ON SOME NONLOCAL PROBLEMS FOR THE POISSON EQUATION

The paper studies the existence of solutions to a nonlocal boundary value problem for the Poisson equation in the unit ball. The boundary condition is nonlocal and relates the value of the unknown function to its normal derivative at diametrically opposite points on the unit sphere. A necessary and sufficient solvability condition is derived in the form of an orthogonality relation for the prescribed right-hand side and the boundary function. Uniqueness of the solution is proved up to an additive constant. An integral representation of the solution is derived using a specially constructed Green’s function for the nonlocal problem, which is expressed through the Green’s functions of related auxiliary local boundary value problems. In addition, the corresponding spectral problem is investigated. For its analysis, two auxiliary spectral problems with Neumann and Robin boundary conditions are introduced, and their eigenvalues and eigenfunctions are described in detail. It is demonstrated that this approach yields a complete orthonormal set of eigenfunctions for the original nonlocal problem, which forms a basis L₂(Ω).

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