Extended-domain-Lavrentiev's regularization for the Cauchy problem

We investigate new issues pertaining to the regularization of the ill-posed Cauchy problem for the Laplace equation. We show how the variational formulation proposed in Ben Belgacem and El Fekih ( 2005 Inverse Problems 21 1915–36 ) allows for a meaningful interpretation of the general source condition, currently used in the analysis of ill-posed problems. This assumption turns out to be closely connected to the possibility for the exact Cauchy solution to be harmonically extended to a larger domain. Then, we turn to the consolidation of the Lavrentiev method owing to the extension of the computational domain. The study we conduct and the numerical examples we present display an important improvement of the regularization capabilities and accuracy, especially for thin domains and when the data are substantially damaged by noise.