Geodesic
On Cauchy problem solution for a harmonic function in a simply connected domain
Problemy analiza, Tome 12 (2023) no. 2, pp. 87-96.

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Here we present an investigation of the Cauchy problem solvability for the Laplace equation in a simply connected plane domain. The investigation is reduced to solution of two singular integral equations. If the problem is resolvable, its solution can be restored via the integral Cauchy formula. Examples of the solvable and unsolvable problems are presented. The construction involves the auxiliary approximate conformal mapping.
Keywords: Cauchy problem, integral equation, holomorphic function, spline, approximate solution. 
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E. A. Shirokova; P. N. Ivanshin. On Cauchy problem solution for a harmonic function in a simply connected domain. Problemy analiza, Tome 12 (2023) no. 2, pp. 87-96. http://geodesic.mathdoc.fr/item/PA_2023_12_2_a5/

[1] Gakhov F. D., Boundary value problems, Pergamon Press, 1966 | MR | Zbl

[2] Goluzin G. M., Krylov V. I., “Generalized Carleman formula and its applications to analytic continuation of functions”, Matem. Sbornik, 40:2 (1933), 144–149 | Zbl

[3] Ivanshin P. N., Shirokova E. A., “On approximate conformal mapping of a disk and an annulus with radial and circular slits onto multiply connected domains”, J. of Appl. Math. and Comp. Mech., 19:2 (2020), 73–84 | DOI | MR

[4] Khasanov A. B., Tursunov F. R., “Of Cauchy problem for Laplace equation”, Ufa Math. J., 11:4 (2019), 91–107 | DOI | MR | Zbl

[5] Khasanov A. B., Tursunov F. R., “On Cauchy problem for the three-dimensional Laplace equation”, Rus. Math., 65:2 (2021), 49–64 | DOI | MR | Zbl

[6] Lavrent'ev M. M., Romanov V. G., Shishatskii S. P., Ill-posed Problems of Mathematical Physics and Analysis, Transl. of Math. Monographs, 64, Amer. Math. Soc., Providence, R.I., 1986 | DOI | MR | Zbl

[7] Markushevich A. I., Theory of analytic functions, GTTL, M., 1950 | MR

[8] Muskhelishvili N. I., Singular Integral Equations, Birkhauser, 2011 | DOI | MR

[9] Schur I., “Bemerkungen zur Theorie der Beschraenkten Bilinearformen mit unendlich vielen Veraenderlichen”, J. Reine Angew. Math., 140 (1911), 1–28 | DOI | MR

[10] Shefer Yu. L., Shlapunov A. A., “On regularization of the Cauchy problem for elliptic systems in weighted Sobolev spaces”, J. of Inv. and Ill-posed Probl., 27:6 (2019), 815–835 | DOI | MR

[11] Shlapunov A. A., “On Iterations of Non-Negative Operators and Their Applications to Elliptic Systems”, Math. Nachrichten, 218:1 (2000), 165–174 | DOI | MR | Zbl

[12] Tarkhanov N., The Cauchy Problem for Solutions of Elliptic Equations, Akademie-Verlag, Berlin, 1995 | MR | Zbl