  We are interested in the classical ill-posed Cauchy problem for the Laplace
equation. One method to approximate the solution associated with compatible
data consists in considering a family of regularized well-posed problems
depending on a small parameter $\varepsilon>0$. In this context, in order to
prove convergence of finite elements methods, it is necessary to get regularity
results of the solutions to these regularized problems which hold uniformly in
$\varepsilon$. In the present work, we obtain these results in smooth domains
and in 2D polygonal geometries. In presence of corners, due the particular
structure of the regularized problems, classical techniques \`a la Grisvard do
not work and instead, we apply the Kondratiev approach. We describe the
procedure in detail to keep track of the dependence in $\varepsilon$ in all the
estimates. The main originality of this study lies in the fact that the limit
problem is ill-posed in any framework.
