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About stability and regularization of ill-posed elliptic Cauchy problems: The case of Lipschitz domains

Laurent Bourgeois 1 Jérémi Dardé 1, 2
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
CNRS - Centre National de la Recherche Scientifique : UMR7231, UMA - Unité de Mathématiques Appliquées, Inria Saclay - Ile de France
Abstract : This article is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for Laplace's equation in domains with Lipschitz boundary. It completes the results obtained by Bourgeois [Conditional stability for ill-posed elliptic Cauchy problems: The case of C1,1 domains (part I), Rapport INRIA 6585, 2008] for domains of class C1,1. This estimate is established by using an interior Carleman estimate and a technique based on a sequence of balls which approach the boundary. This technique is inspired by Alessandrini et al. [Optimal stability for inverse elliptic boundary value problems with unknown boundaries, Annali della Scuola Normale Superiore di Pisa 29 (2000), pp. 755-806]. We obtain a logarithmic stability estimate, the exponent of which is specified as a function of the boundary's singularity. Such stability estimate induces a convergence rate for the method of quasi-reversibility introduced by Lattés and Lions [Méthode de Quasi-Réversibilité et Applications, Dunod, Paris, 1967] to solve the Cauchy problems. The optimality of this convergence rate is tested numerically, precisely a discretized method of quasi-reversibility is performed by using a nonconforming finite element. The obtained results show very good agreement between theoretical and numerical convergence rates. © 2010 Taylor & Francis.
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Submitted on : Friday, August 30, 2013 - 3:34:59 PM
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Laurent Bourgeois, Jérémi Dardé. About stability and regularization of ill-posed elliptic Cauchy problems: The case of Lipschitz domains. Applicable Analysis, Taylor & Francis, 2010, 89 (11), pp.1745-1768. ⟨10.1080/00036810903393809⟩. ⟨hal-00849579⟩



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