On the smoothness of a semi-periodic boundary value problem for a three-dimensional equation of the second kind, second order mixed type in an unbounded domain

S. Z. Djamalov ; B. K. Sipatdinova

Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 50 (2025) no. 1, pp. 9-21

Résumé

In the work of A.V.Bitsadze it is shown that the Dirichlet problem for a mixedtype equation is incorrect. The question naturally arises: is it possible to replace the conditions of the Dirichlet problem with other conditions covering the entire boundary, which ensure the correctness of the problem? For the first time such boundary value problems (non-local boundary value problems) for a mixed-type equation were proposed and studied in the works of F.I. Frankl. As problems for mixed-type equations of the second kind in bounded domains, which are close in formulation to those under study, are investigated in the work of S. Dzhamalov. For mixed-type equations of the second kind of the second order in unbounded domains, semi-periodic boundary value problems in the three-dimensional case have been practically not investigated. In this paper, we investigate the uniqueness, existence and smoothness of a generalized solution to a semiperiodic boundary value problem for a mixed-type equation of the second kind, second order in an unbounded domain. In this paper, we prove the uniqueness of a generalized solution to the problem using the energy integral method. To prove the existence and smoothness of a generalized solution to the problem, the methods of "$\epsilon$-regularization" and a priori estimates using the Fourier transform were used.

Détail

Keywords: mixed-type equation of the second kind, semi-periodic boundary value problem, energy integral, uniqueness of solution, "$\epsilon$-regularization"methods, a priori estimates, existence and smoothness of a generalized solution.
Mots-clés : Fourier transform, anisotropic Sobolev space