Geodesic
Holmgren problem for multudimensional elliptic equation with two singular coefficients
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 63 (2020), pp. 47-59 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Fundamental solutions of the two-dimensional elliptic equation were known in the first half of the last century and they were successfully used in solving the basic boundary value problems and constructing the theory of potential for this equation. Relatively few papers have been devoted to the study of boundary value problems for multidimensional (greater than two-dimensional) elliptic equations with singular coefficients. For example, main boundary value problems for twodimensional and three-dimensional elliptic equations with two singular coefficients in finite and infinite domains have been studied by many authors; however, the study of the Holmgren problem was limited to the two-dimensional case. This work is devoted to finding a unique solution to the Holmgren problem for a multidimensional elliptic equation with two singular coefficients in a quarter of a ball. Using the “abc” method, the uniqueness for the solution of the Holmgren problem is proved. Applying the method of Green's function, we are able to find the solution of the problem in an explicit form. Moreover, the decomposition formula, formula of differentiation, and some adjacent relations for Appell's hypergeometric functions were used in order to find the explicit solution for the formulated problem.
Keywords: multidimensional elliptic equation with two singular coefficients, Holmgren problem, fundamental solution, Gauss–Ostrogradsky formula, Green function. 
@article{VTGU_2020_63_a4,
     author = {T. G. Ergashev and N. J. Komilova},
     title = {Holmgren problem for multudimensional elliptic equation with two singular coefficients},
     journal = {Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika},
     pages = {47--59},
     year = {2020},
     number = {63},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTGU_2020_63_a4/}
}
TY  - JOUR
AU  - T. G. Ergashev
AU  - N. J. Komilova
TI  - Holmgren problem for multudimensional elliptic equation with two singular coefficients
JO  - Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
PY  - 2020
SP  - 47
EP  - 59
IS  - 63
UR  - http://geodesic.mathdoc.fr/item/VTGU_2020_63_a4/
LA  - ru
ID  - VTGU_2020_63_a4
ER  - 
%0 Journal Article
%A T. G. Ergashev
%A N. J. Komilova
%T Holmgren problem for multudimensional elliptic equation with two singular coefficients
%J Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
%D 2020
%P 47-59
%N 63
%U http://geodesic.mathdoc.fr/item/VTGU_2020_63_a4/
%G ru
%F VTGU_2020_63_a4
T. G. Ergashev; N. J. Komilova. Holmgren problem for multudimensional elliptic equation with two singular coefficients. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 63 (2020), pp. 47-59. http://geodesic.mathdoc.fr/item/VTGU_2020_63_a4/

[1] L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Dover Publications Inc., New York, 1958 | MR

[2] F. I. Frankl, Selected works on gas dynamics, Nauka, M., 1973

[3] M. M. Smirnov, Degenerate elliptic and hyperbolic equations, Nauka, M., 1966

[4] R. P. Gilbert, Function Theoretic Methods in Partial Differential Equations, Academic Press, New York–London, 1969, 308 pp. | MR | Zbl

[5] A. V. Bitsadze, Some classes of partial differential equations, Nauka, M., 1981

[6] C. Agostinelli, “Integrazione dell'equazione differenziale $u_{xx}+u_{yy}+u_{zz}+x^{-1}u_x =f$ e problema analogo a quello di Dirichlet per un campo emisferico”, Atti della Accademia Nazionale dei Lincei, 26:6 (1937), 7–8

[7] M. N. Olevskii, “Dirichlet problem solutions related to the equation $\Delta u + px_n^{-1}u_{{x_n}}=f$ for a hemispherical region”, Doklady Akademii nauk SSSR Reports of the Academy of Sciences of USSR, 64:6 (1949), 767–770

[8] T. G. Ergashev, “Holmgren problem for a multidimensional elliptic equation with one singular coefficient”, Byulleten' Instituta matematiki — Bulletin of the Institute of Mathematics, 2019, no. 2, 23–32

[9] M. S. Salakhiddinov, A. Hasanov, “On the theory of the multidimensional Gellerstedt equation”, Uzbekskiy matematicheskiy zhurnal — Uzbek Mathematical Journal, 2007, no. 3, 95–109 | Zbl

[10] A. K. Urinov, “On fundamental solutions for the some type of the elliptic equations with singular coefficients”, Nauchnyy vestnik Ferganskogo gosudarstvennogo universiteta — Scientific Records of the Fergana State University, 2006, no. 1, 5–11

[11] R. M. Mavlyaviev, I. B. Garipov, “Fundamental solution of multidimensional axisymmetric Helmholtz equation”, Complex Variables and Elliptic Equations, 62:3 (2017), 287–296 | DOI | MR | Zbl

[12] I. T. Nazipov, “Solution of the special Tricomi problem for a singular mixed-type equation by the method of integral equations”, Russian Mathematics, 55:3 (2011), 61–76 | DOI | MR | Zbl

[13] M. S. Salakhiddinov, A. Hasanov, “On a boundary value problem for the generalized Tricomi equation”, Izvestiya Akademii Nauk UzSSR. Ser. fiz.-mat. nauk — Bulletin of the Academy of Sciences of the Uzbek SSR. Ser. of phys. and math. sciences, 1979, no. 6, 29–33 | Zbl

[14] E. T. Karimov, J. J. Nieto, “The Dirichlet problem for a 3D elliptic equation with two singular coefficients”, Computers and Mathematics with Applications, 62 (2011), 214–224 | DOI | MR | Zbl

[15] T. G. Ergashev, A. Hasanov, “Fundamental solutions of the bi-axially symmetric Helmholtz equation”, Uzbek Mathematical Journal, 2018, no. 1, 55–64 | MR

[16] J. L. Bkrchnall, T. W. Chakndy, “Expansions of Appell's double hypergeometric functions”, Quart. J. Math. (Oxford). Ser. 11, 1940, 249–270 | MR

[17] A. Hasanov, “On a mixed problem for the equation $\mathrm{sgn} y|y|^m u_{xx}+x^nu_{yy}=0$”, Izvestiya Akademii Nauk UzSSR. Ser. fiz.-mat. nauk — Bulletin of the Academy of Sciences of the Uzbek SSR. Ser. of phys. and math. sciences, 1982, no. 2, 28–32 | MR | Zbl

[18] D. Amanov, “Some boundary value problems for a degenerate elliptic equation in an unbounded domain”, Izvestiya Akademii Nauk UzSSR. Ser. fiz.-mat. nauk — Bulletin of the Academy of Sciences of the Uzbek SSR. Ser. of phys. and math. sciences, 1984, no. 1, 8–13 | MR | Zbl

[19] D. Amanov, “A boundary value problem for the equation $\mathrm{sgn} y|y|^m u_{xx}+x^nu_{yy}=0$”, Izvestiya Akademii Nauk UzSSR. Ser. fiz.-mat. nauk — Bulletin of the Academy of Sciences of the Uzbek SSR. Ser. of phys. and math. sciences, 1984, no. 2, 8–10 | MR | Zbl

[20] M. S. Salakhiddinov, E. T. Karimov, “Spatial boundary problem with the DirichletNeumann condition for a singular elliptic equation”, Applied Mathematics and Computation, 219 (2012), 3469–3476 | DOI | MR

[21] H. M. Srivastava, A. Hasanov, J. Cho, “Double-layer potentials for a generalized bi-axially symmetric Helmholtz equation”, Sohag J. Math., 2:1 (2015), 1–10

[22] Berdyshev A.S, A. Hasanov, T. G. Ergashev, “Double-layer potentials for a generalized bi-axially symmetric Helmholtz equation. II”, Complex Variables and Elliptic Equations, 2019, 1–19 | DOI | MR

[23] T. G. Ergashev, “Third double-layer potential for a generalized bi-axially symmetric Helmholtz equation”, Ufa Mathematical Journal, 10:4 (2018), 111–121 | DOI | MR

[24] T. G. Ergashev, “The fourth double-layer potential for a generalized biaxially symmetric Helmholtz equation”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika I mekhanika — Tomsk State University Journal of Mathematics and Mechanics, 2017, no. 50, 45–56 | DOI

[25] T. G. Ergashev, “On fundamental solutions for multidimensional Helmholtz equation with three singular coefficients”, Computers and Mathematics with Applications, 77 (2019), 69–76 | DOI | MR

[26] A. K. Urinov, T. G. Ergashev, “Confluent hypergeometric functions of many variables and their application to finding fundamental solutions of the generalized Helmholtz equation with singular coefficients”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika I mekhanika — Tomsk State University Journal of Mathematics and Mechanics, 2018, no. 55, 45–56 | DOI

[27] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, v. 1, McGraw-Hill Book Company, New York–Toronto–London, 1953 | MR