中文摘要：

本文主要研究Laplace方程的Cauchy问题，该问题在很多领域有广泛的应用．众所周知，Laplace方程的Cauchy问题是严重不适定问题，即其解不连续依赖于所给的Cauchy数据．本文应用一个高阶Tikhonov正则化方法求解矩形区域上的Laplace方程的Cauchy问题，在对精确解的适当的先验界假设和正则化参数选取下，得到了相应的收敛性估计，数值结果表明所提的方法是高效稳定的．

英文摘要：

In this thesis, we propose a modified Tikhonov regularization method to solve the Cauchy problem of Laplace equation. It is well-known that the Cauchy problem of Laplace equation is severely ill-posed, i.e., the solution do not depend con- tinuously on the given Cauchy data. Convergence estimates for the regularized solution are obtained under different a-priori bound assumptions for the exact solution. Some numerical results are given to show the effectiveness of our proposed method.