Advertisement

Archive for Rational Mechanics and Analysis

, Volume 195, Issue 3, pp 953–990 | Cite as

Carleman Estimate for Elliptic Operators with Coefficients with Jumps at an Interface in Arbitrary Dimension and Application to the Null Controllability of Linear Parabolic Equations

  • Jérôme Le RousseauEmail author
  • Luc Robbiano
Article

Abstract

In a bounded domain of R n+1, n ≧ 2, we consider a second-order elliptic operator, \({A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}\), where the (scalar) coefficient c(x) is piecewise smooth yet discontinuous across a smooth interface S. We prove a local Carleman estimate for A in the neighborhood of any point of the interface. The “observation” region can be chosen independently of the sign of the jump of the coefficient c at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions and the Calderón projector technique. Following the method of Lebeau and Robbiano (Comm Partial Differ Equ 20:335–356, 1995) we then prove the null controllability for the linear parabolic initial problem with Dirichlet boundary conditions associated with the operator \({{\partial_t - \nabla_x \cdot (c(x) \nabla_x)}}\) .

Keywords

Pseudodifferential Operator Trace Formula Transmission Condition Principal Symbol Interpolation Inequality 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alessandrini G., Escauriaza L.: Null-controllability of one-dimensional parabolic equations. ESAIM Control Optim. Calc. Var. 14, 284–293 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Benabdallah A., Dermenjian Y., Le Rousseau J.: Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem. J. Math. Anal. Appl. 336, 865–887 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Benabdallah A., Dermenjian Y., Le Rousseau J.: On the controllability of linear parabolic equations with an arbitrary control location for stratified media. C. R. Acad. Sci. Paris, Ser. I. 344, 357–362 (2007)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bellassoued M.: Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization. Asymptotic Anal. 35, 257–279 (2003)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Doubova A., Osses A., Puel J.-P.: Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients. ESAIM: Control Optim. Calc. Var. 8, 621–661 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fernández-Cara E., Zuazua E.: On the null controllability of the one-dimensional heat equation with BV coefficients. Comput. Appl. Math. 21, 167–190 (2002)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Fursikov, A., Imanuvilov, O. Yu.: Controllability of evolution equations, vol. 34. Lecture notes. Seoul National University, Korea, 1996Google Scholar
  8. 8.
    Hörmander L.: Linear Partial Differential Operators. Springer, Berlin (1963)zbMATHGoogle Scholar
  9. 9.
    Hörmander L.: The Analysis of Linear Partial Differential Operators, vol. IV. Springer, Berlin (1985)Google Scholar
  10. 10.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. III. Springer, Berlin, 1985 (Second printing 1994)Google Scholar
  11. 11.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, 2nd edn., vol. I. Springer, Berlin, 1990Google Scholar
  12. 12.
    Le Rousseau J.: Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients. J. Differ. Equ. 233, 417–447 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lions, J.-L.: Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, vol. 1. Masson, Paris, 1988Google Scholar
  14. 14.
    Lebeau G., Robbiano L.: Contrôle exact de l’équation de la chaleur. Comm. Partial Differ. Equ. 20, 335–356 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lebeau G., Robbiano L.: Stabilisation de l’équation des ondes par le bord. Duke Math. J. 86, 465–491 (1997)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Lebeau G., Zuazua E.: Null-controllability of a system of linear thermoelasticity. Arch. Ration. Mech. Anal. 141, 297–329 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Martinez A.: An Introduction to Semiclassical and Microlocal Analysis. Springer, Berlin (2002)zbMATHGoogle Scholar
  18. 18.
    Miller L.: On the controllability of anomalous diffusions generated by the fractional Laplacian. Math. Control Signals Syst. 3, 260–271 (2006)CrossRefGoogle Scholar
  19. 19.
    Robbiano L.: Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptotic Anal. 10, 95–115 (1995)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Taylor M.E.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)zbMATHGoogle Scholar
  21. 21.
    Zuily C.: Uniqueness and Non Uniqueness in the Cauchy Problem. Progress in mathematics. Birkhauser, Boston (1983)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Laboratoire d’Analyse Topologie Probabilités, CNRS UMR 6632Université de ProvenceMarseilleFrance
  2. 2.Laboratoire Mathématiques et Applications, Physique Mathématique d’Orléans, CNRS UMR 6628, Fédération Denis Poisson, FR CNRS 2964Université d’OrléansOrléans cedex 2France
  3. 3.Laboratoire de Mathématiques de Versailles, CNRS UMR 8100Université de Versailles Saint-QuentinVersaillesFrance

Personalised recommendations