About stability and regularization of ill-posed elliptic Cauchy problems : the case of C1,1 domains
Bourgeois, Laurent
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010), p. 715-735 / Harvested from Numdam

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: COCV 9 (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.

Publié le : 2010-01-01
DOI : https://doi.org/10.1051/m2an/2010016
Classification:  35A15,  35N25,  35R25,  35R30
@article{M2AN_2010__44_4_715_0,
     author = {Bourgeois, Laurent},
     title = {About stability and regularization of ill-posed elliptic Cauchy problems : the case of C1,1 domains},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {44},
     year = {2010},
     pages = {715-735},
     doi = {10.1051/m2an/2010016},
     mrnumber = {2683580},
     zbl = {1194.35497},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2010__44_4_715_0}
}
Bourgeois, Laurent. About stability and regularization of ill-posed elliptic Cauchy problems : the case of C1,1 domains. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010) pp. 715-735. doi : 10.1051/m2an/2010016. http://gdmltest.u-ga.fr/item/M2AN_2010__44_4_715_0/

[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. Pisa 29 (2000) 755-806. | Numdam | Zbl 1034.35148

[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. | Zbl 1094.35134

[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).

[4] A.L. Bukhgeim, Extension of solutions of elliptic equations from discrete sets. J. Inv. Ill-Posed Problems 1 (1993) 17-32. | Zbl 0820.35020

[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. | Zbl 0022.34201

[6] J. Cheng, M Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation. M3AS 18 (2008) 107-123. | Zbl 1155.35108

[7] M.C. Delfour and J.-P. Zolésio, Shapes and geometries. SIAM, USA (2001). | Zbl 1002.49029

[8] C. Fabre and G. Lebeau, Prolongement unique des solutions de l'équation de Stokes. Comm. Part. Differ. Equ. 21 (1996) 573-596. | Zbl 0849.35098

[9] A. Fursikov and O. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34. Research Institute of Mathematics, Seoul National University, South Korea (1996). | Zbl 0862.49004

[10] P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, USA (1985). | Zbl 0695.35060

[11] L. Hormander, Linear Partial Differential Operators. Fourth Printing, Springer-Verlag, Germany (1976).

[12] T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation. Inv. Prob. 20 (2004) 697-712. | Zbl 1086.35080

[13] V. Isakov, Inverse problems for partial differential equations. Springer-Verlag, Berlin, Germany (1998). | Zbl 1092.35001

[14] F. John, Continuous dependence on data for solutions of pde with a prescribed bound. Commun. Pure Appl. Math. 13 (1960) 551-585. | Zbl 0097.08101

[15] M.V. Klibanov, Estimates of initial conditions of parabolic equations and inequalities via lateral data. Inv. Prob. 22 (2006) 495-514. | Zbl 1094.35139

[16] M.V. Klibanov and A.A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. VSP (2004). | Zbl 1069.65106

[17] R. Lattès and J.-L. Lions, Méthode de quasi-réversibilité et applications. Dunod, France (1967). | Zbl 0159.20803

[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). | Zbl 0593.35003

[19] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur. Commun. Partial Differ. Equ. 20 (1995) 335-356. | Zbl 0819.35071

[20] L.E. Payne, On a priori bounds in the Cauchy problem for elliptic equations. SIAM J. Math. Anal. 1 (1970) 82-89. | Zbl 0199.16603

[21] K.-D. Phung, Remarques sur l'observabilité pour l'équation de Laplace. ESAIM: COCV 9 (2003) 621-635. | Numdam | Zbl 1076.93009

[22] L. Robbiano, Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques. Commun. Partial Differ. Equ. 16 (1991) 789-800. | Zbl 0735.35086

[23] D.A. Subbarayappa and V. Isakov, On increased stability in the continuation of the Helmholtz equation. Inv. Prob. 23 (2007) 1689-1697. | Zbl 1127.35082

[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. | Zbl 1185.65173