Geodesic
On the Cauchy problem for the three-dimensional Laplace equation
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2021), pp. 56-73

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In the work, using the Carleman function, they are restored by Cauchy's data on the part of the boundary of the region is a harmonic function, and its derivatives. It is shown that the effective construction of the Carleman function is equivalent to the construction of a regularized solution of the Cauchy problem. It is assumed that a solution to the problem exists and is continuously differentiable in a closed domain with precisely given Cauchy data. For this case, 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. Estimates of the stability of the solution of the Cauchy problem in the classical sense are obtained.
Keywords: Cauchy problem, ill-posed problems, Carleman function, regularized solutions, regularization, continuation formulas. 
@article{IVM_2021_2_a4,
     author = {A. B. Khasanov and F. R. Tursunov},
     title = {On the {Cauchy} problem for the three-dimensional {Laplace} equation},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {56--73},
     publisher = {mathdoc},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2021_2_a4/}
}
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A. B. Khasanov; F. R. Tursunov. On the Cauchy problem for the three-dimensional Laplace equation. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2021), pp. 56-73. http://geodesic.mathdoc.fr/item/IVM_2021_2_a4/