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, Volume 44 (2010) no. 4, p. 715-735

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.

DOI : https://doi.org/10.1051/m2an/2010016
Classification:  35A15,  35N25,  35R25,  35R30
Keywords: Carleman estimate, distance function, elliptic Cauchy problems, conditional stability, quasi-reversibility
@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},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {4},
     year = {2010},
     pages = {715-735},
     doi = {10.1051/m2an/2010016},
     zbl = {1194.35497},
     mrnumber = {2683580},
     language = {en},
     url = {http://www.numdam.org/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, Volume 44 (2010) no. 4, pp. 715-735. doi : 10.1051/m2an/2010016. http://www.numdam.org/item/M2AN_2010__44_4_715_0/

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