Abstract
The work studies the solvability of a nonlocal boundary value problem for the Laplace equation. The nonlocal condition is introduced using transformations in the Rn space carried out by some orthogonal matrices. Examples and properties of such matrices are given. To study the main problem, an auxiliary nonlocal Dirichlet-type problem for the Laplace equation is first solved. This problem is reduced to a vector equation whose elements are the solutions of the classical Dirichlet probem. Under certain conditions for the boundary condition coefficients, theorems on uniqueness and existence of a solution to a problem of Dirichlet type are proved. For this solution an integral representation is also obtained, which is a generalization of the classical Poisson integral. Further, the main problem is reduced to solving a non-local Dirichlet-type problem. Theorems on existence and uniqueness of a solution to the problem under consideration are proved. Using well-known statements about solutions of a boundary value problem with an oblique derivative for the classical Laplace equation, exact orders of smoothness of a problem's solution are found. Examples are also given of the cases where the theorem conditions are not fulfilled. In these cases the solution is not unique.
Highlights
Для функций u (x), u(Sx), ..., u(Sl−1x) получим систему алгебраических уравнений (2.3) – (2.4) с матрицей A:
The work studies the solvability of a nonlocal boundary value problem for the Laplace equation
To study the main problem, an auxiliary nonlocal Dirichlet-type problem for the Laplace equation is first solved. This problem is reduced to a vector equation whose elements are the solutions of the classical Dirichlet probem
Summary
Что если x ∈ Ω или x ∈ ∂Ω, то для любого натурального числа k имеет место включение Skx ∈ Ω, или Skx ∈ ∂Ω. В случае a1 = 1, aj = 0, j = 2, 3, ..., l, получим известную задачу с наклонной производной [2]. Что вырождающиеся краевые задачи с наклонной производной для эллиптических уравнений исследованы в работах многочисленных авторов Что для уравнения Лапласа в двумерном случае краевые задачи с отображениями из примера 1.2 изучены в работе [8], а в работах [9,10] аналогичные задачи исследованы для полигармонического уравнения. В работах [11,12] для нелокального уравнения Лапласа с отображениями S исследованы вопросы разрешимости основных краевых задач
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