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In this paper we investigate the vacuum polarization effects associated witha massive fermionic field due to the non-trivial topology of the globalmonopole spacetime and boundary conditions imposed on this field. Specificallywe investigate the vacuum expectation values of the energy-momentum tensor andfermionic condensate admitting that the field obeys the MIT bag boundarycondition on two concentric spherical shells. In order to develop thisanalysis, we use the generalized Abel-Plana summation, which allows to extractfrom the vacuum expectation values the contribution coming from a single spheregeometry and to present the second sphere induced part in terms ofexponentially convergent integrals. In the limit of strong gravitational fieldcorresponding to small values of the parameter describing the solid angledeficit in global monopole geometry, the interference part in the expectationvalues are exponentially suppressed. The vacuum forces acting on spheres arepresented as the sum of self-action and interaction terms. Due to the surfacedivergences, the first one is divergent and needs additional renormalization,while the second one is finite for all non-zero distances between the spheres.By making use of zeta function renormalization technique, the total Casimirenergy is evaluated in the region between two spheres. It is shown that theinteraction part of the vacuum energy is negative and the interaction forcesbetween the spheres are attractive. Asymptotic expressions are derived invarious limiting cases. As a special case we discuss the fermionic vacuumdensities for two spherical shells on background of the Minkowski spacetime.Comment: 20 pages, 1 figure, discussion on the zeta function for a single  sphere is extende

Spinor Casimir effect for concentric spherical shells in the global monopole spacetime