It is well known that Cauchy problem for Laplace equations is an ill-posed problem in Hadamard's sense. Small deviations in Cauchy data may lead to large errors in the solutions. It is observed that if a bound is imposed on the solution, there exists a conditional stability estimate. This gives a reasonable way to construct stable algorithms. However, it is impossible to have good results at all points in the domain. Although numerical methods for Cauchy problems for Laplace equations have been widely studied for quite a long time, there are still some unclear points, for example, how to evaluate the numerical solutions, which means whether we can approximate the Cauchy data well and keep the bound of the solution, and at which points the numerical results are reliable? In this paper, we will prove the conditional stability estimate which is quantitatively related to harmonic measures. The harmonic measure can be used as an indicate function to pointwisely evaluate the numerical result, which further enables us to find a reliable subdomain where the local convergence rate is higher than a certain order.

翻译：众所周知，拉普拉斯方程的柯西问题在阿达马意义下是一个不适定问题。柯西数据的微小偏差可能导致解的较大误差。研究表明，若对解施加有界性约束，则存在条件稳定性估计，这为构造稳定算法提供了合理途径。然而，在整个区域内获得良好的计算结果仍不可能。尽管拉普拉斯方程柯西问题的数值方法已被广泛研究相当长时间，但仍存在若干未明确的问题，例如：如何评估数值解？即我们能否既良好逼近柯西数据又保持解的有界性？以及在哪些点上数值结果是可靠的？本文将通过定量关联调和测度的方式证明条件稳定性估计。该调和测度可作为指示函数逐点评估数值结果，从而进一步使我们能够确定一个局部收敛阶数高于特定阶数的可靠子区域。