  In this paper we consider a non-local problem for a Laplace operator in a
multidimensional bounded symmetric domain. The investigated problem is an
analogue of the classical periodic boundary value problems in the case of
non-rectangular domain. We prove self-adjointness of the problem and show a
method of constructing eigenfunctions. We obtain an analogue of the Rayleigh
type inequality and some spectral inequalities for the first eigenvalue of the
nonlocal problem.
