Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Bers, F. John, and M. Schechter, Partial Differential Equations [Russian translations], Mir, Moscow (1966).
F. Brackx, K. Delanghe, and F. Sommen, Clifford Analysis, Pitman, Boston–London–Melbourne (1982).
M. M. Dzharbashyan, Integral Transformations and Representations of Functions in Complex Domain [in Russian], Nauka, Moscow (1966).
L. A. Eisenberg, Carleman’s Formulas in Complex Analysis. First Applications [in Russian], Nauka, Novosibirsk (1990).
L. A. Eisenberg and N. N. Tarkhanov, “Abstract Carlemans Formula,” Rep. Acad. Sci. USSR, 298, No. 6, 1292–1296 (1988).
J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations of Hyperbolic Type [Russian translation], Nauka, Moscow (1978).
T. I. Ishankulov, “On generalized analytic continuation into the domain for functions defined on a part of its boundary,” Siberian Math. J. , 41, No. 6, 1350–1356 (2000).
V. K. Ivanov, “The Cauchy problem for the Laplace equations in infinite strip,” Differ. Equ., 1, No. 1, 131–136 (1965).
M. M. Lavrent’ev, “On the Cauchy problem for second-order linear elliptic equations,” Rep. Acad. Sci. USSR,112, No. 2, 195–197 (1957).
M. M. Lavrent’ev, On Some Ill-Posed Problems of Mathematical Physics [in Russian], VTS SO AN SSSR, Novosibirsk (1962).
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, No. 2, 43–53 (2004).
O. Makhmudov, I. Niyozov, and N. Tarkhanov, “The Cauchy problem of couple-stress elasticity,” Contemp. Math., 455, 297–310 (2008).
S. N. Mergelyan, “Harmonic approximation and approximate solution of the Cauchy problem for the Laplace equation,” Progr. Math. Sci., 11, No. 5, 3–26 (1956).
L. F. Nikiforov and V. B. Uvarov, Foundations of the Theory of Special Functions [in Russian], Nauka, Moscow (1974).
E. I. Obolashvili, “Spatial analog of generalized analytic functions,” Rep. Acad. Sci. Georgian SSR, 73, No. 1, 20–24 (1974).
E. I. Obolashvili, “Generalized Cauchy–Riemann system in multidimensional Euclidean space,” Proc. Int. Conf. Compl. Anal. Appl. Part. Differ. Equ., Galle, Germany, 18–24 Oct. 1976, pp. 36–39 (1977).
E. I. Obolashvili, “Generalized Cauchy–Riemann system in multidimensional space,” Proc. Tbilisi Math. Inst., 58, 168–173 (1978).
E. N. Sattorov, “Regularization of solution of the Cauchy problem for the generalized Moisil–Theodorescu system,” Differ. Equ., 44, No. 8, 1100–1110 (2008).
E. N. Sattorov, “On continuation of solutions of the generalized Cauchy–Riemann system in the space,” Math. Notes, 85, No. 5, 768–781 (2009).
E. N. Sattorov, “Regularization of solution of the Cauchy problem for the Maxwell system of equations in infinite domain,” Math. Notes, 86, No. 6, 445–455 (2009).
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., 1, 72–84 (2011).
E. N. Sattorov and Dzh. A. Mardonov, “The Cauchy problem for the Maxwell system of equations,” Siberian Math. J., 44, No. 4, 851–861 (2003).
E. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces [Russian translation], Mir, Moscow (1974).
N. N. Tarkhanov, “On the Carleman matrix for elliptic systems,” Rep. Acad. Sci. USSR, 284, No. 2, 294–297 (1985).
N. N. Tarkhanov, Cauchy Problem for Solutions of Elliptic Equations, Akademie-Verlag, Berlin (1995).
A. N. Tikhonov, “On solution of ill-posed problems and the regularization method,” Rep. Acad. Sci. USSR, 151, No. 3, 501–504 (1963).
F. Tricomi, Lezioni Sulle Equazioni a Derivate Parziali [Russian translation], IL, Moscow (1957).
I. N. Vekua, Generalized Analytic Functions [in Russian], Fizmatgiz, Moscow (1988).
V. S. Vladimirov and I. V. Volovich, “Superanalysis I. Differential Calculus,” Theor. Math. Phys., 59, No. 1, 3–27 (1984).
V. S. Vladimirov and I. V. Volovich, “Superanalysis II. Integral Calculus,” Theor. Math. Phys., 60, No. 2, 169–198 (1984).
Sh. Yarmukhamedov, “On the Cauchy problem for the Laplace equation,” Rep. Acad. Sci. USSR, 235, No. 2, 281–283 (1977).
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, 34–40 (1980).
Sh. Yarmukhamedov, “On continuation of solution of the Helmholtz equation,” Rep. Russ. Acad. Sci., 357, No. 3, 320–323 (1997).
Sh. Yarmukhammedov, “Carleman’s function and the Cauchy problem for the Laplace equation,” Siberian Math. J., 45, No. 3, 702–719 (2004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 65, No. 1, Contemporary Problems in Mathematics and Physics, 2019.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Sattorov, E.N., Ermamatova, F.E. Carleman’s Formula for Solutions of Generalized Cauchy–Riemann Systems in Multidimensional Spatial Domains. J Math Sci 265, 90–102 (2022). https://doi.org/10.1007/s10958-022-06047-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-022-06047-9