Article
Ufa Mathematical Journal
Volume 11, Number 4, pp. 91-107
Of Cauchy problem for Laplace equation
Khasanov A.B., Tursunov F.R.
DOI:10.13108/2019-11-4-91
Download PDF
Article on MathNetAbstact
The paper is devoted to studying
the continuation of a solution and estimates of the stability in 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 approximate 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.