About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1,1 domains
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Revised: 11 June 2009
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.
Mathematics Subject Classification: 35A15 / 35N25 / 35R25 / 35R30
Key words: Carleman estimate / distance function / elliptic Cauchy problems / conditional stability / quasi-reversibility
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