Scientific Journal of Samarkand University


The article studies the problem of continuation of the solution and the stability estimate of the Cauchy problem for the Laplace equation in a domain G by its known values on the smooth part S of the boundary G  . The considered problem belongs to the problems of mathematical physics, in which there is no continuous dependence of solutions on the initial data. It is assumed that the solution to the problem exists and is continuously differentiable in a closed domain with exac tly given Cauchy data. For this case, an explicit formula for the continuation of the solution is established, as well as a regularization formula for the case when, under these conditions, instead of the Cauchy data, their approximations are given with a given error in the uniform metric. We obtain estimates for the stability of the solution of the Cauchy problem in the classical sense.

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