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About stability and regularization of ill-posed elliptic Cauchy problems: the case of \(C^{1,1}\) domains. (English) Zbl 1194.35497
Summary: This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace equation in domains with \(C^{1,1}\) boundary. It is an extension of an earlier result of K.-D. Phung [ESAIM, Control Optim. Calc. Var. 9, 621–635 (2003; Zbl 1076.93009)] for domains of class \(C^{\infty}\). 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 [R. Lattès and J.-L. Lions, Méthode de quasi-réversibilité et applications. Travaux et Recherches Mathématiques, 15. Paris: Dunot (1967; Zbl 0159.20803)] to solve ill-posed Cauchy problems.

MSC:
35R30 Inverse problems for PDEs
35R25 Ill-posed problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35N25 Overdetermined boundary value problems for PDEs and systems of PDEs
35B35 Stability in context of PDEs
35J20 Variational methods for second-order elliptic equations
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