The search session has expired. Please query the service again.

Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions

Jérôme Le Rousseau[1]; Nicolas Lerner[2]

  • [1] MAPMO, UMR CNRS 6628, Route de Chartres, Université d’Orléans B.P. 6759 – 45067 Orléans cedex 2 France
  • [2] Projet analyse fonctionnelle, Institut de Mathématiques de Jussieu, UMR CNRS 7586, Université Pierre-et-Marie-Curie (Paris 6), Boîte 186 - 4, Place Jussieu - 75252 Paris cedex 05, France

Journées Équations aux dérivées partielles (2010)

  • page 1-23
  • ISSN: 0752-0360

Abstract

top
We consider a second-order selfadjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight-function such that a Carleman estimate holds true. We moreover prove that the conditions imposed on the weight function are necessary.

How to cite

top

Le Rousseau, Jérôme, and Lerner, Nicolas. "Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions." Journées Équations aux dérivées partielles (2010): 1-23. <http://eudml.org/doc/116379>.

@article{LeRousseau2010,
abstract = {We consider a second-order selfadjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight-function such that a Carleman estimate holds true. We moreover prove that the conditions imposed on the weight function are necessary.},
affiliation = {MAPMO, UMR CNRS 6628, Route de Chartres, Université d’Orléans B.P. 6759 – 45067 Orléans cedex 2 France; Projet analyse fonctionnelle, Institut de Mathématiques de Jussieu, UMR CNRS 7586, Université Pierre-et-Marie-Curie (Paris 6), Boîte 186 - 4, Place Jussieu - 75252 Paris cedex 05, France},
author = {Le Rousseau, Jérôme, Lerner, Nicolas},
journal = {Journées Équations aux dérivées partielles},
keywords = {Carleman estimate; elliptic operator; non-smooth coefficient; sharp condition; quasi-mode},
language = {eng},
month = {6},
pages = {1-23},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions},
url = {http://eudml.org/doc/116379},
year = {2010},
}

TY - JOUR
AU - Le Rousseau, Jérôme
AU - Lerner, Nicolas
TI - Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions
JO - Journées Équations aux dérivées partielles
DA - 2010/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 23
AB - We consider a second-order selfadjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight-function such that a Carleman estimate holds true. We moreover prove that the conditions imposed on the weight function are necessary.
LA - eng
KW - Carleman estimate; elliptic operator; non-smooth coefficient; sharp condition; quasi-mode
UR - http://eudml.org/doc/116379
ER -

References

top
  1. S. Alinhac, Non-unicité pour des opérateurs différentiels à caractéristiques complexes simples, Ann. Sci. École Norm. Sup. (4) 13 (1980), 385-393 Zbl0456.35002MR597745
  2. A. Benabdallah, Y. Dermenjian, J. Le Rousseau, 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 (2007), 865-887 Zbl1189.35349MR2352986
  3. P. Buonocore, P. Manselli, Nonunique continuation for plane uniformly elliptic equations in Sobolev spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), 731-754 Zbl1072.35049MR1822406
  4. A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations., Amer. J. Math. 80 (1958), 16-36 Zbl0080.30302MR104925
  5. T. Carleman, Sur un problème d’unicité pur les systèmes d’équations aux dérivées partielles à deux variables indépendantes, Ark. Mat., Astr. Fys. 26 (1939) Zbl0022.34201
  6. A. Doubova, A. Osses, J.-P. Puel, Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients, ESAIM Control Optim. Calc. Var. 8 (2002), 621-661 Zbl1092.93006MR1932966
  7. L. Hörmander, On the uniqueness of the Cauchy problem, Math. Scand. 6 (1958), 213-225 Zbl0088.30201MR104924
  8. L. Hörmander, Linear partial differential operators, (1963), Academic Press Inc., Publishers, New York Zbl0108.09301MR161012
  9. L. Hörmander, The analysis of linear partial differential operators. IV, 275 (1994), Springer-Verlag, Berlin Zbl0612.35001MR1481433
  10. O. Yu. Imanuvilov, J.-P. Puel, Global Carleman estimates for weak solutions of Elliptic nonhomogeneous Dirichlet problems, Int. Math. Res. Not. 16 (2003), 883-913 Zbl1146.35340MR1959940
  11. J. Le Rousseau, Carleman estimates and controllability results for the one-dimensional heat equation with B V coefficients, J. Differential Equations 233 (2007), 417-447 Zbl1128.35020MR2292514
  12. J. Le Rousseau, G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, Preprint (2009) Zbl1262.35206
  13. J. Le Rousseau, N. Lerner, Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions, in prep. (2010) 
  14. J. Le Rousseau, L. Robbiano, Carleman estimate for elliptic operators with coefficents with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations, Arch. Rational Mech. Anal. 195 (2010), 953-990 Zbl1202.35336MR2591978
  15. J. Le Rousseau, L. Robbiano, Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces, Invent. Math, to appear (2010) Zbl1218.35054
  16. N. Lerner, Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators, (2010), Birkhäuser, Basel Zbl1186.47001MR2599384
  17. K. Miller, Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients, Arch. Rational Mech. Anal. 54 (1974), 105-117 Zbl0289.35046MR342822
  18. A. Pliś, On non-uniqueness in Cauchy problem for an elliptic second order differential equation, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 (1963), 95-100 Zbl0107.07901MR153959
  19. F. Schulz, On the unique continuation property of elliptic divergence form equations in the plane, Math. Z. 228 (1998), 201-206 Zbl0905.35020MR1630571

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.