Abstract
In this paper we prove the unique solvability and smoothness of the solution of a nonlocal boundary-value problem for a multidimensional mixed type second-order equation of the second kind in Sobolev space Wℓ2(Q), (2≤ℓ is an integer). First, we have studied the unique solvability of the problems in the space W22(Q). Solution uniqueness for a nonlocal boundary-value problem for a mixed-type equation of the second kind is proved by the methods of a priori estimates.Further, to prove the solution existence in the space W22(Q), the Fourier method is used. In other words, the problem under consideration is reduced to the study of unique solvability of a nonlocal boundary value problem for an infinite number of systems of second-order equations of mixed type of the second kind. For the unique solvability of the problems obtained, the ``ε-regularization'' method is used, i.e, the unique solvability of a nonlocal boundary-value problem for an infinite number of systems of composite-type equations with a small parameter was studied by the methods of functional analysis. The necessary a priori estimates were obtained for the problems under consideration. Basing on these estimates and using the theorem on weak compactness as well as the limit transition, solutions for an infinite number of systems of second-order equations of mixed type of the second kind are obtained. Then, using Steklov-Parseval equality for solving an infinite number of systems of second-order equations of mixed type of the second kind, the unique solvability of original problem was obtained. At the end of the paper, the smoothness of the problem's solution is studied.
Highlights
In this paper we prove the unique solvability and smoothness of the solution of a nonlocal boundary-value problem for a multidimensional mixed type second-order equation of the second kind in Sobolev space W 2(Q), (2 ≤ is an integer)
We have studied the unique solvability of the problems in the space W22(Q)
Solution uniqueness for a nonlocal boundaryvalue problem for a mixed-type equation of the second kind is proved by the methods of a priori estimates.Further, to prove the solution existence in the space W22(Q), the Fourier method is used
Summary
Пусть Ω – ограниченная односвязная область в пространстве Rn, n ∈ N с гладкой границей ∂Ω. Впервые нелокальные краевые задачи (1.2)–(1.3) для уравнения смешанного типа второго рода (1.1) в случае a(x, t) = 0 были исследованы функциональными методами в некоторых весовых и негативных пространствах в работах [3]–[4]. Далее в работах [5]–[6] в случае K(x, 0) = K(x, T ) = 0, a(x, t) = 0, γ – постоянное число, отличное от нуля, и при выполнении некоторых сравнительно сильных ограничений на коэффициенты уравнения (1.1) была доказана корректность решения задачи (1.1)–(1.3) в пространствах Соболева. Что в работах [7]–[9] в случае, когда K(x, 0) ≤ 0 ≤ K(x, T ), a(x, t) = 0, γ – постоянное число, отличное от нуля, доказаны однозначная разрешимость и гладкость решения задачи (1.1)–(1.3) в пространствах Соболева.
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