zbMATH — the first resource for mathematics

About stability and regularization of ill-posed elliptic Cauchy problems: the case of \(C^{1,1}\) domains. (English) Zbl 1194.35497
Summary: This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace equation in domains with \(C^{1,1}\) boundary. It is an extension of an earlier result of K.-D. Phung [ESAIM, Control Optim. Calc. Var. 9, 621–635 (2003; Zbl 1076.93009)] for domains of class \(C^{\infty}\). 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 [R. Lattès and J.-L. Lions, Méthode de quasi-réversibilité et applications. Travaux et Recherches Mathématiques, 15. Paris: Dunot (1967; Zbl 0159.20803)] to solve ill-posed Cauchy problems.

35R30 Inverse problems for PDEs
35R25 Ill-posed problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35N25 Overdetermined boundary value problems for PDEs and systems of PDEs
35B35 Stability in context of PDEs
35J20 Variational methods for second-order elliptic equations
Full Text: DOI EuDML
[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. · Zbl 1034.35148 · numdam:ASNSP_2000_4_29_4_755_0 · eudml:84427
[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 · doi:10.1088/0266-5611/22/2/002
[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 Problems1 (1993) 17-32. Zbl0820.35020 · Zbl 0820.35020 · doi:10.1515/jiip.1993.1.1.17
[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 · Zbl 0022.34201
[6] J. Cheng, M Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation. M3AS18 (2008) 107-123. Zbl1155.35108 · Zbl 1155.35108 · doi:10.1142/S0218202508002620
[7] M.C. Delfour and J.-P. Zolésio, Shapes and geometries. SIAM, USA (2001).
[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 · doi:10.1080/03605309608821198
[9] A. Fursikov and O. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series34. Research Institute of Mathematics, Seoul National University, South Korea (1996). · Zbl 0862.49004
[10] P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, USA (1985). Zbl0695.35060 · 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 · doi:10.1088/0266-5611/20/3/004
[13] V. Isakov, Inverse problems for partial differential equations. Springer-Verlag, Berlin, Germany (1998). · Zbl 0908.35134
[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 · Zbl 0097.08101 · doi:10.1002/cpa.3160130402
[15] M.V. Klibanov, Estimates of initial conditions of parabolic equations and inequalities via lateral data. Inv. Prob.22 (2006) 495-514. Zbl1094.35139 · Zbl 1094.35139 · doi:10.1088/0266-5611/22/2/007
[16] M.V. Klibanov and A.A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. VSP (2004). Zbl1118.35070 · Zbl 1118.35070 · doi:10.1007/s00032-004-0033-6
[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).
[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 · doi:10.1080/03605309508821097
[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 · doi:10.1137/0501008
[21] K.-D. Phung, Remarques sur l’observabilité pour l’équation de Laplace. ESAIM: COCV9 (2003) 621-635. Zbl1076.93009 · Zbl 1076.93009 · doi:10.1051/cocv:2003030 · numdam:COCV_2003__9__621_0 · eudml:244786
[22] L. Robbiano, Théorème d’unicité adapté au contrôle des solutions des problèmes hyperboliques. Commun. Partial Differ. Equ.16 (1991) 789-800. Zbl0779.93057 · Zbl 0779.93057 · numdam:JEDP_1991____A7_0 · eudml:93238
[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 · doi:10.1088/0266-5611/23/4/019
[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 · doi:10.1137/070684793
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.