A boundary-value problem for a class of four-dimensional degenerate elliptic equations

A. S. Berdyshev ; A. Hasanov ; A. Ryskan

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 5, Tome 194 (2021), pp. 55-70.

Résumé

The solvability of the problem with mixed Neumann–Dirichlet conditions for a degenerate four-dimensional elliptic equation are studied. The uniqueness of a solution to the problem is proved by the method based on the energy integral.

Détail

Keywords: degenerate four-dimensional elliptic equation, boundary-value problem with Neumann–Dirichlet conditions, Gellerstedt equation in four variables, fundamental solution, Lauricella and Gauss hypergeometric functions.

Bibliographie

[1] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii. Gipergeometricheskie funktsii. Funktsii Lezhandra, Nauka, M., 1973

[2] Bitsadze A. V., Nekotorye klassy uravnenii v chastnykh proizvodnykh, Nauka, M., 1981

[3] Gradshtein I. S., Ryzhik I. M., Tablitsy integralov, summ, ryadov i proizvedenii, Fizmatgiz, M., 1963 | MR

[4] Kogan M. N., “O magnitogidrodinamicheskikh techeniyakh smeshannogo tipa”, Prikl. mat. mekh., 25:1 (1961), 132–137 | Zbl

[5] Smirnov M. M., Vyrozhdayuschiesya ellipticheskie i giperbolicheskie uravneniya, Nauka, M., 1966

[6] Smirnov M. M., Uravneniya smeshannogo tipa, Vysshaya shkola, M., 1985 | MR

[7] Frankl F. I., Izbrannye trudy po gazovoi dinamike, Nauka, M., 1973

[8] Agarwal P., Karimov E., Mamchuev M., Ruzhansky M., “On boundary-value problems for a partial differential equation with Caputo and Bessel operators”, Recent Applications of Harmonic Analysis to Function Spaces, differential Equations, and Data Science, Birkhäuser, Basel, 2017, 707–718 | DOI | MR | Zbl

[9] Agmon S., “The fundamental solution and Tricomi's problem for a class of equations of mixed type”, Proc. Int. Cong. Math. Amsterdam, II (1954)

[10] Altin A., “Solutions of type for a class of singular equations”, International Journal of Mathematical Science, 5:3 (1982), 613–619 | DOI | MR | Zbl

[11] Appell P., Kampe de Feriet J., Fonctions Hypergeometriques et Hyperspheriques. Polynomes d'Hermite, Gauthier-Villars, Paris, 1926

[12] Barros-Neto J. J., Gelfand I. M., “Fundamental solutions for the Tricomi operator, I”, Duke Math. J., 98:3 (1999), 465–483 | DOI | MR | Zbl

[13] Barros-Neto J. J., Gelfand I. M., “Fundamental solutions for the Tricomi operator, II”, Duke Math. J., 111:3 (2002), 561–584 | DOI | MR | Zbl

[14] Barros-Neto J. J., Gelfand I. M., “Fundamental solutions for the Tricomi operator ,III”, Duke Math. J., 128:1 (2005), 119–140 | DOI | MR | Zbl

[15] Bers L., Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Wiley, New York, 1958 | MR | Zbl

[16] Erdelyi A., Magnus W., Oberhettinger F., Tricomi F. G., Higher Transcendental Functions, McGraw-Hill, New York–Toronto–London, 1953 | MR

[17] Fryant A. J., “Growth and complete sequences of generalized bi-axially symmetric potentials”, J. Differ. Equations., 31:2 (1979), 155–164 | DOI | MR | Zbl

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

[19] Golberg M. A., Chen C. S., “The method of fundamental solutions for potential, Helmholtz and diffusion problems”, Boundary Integral Methods: Numerical and Mathematical Aspects, ed. Golberg M. A., WIT Press, Boston, 1998, 103–176 | MR

[20] Hasanov A., Srivastava H. M., “Some decomposition formulas associated with the Lauricella function and other multiple hypergeometric functions”, Appl. Math. Lett., 19:2 (2006), 113–121 | DOI | MR | Zbl

[21] Hasanov A., Karimov E. T., “Fundamental solutions for a class of three-dimensional elliptic equations with singular coefficients”, Appl. Math. Lett., 22 (2009), 1828–1832 | DOI | MR | Zbl

[22] Itagaki M., “Higher order three-dimensional fundamental solutions to the Helmholtz and the modified Helmholtz equations”, Eng. Anal. Bound. Elem., 15 (1995), 289–293 | DOI

[23] Karimov E. T., “On a boundary problem with Neumann's condition for 3D singular elliptic equations”, Appl. Math. Lett., 23 (2010), 517–522 | DOI | MR | Zbl

[24] Kitagawa T., “On the numerical stability of the method of fundamental solution applied to the Dirichlet problem”, Jpn. J. Appl. Math., 5 (1988), 123–133 | DOI | MR | Zbl

[25] Luke Y. L., The Special Functions and Their Approximations, Academic Press, New York–London, 1969 | MR | Zbl

[26] Mathai A. M., Saxena R. K., Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, Springer-Verlag, Berlin –Heidelberg, 1973 | MR | Zbl

[27] Niukkanen A. W., “Generalised hypergeometric series $NF(x_1,\dots,x_N)$ arising in physical and quantum chemical applications”, J. Phys. A: Math. Gen., 16 (1983), 1813–1825 | DOI | MR | Zbl

[28] Rassias J. M., “A maximum principle in $\mathbb{R}^{n+1}$”, J. Math. Anal. Appl., 85:1 (1982), 106–113 | DOI | MR | Zbl

[29] Salakhitdinov M. S., Hasanov A., “A solution of the Neumann–Dirichlet boundary-value problem for generalized bi-axially symmetric Helmholtz equation”, Complex Var. Elliptic Equations., 53:4 (2008), 355–364 | DOI | MR | Zbl

[30] Sneddon I. N., Special Functions of Mathematical Physics and Chemistry, Longman, London–New York, 1980 | MR | Zbl