Geodesic
About stability and regularization of ill-posed elliptic Cauchy problems : the case of C1,1 domains
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 44 (2010) no. 4, pp. 715-735

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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.

DOI : 10.1051/m2an/2010016
Classification : 35A15, 35N25, 35R25, 35R30
Keywords: Carleman estimate, distance function, elliptic Cauchy problems, conditional stability, quasi-reversibility 
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     title = {About stability and regularization of ill-posed elliptic {Cauchy} problems : the case of {C1,1} domains},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {715--735},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {4},
     year = {2010},
     doi = {10.1051/m2an/2010016},
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     zbl = {1194.35497},
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     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2010016/}
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Bourgeois, Laurent. About stability and regularization of ill-posed elliptic Cauchy problems : the case of C1,1 domains. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 44 (2010) no. 4, pp. 715-735. doi: 10.1051/m2an/2010016

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