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

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

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topBourgeois, 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

top- 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
- 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
- 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
- A.L. Bukhgeim, Extension of solutions of elliptic equations from discrete sets. J. Inv. Ill-Posed Problems1 (1993) 17–32. Zbl0820.35020
- 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
- J. Cheng, M Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation. M3AS18 (2008) 107–123. Zbl1155.35108
- M.C. Delfour and J.-P. Zolésio, Shapes and geometries. SIAM, USA (2001). Zbl1002.49029
- C. Fabre and G. Lebeau, Prolongement unique des solutions de l'équation de Stokes. Comm. Part. Differ. Equ.21 (1996) 573–596. Zbl0849.35098
- A. Fursikov and O. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series34. Research Institute of Mathematics, Seoul National University, South Korea (1996). Zbl0862.49004
- P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, USA (1985). Zbl0695.35060
- L. Hormander, Linear Partial Differential Operators. Fourth Printing, Springer-Verlag, Germany (1976). Zbl0321.35001
- T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation. Inv. Prob.20 (2004) 697–712. Zbl1086.35080
- V. Isakov, Inverse problems for partial differential equations. Springer-Verlag, Berlin, Germany (1998). Zbl0908.35134
- F. John, Continuous dependence on data for solutions of pde with a prescribed bound. Commun. Pure Appl. Math.13 (1960) 551–585. Zbl0097.08101
- M.V. Klibanov, Estimates of initial conditions of parabolic equations and inequalities via lateral data. Inv. Prob.22 (2006) 495–514. Zbl1094.35139
- M.V. Klibanov and A.A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. VSP (2004). Zbl1069.65106
- R. Lattès and J.-L. Lions, Méthode de quasi-réversibilité et applications. Dunod, France (1967). Zbl0159.20803
- M.M. Lavrentiev, V.G. Romanov and S.P. Shishatskii, Ill-posed problems in mathematical physics and analysis. Amer. Math. Soc., Providence, USA (1986).
- G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur. Commun. Partial Differ. Equ.20 (1995) 335–356. Zbl0819.35071
- L.E. Payne, On a priori bounds in the Cauchy problem for elliptic equations. SIAM J. Math. Anal.1 (1970) 82–89. Zbl0199.16603
- K.-D. Phung, Remarques sur l'observabilité pour l'équation de Laplace. ESAIM: COCV9 (2003) 621–635.
- L. Robbiano, Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques. Commun. Partial Differ. Equ.16 (1991) 789–800. Zbl0735.35086
- D.A. Subbarayappa and V. Isakov, On increased stability in the continuation of the Helmholtz equation. Inv. Prob.23 (2007) 1689–1697. Zbl1127.35082
- 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