On Cauchy problem for Laplace equation

Abstract

The paper is devoted to studying the continuation of a solution and stability estimates for the Cauchy problem for the Laplace equation in a domain $G$ by its known values on the smooth part $S$ of the boundary $\partial G$. The considered issue is among the problems of mathematical physics, in which there is no continuous dependence of solutions on the initial data. While solving applied problems, one needs to find not only an approximate solution, but also its derivative. In the work, given the Cauchy data on a part of the boundary, by means of Carleman function, we recover not only a harmonic function, but also its derivatives. If the Carleman function is constructed, then by employing the Green function, one can find explicitly the regularized solution. We show that an effective construction of the Carleman function is equivalent to the constructing of the regularized solution to the Cauchy problem. We suppose that the solutions of the problem exists and is continuously differentiable in a closed domain with exact given Cauchy data. In this case we establish an explicit formula for continuation of the solution and its derivative as well as a regularization formula for the case, when instead of Cauchy initial data, their continuous approximations are prescribed with a given error in the uniform metrics. We obtain stability estimates for the solution to the Cauchy problem in the classical sense.