  We investigate the vacuum expectation values of the energy-momentum tensor
and the fermionic condensate associated with a massive spinor field obeying the
MIT bag boundary condition on a spherical shell in the global monopole
spacetime. In order to do that it was used the generalized Abel-Plana summation
formula. As we shall see, this procedure allows to extract from the vacuum
expectation values the contribution coming from to the unbounded spacetime and
explicitly to present the boundary induced parts. As to the boundary induced
contribution, two distinct situations are examined: the vacuum average effect
inside and outside the spherical shell. The asymptotic behavior of the vacuum
densities is investigated near the sphere center and surface, and at large
distances from the sphere. In the limit of strong gravitational field
corresponding to small values of the parameter describing the solid angle
deficit in global monopole geometry, the sphere-induced expectation values are
exponentially suppressed. As a special case we discuss the fermionic vacuum
densities for the spherical shell on background of the Minkowski spacetime.
Previous approaches to this problem within the framework of the QCD bag models
have been global and our calculation is a local extension of these
contributions.
