About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1,1 domains

Laurent Bourgeois

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 4, page 715-735
  • ISSN: 0764-583X

Abstract

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This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with C1,1 boundary. It is an extension of an earlier result of [Phung, ESAIM: COCV9 (2003) 621–635] for domains of class C∞. Our estimate is established by using a Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces a nearly optimal convergence rate for the method of quasi-reversibility introduced in [Lattès and Lions, Dunod (1967)] to solve the ill-posed Cauchy problems.

How to cite

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Bourgeois, Laurent. "About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1,1 domains." ESAIM: Mathematical Modelling and Numerical Analysis 44.4 (2010): 715-735. <http://eudml.org/doc/250773>.

@article{Bourgeois2010,
abstract = { This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with C1,1 boundary. It is an extension of an earlier result of [Phung, ESAIM: COCV9 (2003) 621–635] for domains of class C∞. Our estimate is established by using a Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces a nearly optimal convergence rate for the method of quasi-reversibility introduced in [Lattès and Lions, Dunod (1967)] to solve the ill-posed Cauchy problems. },
author = {Bourgeois, Laurent},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Carleman estimate; distance function; elliptic Cauchy problems; conditional stability; quasi-reversibility},
language = {eng},
month = {6},
number = {4},
pages = {715-735},
publisher = {EDP Sciences},
title = {About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1,1 domains},
url = {http://eudml.org/doc/250773},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Bourgeois, Laurent
TI - About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1,1 domains
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/6//
PB - EDP Sciences
VL - 44
IS - 4
SP - 715
EP - 735
AB - This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with C1,1 boundary. It is an extension of an earlier result of [Phung, ESAIM: COCV9 (2003) 621–635] for domains of class C∞. Our estimate is established by using a Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces a nearly optimal convergence rate for the method of quasi-reversibility introduced in [Lattès and Lions, Dunod (1967)] to solve the ill-posed Cauchy problems.
LA - eng
KW - Carleman estimate; distance function; elliptic Cauchy problems; conditional stability; quasi-reversibility
UR - http://eudml.org/doc/250773
ER -

References

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  1. G. Alessandrini, E. Beretta, E. Rosset and S. Vessella, Optimal stability for inverse elliptic boundary value problems with unknown boundaries. Ann. Scuola Norm. Sup. Pisa29 (2000) 755–806.  Zbl1034.35148
  2. L. Bourgeois, Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation. Inv. Prob.22 (2006) 413–430.  Zbl1094.35134
  3. L. Bourgeois and J. Dardé, Conditional stability for ill-posed elliptic Cauchy problems: the case of Lipschitz domains (part II). Rapport INRIA 6588, France (2008).  Zbl1206.35252
  4. A.L. Bukhgeim, Extension of solutions of elliptic equations from discrete sets. J. Inv. Ill-Posed Problems1 (1993) 17–32.  Zbl0820.35020
  5. T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes. Ark. Mat. Astr. Fys.26 (1939) 1–9.  Zbl0022.34201
  6. J. Cheng, M Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation. M3AS18 (2008) 107–123.  Zbl1155.35108
  7. M.C. Delfour and J.-P. Zolésio, Shapes and geometries. SIAM, USA (2001).  Zbl1002.49029
  8. C. Fabre and G. Lebeau, Prolongement unique des solutions de l'équation de Stokes. Comm. Part. Differ. Equ.21 (1996) 573–596.  Zbl0849.35098
  9. A. Fursikov and O. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series34. Research Institute of Mathematics, Seoul National University, South Korea (1996).  Zbl0862.49004
  10. P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, USA (1985).  Zbl0695.35060
  11. L. Hormander, Linear Partial Differential Operators. Fourth Printing, Springer-Verlag, Germany (1976).  Zbl0321.35001
  12. T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation. Inv. Prob.20 (2004) 697–712.  Zbl1086.35080
  13. V. Isakov, Inverse problems for partial differential equations. Springer-Verlag, Berlin, Germany (1998).  Zbl0908.35134
  14. F. John, Continuous dependence on data for solutions of pde with a prescribed bound. Commun. Pure Appl. Math.13 (1960) 551–585.  Zbl0097.08101
  15. M.V. Klibanov, Estimates of initial conditions of parabolic equations and inequalities via lateral data. Inv. Prob.22 (2006) 495–514.  Zbl1094.35139
  16. M.V. Klibanov and A.A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. VSP (2004).  Zbl1069.65106
  17. R. Lattès and J.-L. Lions, Méthode de quasi-réversibilité et applications. Dunod, France (1967).  Zbl0159.20803
  18. M.M. Lavrentiev, V.G. Romanov and S.P. Shishatskii, Ill-posed problems in mathematical physics and analysis. Amer. Math. Soc., Providence, USA (1986).  
  19. G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur. Commun. Partial Differ. Equ.20 (1995) 335–356.  Zbl0819.35071
  20. L.E. Payne, On a priori bounds in the Cauchy problem for elliptic equations. SIAM J. Math. Anal.1 (1970) 82–89.  Zbl0199.16603
  21. K.-D. Phung, Remarques sur l'observabilité pour l'équation de Laplace. ESAIM: COCV9 (2003) 621–635.  
  22. L. Robbiano, Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques. Commun. Partial Differ. Equ.16 (1991) 789–800.  Zbl0735.35086
  23. D.A. Subbarayappa and V. Isakov, On increased stability in the continuation of the Helmholtz equation. Inv. Prob.23 (2007) 1689–1697.  Zbl1127.35082
  24. T. Takeuchi and M. Yamamoto, Tikhonov regularization by a reproducing kernel Hilbert space for the Cauchy problem for a elliptic equation. SIAM J. Sci. Comput.31 (2008) 112–142.  Zbl1185.65173

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