On the correctness of boundary value problems for the mixed type equation of the second kind

K. B. Sabitov ; I. P. Egorova

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 23 (2019) no. 3, pp. 430-451.

Résumé

In this paper, the intervals of change in the exponent of the degree of degeneration of a mixed-type equation with characteristic degeneration are established. The first boundary problem and the modified boundary problem (analogue of the Keldysh problem) with the conditions of periodicity are correctly set. In the case of the first problem, a criterion for the uniqueness of its solution is manifested. It is shown that the solution of the analogue of the Keldysh problem is unique up to a term of a linear function. Solutions are constructed as the sum of series of eigenfunctions of the corresponding one-dimensional spectral problem. In justifying the convergence of a series representing the solution of the first boundary-value problem, the problem of small denominators of a more complex structure arises in the class of regular solutions of this equation than in previously known works. The estimate on separation from zero is established with the corresponding asymptotic. Based on this estimate, sufficient conditions are found for the boundary functions to substantiate the uniform convergence of the series and their derivatives up to the second order inclusive.

Détail

Keywords: equations of mixed type, characteristic degeneration, boundary-value problems, periodicity conditions, spectral method, uniqueness, small denominators
Mots-clés : existence.

Bibliographie

[1] Keldysh M. V., “On some cases of degenerate elliptic equations on the boundary of a domain”, Dokl. Akad. Nauk SSSR, 77:2 (1951), 181–184 (In Russian) | Zbl

[2] Karol I. L., “On a certain boundary value problem for an equation of mixed elliptic-parabolic type”, Dokl. Akad. Nauk SSSR, 88:2 (1953), 197–200 (In Russian) | MR | Zbl

[3] Frankl F. I., “A generalization of the Tricomi problem and its application to solution of the direct problem of the theory of the Laval nozzle”, Mat. Sb. (N.S.), 54(96):2 (1961), 225–236 (In Russian) | MR | Zbl

[4] Sabitov K. B., “On the uniqueness of the solution of the Tricomi problem for a single equation of mixed type and second kind”, Partial differential equations: differential equations and numerical mathematics, Work Collect. Conf., Novosibirsk, 1980, 48–50 (In Russian) | Zbl

[5] Sabitov K. B., “Formulation of boundary-value problems for an equation of mixed type with degeneracy of second kind on the boundary of an infinite domain”, Sib. Math. J., 21:4 (1980), 591–594 | DOI | MR | Zbl

[6] Sabitov K. B., “A problem of Tricomi type for equations of mixed type with strong characteristic degeneration”, Differ. Uravn., 20:2 (1984), 333–337 (In Russian) | MR | Zbl

[7] Frankl F. I., “On the problems of Chaplygin for mixed sub- und supersonic flows”, Izv. Akad. Nauk SSSR, Ser. Mat., 9:2 (1945), 121–142 (In Russian) | Zbl

[8] Frankl F. I., Izbrannye trudy po gazovoi dinamike [Selected Works on Gas Dynamics], Nauka, Moscow, 1973, 703 pp. (In Russian)

[9] Shabat B. V., “Examples of solving the Dirichlet problem for equations of mixed type”, Dokl. Akad. Nauk SSSR, 112:3 (1957), 386–389 (In Russian) | MR | Zbl

[10] Bitsadze A. V., “Incorrectness of Dirichlet's problem for the mixed type of equations in mixed regions”, Dokl. Akad. Nauk SSSR, 122:2 (1958), 167–170 (In Russian) | MR | Zbl

[11] Nakhushev A. M., “An uniqueness condition of the Dirichlet problem for an equation of mixed type in a cylindrical domain”, Differ. Uravn., 6:1 (1970), 190–191 (In Russian) | MR | Zbl

[12] Soldatov A. P., “Dirichlet-type problems for the Lavrent'ev–Bitsadze equation. I. Uniqueness theorems”, Dokl. Math., 48:2 (1994), 410–414 | MR | Zbl

[13] Soldatov A. P., “Dirichlet-type problems for the Lavrent'ev–Bitsadze equation. II. Existence theorems”, Dokl. Math., 48:3 (1994), 433–437 | MR | Zbl

[14] Cannon J. R., “A Dirichlet problem for an equation of mixed type with a discontinuous coefficient”, Ann. Mat. Pura Appl., IV. Ser., 61 (1963), 371–377 | DOI | MR | Zbl

[15] Hacev M. M., “Dirichlet's problem for a generalized Lavrent'ev–Bitsadze equation in a rectangular region”, Differ. Equ., 14:1 (1978), 96–99 | MR | Zbl | Zbl

[16] Khachev M. M., Pervaia kraevaia zadacha dlia lineinykh uravnenii smeshannogo tipa [The First Boundary Value Problem for Linear Equations of Mixed Type], Elbrus, Nalchik, 1998, 169 pp. (In Russian)

[17] Sokhadze R. I., “The first boundary value problem for an equation of mixed type in a rectangle”, Differ. Uravn., 19:1 (1983), 127–134 (In Russian) | MR | Zbl

[18] Sabitov K. B., “Dirichlet problem for mixed-type equations in a rectangular domain”, Dokl. Math., 75:2 (2007), 193–196 | DOI | MR | Zbl

[19] Moiseev E. I., “On the solution of a nonlocal boundary value problem by the spectral method”, Differ. Equ., 35:8 (1999), 1105–1112 | MR | Zbl

[20] Sabitov K. B., Suleimanova A. Kh., “The Dirichlet problem for a mixed-type equation of the second kind in a rectangular domain”, Russian Math. (Iz. VUZ), 51:4 (2007), 42–50 | DOI | MR | Zbl

[21] Sabitov K. B., Sidorenko O. G., “Problem with periodicity conditions for a degenerating equation of mixed type”, Differ. Equ., 46:1 (2010), 108–116 | DOI | MR | Zbl

[22] Egorova I. P., “The problem with periodicity conditions for the equations of mixed type with characteristic degeneracy”, Vestnik Samarskogo Gosudarstvennogo Universiteta. Estestvenno-Nauchnaya Seriya, 2009, no. 8(74), 15–27 (In Russian)

[23] Arnol'd V. I., “Small denominators and problems of stability of motion in classical and celestial mechanics”, Russian Math. Surveys, 18:6 (1963), 85–191 | DOI | MR | Zbl

[24] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F. G., Higher transcendental functions, v. 2, Bateman Manuscript Project, McGraw-Hill Book Co., Inc., New York, Toronto, London, 1953, xvii+396 pp. | MR | Zbl