Carleman's formula for solutions of the generalized Cauchy--Riemann system in multidimensional spatial domain

E. N. Sattorov ; F. E. Ermamatova

Contemporary Mathematics. Fundamental Directions, Contemporary problems in mathematics and physics, Tome 65 (2019) no. 1, pp. 95-108.

Résumé

In this paper, we consider the restoration problem for solutions of the generalized Cauchy–Riemann system in a multidimensional spatial domain using their values on a piece of the boundary of the domain, i. e., the Cauchy problem. We construct an approximate solution of this problem based on the Carleman matrix method.

Bibliographie

[1] J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations of Hyperbolic Type, Russian translation, Nauka, M., 1978

[2] L. A. Eisenberg, Carleman's Formulas in Complex Analysis. First Applications, Nauka, Novosibirsk, 1990 (in Russian) | MR

[3] L. A. Eisenberg, N. N. Tarkhanov, “Abstract Carlemans Formula”, Rep. Acad. Sci. USSR, 298:6 (1988), 1292–1296 (in Russian)

[4] L. Bers, F. John, M. Schechter, Partial Differential Equations, Russian translations, Mir, M., 1966

[5] I. N. Vekua, Generalized Analytic Functions, Fizmatgiz, M., 1988 (in Russian)

[6] V. S. Vladimirov, I. V. Volovich, “Superanalysis I. Differential Calculus”, Theor. Math. Phys., 59:1 (1984), 3–27 (in Russian) | MR | Zbl

[7] V. S. Vladimirov, I. V. Volovich, “Superanalysis II. Integral Calculus”, Theor. Math. Phys., 60:2 (1984), 169–198 (in Russian) | MR | Zbl

[8] M. M. Dzharbashyan, Integral Transformations and Representations of Functions in Complex Domain, Nauka, M., 1966 (in Russian)

[9] V. K. Ivanov, “The Cauchy problem for the Laplace equations in infinite strip”, Differ. Equ., 1:1 (1965), 131–136 (in Russian) | MR | Zbl

[10] T. I. Ishankulov, “On generalized analytic continuation into the domain for functions defined on a part of its boundary”, Sib. Math. J., 41:6 (2000), 1350–1356 (in Russian) | MR | Zbl

[11] M. M. Lavrent'ev, “On the Cauchy problem for second-order linear elliptic equations”, Rep. Acad. Sci. USSR, 112:2 (1957), 195–197 (in Russian) | Zbl

[12] M. M. Lavrent'ev, On Some Ill-Posed Problems of Mathematical Physics, VTS SO AN SSSR, Novosibirsk, 1962 (in Russian)

[13] O. I. Makhmudov, “The Cauchy problem for a system of equations from the elasticity and thermoelasticity theory in the space”, Bull. Higher Edu. Inst. Ser. Math., 501:2 (2004), 43–53 (in Russian)

[14] S. N. Mergelyan, “Harmonic approximation and approximate solution of the Cauchy problem for the Laplace equation”, Progr. Math. Sci., 11:5 (1956), 3–26 (in Russian) | MR

[15] L. F. Nikiforov, V. B. Uvarov, Foundations of the Theory of Special Functions, Nauka, M., 1974 (in Russian) | MR

[16] E. I. Obolashvili, “Spatial analog of generalized analytic functions”, Rep. Acad. Sci. Georgian SSR, 73:1 (1974), 20–24 (in Russian) | MR

[17] E. I. Obolashvili, “Generalized Cauchy–Riemann system in multidimensional Euclidean space”, Sb. dokl. Mezhd. konf. po kompl. analizu i ego primeneniyam k uravn. s chastn. proizvodnymi (Galle, GDR, 18–24 Oct. 1976), Galle, 1977, 36–39 (in Russian)

[18] E. I. Obolashvili, “Generalized Cauchy–Riemann system in multidimensional space”, Proc. Tbilisi Math. Inst., 58 (1978), 168–173 (in Russian)

[19] E. N. Sattorov, “Regularization of solution of the Cauchy problem for the generalized Moisil–Theodorescu system”, Differ. Equ., 44:8 (2008), 1100–1110 (in Russian) | MR | Zbl

[20] E. N. Sattorov, “On continuation of solutions of the generalized Cauchy–Riemann system in the space”, Math. Notes, 85:5 (2009), 768–781 (in Russian) | DOI | MR | Zbl

[21] E. N. Sattorov, “Regularization of solution of the Cauchy problem for the Maxwell system of equations in infinite domain”, Math. Notes, 86:6 (2009), 445–455 (in Russian) | DOI | MR | Zbl

[22] E. N. Sattorov, “On restoration of solutions of the generalized Moisil–Theodorescu system in a spatial domain by their values of a part of the boundary”, Bull. Higher Edu. Inst. Ser. Math., 2011, no. 1, 72–84 (in Russian) | MR | Zbl

[23] E. N. Sattorov, Dzh. A. Mardonov, “The Cauchy problem for the Maxwell system of equations”, Sib. Math. J., 44:4 (2003), 851–861 (in Russian) | MR | Zbl

[24] E. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Russian translation, Mir, M., 1974

[25] N. N. Tarkhanov, “On the Carleman matrix for elliptic systems”, Rep. Acad. Sci. USSR, 284:2 (1985), 294–297 (in Russian) | MR | Zbl

[26] A. N. Tikhonov, “On solution of ill-posed problems and the regularization method”, Rep. Acad. Sci. USSR, 151:3 (1963), 501–504 (in Russian) | Zbl

[27] F. Tricomi, Lezioni sulle equazioni a derivate parziali, Russian translation, IL, M., 1957

[28] Sh. Yarmukhamedov, “On the Cauchy problem for the Laplace equation”, Rep. Acad. Sci. USSR, 235:2 (1977), 281–283 (in Russian) | MR | Zbl

[29] Sh. Yarmukhamedov, “On analytic continuation of a holomorphic vector by its boundary values on a part of the boundary”, Bull. Acad. Sci. Uzbek SSR. Ser. Phys.-Math. Sci., 6 (1980), 34–40 (in Russian)

[30] Sh. Yarmukhamedov, “On continuation of solution of the Helmholtz equation”, Rep. Russ. Acad. Sci., 357:3 (1997), 320–323 (in Russian) | MR | Zbl

[31] Sh. Yarmukhammedov, “Carleman's function and the Cauchy problem for the Laplace equation”, Sib. Math. J., 45:3 (2004), 702–719 (in Russian) | MR

[32] Brackx F., Delanghe K., Sommen F., Clifford analysis, Pitman, Boston–London–Melbourne, 1982 | MR | Zbl

[33] Makhmudov O., Niyozov I., Tarkhanov N., “The Cauchy problem of couple-stress elasticity”, Contemp. Math., 455, 2008, 297–310 | DOI | MR | Zbl

[34] Tarkhanov N. N., Cauchy problem for solutions of elliptic equations, Akademie-Verlag, Berlin, 1995 | MR | Zbl