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Conditional stability for ill-posed elliptic Cauchy problems : the case of Lipschitz domains (part II)
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.
Cited literature [12 references]
https://hal.inria.fr/inria-00324166
Contributor : Laurent Bourgeois <>
Submitted on : Wednesday, September 24, 2008 - 11:26:53 AM
Last modification on : Thursday, January 14, 2021 - 11:56:03 AM
Long-term archiving on: : Tuesday, June 28, 2011 - 4:48:11 PM
Contributor : Laurent Bourgeois <>
Submitted on : Wednesday, September 24, 2008 - 11:26:53 AM
Last modification on : Thursday, January 14, 2021 - 11:56:03 AM
Long-term archiving on: : Tuesday, June 28, 2011 - 4:48:11 PM
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RR-6588.pdf
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