Abstract
We study a boundary value problem with periodicity conditions and with a nonlocal Dezin condition for a mixed elliptic-hyperbolic equation in a rectangular domain with power-law degeneracy on the transition line. Necessary and sufficient conditions for the uniqueness of the solution are established, the uniqueness of the solution being proved based on the completeness of the system of eigenfunctions of a one-dimensional eigenvalue problem. The solution is constructed in the form of a series. The problem of small denominators occurs when justifying the convergence of the series. Under some conditions imposed on the given parameters and functions, the convergence of the series is proved in the class of regular solutions.
Similar content being viewed by others
References
Dezin, A.A., On the solvable extensions of partial differential operators, Outlines Joint Sov.-Am. Symp. Part. Diff. Equat., Novosibirsk, 1963, pp. 65–66.
Nakhusheva, Z.A., On a nonlocal problem of A.A. Dezin for the Lavrent’ev-Bitsadze equation, Differ. Equations, 2009, vol. 45, no. 8, pp. 1223–1228.
Nakhusheva, Z.A., Nelokal’nye kraevye zadachi dlya osnovnykh i smeshannykh tipov differetsial’nykh uravnenii (Nonlocal Boundary Value Problems for Basic and Mixed Types of Differential Equations), Nal’chik, 2011.
Sabitov, K.B. and Novikova, V.A., Nonlocal Dezin’s problem for Lavrent’ev-Bitsadze equation, Russ. Math., 2016, vol. 60, no. 6, pp. 52–62.
Sabitov, K.B. and Gushchina, V.A., A.A. Dezin’s problem for inhomogeneous Lavrent’ev-Bitsadze equation, Russ. Math., 2017, vol. 61, no. 3, pp. 31–43.
Arnol’d, V.I., Small denominators. I: Mappings of the circumference onto itself, Izv. Akad. Nauk SSSR. Ser. Mat., 1961, vol. 25, pp. 21–86; English transl.: Am. Math. Soc. Transi. II. Ser., 1965, vol. 46, pp. 213–284.
Sabitov, K.B., Dirichlet problem for mixed-type equations in a rectangular domain, Dokl. Math., 2007, vol. 75, no. 2, pp. 193–196.
Sabitov, K.B. and Sidorenko, O.G., Problem with periodicity conditions for a degenerating equation of mixed type, Differ. Equations, 2010, vol. 46, no. 1, pp. 108–116.
Moiseev, E.I., On the solution of a nonlocal boundary value problem by the spectral method, Differ. Equations, 1999, vol. 35, no. 8, pp. 1105–1112.
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2019, published in Differentsial’nye Uravneniya, 2019, Vol. 55, No. 10, pp. 1426–1431.
Rights and permissions
About this article
Cite this article
Sabitov, K.B. Dezin Problem for an Equation of the Mixed Type with a Power-Law Degeneracy. Diff Equat 55, 1384–1389 (2019). https://doi.org/10.1134/S0012266119100136
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266119100136