On the Cauchy Problem for the Three-Dimensional Laplace Equation

Abstract

In the present work, using the Carleman function, a harmonic function and its derivatives are restored by the Cauchy data on a part of the boundary of the region. It is shown that the effective construction of the Carleman function is equivalent to the construction of a regularized solution to the Cauchy problem. We assume that a solution to the problem exists and is continuously differentiable in a closed domain with precisely given Cauchy data. For this case, we establish an explicit formula for the continuation of the solution and its derivative, as well as the regularization formula for the case when under the indicated conditions instead of the initial Cauchy data their continuous approximations are given with a given error in the uniform metric. The stability estimates for the solution to the Cauchy problem in the classical sense are obtained.