Some spectral properties and isoperimetric inequalities for a nonlocal   Laplacian problem
  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. 
Makhmud A. Sadybekov, Berikbol T. Torebek
math.AP