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.
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REFERENCES
Tikhonov, A.N. “On the Stability of Inverse Problems”, C. R. (Doklady) Acad. Sci. URSS (N.S.) 39 (5), 176–179 (1943).
Carleman, T. Les Functions quasi analytiques (Paris, 1926).
Goluzin, G.M., Krylov, V.I. “Generalized Carleman's Formula and its Application to the Analytic Continuation of Functions”, Matem. Sb. 40, 144–149 (1933).
Aizenberg, L.A. Carleman's Formulas in Complex Analysis (Nauka, Novosibirsk, 1990) [in Russian].
Lavrent'ev, M.M. “On the Cauchy Problem for Laplace Equation”, (Russian) Izv. Akad. Nauk SSSR. Ser. Mat. 20 (6), 819–842 (1956).
Lavrent'ev, M.M. On Some Ill-posed Problems of Mathematical Physics (Izdat. SO AN SSSR, Novosibirsk, 1962) [in Russian].
Yarmukhamedov, Sh.Ya. “On the Harmonic Continuation of Differentiable Functions Defined on a Boundary Segment”, Siberian Math. J. 43 (1), 183–193 (2002).
Yarmukhamedov, Sh.Ya. “Representation of a Harmonic Function as Potentials, and the Cauchy Problem”, Math. Notes 83 (5–6), 693–706 (2008).
Yarmukhamedov, Sh.Ya. On the Cauchy Problem for the Laplace Equation, Doctoral Dissertation in Mathematics and Physics (Novosibirsk, 1983).
Hadamard, J. Lectures on the Cauchy Problem in Linear Partial Differential Equations (Yale Univ. Press, New Haven, 1923).
Alessandrini, G., Rondi, L., Rosset, E., Vessella, S. “The Stability for the Cauchy Problem for Elliptic Equations”, Inverse Probl. 25, 1–47 (2009).
Tikhonov, A.N., Arsenin, V.Y. Solution of Ill-posed Problems (Winston and Sons, Washington, 1977).
Engl, E.H., Hanke, M., Neubauer, A. Regularization of Inverse Problems (Math. and Its Appl. Kluwer, Dordrecht, 1996).
Klibanov, M.V., Santosa, F. “A Computational Quasi-reversibility Method for Cauchy Problems for Laplace's Equation”, SIAM J. Appl. Math. 51, 1653–1675 (1991).
Takeuchi, T., Yamamoto, M. “Tikhonov Regularization by a Reproducing Kernel Hilbert Space for the Cauchy Problem for an Elliptic Equation”, SIAM J. Sci. Comput. 31 (1), 112–142 (2008).
Marin, L. “Relaxation Procedures for an Iterative MFS Algorithm for Two-dimensional Steady-state Isotropic Heat Conduction Cauchy Problems”, Eng. Anal. Bound. Elem. 35, 415–429 (2011).
Klibanov, M.V. “Carleman Estimates and Inverse Problems in the Last Two Decades” (in: Surveys on Solution Methods for Inverse Prob., Springer, pp. 119–146 (2000)).
Khasanov, A.B., Tursunov, F.R. “On Cauchy Problem for Laplace Equation”, Ufa Math. J. 11 (4), 91–107 (2019).
Ikehata, M. “Inverse Conductivity Problem in the Infinite Slab”, Inverse Probl. 17 (3), 437–454 (2001).
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Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 2, pp. 56–73.
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Khasanov, A.B., Tursunov, F.R. On the Cauchy Problem for the Three-Dimensional Laplace Equation. Russ Math. 65, 49–64 (2021). https://doi.org/10.3103/S1066369X21020055
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DOI: https://doi.org/10.3103/S1066369X21020055