On Periodic Boundary Value Problems With an Oblique Derivative for a Second Order Elliptic Equation

Abstract

In this paper, we study solvability of new classes of nonlocal boundary value problems for a second-order elliptic type equation. The considered problems are multidimensional analogues (in the case of circular domains) of classical periodic boundary value problems in rectangular domains.To study the main problem, first, an auxiliary boundary value problem with inclined derivative is considered for the second order elliptic equation. The main problems are solved by reducing them to a sequential solution of the Dirichlet problem and the problem with inclined derivative. Theorems on the existence and uniqueness of a solution of considered problems are proved.

On Periodic Problems for the Nonlocal Poisson Equation in the Circle

“…when a 0 = 1 and a j = 0, j = 1, 2, 3 were studied in [13], [14]. Later, some generalizations of these problems with conditions of the Dirichlet, Neumann, and Robin types, as well as of the Samarskii-Ionkin type, were studied in [6], [7], [10], [11], [15], [19]. Further, in [3], [4] for a nonlocal Laplace operator with involutively transformed arguments in rectangular domains, problems of the Cauchy and Dirichlet types were studied.…”