ON THE SCHWARZ PROBLEM FOR THE MOISIL–TEODORESCU SYSTEM IN A SPHERICAL LAYER AND IN THE INTERIOR OF A TORUS
DOI:
https://doi.org/10.26577/JMMCS.2022.v114.i2.04Keywords:
Cauchy–Riemann system, Moisil–Teodorescu system, Schwartz problem, spherical layer, torus interior, solvability of the problemAbstract
Doubly connected regions play a significant role in fluid mechanics. For example, the flow created by a long solid cylinder moving in the direction of the normal to its axis occurs precisely in a doubly connected region.
In this paper, well-posed problems for the Moisil--Teodorescu system are presented in the case of a spherical layer and the interior of a torus. The Moisil--Teodorescu elliptic system is an example of a multidimensional generalized Cauchy--Riemann system. The results of this work show a significant difference between the well-posed problem in a spherical layer and a similar problem in a torus.
References
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[2] Hatcher A., Algebraic Topology, Cambridge University Press. New Yourk. 2002.
[3] Simmons G.F., Introduction to Topology and Modern Analysis, McGraw Hill Book Company. 1968.
[4] Agmon S., Douglis A., Nirenberg L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure and Appl. Math., 17 (1964), 35—92.
[5] Shevchenko V.I., Some boundary value problem for holomorphic vector, Sborn. Math. Phys. Kiev, 8 (1970), 172—187. (in Russian)
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[7] Soldatov A.P., The Riemann-Hilbert problem for the Moisil-Theodorescu system, Izv. RAN, Ser. Math. (2018) (in
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[8] Polunin V.A., Soldatov A.P., Riemann-Hilbert problem for the Moisil-Teodorescu system in multiply connected domains, Electronic Journal of Differential Equations, 310 (2016), 1–5.
[9] Polunin V.A., Soldatov A.P., On the integral represenation of solutions of the Moisil-Theodorescu system in multiply connected domains, Dokl. RAN, 475:4 (2017), 192–201.
[10] Moisil Gr. C., Theodoresco N., Fonctions holomorphes dans l’espace, Mathematica, 5 (1931), 142-153.
[11] Stein I., Weiss G., Introduction to harmonic analysis on Euclidean spaces, Moscow. Mir. 1974.
[12] Bitsadze A.V., Spatial analogue of the Cauchy-type integral and some of its applications, Izv. USSR Academy of Sciences, ser. math., 17:6 (1953), 525-538.
[13] Bitsadze A.V., Boundary value problems for second order elliptic equations, Moscow. Mir. 1966.
[14] Polunin V.A., Soldatov A.P., Three-dimensional analogue of the Cauchy-type integral, Differ. equations, 47:3 (2011), 366-375.
[15] Nazarov S., Plamenevsky B.A., Elliptic problems in domains with piecewise smooth boundaries. Walter de Gruyter, 2011.
[16] Kozhanov A.I., Koshanov B.D., Sultangazieva Zh.B., New boundary value problems for fourth-order quasi-hyperbolic equations, Siberian Electronic Mathematical Reports, 16 (2019), 1383-1409. http://dx.doi.org/10.33048/semi.2019.16.098
[17] Koshanov B.D., Soldatov A.P., Boundary value problem with normal derivatives for a higher order elliptic equation on the plane, Differential Equations, 52:12 (2016), 1594-1609. https://doi.org/10.1134/S0012266116120077
[18] Koshanov B.D., Soldatov A.P., On the Solvability of the Boundary Value Problems for the Elliptic Equation
of High Order on a Plane, Bulletin of the Karaganda University. Mathematics series, 91:3 (2018), 24-31.
https://doi.org/10.31489/2018M3/24-30
[19] Koshanov B.D., Equivalence of the Fredholm solvability condition for the Neumann problem to the
complementarity condition, Journal of Mathematics, Mechanics and Computer Science, 109:1 (2021), 34-54.
https://doi.org/10.26577/JMMCS.2021.v111.i3.04
[2] Hatcher A., Algebraic Topology, Cambridge University Press. New Yourk. 2002.
[3] Simmons G.F., Introduction to Topology and Modern Analysis, McGraw Hill Book Company. 1968.
[4] Agmon S., Douglis A., Nirenberg L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure and Appl. Math., 17 (1964), 35—92.
[5] Shevchenko V.I., Some boundary value problem for holomorphic vector, Sborn. Math. Phys. Kiev, 8 (1970), 172—187. (in Russian)
[6] Yao Y., The Riemann boundary value problem for the Moisil-Theodorescu system, In Tournal-Sichuan Normal University. Natural Science Edition, 29 (2006), 188—191.
[7] Soldatov A.P., The Riemann-Hilbert problem for the Moisil-Theodorescu system, Izv. RAN, Ser. Math. (2018) (in
Russian).
[8] Polunin V.A., Soldatov A.P., Riemann-Hilbert problem for the Moisil-Teodorescu system in multiply connected domains, Electronic Journal of Differential Equations, 310 (2016), 1–5.
[9] Polunin V.A., Soldatov A.P., On the integral represenation of solutions of the Moisil-Theodorescu system in multiply connected domains, Dokl. RAN, 475:4 (2017), 192–201.
[10] Moisil Gr. C., Theodoresco N., Fonctions holomorphes dans l’espace, Mathematica, 5 (1931), 142-153.
[11] Stein I., Weiss G., Introduction to harmonic analysis on Euclidean spaces, Moscow. Mir. 1974.
[12] Bitsadze A.V., Spatial analogue of the Cauchy-type integral and some of its applications, Izv. USSR Academy of Sciences, ser. math., 17:6 (1953), 525-538.
[13] Bitsadze A.V., Boundary value problems for second order elliptic equations, Moscow. Mir. 1966.
[14] Polunin V.A., Soldatov A.P., Three-dimensional analogue of the Cauchy-type integral, Differ. equations, 47:3 (2011), 366-375.
[15] Nazarov S., Plamenevsky B.A., Elliptic problems in domains with piecewise smooth boundaries. Walter de Gruyter, 2011.
[16] Kozhanov A.I., Koshanov B.D., Sultangazieva Zh.B., New boundary value problems for fourth-order quasi-hyperbolic equations, Siberian Electronic Mathematical Reports, 16 (2019), 1383-1409. http://dx.doi.org/10.33048/semi.2019.16.098
[17] Koshanov B.D., Soldatov A.P., Boundary value problem with normal derivatives for a higher order elliptic equation on the plane, Differential Equations, 52:12 (2016), 1594-1609. https://doi.org/10.1134/S0012266116120077
[18] Koshanov B.D., Soldatov A.P., On the Solvability of the Boundary Value Problems for the Elliptic Equation
of High Order on a Plane, Bulletin of the Karaganda University. Mathematics series, 91:3 (2018), 24-31.
https://doi.org/10.31489/2018M3/24-30
[19] Koshanov B.D., Equivalence of the Fredholm solvability condition for the Neumann problem to the
complementarity condition, Journal of Mathematics, Mechanics and Computer Science, 109:1 (2021), 34-54.
https://doi.org/10.26577/JMMCS.2021.v111.i3.04
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Koshanov, B. D., Baiarystanov, A. O., Dosmagulova, K. A., Kuntuarova, A. D., & Sultangazieva, Z. B. (2022). ON THE SCHWARZ PROBLEM FOR THE MOISIL–TEODORESCU SYSTEM IN A SPHERICAL LAYER AND IN THE INTERIOR OF A TORUS. Journal of Mathematics, Mechanics and Computer Science, 114(2). https://doi.org/10.26577/JMMCS.2022.v114.i2.04
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