The spatial problem of the theory of elasticity is solved for a layer with several infinite cylindrical cavities disjoint to each other and the surface of the layer. Stress values are preset for the cavities and for the upper boundary of the layer; displacements are preset on the lower boundary of the layer. The problem is solved using the generalized Fourier method with respect to the system of Lame’s equations. If the boundary conditions are satisfied, we are led to infinite systems of linear algebraic equations that are solved by the reduction method. As a result, values of displacements and stresses at various points of the elastic layer are obtained. A computational investigation is carried out for a concrete layer adhered to a rigid base and weakened by two unloaded cavities. A normal stress value is preset on the upper boundary of the layer. The analysis of the stress-strain state of the layer in the vicinity of the load application, as well as in the vicinity of the left cavity located closer to the load, is carried out. It is compared to the option when the second cavity is absent. The proposed method can be used to calculate structures and parts with similar design models, and the stress state analysis can be made to select the geometric characteristics of the designed structure.