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

Published online by Cambridge University Press:  23 February 2010

Laurent Bourgeois
Laurent Bourgeois*
Affiliation:
Laboratoire POEMS, ENSTA, 32 Boulevard Victor, 75739 Paris Cedex 15, France. laurent.bourgeois@ensta.fr
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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 C 1,1 boundary. It is an extension of anearlier 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.

Keywords

Carleman estimatedistance functionelliptic Cauchy problemsconditional stabilityquasi-reversibility
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 4 , July 2010 , pp. 715 - 735
Copyright
© EDP Sciences, SMAI, 2010

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