Local Convergence of the Lavrentiev Method for the Cauchy Problem via a Carleman Inequality

We invenstigate a local analysis of the Lavrentiev regularization method applied to the ill-posed Cauchy problem. A convergence analysis of the approximated solution is carried out in a sub region away from the incomplete boundary, which is the source of the instability of the Cauchy solution. The use of a Carleman estimate with boundary condition by D.Tataru (see [4]) brings about super-convergence results, which is similar to that derived with a supplementary source condition assumed on the exact Cauchy solution. The achieved results look similar to those established for the Quasi-Reversibility method (by Cao, Klibanov and Pereverev).