics, Sichuan University, Chengdu, China

    Xiaoyu Fu, Qi Lü & Xu Zhang

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  1. Xiaoyu Fu
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Fu, X., Lü, Q., Zhang, X. (2019). Introduction. In: Carleman Estimates for Second Order Partial Differential Operators and Applications. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-29530-1_1

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