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Conditional stability for ill-posed elliptic Cauchy problems : the case of Lipschitz domains (part II)

Laurent Bourgeois 1 Jérémi Dardé 1
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
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 Lipschitz boundary. It completes the results obtained in \cite{bourgeois1} for domains of class $C^{1,1}$. 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 from \cite{alessandrini}. 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 in \cite{lions} 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.
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Laurent Bourgeois, Jérémi Dardé. Conditional stability for ill-posed elliptic Cauchy problems : the case of Lipschitz domains (part II). [Research Report] RR-6588, INRIA. 2008. ⟨inria-00324166⟩

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