Skip to main content
SpringerLink
Log in

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

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

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 \({{\partial_t - \nabla_x \cdot (c(x) \nabla_x)}}\) .

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alessandrini G., Escauriaza L.: Null-controllability of one-dimensional parabolic equations. ESAIM Control Optim. Calc. Var. 14, 284–293 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  4. Bellassoued M.: Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization. Asymptotic Anal. 35, 257–279 (2003)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  7. Fursikov, A., Imanuvilov, O. Yu.: Controllability of evolution equations, vol. 34. Lecture notes. Seoul National University, Korea, 1996

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

    MATH  Google Scholar 

  9. Hörmander L.: The Analysis of Linear Partial Differential Operators, vol. IV. Springer, Berlin (1985)

    Google Scholar 

  10. Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. III. Springer, Berlin, 1985 (Second printing 1994)

  11. Hörmander, L.: The Analysis of Linear Partial Differential Operators, 2nd edn., vol. I. Springer, Berlin, 1990

  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)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lions, J.-L.: Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, vol. 1. Masson, Paris, 1988

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  16. Lebeau G., Zuazua E.: Null-controllability of a system of linear thermoelasticity. Arch. Ration. Mech. Anal. 141, 297–329 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Martinez A.: An Introduction to Semiclassical and Microlocal Analysis. Springer, Berlin (2002)

    MATH  Google Scholar 

  18. Miller L.: On the controllability of anomalous diffusions generated by the fractional Laplacian. Math. Control Signals Syst. 3, 260–271 (2006)

    Article  Google Scholar 

  19. Robbiano L.: Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptotic Anal. 10, 95–115 (1995)

    MATH  MathSciNet  Google Scholar 

  20. Taylor M.E.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)

    MATH  Google Scholar 

  21. Zuily C.: Uniqueness and Non Uniqueness in the Cauchy Problem. Progress in mathematics. Birkhauser, Boston (1983)

    MATH  Google Scholar 

Download references

Author information

Author notes

  1. Jérôme Le Rousseau

    Present address: Laboratoire Mathématiques et Applications, Physique Mathématique d’Orléans, CNRS UMR 6628, Fédération Denis Poisson, FR CNRS 2964, Université d’Orléans, B.P. 6759, 45067, Orléans cedex 2, France

Authors and Affiliations

  1. Laboratoire d’Analyse Topologie Probabilités, CNRS UMR 6632, Université de Provence, Marseille, France

    Jérôme Le Rousseau

  2. Laboratoire de Mathématiques de Versailles, CNRS UMR 8100, Université de Versailles Saint-Quentin, 45 Avenue des États-Unis, 78035, Versailles, France

    Luc Robbiano

Authors
  1. Jérôme Le Rousseau
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luc Robbiano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jérôme Le Rousseau.

Additional information

Communicated by P.-L. Lions

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rousseau, J.L., Robbiano, L. 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. Arch Rational Mech Anal 195, 953–990 (2010). https://doi.org/10.1007/s00205-009-0242-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-009-0242-9

Keywords

Search

Navigation