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In this paper we consider a non-local problem for a Laplace operator in a two-dimensional bounded symmetric domain with center at the origin. Investigated problem is an analogue of the classical periodic boundary value problems in the case of non-rectangular region. 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. [ABSTRACT FROM AUTHOR]

In this paper, we study some boundary value problems for fractional analogue of Helmholtz equation with an involution perturbation in a rectangular domain. Theorems on existence and uniqueness of a solution of the considered problems are proved by spectral method. [ABSTRACT FROM AUTHOR]

In this paper, a generalized third boundary value problem for the biharmonic equation in the unit ball with boundary operators up to third order containing normal derivatives and Laplacian is investigated. The existence and uniqueness theorems are proved. Some particular cases of general problem are considered. [ABSTRACT FROM AUTHOR]

The paper is devoted to the investigation of questions about constructing the explicit form of the Green's function of the Robin problem. For constructing this function we use the representation of the fundamental solution of the Laplace equation in the form of a series. An integral representation of the Green function is obtained and for some values of the parameters, the problem is presented in elementary functions. [ABSTRACT FROM AUTHOR]