On some analogues of periodic problems for Laplace equation with an oblique derivative under boundary conditions

Abstract

In this paper, we study solvability of new classes of nonlocal boundary value problems for the Laplace equation in a ball. The considered problems are multidimensional analogues (in the case of a ball) of classical periodic boundary value problems in rectangular regions. To study the main problem, first, for the Laplace equation, we consider an auxiliary boundary value problem with an oblique derivative. This problem generalizes the well-known Neumann problem and is conditionally solvable. The main problems are solved by reducing them to sequential solution of the Dirichlet problem and the problem with an oblique derivative. It is proved that in the case of periodic conditions, the problem is conditionally solvable; and in this case the exact condition for solvability of the considered problem is found. When boundary conditions are specified in the anti-periodic conditions form, the problem is certainly solvable. The obtained general results are illustrated with specific examples.

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

“…Note that similar problems for the Laplace and Poisson equations with normal derivatives of integer and fractional orders were studied in [1], [2], [3]. Moreover, in [4] similar problem was studied for a boundary operator with inclined derivative without degeneracy. We also note that degenerate boundary value problems with inclined derivative were studied in [5], [6], [7], [8].…”