In this thesis, we study Cauchy problems for elliptic and parabolic equations. These include the stationary Stokes system and the heat equation. Data are given on a part of the boundary of a bounded domain. The aim is to reconstruct the solution from these data. These problems are ill-posed in the sense of J. Hadamard.

We propose iterative regularization methods, which require solving of a sequence of well-posed boundary value problems for the same operator. Methods based on this idea were _rst proposed by V. A. Kozlov and V. G. Maz'ya for a certain class of equations which do not include the above problems. Regularizing character is proved and stopping rules are proposed.

The regularizing character for the heat equation is proved in a certain weighted L² space. In each iteration the Zaremba problem for the heat equation is solved. We also prove well-posedness of this problem in a weighted Sobolev space. This result is of independent interest and is presented as a separate paper.