Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-16T09:22:26.776Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  23 February 2010

Laurent Bourgeois
Laurent Bourgeois*
Affiliation:
Laboratoire POEMS, ENSTA, 32 Boulevard Victor, 75739 Paris Cedex 15, France. [email protected]
Get access

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 C 1,1 boundary. It is an extension of anearlier 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.

Keywords

Carleman estimatedistance functionelliptic Cauchy problemsconditional stabilityquasi-reversibility
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 4 , July 2010 , pp. 715 - 735
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alessandrini, G., Beretta, E., Rosset, E. and Vessella, S., Optimal stability for inverse elliptic boundary value problems with unknown boundaries. Ann. Scuola Norm. Sup. Pisa 29 (2000) 755–806.
Bourgeois, L., Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation. Inv. Prob. 22 (2006) 413–430. CrossRef
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).
Bukhgeim, A.L., Extension of solutions of elliptic equations from discrete sets. J. Inv. Ill-Posed Problems 1 (1993) 17–32.
Carleman, T., 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.
Cheng, J., Choulli, M and Lin, J., Stable determination of a boundary coefficient in an elliptic equation. M3AS 18 (2008) 107–123.
M.C. Delfour and J.-P. Zolésio, Shapes and geometries. SIAM, USA (2001).
Fabre, C. and Lebeau, G., Prolongement unique des solutions de l'équation de Stokes. Comm. Part. Differ. Equ. 21 (1996) 573–596. CrossRef
A. Fursikov and O. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34. Research Institute of Mathematics, Seoul National University, South Korea (1996).
P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, USA (1985).
L. Hormander, Linear Partial Differential Operators. Fourth Printing, Springer-Verlag, Germany (1976).
Hrycak, T. and Isakov, V., Increased stability in the continuation of solutions to the Helmholtz equation. Inv. Prob. 20 (2004) 697–712. CrossRef
V. Isakov, Inverse problems for partial differential equations. Springer-Verlag, Berlin, Germany (1998).
John, F., Continuous dependence on data for solutions of pde with a prescribed bound. Commun. Pure Appl. Math. 13 (1960) 551–585. CrossRef
Klibanov, M.V., Estimates of initial conditions of parabolic equations and inequalities via lateral data. Inv. Prob. 22 (2006) 495–514. CrossRef
M.V. Klibanov and A.A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. VSP (2004).
R. Lattès and J.-L. Lions, Méthode de quasi-réversibilité et applications. Dunod, France (1967).
M.M. Lavrentiev, V.G. Romanov and S.P. Shishatskii, Ill-posed problems in mathematical physics and analysis. Amer. Math. Soc., Providence, USA (1986).
Lebeau, G. and Robbiano, L., Contrôle exact de l'équation de la chaleur. Commun. Partial Differ. Equ. 20 (1995) 335–356. CrossRef
Payne, L.E., On a priori bounds in the Cauchy problem for elliptic equations. SIAM J. Math. Anal. 1 (1970) 82–89. CrossRef
Phung, K.-D., Remarques sur l'observabilité pour l'équation de Laplace. ESAIM: COCV 9 (2003) 621–635. CrossRef
Robbiano, L., Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques. Commun. Partial Differ. Equ. 16 (1991) 789–800. CrossRef
Subbarayappa, D.A. and Isakov, V., On increased stability in the continuation of the Helmholtz equation. Inv. Prob. 23 (2007) 1689–1697. CrossRef
Takeuchi, T. and Yamamoto, M., Tikhonov regularization by a reproducing kernel Hilbert space for the Cauchy problem for a elliptic equation. SIAM J. Sci. Comput. 31 (2008) 112–142. CrossRef