Quasi-reversibility Regularization Methods For The Cauchy Problem Of Elliptic Equations

In this thesis, we propose a quasi-reversibility regularization method to solve the Cauchy problem of elliptic equations, including the Cauchy problem for the Laplace equation and the Cauchy problem for the Helmholtz equation. It is well-know that the Cauchy problem of elliptic equations is severely ill-posed, i.e., the solutions do not depend continuously on the given Cauchy data. Conver-gence estimates for the regularized solutions are obtained under a-priori bound assumptions for the exact solution and suitable choices of regularization param-eters. Some numerical results are given to show the effectiveness of the proposed method. 2...